Numerical-analytic methods in the theory of boundary-value problems

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Numerical-Analytic Methods in the Theory of Boundary-Value Problems

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Numerical-Analytic Methods in the Theory of Boundary-Value Problems M. Ronto University of Miskolc, Hungary

A. M. Samoilenko National Academy of Sciences , Ukraine

World Scientific Singapore & NewJersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NUMERICAL-ANALYTIC METHODS IN THE THEORY OF BOUNDARY-VALUE PROBLEMS Copyright m 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981 -02-3676-X

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Printed in Singapore by Uto-Print

PREFACE

Formulation and development of constructive methods is one of new directions in the contemporary mathematical analysis and simulation. In spite of the fact that investigations in this field were carried out for only several decades, the class of constructive mathematical methods draws more and more attention. It is likely that there is no accepted definition which would strictly restrict the class of constructive methods. Nevertheless, this term becomes more and more common. Apparently, the most natural way is to regard constructive methods as certain methods for the construction of solutions of different classes of equations and investigation of the existence and properties of exact and approximate solutions. Furthermore, the main characteristic of constructive methods is the fact that they enable one to completely solve the problem (up to the numerical values) and practically verify the theoretical background and conditions that guarantee the applicability of these methods to specific classes of problems. Apparently, for the first time, the constructive side of methods attracted attention in the fields of mathematics that are now known as the theory of nonlinear oscillations and nonlinear mechanics. It is widely recognized that asymptotic methods using different averaging schemes became, in fact, one of the main tools of constructive investigation and construction of solutions of various problems of nonlinear mechanics. The monograph [GrR3], for example, convincingly points to this fact. In the monograph mentioned, the constructivity of methods is investigated with the use of asymptotic decompositions, the averaging principle, various iterative versions of the method of LyapunovPoincare" series in powers of a small parameter, and the iteration method with accelerated convergence. These investigations are mainly aimed at studying periodic and quasiperiodic solutions. It is clear that there is an urgent need in developing constructive methods for other branches of the theory of differential equations, in particular, for the theory of boundaryvalue problems for ordinary differential equations. The analysis of the contemporary state of methods for the investigation of boundary-value problems convincingly shows that the classes of analytic, functional-analytic, numerical, and numerical-analytic methods are most often used in this field. Obviously, each group of these methods has advantages and disadvantages. However, it should be noted that, in the theory of boundaryvalue problems, the numerical-analytic methods compare favorably with other methods by their constructivity both at the stage of construction of solutions and in the course of v

vi

Preface

investigation of the principal qualitative problems such as establishing the existence of solutions, the verification of convergence of approximate solutions to exact solutions, and obtaining error estimates for approximate solutions that can be practically verified. All problems indicated above demonstrate that the numerical-analytic methods open good prospects of further development of constructive methods for the investigation of solutions of boundary-value problems for ordinary differential equations. I am pleased to introduce this book to readers. It is written by the well-known experts in the field of numerical-analytic methods and is a natural continuation of their works [SaRI], [SaR2], and [SaR4]. The present monograph is characterized by the perfect combination of the most established elements of the theory of the numerical-analytic method of successive approximations, which has proved to be very efficient; new directions of research are also described. In view of the small volume of this monograph, the authors consider a narrow range of problems concerning only the generalization of nonlinear, boundary-value problems for ordinary differential equations to new classes. One of the advantages of this work is the fact that the authors demonstrate the possibility of application of the numerical-analytic schemes under consideration not only to classic boundary-value problems, but also to nonstandard boundary-value problems, e.g., boundary-value problems with parameters in boundary conditions or pulse influence. Another distinctive feature of this book is the development of the idea of possibility and expediency of combining various numerical-analytic methods for the investigation of periodic solutions and solutions of nonlinear boundary-value problems of the general form. Clearly, this monograph contains an interesting and necessary material, which will be interesting for experts in the theory of boundary-value problems and nonlinear oscillations and will contribute to the further development of constructive numerical-analytic methods for the investigation of periodic boundary-value problems and boundary-value problems of general form. Academician Yu. A. Mitropolsky

comm Preface

v

Introduction

1

Chapter 1. NUMERICAL-ANALYTIC METHOD OF SUCCESSIVE APPROXIMATIONS FOR TWO-POINT BOUNDARY-VALUE PROBLEMS

19

1. Abstract Scheme of the Method

19

2. Choice of the Form of Successive Approximations and Their Uniform Convergence

26

3. Sufficient Conditions for the Existence of Solutions

41

4. Necessary Conditions for the Solvability of the Boundary-Value Problem

45

5. Error of Calculation of the Initial Value of a Solution

52

6. Special Types of Successive Approximations and Estimates

64

7. Numerical-Analytic Method in the Case of Nonlinear Two-Point Boundary Conditions

72

8. Boundary-Value Problems with Small Parameter

81

Chapter 2. MODIFICATION OF THE NUMERICAL-ANALYTIC METHOD FOR TWO-POINT BOUNDARY-VALUE PROBLEMS

89 89

9. Periodic Boundary-Value Problem 10. Theorems on the Properties of Determining Functions of a Periodic Boundary-Value Problem

98

11. Solvability of the Approximate Determining Equation and the Error of the Initial Value of a Periodic Solution

105

vu

viii

Contents

12. Modification of the Method for Two-Point Problems 115 13. Relationship between Exact and Approximate Determining Equations

123

14. Determination of Initial Values of Solutions of Two-Point Boundary-Value Problems 131 15. Realization of the Method for Systems of Two Equations 142 Chapter 3. NUMERICAL -ANALYTIC METHOD FOR BOUNDARY -VALUE PROBLEMS WITH PARAMETERS IN BOUNDARY CONDITIONS

149

16. Successive Approximations for Problems with One Parameter in Linear Boundary Conditions 150 17. Sufficient Solvability Conditions and Determination of the Initial Value of a Solution of the Boundary-Value Problem with Parameter 158 18. Boundary-Value Problems with Nonfixed Right Boundary 168 19. Solvability Theorems for Problems with Nonfixed Right Boundary 177 20. The Case of Several Parameters in Boundary Conditions 193 21. Linear Dependence of Boundary Conditions on Two Parameters 209 Chapter 4. COLLOCATION METHOD FOR BOUNDARY-VALUE PROBLEMS WITH IMPULSES 221 22. Green Function of a Homogeneous Two-Point Boundary-Value Problem

222

23. Inhomogeneous Linear Impulsive Boundary-Value Problems 229 24. Convergence of the Algebraic Collocation Method for Nonlinear Systems 234 25. Method of Trigonometric Collocation for Periodic Systems 245 26. Practical Solution of Impulsive Problems 250 27. Green Function for a Three-Point Boundary-Value Problem with Single Impulse Influence 257 28. Semihomogeneous Linear Two-Point Boundary-Value Problem with m-Impulse Influence

271

Contents

ix

29. Inhomogeneous Linear m-Impulse Two-Point Boundary-Value Problem

279

30. Multipoint m-Impulse Boundary-Value Problem

280

31. Equations with Piecewise-Continuous Right-Hand Side

286

32. Construction of Solutions of Two-Impulse Systems

308

Appendix. THE THEORY OF THE NUMERICAL -ANALYTIC METHOD : ACHIEVEMENTS AND NEW TRENDS OF DEVELOPMENT

317

Al. History of the Method of Periodic Successive Approximations

317

A2. Relation to Other Investigations

329

A3. Application of Modifications of the Numerical-Analytic Method to the Investigation of Various Boundary-Value Problems

339

REFERENCES

415

IIITRODUCTIOII

The development of the theory of boundary-value problems for ordinary differential equations was initiated by the works of G. Floquet [Flo], A. M. Lyapunov [Lyal], [Lya2], S. N. Bernstein [Ber], G. D. Birkhoff [Bir], E. L. Bunitsky [Bunl], [Bun2], D.Jackson [Jacl], [Jac2], G. A. Bliss [Blil], [Bli2], G. D. Birkhoff and R. E. Langer [BiL], and by E. Kamke [Kam], which appeared at the end of the 19th century and in the first decades of the 20th century. The analysis of bibliography shows that boundary-value problems have always attracted particular attention. It is natural that, at the first stage, the theory of linear boundary-value problems was developed. Solvability conditions for homogeneous and inhomogeneous boundary-value problems were studied for scalar second- and higher-order differential equations and for systems of first-order differential equations. Periodic solutions of differential equations with periodic coefficients were investigated, i.e., in fact, boundary-value problems with periodic boundary conditions were considered. On the basis of the properties of differential operators generated by given differential equations and boundary conditions, spectral properties of boundary-value problems and their relation to the corresponding conjugate problems were investigated. Much attention was given to the construction of Green functions and generalized Green functions for various differential operators and to the investigation of their properties. The theory of nonlinear boundary-value problems was also gradually developed in depth. Methods that enable one to study the problem of unique existence of solutions, analyze the oscillation properties of solutions, construct approximate solutions, and estimate errors were developed. At present, the theory of boundary-value problems possesses a fairly powerful stock of diverse methods. These methods can conventionally be classified into several main groups, namely, analytic methods, functional-analytic methods, numerical methods, and numerical-analytic methods. It should be noted that analytic and functional-analytic methods are mainly aimed at the investigation of qualitative problems of unique existence of solutions and their continuous dependence on parameters. The group of numerical methods is aimed at the direct calculation of numerical values of approximate solutions. Finally, numerical-analytic methods are fairly universal and can be used for both the investigation of the problem of existence and practical construction of solutions. Numerous profound monographs, theses, surveys, and articles that reflect the state of the theory and methods for the investigation of various boundary-value problems in the 1

2 Introduction directions indicated above were written by many mathematicians from different countries. We do not even pretend to give a complete list of the most important works; we only mention works in which a fairly complete bibliography can be found and which reflect the most typical approaches , as well as investigations of individual scientific schools. Apparently , the problems that were first comprehensively investigated were periodic boundary-value problems, which play an extremely important role in the theory of oscillations and nonlinear mechanics . In fact, all methods for studying periodic solutions are, at the same time , methods for the investigation of the corresponding periodic boundaryvalue problems . This is related to the known fact [Har] that a T-periodic solution x(t) of the differential equation x = f(t,x), f(t+T,x) = f(t,x),

(0.1)

satisfies the periodic boundary condition x(0) = x(T)

(0.2)

and, conversely , if x(t ) is defined for t E [0, TI and is a solution of the periodic boundary-value problem (0.1), (0.2), then it is the restriction of a T-periodic solution of equation (0.1) defined for t E (- -, oo) to the segment [0, T]. Among the most powerful and universal analytic methods in the theory of oscillations, one should mention asymptotic methods of nonlinear mechanics, the method of a small parameter, and averaging methods . A notable contribution to the development of these methods was made by mathematicians of the Kiev scientific school. Asymptotic methods created for various types of equations in the fundamental works of N. M. Krylov and N . N. Bogolyubov [KrB], N. N. Bogolyubov and Yu. A . Mitropolsky [BoM], and by Yu. A. Mitropolsky [Mit] were developed and generalized in the monographs of the following authors : N. N. Bogolyubov , Yu. A. Mitropolsky , and A. M. Samoilenko [BMS ], V. I. Arnold [Am], A. N. Filatov [Fil], Yu. A. Mitropolsky and 0. B. Lykova [MiLy ], E. F. Mishchenko and N. Kh . Rozov [MiR], Yu. A. Mitropolsky and B. I. Moseenkov [MiMo], Yu. A. Mitropolsky and D. I. Martynyuk [MiM], N. N. Moiseev [Moi], S . A. Lomov [Lom ], N.J. Shkil, A. N. Voronoi , and V. N. Leifura [SVL], A. V. Skorokhod [Sko], V. S. Korolyuk , N. S. Bratiichuk, and V . Pirdzhanov [KBP], A. M. Samoilenko [Sams] , Yu. A. Mitropolsky and A. K. Lopatin [MiLl, Yu. A. Mitropolsky, A. M. Samoilenko , and D. I. Martynyuk [MSM]. In the theory of oscillations and related fields , analytic methods were developed and studied in the monographs of L. Cesari [Ces2], Ph. Hartman [Har], V. A. Pliss [Pli2], V. Vazov [Vaz ], A. D. Myshkis [Mysi], A . B. Vasil ' eva and V.F. Butuzov [VaB], K. G. Valeev and 0. A. Zhautykov [VaZ], A. I. Pavlyuk, V. M. Burym, and Yu. A. Pasenchenko [PBP], I. G. Malkin [Mal], R. Z. Khasminsky [Kha], A. P. Proskuryakov [Pro], Yu. A. Mitropolsky , A. M. Samoilenko, and V. L. Kulik [MSK], A. A. Martynyuk [Mara], E. F. Tsarkov [Tsa], and D. G. Korenevsky [Kor].

Introduction 3 The application of a combination of the Lyapunov method, the Poincare method of small parameter, and the averaging method to nonautonomous quasilinear systems and autonomous equations was considered in the monograph of V. M. Starzhinsky [Sta]. Sufficient existence conditions and asymptotics of periodic solutions of differential and integro-differential equations with small parameter are presented in the book of Ya. V. Bykov and D. Ruzikulov [ByR]. Some practical aspects of the use of analytic methods based on the method of perturbations, averaging, and iterations in the theory of nonlinear oscillations are described in the works by T. Hayashi [Hay], M. Farkas [Farl], M. Farkas and H. I. Freedman [FaF], Yu. V. Trubnikov and A. I. Perov [TrP], and V. Volterra [Vol]. The analytic theory of linear periodic boundary-value problems is developed in the monographs of N. P. Erugin [Eru], V. A. Yakubovich and V. M. Starzhinsky [YaS], M. V. Fedoryuk [Fed], S. F. Feshchenko, N. I. Shkil, and L. D. Nikolenko [FSN], and G. S. Zhukova [Zhu]. Analytic methods were successfully used in the investigation of the interrelation between the problems of existence and sign-constancy of Green functions and the problems of existence and uniqueness of solutions of periodic and nonperiodic boundary-value problems in the scalar and vector cases (A. Ya. Khokhryakov [Kho], A. I. Perov [Perol], A. V. Kibenko [Kib2]). Some applied problems in the theory of nnlinear oscillations were investigated in the monograph of E. N. Rozenvasser [Roz] with the use of the methods of integral equations. Numerous works are devoted to the investigation of nonperiodic boundary-value problems, in particular, with nonseparable boundary conditions. In these works, the problem of unique existence of solutions and the properties of the Green function are mainly analyzed (see, e.g., the works by G. N. Zhevlakov, Yu. V. Komlenko, and E. L. Tonkov [ZKT], V. M.Zubov [Zub], and Yu. V. Pokomyi [Pokl]). A considerable contribution to the development of one of the basic directions in analytic methods for the investigation of boundary-value problems, namely, methods based on a priori estimates and differential inequalities, was made by the mathematicians of the Riga school. The results of these investigations with detailed bibliographical notes are presented in the monographs of V. V. Gudkov, Yu. A. Klokov, A. Ya. Lepin, and V. D. Ponomarev [GKLP], N. I. Vasil'ev and Yu. A. Klokov [VaK], and A. Ya. Lepin and L. A. Lepin [LeL]. Two-point boundary-value problems for scalar second-order equations and for systems of first-oder and second-order equations whose right-hand sides satisfy either conditions of continuity or the Caratheodory conditions were studied in [GKLP]. With the extensive use of the method of a priori estimates, the notion of lower and upper solutions introduced by M. Nagumo [Nag], and the inclination functions, necessary and sufficient conditions for the unique solvability of the boundary-value problems were obtained. In the same work, quasilinear multipoint boundary-value problems for systems of equations of the following form are also considered: x = A(t)x + g( t,x), t r= [a, b], m

I A.x(t,) = g (x ( tt ), ... , x(tm)),

1=1

4 Introduction n

t r Car (I X Rn), ( E C" (Rnm) Recall that sufficient conditions for the solvability of the scalar two-point problem z = f (t, x, x ), x (a) = A, x (b) = B under the assumption of analyticity of the righthand side f (t, x, i) were established in [Ber]; in [Nag], they were generalized to the case where the right-hand side is a continuous function. Further results in this direction are described, in particular, in the works of Z. Opial [Opi] and I. T. Kiguradze [Kigl], [Kig2]. The results obtained in [GKLP] were generalized and developed in [VaK] and [LeL]. In these works, on the basis of the method of a priori estimates, parallel with the problem of existence, the problem of continuous dependence of solutions on the data of problems and boundary-value problems with nonsummable singularities were considered. Boundary-value problems with a condition at infinity were studied in detail by Yu. A. Klokov [Klo]. Some problems in the theory of linear boundary-value problems with a condition at infinity for equations in a Banach space were studied by G. V. Radzievsky [Rad]. In analytic methods for the investigation of boundary-value problems, one can also use the estimates obtained by Ya. D. Mamedov, S. Ashirov, and S. Atdaev [MAA]. Fundamental results in the theory of nonlinear singular differential equations with functional two-point and multipoint boundary conditions are presented in the known monograph of I. T. Kiguradze [Kig2] and in the works of I. T. Kiguradze [Kig3], I. T. Kiguradze and B. L. Shekhter [KiS], T. A. Chanturiya [Cha], M. A. Kakabadze [Kak], B. L. Shekhter [She], Sh. M. Gelashvili and I. T. Kiguradze [GeK], M. T. Ashordiya [Ash], and D. G. Bitsadze and I. T. Kiguradze [BiK]. An independent direction in the development of the theory of boundary-value problems for differential systems generalized in the Kurzweil sense was developed by J. Kurzweil [Kur], S. Schwabik, M. Tvrdy, and O. Vejvoda [STV], and S. Schwabik [Sch]. Important and interesting results for boundary-value problems in the case of functional differential equations were obtained by N. V. Azbelev [Azbl], [Azb2], N. V. Azbelev and V. P. Maksimov [AzM], N. V. Azbelev and L. F. Rakhmatullina [AzR], L. F. Rakhmatullina [Rak], V. P. Maksimov [Mak], and M. E. Drakhlin [Dra] and, for the problem of existence of periodic solutions of functional differential equations with unbounded delay, by J. Mawhin [Maw2], T. A. Burton and L. Hatvani [BuH], and L. Hatvani and T. Krisztin [HaK]. Some aspects of the theory of boundary-value problems for differential-difference and functional differential equations were also developed by A. N. Sharkovsky, Yu. A. Maistrenko, and E. Yu. Romanenko [SMR], G. P. Pelyukh and A. N. Sharkovsky [PeS], G. A. Kamensky and A. D. Myshkis [KaM], M. I. Kamensky [Kam], and V. G. Kurbatov [Kur]. The idea of a priori estimates and differential inequalities, realized for some problems in the monograph of S. A. Chaplygin [Chap], was further developed by K. W. Chang and F. A. Howes [ChH] for the investigation of the problem of existence and asymptotic behavior of certain classes of singularly perturbed boundary-value problems. A combi-

Introduction 5 nation of asymptotic and qualitative methods was extensively used by V. V. Strygin and V. A. Sobolev [ StS] for the investigation of singularly perturbed systems. Analytic methods for the investigation of differential equations with discontinuous right-hand side and differential inclusions were developed by A. F. Filippov [Fili] and A. A. Tolstonogov [Toll. The investigation and solution of problems related to boundary-value problems were also performed in the monographs of F. V . Atkinson [Atk] and A. Kufner and S. Fucik [KuF] and in the works of Yu. V. Pokornyi [Pok2], A. I. Perov and A. V. Kibenko [PeK], Yu. V. Komlenko [ Kom], R. E. Gaines and J. Mawhin [GaM1] , and A. D. Wood and F. D. Zaman [WoZ]. The analytic theory of periodic solutions and boundary-value problems, the problems of dichotomy , reducibility , and branching were considered in the works of E. I. Grudo [Gru], N. A. Izobov [Izo ], V. M. Millionshchikov [Mil], Z. P. Ordynskaya [Ord], A. M. Samoilenko and R. I. Petrishin [Safe] , A. M. Samoilenko and Yu. V. Teplinsky [SaT1], and V. I.Tkachenko [Tka]. Some questions related to finding the initial conditions of periodic solutions were considered in the book by V. I. Mironenko [Mir]. The class of bilateral processes of successive approximations monotonically convergent to the required solutions can also be attributed to analytic methods . The theory of such methods for periodic solutions is presented in the monograph of N. S. Kurpel and B. A. Shuvar [KuS]. The monograph of V. I. Fushchich, V. N. Shtelen, and N. I. Serov [FSS] also enriched the arsenal of analytic methods. In this book, on the basis of symmetry analysis , the authors reduced certain multidimensional problems of mathematical physics to ordinary differential equations and constructed their exact solutions. The group of functional- analytic methods widely uses the apparatus of functional analysis , topological notions , and the theory of approximate methods for solving operator equations. The foundation for these methods, in their contemporary conception , for periodic boundary-value problems was laid by M. A. Krasnoselsky [KVZRS], L. Cesari [CesI], and J. Mawhin [Maw4] . For periodic and multipoint boundary-value problems in the resonance and nonresonance cases , these methods were developed by Fam Ki Anh [Fam], A. I. Kolosov [Kol ], and V. Ya. Derr [Der] . The method of point mappings, which is widely used in the theory of oscillations , can also be attributed to functional -analytic methods; this method is presented in the monographs of A. A. Andronov, A. A. Vitt, and S. E. Khaikin [AVK], Yu . I. Neimark [Neil, and N. V. Butenin, Yu. I. Neimark, and N. A. Fufaev [ BNF]. For differential equations, these methods were also developed by A. D. Bryuno [Bry], K. S. Sibirsky [Sib], and N. I. Vulpe [Vul]. A very important class of functional-analytic methods is formed of the methods that use the properties of topological degree , the coincidence degree, and the continuation theorem [Maw4]. With the use of this apparatus, interesting results for periodic and general boundary -value problems were obtained by A. Capietto, J. Mawhin, and F.Zanolin [CMZI]-[CMZ4] and A. Capietto and F. Zanolin [CaZ]. Under the assumption of the existence of solutions, the group of numerical methods gives practical algorithms for approximate construction of solutions of boundary-value problems . One part of numerical methods is aimed at the determination of the initial

6 Introduction values of the required solutions , i.e., at the reduction of boundary -value problems to Cauchy initial-value problems. The other part of numerical methods is aimed at finding solutions of boundary-value problems on the entire domain of variation of the independent variable. The development and justification of various numerical schemes for solving boundary-value problems such as the sweep method, the shooting method , the immersion method, the method of reduction to Cauchy problems, the finite-difference and quadrature-difference methods, the method of subdomains, the method of spline approximation, the Newton method, the quasilinearization method, and eigenvalue problems are considered in numerous works, e.g., the works of N. S. Bakhvalov [Bak], J. K. Batcher, J. L. Lambert, A. Prottero, et al. [BLP], B. N. Pshenichnyi and Yu. M. Danilin [PsD], I. Taufer [Tau], J. Casti and R. Calaba [CaK], E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders [DMS], F. P. Vasil'ev [Vas], J. M. Ortega and W. G. Poole [OrP], and G. E. Pukhov [Puk]. Various aspects of numerical methods and their applications were studied by A. A. Abramov and V. B. Andreev [AbA], Z.Aktas and H. J. Stetter [AkS], G. M. Vainikko [Vai2], A. M. Samoilenko and V. A. Ronto [SaRol], M. Ronto, E. P. Semagina, and E. N. Dzhigun [RSD]. Among contemporary methods for the investigation of nonlinear boundary-value problems, an important place belongs to numerical-analytic methods for the investigation of the problem of existence and finding approximate solutions. We understand numerical-analytic methods as methods that enable one to represent the required solution in an analytic form, although some of its parameters or coefficients are determined numerically. According to this definition, first of all, all direct methods of mathematical physics, i .e., all variational and projective methods , including the collocation method, can be attributed to the group of numerical-analytic methods. Recall that, according to S. L. Sobolev [Sobo], direct methods of mathematical physics are defined as methods for solving problems of the theory of integral and differential equations based on the replacement of these equations by algebraic ones . It is clear that, in the sense of this definition, finite-difference methods for solving boundary -value problems can also be attributed to direct methods. We note that the history of development and application of the theory of direct methods and relevant questions are presented in detail in the monograph of A. Yu . Luchka and T. F. Luchka [LuL]. On the basis of projection and projection-iterative methods, the numerical-analytic methods for periodic and more general boundary-value problems were developed by M. Urabe [Ural], [Ura2], A. Yu. Luchka [Luc I], A. I.Perov [Pero2], A. M. Samoilenko and O. D. Nurzhanov [SaN], A. M. Samoilenko and 1. 0. Parasyuk [SaPa], and I. A. Lukovsky, M. Ya. Barnyak, and A. N. Komarenko [LBK]. In the theory of periodic solutions, numerical-analytic schemes on the basis of iterative methods were, apparently, first developed by L. Cesari [Ces2], N. N. Bogolyubov, Jr., and B. I. Sadovnikov [BoS], J. K. Hale [Hall], [Ha12], and A. M. Samoilenko [Saml][Sam3] . Later, numerical-analytic methods for periodic and general boundary-value problems were developed on the basis of iterative schemes by A. M. Samoilenko and V. A. Ronto [SaRo2], A. M. Samoilenko and V. N. Laptinsky [SaLa], Le lyong Tai [Le13],

Introduction 7 A. A. Boichuk [Boi], E. P. Trofimchuk [TroI], and A. M. Samoilenko and B. P. Tkach [SaTk]. A significant contribution to the development of numerical-analytic and functionalanalytic methods and their application to the numerical solution of nonlinear boundaryvalue problems for ordinary differential equations was made by the mathematicians of the Tartu school. Theoretical problems related to the theory of approximate methods for solving operator equations and discretization and approximation methods were investigated in the works of G. M. Vainikko [Vail], [Vai4], G. M. Vainikko, A. Pedas, and P. Uba [VPU], G. M. Vainikko and A. Yu. Veretennikov [VaV], and M. A. Krasnoselsky, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko [KVZRS]. In these works, as well as in the works G. M. Vainikko [Vai2], [Vai3], [Vai5] and G. M. Vainikko and P. Kh. Miilda [VaM], the general results obtained for operator equations were used and various methods for solving broad classes of initial-value and boundary-value problems in the case of differential, integral , and integro-differential equations were justified. The main attention was given to the Galerkin method, method of mechanical quadratures, method of subdomains, and difference methods . The collocation method for scalar linear and nonlinear differential equations of the in th order in the case of twopoint linear boundary conditions was first justified in [Vail] and [Vai2]; in particular, in the work of G. M. Vainikko [Vai6], these results were generalized to the case of nonlinear boundary conditions for a problem of the form Au := u(m)- f (t,u,u(l),...,u (m_l)) = 0, a 5 t <_ b, Bju (pj(u(a), u(b), u(1)(a), u(')(b)..... u(m-l)(a), u(m_l)(b)) = 0, j = 1,2,...,m, and also to the problem of periodic solutions of an autonomous differential equation of the m th order given in an implicit form as follows:

g(v(T), U(')(T), ... , V(m)(T)) = 0.

A fairly extensive bibliography on the application of numerical and numerical-analytic methods in the theory of periodic and multipoint boundary -value problems is presented in the monograph of A. M. Samoilenko and M. Ronto [SaR4]. One more direction in the development of numerical -analytic methods is based on the combination of the ideas and methods of the theory of approximation of functions, theory of differential and integral equations , and computational mathematics. On the basis of this approach, V. K. Dzyadyk [Dzy] developed an approximation method for the solution of linear differential equations with polynomial coefficients and an approximation -iterative method for the solution of nonlinear differential and integral equations with analytic conditions . Some applications of these methods can be found in the works of V. I. Bil-

8

Introduction

enko [Bil] and V. I. Bilenko, V. N. Konovalov, I. A. Lukovsky, A. Yu. Luchka, G. E. Pukhov, and M. Ronto [BKLLPR]. The determining role in the process of formation and development of constructive aspects of numerical-analytic methods and their adjustment for realization on computers for finding periodic and quasiperiodic solutions of nonlinear differential equations was played by the known monographs of E. A. Grebenikov and Yu.A. Ryabov [GrRl][GrR3], D.K.Lika and Yu.A. Ryabov [LiR], E. A. Grebenikov [Gre], and E. A.Grebennikov, M. N. Kiosa and S. V. Mironov [GKM]. The theory of construction of numerical-analytic periodic and quasiperiodic solutions for a broad class of differential equations with the use of a computer was developed in [GrR3], This theory is based on setting the analytic structure of solutions and finding the numerical coefficients of the symbolic formulas. It was also shown that, for the construction of solutions, one can efficiently use various iteration processes. In this case, it is required that theright-handside of an equation have a polynomial representation with coefficients in the form of Fourier polynomials. In this direction, fairly cumbersome operations such as multiplication and exponentiation of the Fourier polynomials and substitution of a series into another series were formalized for realization on a computer. In [GKM], a constructive theory of regularly perturbed systems of the form ^ = \iX(x, v,u), at

^ = Q)(x) + u K U y . u ) at

was developed, which is understood as the investigation of the properties and construction of explicit solutions. For this purpose, a computational algorithm for the computer realization of the Krylov-Bogolyubov transformation with any given accuracy with respect to n was developed, which, in turn, uses computer-oriented operations over analytic expressions given by power and trigonometric polynomials and entire, fractional-rational, or polyharmonic functions, multiple substitutions of truncated series one into another, and the computer verification of hypotheses about the analytic structure of partial and first integrals. Note that one of the first stages in the development of the class of numerical-analytic methods was the stage of "transformation" of analytic methods into numerical-analytic ones and their "adaptation" for realization on computers. In particular, this approach was developed in connection with the appearance of the possibility of executing symbolic operations on a computer. In a certain sense, a similar stage was also passed by the asymptotic methods of nonlinear mechanics in the works of Yu. V. Blagoveshchensky, Yu. S. Fishman, and V. A. Shcherbov [BFS], K. G. Valeev and E. I. Storozhenko [36], and A.I.Skripnik[174]. This approach produced significant positive results, especially in the theory of oscillations. To a lesser extent, it affected the methods for the construction of solutions of nonperiodic boundary-value problems, but, surely, this was far from exhausting all possibilities in the development of numerical-analytic methods. New special methods have appeared which, by their nature and structure, are numerical-analytic methods. As a rule, these methods are based on properly constructed itera-

Introduction 9 tion schemes and enable one to represent an approximate solution in an analytic form and, using its properties , establish the solvability of the boundary -value problem under consideration . However, in this case, some parameters in the analytic solution , e.g., its initial value, are determined numerically . In the monographs of A. M. Samoilenko and M. Ronto [SaRI ], [SaR2] , and [SaR4], such numerical-analytic methods were justified for the case of periodic boundary conditions (0.2) as well as for two -point or multipoint nonseparable linear boundary conditions Ax(a) + Cx(b) = d,

det C * 0, (0.3)

P

E Bj x(tj)

= d,

(0.4)

1=0

for nonautonomous and autonomous equations and for equations with delay of the form

x =f(t, x), x X

=

x =f(x),

(0.5)

= f(t,x (t),x(t -'L,)),

f(t,x(t), x(t -til), x(t), z(t -ti2))•

(0.6)

In connection with this , the following schemes were developed: I. Numerical-analytic method of successive periodic approximations that enables one to approximately construct solutions of the periodic boundary-value problem (0.1), (0.2) and investigate the problem of existence of solutions with the use of a uniformly convergent sequence of T-periodic functions. II. Numerical-analytic method of successive approximations that enables one to establish the existence of solutions of the two-point boundary-value problems (0.1), (0.3), (0.4) and find approximate solutions with the use of a uniformly convergent sequence of functions that satisfy the boundary conditions (0.3), (0.4). III. Method of trigonometric collocation for finding T-periodic solutions of equations (0.1), (0.6) in the form of trigonometric polynomials. IV. Method of algebraic collocation for finding solutions of the two-point and multipoint boundary-value problems (0.1), (0.6), (0.3), (0.4) in the form of algebraic polynomials. V. Numerical-analytic method of successive trigonometric approximations for the boundary-value problem (0.1), (0.2) in the case where successive approximations are constructed in the form of trigonometric polynomials.

Introduction

10

VI. Numerical-analytic method of algebraic approximations for the boundary-value problem (0.1), (0.3) in the case where successive approximations have the form of algebraic polynomials satisfying the boundary conditions (0.3). VII. Methods for approximate construction and numerical solution of the determining equations that specify the initial values of solutions of the linear equations x = A(t)x + f (t) and the nonlinear systems (0.1) in the case of two -point and multipoint linear and nonlinear boundary conditions of the form (0.3), (0.4), and

L(x(to ), x(t1),...,x(tP)) = 0.

(0.7)

In recent years , some of the methods listed above were enriched with new generalizations and applications to new classes of problems . Thus , in the paper by A. M . Samoilenko and Kh. Ovezdurdyev [SaO], the numerical -analytic method I was extended to the differential equation f (t, x), x >_ 0, X

=

1f2 (t, x),

x < 0,

with discontinuous right-hand side . In the works of N. A. Evkhuta and P. P. Zabreiko [EvZ1]-[EvZ3] and N. A. Evkhuta [Evk], this method was generalized for the investigation of periodic solutions of differential equations in Banach spaces of the form dx =Ax +f(t,x), dt

(0.8)

where A is an operator acting in a Banach space . Equation (0.8) encompasses infinite systems of differential equations , integro-differential equations, and partial differential equations of parabolic and hyperbolic type. The justification of method I for the investigation of periodic solutions of operator differential equations of the form \ dx 'x t, x, d , Axl , -f(t,x,Ax) d d t2 = f( dt

d =.f(x,Ax)

was given by G. D . Zavalykut and O. D. Nurzhanov [ZaN] and G . D. Nefedova [Nef]. In the papers of D. D. Bainov and G. H. Sarafova [BaS2] , G. H. Sarafova and D. D. Bainov [SaB1]-[SaB3], and M. G. Gospodinov [Gos], the numerical -analytic method was used for the investigation of periodic solutions of differential equations with "maxima" and integro-differential equations.

Introduction

11

The numerical - analytic method was further generalized to the case of differential, functional differential , and difference equations by M. Kwapisz [Kwal ]-[Kwa6] and A. Augustynowicz and M. Kwapisz [AuK]. Note that, at present, the theory of differential equations with impulse influence is being extensively developed. Most recent results obtained in this area are presented in the monographs of S. T. Zavalishchin, A.N. Sesekin , and S. E. Drozdenko [ZSD], S. T. Zavalishchin and V. I. Sukhanov [ ZaS], and A. M. Samoilenko and N. A. Perestyuk [SaP3] and in the papers of N. A. Perestyuk [Per2] , M. Levi [Lev], E. Yu. Mamsa [Mam], A. M. Samoilenko , N. A. Perestyuk, and S . I. Trofimchuk [SPIT], and M. Ronto and S. I. Trofimchuk [RoTl]. A notable contribution to the theory of differential equations with impulse influence was made by A. B. Dishliev and D. D. Bainov [DiB], S . G. Hristova and D. D. Bainov [HrB 1 , HrB2] , P. S. Simeonov and D. D. Bainov [ SiB], G. K. Kulev and D. D. Bainov [KuB], D. D. Bainov and S. D. Milusheva [BaM], N. V. Milev and D . D. Bainov [MiB], D. D. Bainov, S. I. Kostadinov , and A. D. Myshkis [BKM], and by G. H. Sarafova and D. D. Bainov [SaB4]. At present, much attention is also given to another field of the theory of boundaryvalue problems , namely , to the investigation of solutions of problems with parameters either in the differential equation or in the boundary conditions . Two-point boundaryvalue problems for differential equations with a parameter of control on the right-hand side were studied by the functional-analytic method by A. V. Kibenko and A. I. Perov [KiP], A. V. Kibenko [Kibi] , and M. Feitkan [Feci ], [Fed2]. Solvability conditions and algorithms for the construction of solutions of such problems and equations with delay were developed by V. V. Mosyagin [Mosy] , I. A. Goma [Gom], and Z. B. Seidov [Sei]. Multipoint boundary-value problems with parameters were studied by N. S . Kurpel and A. G. Marusyak [KuM]. A fairly universal method for a two -point boundary-value problem for the equation z = f (t, x, A.)

(0.9)

was analysed in the paper of Yu . S. Eidelman [Eid], where, instead of (0.9), the following second-order equation obtained by eliminating the parameter A. from system (0.9) was considered:

of x = ax 1_ .i +at

(0.10)

For the solution of boundary-value problems with parameters in the case of differential and integro-differential equations , the following well-known approximate methods are also used : the Newton method (0. Khosabekov [Khos]) and the method of averaging functional corrections (K. T. Akhmedov, N. A. Svarichevskaya, and M. A. Yagubov [ASY]). Solutions of boundary -value problems with parameters were investigated with the use of the numerical- analytic method of successive periodic approximations and its modifica-

12 Introduction tions combined with bilateral and projection-iterative methods by P. 1. Sobkovich [Sob2] and with the use of projection -iterative methods by A. Yu . Luchka [Luc2], [Luca]. Boundary-value problems with parameters are of independent interest . At the same time, it was shown by D. S . Dzhumbaev [Dzh] and N. S. Kurpel and A. G. Marusyak [KuM] that some multipoint and periodic boundary -value problems without parameters can be solved by their transformation to problems with parameters . In [Sob2], this approach was used in the case of a multipoint boundary - value problem of the general form (0.1), (0.7 ), for which a numerical-analytic method for the investigation of solutions was presented. Note that, in recent years, some of the numerical- analytic methods I-VII listed above have been successfully applied to rather complicated objects such as the boundary-value problems with parameters and boundary - value problems with impulse influence. Thus, A. M. Samoilenko , M. Ronto, and V. A. Ronto [SRR], M. Ronto and V. A. Ronto [RoR], M. Ronto and I. I. Korol [RoK ], and M. Ronto [Ron3] generalized method II for the investigation of existence and construction of solutions of boundary -value problems in the case where the boundary conditions linearly or nonlinearly depend on the parameter. A. T. Alymbaev .[Aly3] analyzed the possibilities of method I in the case where the Newton iteration process is used as a method of successive approximations . A. M. Samoilenko and M. Ronto [SaR5 ] considered one of possible approaches for the acceleration of convergence of successive approximations in methods I and II, which is based on the combination of these methods with the collocation method. Efficient algorithms for the construction of solutions of broad classes of nonlinear boundary-value problems for impulsive systems of differential equations on the basis of methods III and IV were justified by A. M . Samoilenko, M. Ronto, and 0.0. Kurbanbaev [SRK1], [SRK2], M. Ronto and A. Tuzson [RoTu], and A. Tuzson [Tuz]. Fairly general nonlinear multipoint boundaryvalue problems with restrictions given by linear functionals were studied by M. Ronto and T. V. Putyatina [RoPl] , [RoP2] and T . V. Putyatina [Put] on the basis of method IV. In the paper of A. M . Samoilenko and Le Lyong Tai [SaL2], method II was justified for the case of nonlinear differential equations with nonlinear boundary conditions. A numerical- analytic method of successive approximations for the investigation of periodic solutions of countable systems of second - order differential equations was developed by M. Ronto and 0 . M. Martynyuk [Rolla] . In the paper of A. M . Samoilenko, M. Ronto, and S. V. Martynyuk [ SRM], method II was modified for the investigation of solutions of nonlinear systems of differential equations with boundary conditions that involve definite integrals. Various classes of three -point boundary-value problems were considered in the works of M. Ronto and T. V. Savina [RoS] and M. Ronto and R. Tegen [RoTB ] with the use of the numerical -analytic method . The problems related to the investigation of convergence of the numerical- analytic method and the extension of its applicability were studied by M. Ronto and F. Mdszaros [RoM]. Some of methods VII were extended to new problems and, in the paper of M. Ronto and L. I. Karandzhulov [RoK], the accuracy of these methods for finding the initial values of solutions of linear boundary -value problems for integro-differential equations was increased.

Introduction

13

A periodic boundary -value problem for nonlinear systems of differential equations whose right-hand sides satisfy the Caratheodory conditions was studied by M. Ronto and S. I. Trofimchuk [RoT2] with the use of the numerical-analytic method. In this case, the method was specified, in particular, for differential equations with sublinear growth. Note that the application of the technique of the numerical -analytic method developed in this paper to autonomous systems gives interesting estimates for the lower bound of the period of T- periodic solutions. These results are close to those obtained in the known works of F. A. Yorke [Yor], A. Lasota and F. Yorke [LaY], S. Busenberg, D. Fisher, and M. Martelli [BFMI ], [BFM2], S. Busenberg and M. Martelli [BuM ], and D. Fisher [Fis]. The numerical- analytic method for the investigation of T-periodic solutions of differential and difference equations in partially ordered Banach spaces was justified by M. Ronto, A. N. Ronto, and S. I. Trofimchuk [RRT]. Here, both for differential and difference equations , main attention was given to the improvement of conditions of convergence of the method. Furthermore , sharper conditions were obtained for the convergence of the numerical -analytic method , and the domain of its applicability was extended as compared with the corresponding results in [RoM], [Kwa3], [Kwa4 ], and [SaR4]. Furthermore, similarly to [RoT2], the connection with the problem of the lower bound of the period of periodic solutions of dynamical systems was discussed. The present monograph summarizes , in a certain sense , these investigations. We describe new promising directions in the development of numerical -analytic methods for the investigation of solutions of nonlinear boundary -value problems in the case of general boundary conditions , problems with parameters of control, and boundary -value problems for impulsive systems. Following the tradition of the previous works [SaRl ], [SaR2], and [SaR4] , we study both periodic and nonperiodic boundary-value problems from the common point of view. In Chapter 1, we construct a numerical-analytic scheme that can be simultaneously applied to the investigation of existence of solutions and to the approximate determination of solutions of the system of differential equations (0.1) with the two-point nonseparable boundary conditions (0.3), where the matrix C is degenerate and det (k1 A + k2 C) # 0 for some fixed real numbers kt and k2. We prove the uniform convergence of approximate solutions to the exact solution and estimate the error of the approximate solution and its initial value . We establish sufficient conditions for the solvability of the boundary -value problem under consideration . We also obtain necessary conditions that allow one to conclude whether some subdomain can contain points that are the required initial conditions . The numerical-analytic method is justified for a nonlinear system of differential equations i = f (t, z) with the two-point nonlinear boundary conditions g (z (0), z ( T)) = 0 and for nonlinear boundary -value problems of the form i = ef(t, z),

Az(0 ) + Bz(T) = ecp (z(0), z(T)),

which contain a small parameter e. In Chapter 2, we justify a new modification of the numerical -analytic method of successive approximations for the investigation of solutions of nonlinear systems of differen-

14 Introduction tial equations of the form (0.1) with the periodic or linear two-point boundary conditions (0.2), (0.3). The fact that the considered schemes of successive approximations admit certain arbitrariness in the choice of zero approximation enables the efficient joint use of different numerical-analytic methods. Thus, for a periodic boundary-value problem, we construct a sequence of periodic functions depending on a parameter x0, which is, in fact, a correction to the initial value obtained by another numerical-analytic method, namely, by the method of trigonometric collocation. It is shown that the use of the approximate solution obtained by the method of trigonometric collocation as the zero iteration allows one to accelerate the convergence of approximate solutions to the exact solution. We analyze in detail the properties of exact and approximate determining equations that determine the required value of the parameter x0. On the basis of the properties of the mth approximation of a periodic solution and the approximate determining equation, we obtain sufficient conditions for the existence of periodic solutions. We also establish necessary conditions that enable one to conclude by the properties of an approximate solution whether the value of the parameter x0 that specifies the exact initial value of a periodic solution is contained in the given domain of definition of the right-hand side of the system of differential equations (0.1). The solvability of the approximate determining equation is investigated. We also estimate the error of an approximate periodic solution constructed with the use of a certain approximation to a solution of the approximate determining equation. Similar problems are also considered for the nonlinear two-point boundary-value problem (0.1), (0.3). In this case, the classical scheme of the numerical-analytic method of successive approximations is combined with the approximations obtained by the method of algebraic collocation. In Chapter 3, we study boundary-value problems for the nonlinear system of differential equations (0.1) in the case where the boundary conditions depend on one or more parameters of control, namely, Ax(0) + ? Cx(T) = d, A.1Ax(0 ) + Cx(?) = d, X1Ax(0) + X2Cx(T) = d,

x1(0) = x10, (0.11)

x1 (0) = x1o, x1(0) = x10,

x2(0) = x20,

x2(0) = x20•

(0.12) (0.13)

For the boundary-value problems (0.1), (0.11)-(0.13), we develop a numerical-analytic method that enables one to investigate the existence of a solution, which is understood as a pair { %, x(t) } that satisfies the corresponding differential equation and given boundary conditions. An algorithm for the approximate construction of solutions is proposed. We establish conditions that guarantee the uniform convergence of approximate solutions and give sufficient conditions for the solvability of problems with parameters in boundary conditions. We study the properties of the exact and approximate determining equations for boundary-value problems with parameters of control. We also consider problems related to the estimation of the error of the solution and its initial value at t = 0. The

Introduction

15

method is illustrated by an example of a two -point boundary-value problem with one parameter in the boundary conditions. In Chapter 4, we investigate two-point and multipoint boundary-value problems for linear and nonlinear differential equations with impulse influence . We consider the case where a given differential equation is subjected to impulse influence at a single fixed point. In this case, first, we introduce and explicitly construct the Green function for the linear homogeneous two-point boundary-value problem x = A(t)x,

t # 't1,

t E [a, b],

x E R", it E (a, b),

S1x(t1+0) + S2x(ti1-0 ) = 0, Blx(a ) + B2x(b) = 0 and then, with the use of this function , we obtain solutions of semihomogeneous boundary-value problems with impulse influence. We justify the method of algebraic collocation for finding solutions of boundaryvalue problems in the case of nonlinear ordinary impulsive differential equations of the form z= f(t,x)+ (p (t), t# 't1, t E [a, b], x,f,cp c- R", S1x(ti1 + 0 ) + S2x('C1-0 ) = q, Blx(a ) + B2x(b) = d. We investigate the convergence of the method of trigonometric collocation for nonlinear periodic impulsive systems, namely, for the periodic boundary-value problem

x = f(t,x) + (p (t), (p (t) = cp (t+T),

t # 'cl,

f(t,x) = f(t+T,x),

x(T1 +0) - x('r1-0) = q,

x(0) = x(T).

We prove the convergence of the approximate solutions obtained by the collocation method to the exact solution and present an estimate of the error. Further, we study multipoint boundary-value problems with finitely many impulse influences . For various classes of multipoint boundary -value problems with impulse influence at fixed times, we introduce and explicitly determine the Green function. In particular, we perform this for the following three-point boundary -value problem with impulse at a single point t = rl: .x = A(t)x, x E R", t # 'cl, t E [a, b], Cx(t1-0) + Dx(t1+0 ) = 0,

't1 E (a,b),

Box(a ) + Blx(tl) + B2x(b) = 0.

detD # 0,

Introduction

16

We demonstrate the dependence of the Green function on the location of the time of impulse influence. We study a two-point linear m -pulse boundary -value problem of the form x = A(t)x+f(t),

x,f E R",

t * ri ,

i = 1,2,...,m,

C;x(tii- 0) + Dix('[;+0 ) = 0, i = 1,2,..., m, B1x(a) + B2x(b) = 0. We also investigate multipoint m-pulse homogeneous and inhomogeneous linear boundary-value problems of the form x = A(t)x+f(t), t ^ ri,

i = 1,2,...,m,

Cix(tii-0) + Dix(tii+0) = d;, i = 1,2,...,m, m+2 A,, x(tk) = dm+1, tk # Ti,

i = 1,2,...,m,

k=0 tk < rk < tk+1, k = 1,2,...,m.

We consider the application of the method of trigonometric collocation to the construction of T-periodic solutions of systems of differential equations (f (t, x), t e [0, ti),

dx=f(t,x)={l

dt f2 (t, x), t E (ti, T],

with periodic boundary conditions and impulse influence x(0) = x(T), S1x(,c+0) + S2x(ti-0) = q. The theoretical propositions are also illustrated by examples. In Appendix, written by the authors together with S. I. Trofimchuk , we analyze the history of development of the theory of the numerical-analytic method of successive approximations for about 30 years . The achievements and modem trends in the development of the method are described . In numerous cases, the results obtained earlier are improved. New, still unresolved, problems connected with the numerical -analytic method and its applications are formulated.

Introduction 17 We present fairly complete bibliographical notes of domestic and foreign publications related to the numerical-analytic method. We present the history of the method and reveal the relationship, unnoticed before, between the numerical-analytic method and the Lyapunov-Schmidt and Cesari-Hale methods for weakly nonlinear systems, the Cesari method for essentially nonlinear systems, and some results obtained by J. Mawhin. We discuss the problem of application of the numerical-analytic method to the investigation of various classes of boundary-value problems for second-order equations, equations with delay, functional differential equations, equations with "maxima," integro-differential equations, and operator differential equations. We analyze the schemes of the numerical-analytic method for autonomous systems, equations with impulse influence, difference equations, multipoint boundary-value problems, abstract differential equations, and equations unresolved with respect to the derivative. We also consider problems related to new trends in the development of the numerical-analytic method. We hope that the numerical-analytic methods considered in the present book will gain further applications to new classes of boundary-value problems. The authors are grateful to Academician Yu. A. Mitropolsky for his interest in these investigations and to Academicians B. N. Pshenichny and I. A. Lukovsky for their remarks made in the process of work on the manuscript of this book.

i. IIUmERICRL-RIIRLYTIC METHOD OF SUCCESSIVE RPPROXIIIlRTIOIIS FOR TWO -POIIIT BounDRRY -VRLUE PROBLEMS In this chapter, we present a numerical -analytic method oriented at the investigation of existence and the construction of a solution of the system of differential equations (0.1) with continuous right -hand side considered with nonseparable two-point boundary conditions of the form (0.3). Moreover, the matrices A and C in the boundary conditions may be degenerate , but we assume that det (k1A + k2C) # 0 for fixed real numbers ki and k2. In Section 1, we discuss the principal concepts of the method in connection with an abstract boundary-value problem. In Section 2, we construct a uniformly convergent sequence of functions each of which satisfies given boundary conditions . In this section, we also establish the relationship between the limit function of this sequence and the solution of the boundary-value problem under consideration. In Section 3, we prove a theorem that establishes sufficient conditions for the solvability of the boundary -value problem. On the basis of the necessary conditions of solvability obtained , in Section 4 we present a practical method for finding subdomains of the domain of definition of the right-hand side of a differential equation the points of which can determine the initial values of the required solution . In Section 5, we study the relationship between the determining equations that, generate , respectively , exact and approximate initial values of solutions of the boundary-value problems under consideration. We also estimate the error of an approximate solution of the boundary -value problem. In Section 6, we specify the form of successive approximations and estimates obtained for certain types of boundary conditions. In Sections 7 and 8, we justify the numerical -analytic method for nonlinear differential equations with two-point nonlinear boundary conditions and in the case of boundary-value problems with small parameter.

1. Abstract Scheme of the Method For systems of differential equations of the normal form x = f(t,x),

x,fE R°, 19

tE [ 0,T],

(1.1)

20 Numerical-Analytic Method of Successive Approximations

Chapter 1

and the higher-order equations x(n) = f (t, X, x, ... , x(n-1)),

(1.2)

the initial- value problems are thoroughly investigated in the case where, as additional conditions determining a particular solution , the initial values of an unknown function and its derivatives are given for a single fixed value of the independent variable t = 0, i.e., x(0) = xo, (1.3)

x(0) = x0, x(0) = x0, 1....,x(n-1)(0) = x0,._1. (1.4)

It is clear that relations (1.3) and (1 .4) are not the only possible form of conditions that select a particular solution of equations (1.1) and (1 .2), respectively. As these conditions, one often considers so-called boundary conditions that connect the values of the unknown function and its derivatives or functionals for several fixed values or all values of the independent variable. For equation ( 1.1), we can represent boundary conditions in the general form as follows:

U(x) = (U1 ( x), ... , Un (x)) = 0,

U1(x) = U.( x(t),x(t1 ),..., x(tP)),

(1.5)

where U; ( x(t), x(t 1 ), ... , x(tP)) are given functionals. If U; (x (t), x(t 1), ... , x (tP)) are linear functionals , then the boundary conditions are linear; otherwise , they are nonlinear . One distinguish two- or multipoint boundary conditions, depending on the number of values tP of the independent variable appearing in (1.5). The problem of finding a particular solution of the differential equation ( 1.1) satisfying the given boundary conditions ( 1.5) is called a boundary -value problem. Several particular cases of the boundary conditions ( 1.5) define the following corresponding classical boundary-value problems for equation (1.1): (i) a two-point boundary- value problem with nonlinear boundary conditions U(x) = (Up (x(0), x(T)), ... , Un (x(0), x(T))) = 0; (ii) a two-point boundary-value problem with linear boundary conditions U(x) = Ax(0)+Cx(0)-d = 0,

Section 1

Abstract Scheme of the Method

21

where A and Care (n x n ) constant matrices and d = (d I, d2, ... , d„) is a constant vector; (iii) the Cauchy-Nicoletti boundary-value problem Ui(x) = xi( ti)-di = 0,

i=1,2,...,n,

0 5 ti <- t2 5 ... <- to = T; (iv) a boundary-value problem of interpolation type U.(x) = xI(ti) - di = 0, i=1 , 2,...,n, 0 -< tl < t2 < ...<- tn= T; (v) a periodic boundary -value problem Ui(x) = xi(0)- xi(T) = 0, i=1 , 2,...,n; (vi) a p-point boundary -value problem with nonlinear boundary conditions Ui(x) = Ui(x( tl),...,x(tp)) = 0, i=1,2,..., n;

(vii) a (p + 1)-point boundary -value problem with linear boundary conditions P U(x) = 1: Bix(ti) - d = 0, 0 = to < t I < ... < tP = T, i=0

where B i are (n x n) constant matrices; (viii)

a (p + 1)-point boundary -value problem with functional boundary conditions, e.g.,

U(x) = E Bi x(ti) + J p(t)x(t)dt - d = 0, i=0 0 where p (t) is a given continuous function; (ix) a two-point boundary-value problem with separated boundary conditions Ui(x) = xi ( 0)-di = 0 ,

i=1,2,...,q,

22 Numerical-Analytic Method of Successive Approximations

Chapter 1

Uj (x)=xj(T)-dj =0, j=q+1,...,n. For the scalar equation ( 1.2), the general form of boundary conditions can be written as follows: U(x) = ( U1(x), ... , Un (x)) = 0, Ui(x) = Ui(x( t),x(t0 ),... , x(n-1)(t0 ), ... , x(tp), ... ,xtn -1l(td),

(1.6)

i=1,2,.. ,n. Problems for the second-order differential equation ( 1.2) are studied the most thoroughly. Thus, for n = 2, particular cases of the boundary conditions (1.6) define the following boundary-value problems with standard names: (i) the first boundary-value problem U1(x) = x(0)-d1 = 0,

U2(x) = x(T)-d2 = 0;

(ii) the mixed boundary -value problem U1(x) = x(0)-d1 = 0,

U2(x) = x(T)-d2 = 0;

(iii) the Neumann boundary-value problem U1(x) = x(0)-d1 = 0,

U2 (x) _ .x(T)-d2 = 0;

(iv) the periodic boundary-value problem U1(x) = x(0)-x(T) = 0,

U2(x) = x(0)-.r(T) = 0;

(v) the boundary -value problem with conditions at infinity U1(x) = x(0)-d1 = 0,

U2(x) = x(T) = oo

or U1(x) = .z(0)-d1 = 0,

U2( x) =

lim x(t) = 0. t-4 00

In the case of arbitrary n, the following problems are known for (1.2):

Abstract Scheme of the Method

Section 1

23

(i) a two-point boundary -value problem with nonlinear boundary conditions Ui(x) = Ui( x(0),...,x(n -1)(0),x(T),...,x(n-1)(T)) = 0, i=1,2,...,n; (ii) a two-point boundary-value problem with linear boundary conditions n

Ui(x) =

where

aii

)(0) +(3iix ( i)(T)j = di, ai;x(i E i=o

i = 1 , 2, ... , n,

[

and

0 ii

are constants;

(iii) a (p + 1)-point boundary -value problem with nonlinear boundary conditions

Ui(x) = Ui( x(t0), ... ,X ( n-1)(t0), ... ,x ( tP), ... ,X(n - 1 )(tp)),

i=1,2,...,n; (iv) a (p + 1)-point boundary-value problem with linear boundary conditions n

Ui(x) _ [ ai9)x(i)(to)+a^t) x(i)(t1)+...+a^P i=o i=1,2,...,n ,

to=0, tP=T,

)x(i)(tp)] = dig

t1,t2,...,tp _1E(0,T);

(v) the de la Vallee-Poussin boundary -value problem U.(x) = x(`- 1)(tk) = 0, i= 1 , 2, ... , nk,

k=0, 1, 2, ... ' p,

P nk = n. k=0

It is clear that the properties of the right-hand sides of the differential equations under consideration and given boundary conditions affect , to a large extent, the possibility of efficient construction of approximate solutions or the investigation of the solvability of boundary-value problems. The variety of types of boundary-value problems and certain universality of the numerical-analytic method considered require the description of the idea of this method for an abstract problem.

24 Numerical-Analytic Method of Successive Approximations

Chapter 1

Let C(D) be the space of continuous vector functions y (x) = (y t (x 1, ... , x,n ^ ... , yn(x1, ... , xn,)) with domain of definition x = (x1, ... , x. ) e D c R m. Denote by K(D) the subspace of functions C(D) satisfying the additional conditions U(y) = (Ul (y), ... , UU(y)) = 0.

(1.7)

Taking into account the aim of application of the method considered , we say that ( 1.7) are abstract boundary conditions. Consider an operator Q with the domain of definition DQ = C(D) and the range of values RQ=QC(D)c C(D). Let the equation y = Qy, DQ = C( D), RQ =QC (D)c C(D ) ,

(1.8)

be defined by the operator Q and let the problem of finding a solution of equation (1.8) satisfying the abstract boundary conditions ( 1.7) be posed, i.e., solutions y belong to K(D). If the operator Q mapped functions satisfying conditions ( 1.7) into functions of the same class, then it would be possible to analyze the problems of solvability of equation (1.8) and the construction of its solutions by considering the operator Q in the space K(D). Moreover, for this purpose , it would be possible to use the general theorems on fixed points and methods for approximate solution of operator equations [Kra2]. However, for numerous equations ( 1.8), the operator Q maps functions y e K(D) into functions Aye C( D) that, generally speaking , do not belong to the space K(D), i.e., do not satisfy the abstract boundary conditions ( 1.7). Consequently, it is impossible to find a solution of equation ( 1.8) satisfying conditions ( 1.7) by considering it in K(D). Therefore, along with the operator Q, we must properly introduce another operator Q so that its domain of definition DQ and range of values RQ = QK( D) coincide with the space K(D). Let the operator Q define the equation z = Qz, DD = K(D), RQ = K(D).

(1.9)

It is natural that, in a certain sense, the operators Q and Q must be close and interrelated, namely, (a) any solution y of equation ( 1.8) satisfying the boundary conditions ( 1.7) is also a solution of equation (1.9); (b) any solution z e K(D) of equation ( 1.9) that satisfies the additional condition

Section 1

Abstract Scheme of the Method

25

Bz = 0 (1.10) defined on solutions of equation ( 1.9) and satisfied by all solutions of problem ( 1.8), (1.7) is also a solution of the original problem ( 1.8), (1.7).

In view of the fact that the operator Q acts in the space K(D), the solutions of equation ( 1.9) can be found by different iterative processes . In particular, in the numerical-analytic method under consideration, as an iterative process, we use the method of successive approximations k=0, 1, ... ,

zk+1 = QZk,

(1.11)

beginning with zero approximation zo chosen properly . Moreover, there is certain arbitrariness in the choice of zo, e.g., it may depend on a free parameter or an arbitrary function. The arbitrariness of choice of zo is used to satisfy the additional condition ( 1.10). Note that condition ( 1.10) can be satisfied only by the limit function z * = lim k- =Zk or by any iteration beginning with a certain number k > ko. For special cases of realization of the numerical -analytic method, for the construction of the limit function z* of sequence ( 1.11) we use the analytic technique of the method of successive approximations, and for the solution of the additional equation ( 1.10) we apply numerical methods. As for the verification of applicability of the method to the proof of existence and construction of solutions of the abstract problem ( 1.8), (1.7), the fact that the successive approximations ( 1.11) converge generates a group of conditions, and the fact that conditions (1.10) are satisfied generates another group of conditions for the operators Q, Q, and B. We can specify the form of these operators . Thus , it follows from [SaRI] that, for the periodic boundary -value problem (0.1), (0.2), we have t Qx = xo + f f(t,x)dt,

0

OX = xo + f f(t, x) - T J f(s, x(s)) ds I dt, 0 o J T

Bx = 1 Bx = T f f(s,x(s))ds = 0. 0 According to [SaR4], for the two-point boundary -value problem (0.1), (0.3), we have

26 Numerical-Analytic Method of Successive Approximations

Chapter 1

t Qx = x0 + J f(t, x)dt,

0

Qx = x0 + J {f(tx) - T J f(s,x(s))ds + T[C-Id-(C-IA+E) xo]}dt, 0 0 T

Bx = TrC-Id-(C-IA+E)x0] -

TJ

f(s,x(s))ds = 0.

0

Consider the differential equation (0.1) with two-point boundary conditions containing the parameter X: Ax(0)+? Cx(T) = d,

x1( 0) = x10, detC# 0.

In this case, the results of [SRR] yield t Qx = x0 + J f(t,x)dt,

0

Qx = xo + J {f(tx) - TJf(s,x(s))ds + T[C-Id-(C-A+2E )xo]}dt, T Bx = XT[C-Id-(C-lA+a.E)xo] - TJf( s,x(s))ds = 0.

0 We now pass to the presentation of results related to the application of this abstract scheme to special classes of boundary-value problems for ordinary differential equations.

2. Choice of the Form of Successive Approximations and Their Uniform Convergence

Consider the two-point boundary-value problem with nonseparable linear boundary conditions: x = f(t,x ),

(2.1)

Section 2

Choice of the Form of Successive Approximations 27 Ax(0)+Cx( T) = d, det (k1A+k2C) * 0; (2.2)

here, x, f, and d are points of the n-dimensional Euclidean space R " and A and C are constant ( n x n) matrices such that det (k1A +k2C) # 0 for certain fixed real numbers k1 and k1, k1 * k2. For this problem, in this section , we construct a uniformly convergent sequence of functions xm (t, x0) satisfying the boundary conditions (2.2) for all m and an arbitrary value of the parameter x0 E R ". We also study the relationship between the limit function of this sequence and the exact solution of the boundary-value problem (2.1), (2.2). Note that the boundary -value problem (2.1), (2.2) is more general than the boundaryvalue problem investigated in [SaR4] , i.e., problem (0.1), (0.3), because, in this case, the matrices A and C can be separately degenerate. We assume that the function f (t, x) is defined and continuous in the domain (t,x)E (0,T]xD,

(2.3)

where D is a closed bounded domain in the space R", is bounded by a vector M = (M1, ... , Mn), M. > 0, and satisfies the Lipschitz condition with a matrix K = { K ii >- 0, i, j = 1, 2, ... , n }, i.e.,

Jf(t,x)- f(t,x")I <- KIx -x'1.

I.f(t,x)I < M,

(2.4)

Here, I.f(t,x)I =

(If1( t,x) I,...,I.fn (t,x)I ),

and the inequality between vectors means the componentwise inequality . Let us restrict the class of boundary-value problems under consideration so that the parameters M, K, A, C, d, T, k1, and k2 , which characterize these problems, and the domain of definition (2.3) satisfy the additional conditions presented below. The set Do of points x0 E R " such that the points x0 + k 1(k 1 A+ k 2 C)-1 [d- (A + C)x0] = z0( x0) are contained in the domain D together with their P-neighbourhoods is not empty, i.e., Do where

R = TM+P,(xo).

*

0,

(2.5)

28 Numerical-Analytic Method of Successive Approximations

Chapter 1

(31(x0) = l(k2-kl)(k1A+k2C)-1[d-(A+C)]xol.

The maximum eigenvalue ? (Q) of the matrix Q = T K does not exceed the unity, q i.e., ?.(Q) < 1. (2.6) where q = it. To construct a sequence of functions satisfying the boundary conditions (2.2), we consider a sequence of functions of the form

X. (t, x0 ) = XO +

f 0

L f(t

, Xm - 1(t, x0))

1T J f(s,xm _1(s,xo))ds dt + a [k1T+(k2-k1)t], To

m=1,2,..., xo ( t,xo) = xo+akIT, (2.7) regarding xO = (x01, ... , x(h) and a = ( (X 1, ... , a„) as parameters. Let us choose a parameter a so that functions (2.7) satisfy the boundary conditions (2.2) for all m = 1, 2, ... and an arbitrary value of another parameter xO e Dp . Substituting (2.7) in ( 2.2), we obtain the following system of linear algebraic equations with respect to a: T(k1A+k2C)a = d-(A+C)xo, Hence, a = T (k1A + k2C)-1 [d - (A + C)xo I. Therefore, all terms of the sequence of functions xm(t,xo ) = xo+k1Hd(xo)

L

+ J f(t^ Xm -1(t, x0)) - T

J f(s, X'.

-1(S,

x0)) dt + T (k2 - kl)Hd(xo),

J

m = 1, 2, ... , xo ( t, x0) = xo+ k1Hd (xo) = zo (xo), (2.8)

Section 2

Choice of the Form of Successive Approximations 29

where H = (k1A+k2C)- I,

d(xo) = d -(A+C)xo,

satisfy the boundary conditions (2.2) for arbitrary x0. To prove the uniform convergence of the sequence of functions (2.8), we need two auxiliary statements. Lemma 2.1 [SaR1]. Let f (t) be a function continuous on the segment tE [0, T]. Then, for all t E [0, T], we have

[f(t)_J f(s) ds] dt

C -T) f I f(s)Ids + t f I f(s)Ids l

o

a,(t)tma lIf(t)I,

5

r

where

al(t) = 2t (1-T),

IaI (t)I < T•

Lemma 2.2 [SaR5]. Let the sequence of functions ap(t)= 1, al (t)= 2t(I - T), . , which are continuous for tE [ 0, T], be defined by the recurrence relation ( t ) ,---,

am+I(t) = (i_

) f am(s)ds + T Jam(s) ds, m=0, 1,2,... (2.9) 0 t

Then there exist positive constants qm <- m such that m

am+1(t) 5 Tmgal(t) 5

al(t) = 3a1(t) =

nm al(t),

3 t(1 T)

(2.10)

(2.10)

for all m = 0, 1, 2, ... Proof of Lemma 2.2. In view of the fact that a I (t) 5 T/2, it follows from (2.9) that

30

Chapter 1

Numerical-Analytic Method of Successive Approximations

a2(t) = al (t) r6 +a3t)] < 3 al(t) = Tal(t), 2 a3(t) = al (t)+ 1 20

5a1(t)+15 al(t) T2

T2

T2

(2.11)

<_ n2 al (t). <_ 10 a 1 (t ) <_ n2a1(t)

Thus, inequality (2.10) holds for m = 0, 1, 2. For t e [ 0, T], it follows from relations (2.9) and (2 . 11) that

T2

a4(t) <_ 10a2

T2 T2 ( t), ..., am(t) <_ l0a,n-2(t) < -j

am - 2(t)•

By induction, we can conclude from the last relation that

am+1(t) <-mal (t), m = 0, 1, 2,... ,

which completes the proof of Lemma 2.2. Note that estimate (2.10) differs from the corresponding estimate in [SaR4, p. 14], where it was obtained separately for m ? 4 and m = 0, 1, 2, 3 with the constants q0 = 1, q, = 1, q2 = 1 , and q3 = 1 , by the property that qm = m beginning with m = 3 10 30 7c 0. In this case, the function al(t) = 3 a l (t) appears in estimate (2.10) instead of the function a l (t). Estimates of Lemmas 2 . 1 and 2.2 in [RoM] were improved as follows: Lemma 2.3. Suppose that the function f (t) is continuous in the interval t E [0, T]. Then the following inequality is true for any t E [ 0, T] :

j [ft - T 0

J f(s)ds]dti

0

Proof. It is obvious that

Ial(t)I max f(t)- min f(t) I. 2 \ te[0,T]

te[0,T]

I

Choice of the Form of Successive Approximations

Section 2

[f(t) _

31

rT

Jf(sds]dt

J J [f(ti) - f(s)]dsd t

T 00

TJ f

[f(ti)- f(s)]dsd r + T J J 00 00

[f(r)- f( s)]dsd T.

Furthermore,

]

= zf(t)_Jf(s)ds dt J J [f(ti)-f( s)]dsdti f[

t t = tf f(ti)d-t - tf f(s)ds = 0. 0 0 Therefore,

f 0

1

f(ti)-T J f( s ) ds dti 0

]

< 1 J JIf(ti)- f(s) Idsdti < 1 max If (ti)-f(s)I TO

t

T r E [0,T] se[t,T]

< 2«t(t)(tma]f(t)-t mo]f(t) I. Remark 2.1. It can easily be verified that

1 I max f (t) - min f (t)] <2 L tE[O,T]

max I f (t) 1

tE[0,T] tE[0,T]

and the equality

l r max f (t) - min f(t) 1 = max I f (t) 1

2 tE[0, T] te[0, T] J tE[0,T]

holds if and only if

max f (t) tE[0,T]

max f (t) = max I f (t) I. tE [0,T]

tE[0,T]

32 Numerical-Analytic Method of Successive Approximations

Chapter 1

Example 2.1. Consider the function f(t,x) = lnxsint+10 with the domain of definition

(t, x) E G = [0, 7021 x [e, e2]. It can easily be seen that

max I f (t, x) I = 12, (t,x)EG

and 2 [ max f (t, x) - min f (t, x)l tE[0,T] (t,x)EG

1(12-10) = 1.

Therefore, according to Lemma 2.1, we have

[

J .ft , x)-T j f(s, x)ds dt

]

aI(t)•12 <- 4.12= 9.42

and, by virtue of of Lemma 2.3,

[

f .ft, x) - T J.f(s, x)ds

]

dt <

a^(r)1 :5

4=

0.78.

On the other hand, by direct computation , one can obtain t

2a/2

f lnxsint+10- f (in xsin s + 10)ds dt o 't o

[

= I 1nxsint+10_1nx +51t]dt = (1_t_cost)lnx,

and 2 max II1 -- t-cost lnx^ = 0.42. (t,x)EG \ It `

Section 2

Choice of the Form of Successive Approximations

33

Consequently, it is obvious that the estimate in Lemma 2.3 is better than the estimate in Lemma 2.1. Lemma 2.4. Let a sequence of continuous functions a m + 1(t) be given by relation (2.9). Then there exist positive constants r3 m gm5(1) for which the following inequalities are true:

am+1(t)

3T)

am(t),

3 m

am+i(t) <_ (10T) • a1(t), m=0 ,1,2 , ... , where

at(t) = 10 a, (t), al(t) = 2t(1-T). Proof. By direct calculation, we obtain from the iterative formula (2.9) that

a2(t) = al(t)[-T +Iai(t)], 3 r2 a3(t) = ai(t)I 2 + 15at(t) + 15ai(t)

Taking into account that a t (t) <_ 2 , we get

a2(t) = -al (t) = 10T (9 at(t)) = (iT )i(t), a3(t) = OTat( t)[-+3at(t)]

3OTa2(t) By induction, the last inequality yields

2( I t) + 15ai3 (t)

30Ta12 (t)[T-2a1(t)] 5 (10T) a2(t)

34 Numerical-Analytic Method of Successive Approximations

am+1(t)

am

<( pT

)Um(t),

Chapter 1

m =1,2,3,...,

10

+,(t) :5 3 )a, (t), al(t) = 9 al(t) (10 T

Hence, the required estimates are obtained. Remark 2.2. By comparing the above estimates with those obtained by Samoilenko [Saml, Sam2] and Kwapisz [Kwa3], we get 1 1 4m 10( m/2] . 32(m/2) (3.1)n and 1 qm - (- )m

1 (3 .162 )m'

respectively, instead of

qm

(3)n _ 1 10_ (3.333)m

The following statement on the convergence of successive approximations xm (t, x0) of the form (2.8) is true: Theorem 2.1. Suppose that the right-hand side f (t, x) of system (2.1) is defined and continuous in domain (2.3) and conditions (2.4)-(2.6) are satisfied. Then a sequence of functions xm (t, x O ) of the form (2.8) satisfying the boundary conditions (2.2) converges uniformly as m -4 - ( with respect to the domain (t, x0) e [0, T]x Dp ) to the limit function x*(t, x0). Moreover, x*(t, x0) is a solution of the integral equation x(t) = xp+k1Hd(xo)

+ J [f(tx(t) ) - T J f(s, x(s)) + T(k2 -kl)Hd(xo ) I dt (2.12)

that passes through the point x*(0, x0) = xp + k,Hd (xp) at t = 0; furthermore, x*(t, x0) satisfies the boundary conditions (2.2), i.e., it is a solution of the boundaryvalue problem

Section 2

Choice of the Form of Successive Approximations

35

x = f(t,x)+A(x0), Ax(0)+Cx(T) = d, where

0(x0) = T(k2 -k1)Hd(xo) - T J f(t, x(t))dt. 0 For all m = 1 , 2, ... , the following estimate is true for the deviation of x * (t, x0) from xm (t, x0):

I x*(t, x0) - xm (t, x0) I 5 a1(t)[Qm(E-Q)-1M+KQm-1(E-Q)-'f 1(xo)] = a1(t)W(x0),

(2.13)

m=1,2,.. , where W(x0) = Qm (E-Q)-1M+KQm_l(E-Q)-1 31(x0). Proof. Let us prove that, in the space of continuous vector functions, sequence (2.8) is fundamental and, consequently, uniformly convergent. First, we establish that if x0 e Dp, then all functions xm (t, x0 ) belong to D. Indeed, according to Lemma 2 . 1, relation (2.8) implies that

Ix1(t,x0) - (x0+k 1Hd (xo))I

[

J f(t, zo(xo)) -

T J f(S, zo(xo )) ds]dt

cz1(t)M+31 (x0) 5

T

M+31(x0)

+ I (k2-kl)Hd(xo)I

(2.14)

Therefore, x 1(t, x0) E D if x 0 E Dp. By induction, one can easily establish that all functions (2.8) are also contained in the domain D for all m = 1, 2 , ... , tE [0, T], and arbitrary x0E Dp.

Numerical-Analytic Method of Successive Approximations

36

Chapter 1

Let us estimate the difference of the functions xm + 1(t, x0) - xm (t, x0)

L

= f 0

(f(t, xm (t,

x0 )) -

f(t, xm-1(t,

x0)))

1T J - T J (f(S, xm(S, xo))-.f (S, xm-1 (S, xo )))ds dt 0

_ (1

-T)J[0 f(

S, xm(S,xo))-

f(S,xm- 1(S, xo))]ds

T

- TJ [f(s,xm(S,xO))t

f(s, xm -1(S , xo))]ds.

If we denote rm + 1(t) = I xm + 1(t , xo) - xm (t, xo) I , then, in view of the Lipschitz condition (2.4), the last equality yields

rm +1(t) = K I 1-_ I Jrm (s)ds + \ o

J rm(s)ds .

L

]

r

(2.15)

According to (2.14), we have r1(t)

= I x 1(t, xo)- xo ( t,xo)I = Ix1 ( t,xo)-zo(xo)I

a 1(t )M +R 1(x0).

Therefore, for m = 1 , in view of (2.9), it follows from (2 . 15) that

r2(t) = KM I I 1-T I J a1(s)ds + T Jal(s)ds L\

/0

r

L -T)

+ KR1(xo) (1

f ds+ o

] T ] Ids

<- K[aI ( t)M+al(t)F+1(xo)J-

r

By induction , we can easily obtain rm+l(t) = Km

[am+1(t)M + am (t)P1(xo)l,

m=0,1,2,...,

Section 2

Choice of the Form of Successive Approximations 37

where a m + 1(t) and a m (t) are calculated according to formula (2.9). By virtue of estimate (2.10) in Lemma 2.2, we have

rm+1(t) = al (t)[

Km m M + KKm - 1

Tm-1

$1(x0)]

= &1(t)[QmM + KQm-1 i1(xo )]

(2.16)

for all in = 1, 2, .... Further, in view of (2.16), the relation xm+j(t, x0 ) - xm (t, x0)

= (xm+j ( t,x0 )- xm+j-1 (t, x0)) +... + (xm+1(t,x0)- xm(t,x0))

yields

Ixm+j{t, x0 )- xm(t, x0 )I .5

jrm+i(t) i=1

j-1 al(t) G (

Qm+iM

+ KQm+I-1131(x0))

i=0

S

j-1

j-1

i=0

i=0

at(t) Qm E Q`M + KQm -1 K E Qi 131(x0) . (2.17)

Since Y Q` = (E-Q)-1 i=0

and

lim Qm = 0 m->-

in view of assumption (2.6), we can conclude from (2.17) that , according to the Cauchy criterion, the sequence xm (t, x 0 ) uniformly converges in the domain ( t, x0) E [ 0, T] x Dp as m-°° , namely, lim xm (t, xo) = x*(t,xo). m-.>-

(2.18)

Since all functions xm (t, x0) of sequence (2.8) satisfy the boundary conditions (2.2), the limit function x*(t, x0 ) also satisfies these conditions.

38

Numerical-Analytic Method of Successive Approximations

Chapter 1

For j -4 oo and all m = 1, 2, ... , relation (2.17) implies the required estimate (2.13) for the deviation of x*(t, x0) from xm (t, x0). If we pass to the limit as m -a oo in equality (2.8) and take (2.18) into account, then we can see that the limit function x*(t, x0) is indeed a solution of the integral equation (2.12) that passes through the point xo + k t Hd (xo) at t = 0 and satisfies the boundary conditions (2.2). Theorem 2.1 is proved. Remark 2.3. Estimate (2.13) can be improved if we use Lemmas 2.3 and 2.4 instead of Lemmas 2.1 and 2.2 in the proof of Theorem 2.1. In this case, in condition (2.6), we should set q = 10 instead of q = it, replace a I (t) = 3 a t (t) by al(t) = 19 a t (t), and choose the vector M in the form

M=2

min f (t, x) max f (t, x) (t,x)E[O,T]xD (t,x)e[O,T]xD

instead of (2.6). This remark is also valid in all subsequent calculations involving the vector M and the matrix Q = T K. q Let us show that, by using a properly chosen parameter of control, we can always change of the form of the right-hand side of the differential equation (2.1) so that a solution of the equation obtained that passes through a certain point at t = 0 satisfies, at the same time, the boundary conditions (2.2). Theorem 2.2. If the conditions of Theorem 1.1 are satisfied, then, for any point xo E Dp, one can find the unique value of the parameter of control

n=

10

I (k2 - kl)Hd(xo) - T .f(t, x*(t, xo))dt,

n = Oii,....ji„).

(2.19)

where x*(t, x0) is the limit function of sequence (2.8), such that the solution x = x(t)= x*(t, x0) of the system of differential equations with a parameter on the righthand side of the form z = f(t,x)+ µ

(2.20)

that takes the initial value x0 + k1Hd(xo) for t = 0 satisfies, at the same time, the boundary conditions (2.2), i.e., is a solution of the boundary-value problem (2.20), (2.19), (2.2).

39

Choice of the Form of Successive Approximations

Section 2

Proof. It follows from Theorem 2.1 that the function x(t) = x*(t, x0) that is a solution of the integral equation (2.12) is, at the same time, a solution of the Cauchy problem

x = f(t, x ) + I (kz - k1)Hd(xo) - T J.f(t, x*(t, xo))dt, (2.21) 0 x(O) = xo+k1Hd (xo) (2.22) that satisfies the boundary conditions (2.2). Thus, we have found the value of a parameter t of the form (2.19) for which x(t) _ x*(t, x0 ) is a solution of the boundary-value problem (2.21), (2.2) satisfying the initial condition (2.22). It remains to prove that this value of the parameter is unique , i.e., that for any value of µ different from (2.19), the solutions of the Cauchy problem (2.20), (2.22) do not satisfy the boundary conditions (2.2). Assume the contrary . Let there exist two values t' and µ", such that the solutions x(t, x0, µ') and x(t, x0, µ") of the Cauchy problem (2.20), (2 .22) with µ = µ' and µ = µ" also satisfy the boundary conditions (2.2). Then relation (2.12) yields the following identity for the difference x(t, x0 , µ") - x(t, x0, µ'): x(t, x0, It")-x(t, xo, µ')

= j I o (t, t, x(t, x(, R")) - f (t, x(t, x0, W))) 0 T

T f (f(s,x(s,x

0,p."))- f(s,x(s,x0,µ ')))ds

dt.

0

Hence, as in the case of Lemma 2.1, we get I x(t, x0, µ") -x(t, x0, µ') I r <- K [ (1 t) f Ix(s,xo , T0

x(s,xo ,

ds

T + t f I x(s, xo, x(s, xo, ds . T r

(2.23)

40 Numerical-Analytic Method of Successive Approximations

Chapter 1

Setting

Ix(t,xo,x(t,x0, t')I =r(t)

and

Ir(t)lo

=

(suPl ri (t )I...suPI?(t)IJ r

r

we can obtain from (2.23) that r(t) <- am +1(t)Km

+hIr(t)lo,

(2.24)

for all m = 0, 1, 2, ... ; here, am+1(t) are positive functions determined according to (2.9). In view of (2.10), it follows from (2.24) that

I r(t)1 0 =

a1( t)KQmlr(t)l0

<-

3

2KQmlr(t)Io.

Since all eigenvalues of the matrix Q lie in the unit disk, the last inequality is possible only for I r(t)lo = 0, i.e., for .t" = µ'. The contradiction obtained proves the uniqueness of the parameter of control µ. Let us find necessary and sufficient conditions for the limit function of sequence (2.8) to be a solution of the boundary -value problem (2.1), (2.2). Theorem 2.3. If the conditions of Theorem 2.1 are satisfied, then in order that a solution x = x*(t) of the Cauchy problem (2.1), (2.22) be also a solution of the original boundary-value problem (2.1), (2.2), it is necessary and sufficient that the determining function

A(xo) = 1( k2 - k1)Hd(xo) - T f f(t, x*(t, xo ))dt (2.25) 0 be equal to zero at the point x = x0

A(xo) = 1 (k2 -k1)Hd(xo) - T J.f(t, x*(t, xo))dt = 0, 0

(2.26)

where x* ( t, x0) is the limit function of sequence (2.8). Moreover, in this case, x*(t) = x*(t, x0) and, for all m = 1 , 2, ... , inequality (2.13) holds for the deviation of the exact solution x =x*(t) = x*(t, x0) of the boundary-value problem (2.1), (2.2) from its approximate solution x = xm (t, x0 ) of the form (2.8).

Section 3

Sufficient Conditions for the Existence of Solutions 41

Proof. The sufficiency of condition (2.26) follows immediately from the fact that the function x*(t,x0) satisfying the boundary conditions (2.2) is a solution of the integral equation (2.12) that passes through the point z0(x0) at t = 0 or, in other words, is a solution of the Cauchy problem (2.21), (2.22). Obviously , if (2.26) is true, this condition is sufficient for x* ( t, x0) to be a solution of the boundary-value problem (2.1), (2.2). The necessity of condition (2.26) follows from the fact that if x =x *( t) is a solution of the boundary-value problem (2.1), (2.2) that passes through the point x*(0) = x0 + k,Hd(x0 ), x0 a Dp at t = 0, then the solution x = x(t, x0 , A) of the system x = f(t, x)+ A(x0) with the same initial value x ( 0, x0, A ) = z0(x0 ) satisfies the boundary conditions (2.2) for 0 (x0) = 0 because, in this case, x(t, x0, A = 0) = x*(t). According to Theorem 2.2, a solution of the Cauchy problem (2.20), (2 .22) is, at the same time , a solution of the boundary-value problem (2.20), (2 .2) only for the unique value of a parameter µ of the form (2.19). But we already have this value of the parameter for µ = 0(x0 ) = 0 and, in this case , x*(t) = x*(t, x0), which yields inequality (2.13). Theorem 2.3 is proved. In fact, the last statement gives the following numerical-analytic algorithm for the construction of a solution of the boundary-value problem (2.1), (2.2): For X0 E DO, according to (2.8), we construct a sequence of functions xm (t, x0) depending on x0 as a parameter; then we find the limit function x* (t, x0) of the sequence xm (t, x0) ; on the basis of the limit function x*(t, x0) , we construct a determining function A(x0) of the form (2.25) and find a solution x0 = xo of equality (2.26); then we solve the Cauchy problem for (2.1) with the initial condition x(0) = x0* + k I Hd(x0) on [0, T] . The function obtained is a solution of the boundary-value problem (2.1), (2.2), and, as its approximate solution, we can take xm (t, xp) of the form (2.8). The principal difficulty with the realization of this algorithm is connected with the construction of the function x*(t, x0) in analytic form. It turns out that we can make a conclusion about the existence of a solution of the boundary-value problem (2.1), (2.2) on the basis of the properties of successive approximations xm (t, x0).

3. Sufficient Conditions for the Existence of Solutions

In the present section, we establish sufficient conditions for the solvability of the boundary-value problem (2.1), (2.2) with the use of the properties of successive approximations xm (t, x0) of the form (2.8). First, we briefly recall certain notions necessary for what follows, namely, the notion of rotation of a finite-dimensional vector field and the index of a fixed point of a completely continuous operator and certain their properties [KVZRS, pp. 166-171].

42 Numerical-Analytic Method of Successive Approximations

Chapter 1

Note that Krasnoselsky [Kral ] showed that the notion of rotation of a completely continuous vector field is equivalent to the notion of degree of the Leray -Schauder mapping with respect to zero. If S2 is a bounded domain in the n-dimensional space R" and S is the boundary of the domain S2, on which a continuous vector field 'x without zero elements is defined, then the continuous mapping II (Dx II-' cb(x ) of the boundary S of the domain S2 onto the unit sphere II x II = 1 of the space R" is defined. The degree of this mapping is the rotation y(c, S) of the field (D on S. The rotation can be an arbitrary integer number: 0,±1,±2,.... Two vector fields '0x and (Di x are homotopic on S if there exists a vector func- 0 5 1, x e S, taking tion 4)(0, x) continuous in the collection of variables for 0 < values in R", and such that (D(O, x) = fi0x, 0 (1, x) =

(D1 x,

x E S, and (0, x) * 0,

0<-0<- 1, xES. Lemma 3.1 [KVZRS, p . 166]. If a continuous vector field Ox is defined in the bounded domain f = 0 U S and the rotation y((D, S) # 0 is not equal to zero on the boundary S, then the field (Dx is equal to zero at at least one point of the domain Q. Lemma 3.2 [KVZRS, p. 166]. Homotopic vector fields (D 0x and b x have the same rotation Y((Do, S) = Y((D1, S) The vector field 4) x = x - Q x is completely continuous if Q is a completely continuous operator in the Banach space E. If a completely continuous operator Q is defined in the neighbourhood of a point x0 E E that is an isolated fixed point of the operator Q, then the vector field Cbx = x Qx is not degenerate on all spheres II x - xo II = r of sufficiently small radius r and the rotation of the given field is the same on all these spheres. This common rotation is the index y(x0, Q) of the fixed point x0 of the operator Q. Lemma 3.3 [KVZRS, p. 166]. The index Y(x0, Q) coincides with the rotation of the vector field 4) x = x - Qx on the boundary S of an arbitrary bounded domain S2 in which x0 is the unique fixed point of the operator Q. To investigate the solvability of the boundary-value problem (2.1), (2.2), in addition to the determining equation (2.26) we consider the approximate determining equation

' m(x0 ) = T (k2 - kl)Hd(xo) - T if f(t,xm(t,x0))dt = 0, (3.1) 0 which differs only by the fact that xm (t, x0) appears instead of x*(t, x0).

Section 3

Sufficient Conditions for the Existence of Solutions 43

Let us formulate a statement that establishes sufficient conditions for the solvability of the boundary-value problem (2.1), (2.2). Theorem 3.1. Suppose that the assumptions of Theorem 2.1 and the following conditions are satisfied: (a) There exists a convex bounded domain D 1 c D p such that , for a certain fixed m ? 1, the mapping Am: 4 -* R" generated by (3.1) has a unique singular point x0 = xOm with nonzero index, i.e., the approximate determining equation (3.1) has a unique solution of nonzero index in D1 ; (b) on the boundary S1 of the domain D1, the following inequality is true:

x1" I1m (xo)I > ()2Qw(xo) ,

(3.2)

where W(x0)= Qm(E-Q)-1M+KQm-1(E-Q)-131(x(,) is determined in (2.13). Then the boundary-value problem (2.1), (2.2) has a solution x = x*(t) for which the initial value x*(O) = xo + k1Hd(xo) is determined by the value x0 = xo that belongs to D1, i.e., xo e D1. Proof. According to condition (a), the point x0 = xOm is the unique singular point of mapping (3.1) of nonzero index in the domain D1. Consequently, according to Lemma 3.3, the rotation of the vector field A. (x0) differs from zero on S1. If we show that vector fields A(x0) and A. (x0) of the form (2.25) and (3.1) are homotopic, then, according to Lemma 3.2, the rotation of the vector field A(x0) in Si also differs from zero. Therefore, according to Lemma 3.1, the vector field A(x0) vanishes at at least one point x0 = xo in the domain D1, i.e., the determining equation (2.26) has at least one solution x0 = xO* in D1. Consequently, according to Theorem 2.3, the solution x = x * (t) of the Cauchy problem (2.1), (3.3) x*(0) = xo +k1Hd(xo) is a solution of the boundary-value problem (2.1), (2.2).

(3.3)

44 Numerical-Analytic Method of Successive Approximations

Chapter 1

To complete the proof of this theorem , it remains to establish the homotopy of the vector fields Am (xo ) and A(xo ) . Consider the following family (continuous in the collection of variables) of vector fields continuous everywhere in S1 and connecting the fields P(O, xo ) = A. (x0 ) and P(1, xo ) = A(xo): P(0, xo) = A. (xo ) + 0 [A(xo) - A. (xo)1

050:51.

(3.4)

Let us prove that if 0 <- 0 5 1, xo E S1 , and conditions (3.2) are satisfied, then the vector function P(0, x0) is not equal to zero. Indeed , for m >- 1 , in view of the Lipschitz condition (2.4) and estimate (2.13), relations (2.25) and (3.1) yield T

JI.f(t,x*(t,xo))-f(t, xm(t,xo))Idt

IA(x0)- Am (O)I 5

0 T

TKW(xo ) f a1(t)dt = 9CKW(xo) 0

=

(

)2 Qw(xo) = E(A (Xo), A. (xo))

(3.5)

where W(xo)= Qm (E-Q)-1M+KQm 1(E-Q)-1 11(x(,)• Consequently, according to (3.4), (3.5), and (3 .2), the inequality I P(0, x0) I = I A. (x0 ) I - I A(xo) - A. (x0) I > 0,

(3.6)

is always true on S1, i.e., for 0:- 05 1 the vector function P(0, x0 ) does not vanish on S1, which means the homotopy of the vector fields A (xo) and Am (xo). Theorem 3.1 is proved. Clearly, in practice, to verify condition (a) of Theorem 3.1, one must calculate a function xm (t, x0) depending on xo as a parameter with the use of the recurrence formula (2.8) and then find a solution xo = xom of equation (3.1) and calculate the index of this isolated singular point of the mapping Am. This problem is rather laborious but , in certain cases, one can use fairly simple criteria that enable one to find out whether the value of the index is not equal to zero . Thus, in particular, it is known [Ale] that if A m is a topologic mapping of a neighbourhood D1 of a singular point xom , then the index of this

Section 4

Necessary Conditions for Solvability of Boundary-Value Problem 45

point is equal to ± 1 , and if A (xo) is a continuously differentiable function, then the index is not equal to zero whenever the Jacobian of the function A,,, (xo) at the point xo = xom is not equal to zero, i.e.,

det (DA.(xom)1 ^ 0.

o

J

4. Necessary Conditions for the Solvability of the Boundary -Value Problem

Let us find necessary conditions for the solvability of the boundary-value problem (2.1), (2.2), i.e., conditions necessary for a certain subdomain of the domain Dp to contain a point xo that determines the initial condition x*(O) = xo +k1Hd(xo) (4.1) of the solution x * (0) of the boundary-value problem under consideration at t = 0. Then we describe the numerical algorithm for selecting these points. First, we prove a lemma that estimates the closeness of the limit functions x*(t, xo ) and x*(t, xp) for points xo, xp a Dp and a theorem on the continuous dependence of a determining function A(xo) of the form (2.25) on xo. Lemma 4.1. Suppose that conditions (2.4)-(2.6) are satisfied for the boundaryvalue problem (2.1), (2.2). Then, for arbitrary points xo, xp a Dp, the following inequality holds for the difference between the limit functions x*(t, xp) and of sequences xm(t, xo) and xm(t, xo) of the form (2.8):

x*(t, xp )

Ix*(t,xo)-x*(t,xo)I <- [E+a1(t)K(E-Q)-1](R1+R2)Ixp-xo1, (4.2) where R1 = I E-k1H(A+C)I and

R2 = I (k2 -k1)H(A+C)I

are the matrices whose elements are the absolute values of the elements of the matrices E-k1H(A+C) respectively.

and (k2 -k1)H(A+C),

46 Numerical-Analytic Method of Successive Approximations

Chapter 1

Proof. It follows directly from (2.8) that

x1 (t, xo) - x1(t, x0) = (x0 - x0 ) + kl H[d (xo) - d (xo )

+J

[ it. z0(

x )) - f(t, zo(x ')))

]

- Tf (f(t,zo(xo))-f(t,z0 (xo)))ds dt

+ T(k2_ki)H[d( xu)-d(xp)]. Hence, by using the relations

J [f(t) -T f(s) ds]dt =

(1-!)

J f(s)ds -T J f(s) ds, (4.3)

d(xo)-d (xo) = -(A+C)(xo-xo

zo(xo) - z0(xo )

xo-xo+k1H[d(xo)-d(x)] = [E-k1H(A+C)](xo-xp),

Izo(xo )- zo(xo ) I < Rt1xO -x0 , al(t) = 2t (1-Tl = (l_)

fdt + 0

)

Jdt, t

T

and the Lipschitz condition (2.4), we obtain Ixt(t,xo)-xi(t,xo)I <_ RiIxO-xol+ R2Ixo-x0 (1-

+ K

LL

r) Jlzo(xo)-zo(xo)Ids + T0

t- Jlzo(xo)-zo(xo)Ids

Tt

]

_ (E+al(t)K)R1Ixo-x"I+ R2Ixo-xol. (4.4) Similarly,

1 X20, xo) - X20, xo) I <- RIIxp -xO "I+ R2

IxO -xO l T

r t r + KI(1- r) Jlxl(sl,xo)-x1(sl,x0)Ids + Jlxl(s,xo)-xl(s,xo)Ids tT L T 0 t

Section 4

Necessary Conditions for Solvability of Boundary-Value Problem 47

Taking into account (4.4) and the recurrence relation (2.9), we get 1 x2(t,xo)-x2(t, xo)I

Ri 1

xo - o 1 + R2 Ixo - xo., I x

\t + K (1- t f ((E+a,(t)K)RiIxp- x4'I+R2Ixo-x'I)ds 0

+

1 fT ((E +ai (t)K)RI xo - xoI+R 2I xo - 4j )dsl

Tr

J

= (E+al(t)K+a2(t)K2)Ri1xo-4I+(E+ai(t)K)R2Ixo-4I By induction, according to (4.4) and (4.5), we can obtain

Ixm(t,xo)-xm(t,xo)I <_

I(

E +ai(t)K +... + (Xm(t)Km)Ri + (E + ai (t)K + ... + am-1(t)Km-1)R2 I xo -xoI ] . (4.6)

In view of inequalities (2.10) and (2.6) and Lemma 2. 2, relation (4.6) implies that, for m -+ °°, we have

l x* (t,xo)- x.(t, x0) I 5

[

)

[ E+tsi(t)tK(K )' JRi + (E+&i(t)K(1 K) R2] x-xo

i=o

_ [E+ai(t)K(E-Q)-i] (Ri+R2)1xo-xoI, i.e., (4.2) is true. Lemma 4.1 is proved. Theorem 4.1. If the boundary-value problem (2.1), (2.2) satisfies conditions (2.4)(2.6), then a function 0 (xo) of the form (2.25) is defined, continuous in the domain Dp and, for all xo , xp a Dp, the following estimate is true:

10(x0)-0(x0)1 = LTR2+K(E+ 9 K(E-Q)-tll(Ri+R2)Ixo-x01. (4.7)

48 Numerical-Analytic Method of Successive Approximations

Chapter 1

Proof. For all points x0 E Dp, there exists the limit of the uniformly convergent sequence of functions (2.8), which is also a continuous function . Therefore, if x0 varies in the domain Dp , the function A(x0) is also continuous and bounded and we have

IA(xo)I Relation (2.25) yields

<-

TI (k2-

ki)Hd(xo)I +M.

O(xo) - 0(x0)

£

1 (*2-*,)//
Q Ixo-xoI + TKjlx*(t,xo)-x*(t,xo)Idt. (4.8) 0

Substituting (4.2) in (4.8), we get

D(xo) - 0(xo) I T TR2Ixo-xo'I + 7,K f [E+al(t)K(E-Q)-1](Ri+R2)Ixo-4ldt. (4.9) 0

Since T ttT2 ja1(t)dt = 9, 0 inequality (4.9) yields

IA(xo)-0(xo)I < TR2Ixo-xoI+K(R1+R2)Ixo-xoI 2 + IC9

K2(E-Q)-1(R1+R2)Ixp-xoI

--K(E-Q)-'^R,+R, )]j xp -xoI, [IR2+K(E+ET

which completes the proof. The statement below gives necessary conditions for the solvability of the boundaryvalue problem (2.1), (2.2).

Section 4

Necessary Conditions for Solvability of Boundary -Value Problem 49

Theorem 4.2. Suppose that the boundary-value problem (2.1), (2.2) satisfies conditions (2.4)-(2.6). Then in order that a subdomain D2 c Do contain the point xo = xo that determines the initial value x * (0) = xp + k I Hd (xo*) of the solution x = x * (t) of the boundary-value problem (2.1), (2.2) at t = 0 according to formula (4.1), it is necessary that, for all m and arbitrary io a D2, the following inequality be satisfied:

I A. (x0) I

< xsupz

[

!R2E+i.K(E_Q)-l(R1 T +R2)

)]

x0

- xoI

+ ()3Qw(10) 3 , (4.10)

where W(10) = Qm(E-Q)-IM + KQm-I(E-Q)-IRi(xo), RI(xo) = I(k2 -k1)(k1A+k2C)-'[d-(A+C)xo]I, T

Om(xo ) = T(k2 -k1) Hd(x0 ) - T

f f(t, xm(t, 10))dt, 0

d(xo) = d-(A+C)xo. Proof. Assume that a determining function A(xo) of the form (2.25) vanishes at the point xo = xo, i.e., A(xo ) = 0. Consequently , according to Theorem 2.3, the initial value of a solution of the boundary -value problem (2.1), (2.2) is calculated according to (4.1). Let us use Theorem 4. 1 in the case where xo = xo and xp = xo*. Then it follows from (4.7) that

IA(xo)I = [R2+K(E^

K(E_Q_1)Rl+R2]l1o_x. (4.11)

In view of inequality (3.5), we get

IL (xo)-Am (zo)I s 3J QW(XO), i.e.,

Chapter 1

Numerical-Analytic Method of Successive Approximations

50

2

IAm(Xo)I < IA (xo)I + ( ) QW(xo ).

(4.12)

The combination of (4.12) and (4.11) proves the validity of inequality (4.10), namely, I Am(x0) I

+K^E+ 9 ET- K(E-Q)-1)(Rt LR2

+R2 ]Ixo -xI+ () Qwio ) 2

T R2+KI/5sup E+ 9 K(E-Q)-1)(R1+R2)]Ixo-xoI+ ()2 Qwio.

X o Di L

According to Theorem 4.2, we can indicate an algorithm for approximate determination of the initial value (4.1) of a solution of the boundary -value problem (2.1), (2.2). For this purpose, we represent the set Dp defined, according to (2.5), as the union of finitely many subsets D., i.e., Dp = Ui 1 Di. In every Di, we choose an arbitrary point xo E Di and, for a certain number m, we calculate successive approximations x m (t, xo) according to (2.8) and Am (xo) according to (3.1 ). Then, according to (4.10), we eliminate the subsets Di for an arbitrary point of which the inverse inequality

IOm(x0) I

>

sup x 0 eD,

1

IT

( R2 +KIE+

K(E-Q)-1)(R1+R2)]Ixo-x+()2 Qw(x)

is true because, by virtue of Theorem 4.2, such subsets cannot contain the point xo that determines the initial value of a solution of the boundary-value problem (2.1), (2.2) according to (4.1). The other subsets Di form a certain set Mm, which tends as i, m - oo to the set M(xo) determining the initial values of a solution of the boundary-value problem (2.1), (2.2) according to (4.1). Any point x0 = zoe Mm can be regarded as an approximate value of the point xo = xo . In this case, the estimate

I zo -xoI <_ sup. I xo - xo I (4.13) X0 EM.,

is true and the function xm (t, i0) calculated by using the recurrence formula (2.8) is naturally taken as an approximate solution of the boundary-value problem (2.1), (2.2).

Section 4

Necessary Conditions for Solvability of Boundary- Value Problem

51

Let us estimate the deviation of the exact solution , i.e., the deviation of the limit function x * (t, xo ) from the approximate solution xm (t, io).

Lemma 4.2. If conditions (2.4)-(2.6) are satisfied for the boundary-value problem (2.1), (2.2), xo is a solution of the determining equation (2.26), and

io is an arbi-

trary point in the set Mm satisfying condition (4.13), then, for the deviation of the exact solution x * (t, xo*) from the approximate one xm (t, io) obtained by using (2.8), the following inequality is true: x*(t,xo )xm(t,x0)I

I

r

m-1

l

sup al (t)W(xo) + E+a1(t)KY.Q' I(R1+R2)Ixo-0 XOEM,

i=0 J)

11

(4.14) It

where W(xo) is calculated according to (2.13). Proof. We use the obvious inequality

Ix

*(t,x0)- xm(t,x

o )I : I x

*(t,x0)- xm(t,4

)1 +

I xm(t, xo )-

xm(t,io)I. (4.15)

We estimate the first term on the right-hand side of (4.15) according to (2.13):

I x*(t, 4) -

xm(t,

1

4 ) < al(t)W(xo). (4.16)

The second term is estimated according to (4.6): m-1 Ixm(t,4)-xm(t,io)I < [ E+ai(t)KQ J(Ri +R2)14_ioI. (4.17) i=o

Consequently , in view of (4.13), (4 . 16), and (4. 17), inequality (4.15) yields m-1

Ix *(t,xo)- xm(t,io)I <- al(t)W(xo) + [ E+^i(t)KQ' ) (Ri + R2) 4

i=o

or

-i0 ,

Chapter 1

52 Numerical-Analytic Method of Successive Approximations Ix*(t,x 0) -xm(t, 4)I

< sup, x0EM,,

I

[

CI (t)W(XO) + E+ai(t)KQ' J(Ri +R2)4 _o].

i.e, estimate (4.14) is true.

5. Error of Calculation of the Initial Value of a Solution

Consider the relationship between the solvability of the approximate and exact determining equations given according to (3.1) and (2.26). In Theorem 3.1, we have already indicated sufficient conditions under which the existence of an isolated solution of equation (3.1) implies the existence of a solution of the exact determining equation (2.26). In what follows, we find conditions under which, conversely, the solvability of the exact determining equation implies the solvability of the approximate one. We also establish conditions that guarantee the existence of solutions of the approximate determining equation (3.1). Lemma 5.1. If the boundary-value problem (2.1), (2.2) satisfying conditions (2.4)(2.6) has a solution x =x*(t) passing through the initial point (4.1) at t = 0, and in the ball II x -

411

<_

5,

8>0,

(5.1)

xp is an isolated solution (with nonzero index) of the exact determining equation (2.26), then, for sufficiently large m , the approximate determining equation (3.1) has a solution x 0 = xo m in ball (5.1) and the sequence x o m converges in norm as m -4 °° t o the value xo*:

IIxom-4 11 m- 0. Proof. According to estimate (3.5), we get

I(xo)- m(xo)l < ()

2 Qm(E_

Q)- '[QM+K131(xo)].

Since the maximum eigenvalue of the matrix Q is less than the unity, we have

(5.2)

Section 5

Error of Calculation of the Initial Value of a Solution

53

sup I A. (x0) - 0(x0) m 0. IIxo-xo UU S & This relation, together with the other assumptions , guarantees that all conditions of Theorem 1 in [Vai2] are satisfied , which implies the solvability of equation (3.1) and the validity of the limit relation (5.2). Lemma 5 . 1 is proved. Under certain smoothness conditions imposed on the right -hand side of (2.1), we now estimate the rate of convergence in (5.2) and determine the deviation of x * (t, xo) from xm (t, x0. ). For this purpose, we need the following statement: Lemma 5.2. Suppose that the right-hand side of the system of differential equations (2.1) satisfies inequalities (2.4)-(2.6) in domain (2.3), is a function continuously differentiable with respect to x, and satisfies the relations

af(t, x)

<_ K,

ax

(5.3)

of (t, x(t, xo )) of (t, x"(t, xo ))
ax (t, x0) ax (t, x0 ) axo axo

axo

x, x (t, x0 ), x"(t, x0) E D.

Then the Jacobi matrices

axm(t, xo)

axx(t, xo) atm(xo)

axo '

a0p(xo)

axo axo ' axo

of the vector functions (2.8) and (3.1) satisfy the following inequalities for x0 r: Dp : axp(t, x0) axo axm(t, X0)

< OP(t) <_ (3p, (5.4)

axP(t, x)

R4(t) <- R, (5.5)

axo axo

aA p(x0) axo

where

5 R2+ K(3p,

aAm(xo ) _ DAP(x0) axo axo

<_ KR4,

(5.6)

Chapter 1

54 Numerical -Analytic Method of Successive Approximations

R4(t) = a1(t)R3, R3 = KQP-1(E-Q-1(R1+R2)),

R4 <_ 6 R3,

RP(t) _ (E+a1(t)K(E-Q)-1)(Ri+R2), (5.7) p = (E+E-T 6 K(E-Q)-1(Ri+R2), and p and m are arbitrary natural numbers. Proof. In view of conditions (5.3), Lemma 2. 1, and the relations

xo(t,xo ) = zo(xo ) = xo+k1Hd(xo), (5.8) axo(t,xo) = E-k1H(A+C) axo it follows directly from (2.8) that axl(,xo) a

= (E-k1H(A+C)) - T(k2-k1)H(A+C)

0

[ j f a f(t, zo) a o

= (£-*,«(A + C))

zo

-1

T

a f(s' zo) azo dsl dt

azo axo T 0 azo axo

T(k2-k1 )H(A+C)

C1- j a f(t, z0) ao dt - r af(. 1 -L

T)

ax1(t, xo) axo

o

af(t, zo) azo o

Tr

azo &0

azo

af(t, Zo) azo

axo

<- R1+R2+KR1a1(t) = (E+a1(t)K)R1+R2, ax2(t, xo) axo

< R{ + R2 + (,--L \
J0

t

T

+ T t

af (t, x1(t, x0)) ax1

af (t, X, Q, x0)) ax1

, z0) azo dr,

azo axo

ax, Q, xo) axo

ax1(t, xo)

dt

dt

axo

<- ( E+a1(t)K + a2(t)K2 ) R1+(E+a1 ( t)K)R2.

azo dt axo

Section 5

Error of Calculation of the Initial Value of a Solution

55

According to (5.9), by induction, we obtain axp(t, x0) axo

<

p

l

( p-1

I ai(t)K` R1 + I a;(t)K` R2 =o i=O J

and, in view of estimate (2.10), axp(t, x0) axo

-< (E+ a1(t)K(E - Q)-t)(Ri + R2).

Since al(t) = 3 al (t) !5 6 T, (5.10) we indeed have R p(t) <- Op. If we take (5.8) into account, then , by analogy with (5.9), it follows from (2.8) that ax1(t, x0) _ axo(t, x0) axo axo

k 1 )H(A+C) + (1T)

_ T(k 2

- T af(t,zo)azo dt, j af((t,zo)azodt zo o rj zo 0

o

ax1(t, x0) _ axo(t, x) axo axo

a

<- R2+a1(t)KR1.

Further,

ax1(t, x0) axo - axo

ax2(t, xo)

t of (t, x1(t, x0)) - of ( t, xo(t, x0)) 0

L

axo axo

r of (s, X, (S' x0)) _ of (S, xo(t, xo)) ds dt, To( axo axo whence, in view of (5.3), we get

56

Numerical-Analytic Method of Successive Approximations axm(t, x0 )

ax1(t, x0)

axo - axo

Chapter 1

<- al(t)KR2+a2 ( t)K2R1.

By induction, we obtain

a xp +1(t, xo) a xm(t, x0)

(5.11)

<- aj+1(t)Kj+1R1 +aj(t)KjR2,

axo axo

j = 0, 1, 2, ... Let m > p. Then

axm(t, x0) aXp(t, x0)

axo axo

(axm(t, x0) - axm - 1(t, x0) ax0 ax0

+

aXm

-1(t , xo) - axm _2(t, xo ) + ... + axp+I(t, xo) - axm(t, x0)

axo axo ) ( axo axo

In view of (5.11), the last expression yields

axm(t, x0)

axp(t, x0)

axo axp

<- (^ ai(t)K` R1 +

J

l i=p+l

C

a; (t)K` 1 R2. (5.12)

i=p

By virtue of Lemma 2 .2, it follows from (5.12) that axm(t, x0 )

axm(t, x0)

axp axp

<- a1(t)KQp ( E-Q)-1 R,

+ a1 (

t)KQp-1(E-Q)-1R2

<- a1(t)R3 = R4 ( t), R4(t) <- Ra, i.e., inequality (5.5) is true. Let us prove (5.6). In view of (5.3)-(5.5), relation (3.1) yields aAp(x0) axo

1 1 T af(t,xp(t, x0 )) axp(t,x0) e -T(k2-kl)H (A+G^ - Tf ax dt 0 ax p 0

,

Error of Calculation of the Initial Value of a Solution 57

Section 5

a AP(x0) ax0

a_(x0)

S R2+Kf ,,

aAP(x0 )

ax0 ax0

T

af

Xm (t,

(t, 1 X0 )) - of (t, xP(t, x0)) dt TO r ax0 ax0

< K

axm(t, x0) aXP(t, x0)

< KR4 = 6 K2QP-1(E-Q)-1(R1+R2).

ax0 ax0

Lemma 5.2 is proved. Let us estimate the deviation of the approximate solution xm (r, xom) from the exact solution x * (t, xi). Theorem 5.1. Suppose that the conditions of Lemma 5.2 are satisfied for the boundary-value problem (2.1), (2.2) and, for a certain p < m and all x 0 E Dp, one can indicate a constant (n x n) matrix K1 with positive elements such that

C

_I

aAp(xo)1 axo

)J

< K1 (5.13)

and the norm of the matrix

IIK211

=

I KIKR4 I

=g2<1. (5.14)

Then the following assertions are true: (i) for the difference between the solutions 4 and xom of the exact and approximate determining equations (2.26) and (3.1) of the boundary-value problem (2.1), (2.2), i.e.,

A(xo) = 0 and respectively, the estimates

1 m (xom) = 0 , (5.15)

58

Numerical-Analytic Method of Successive Approximations

Chapter 1

11 xo -XOm II <- 1 lq2 11 K. 1111 E(A(xo), Am (X0)) 11, 4 -xOm I <_ AiKiE(A (x0), Am ( x0)) (5.16) are true and, furthermore,

UU xo -XOm ll 0, 1 xo -XOm 14 0 (5.17) as m --- > 00; here, A 1 is a matrix with positive elements defined by (5.24) and E(A(x0), Am(x0))

()2

Q m+l (E - Q)-' M

+ ()2 KQm(E_ Q ) ' sup I (k2 -k1 )(k1A+k2Q-1[d -(A+C)xo ]I, 3 xoes.

(5.18)

(ii) the approximate solution xm (t, xOm) of the boundary-value problem (2.1), (2.2) determined with the use of the recurrence formula (2.8) converges uniformly as m -4 oo to the exact solution of this problem x * (t) = x*(t, x0), which is the limitfunction of sequence (2.8), and, furthermore, I X*(t)-Xm (t, xOm )'

<- al(t)[Qm(E-Q) -'M + KQm-'(E-Q)-

'I(k'2-kl)(k1A+k2C)-1(d-(A+C)xom)I ]

+ [E +al(t)K(E-Q)-'](R1 +R2)A1KIE(L(xo), E(x*(t, x0*), xm(t, x0m)),

where g(A(x0), Am(x0)) are calculated according to (5.18). Proof. Obviously, Am(4) = Am(x0m+(X0-x0m))

aAm (xOm + ,C(x*- x0m )) * =0 m(xom)+ ax (x0-XOm), 0

,&m(x0))

(5.19)

Section 5

Error of Calculation of the Initial Value of a Solution

59

where t:5 ti <- 1. Therefore , according to (5.15 ), we have aAm (xo. + ti(xo - xom )) (xo-xOm) = Om(x0),

axo

1a'&.(XO.+'C(eO_xO.))1 Om (4) Xp -X0m =

axo

J

.

(5.20)

However, it is easy to see that a0m(xo) -

a/ P(x0)

- (a0 p(xo) aAm(xo))

axo axo I` axo axo

o)-,(DAP(xo ) -aLm (0o)11, = aAp(o0)IE-(aAp(xo)

ax ax axax J J

aom (xo) 1 = E- ( "'P(xo) )-' ( aAp(xo) -a&m (xo) aAp(xo) -'. axo axo J [ axo axo J] t axo

(5.21)

Not e that, by virtue of (5.15 ), we have I A(xo) - Om (xo) I = I A. (xo) l . Therefore, estimating (5.20) in norm and taking (3.5) into account, we get

11

xo

-

x omlI

Lm(xom + r( x0 -x0m ))] -1

<

a xo

Ia

IISm(xo)

(5.22)

where

(3)2QW(Xo) Sm(xo)

=

= R1(xo) =

(

!)2[

Qm+1(E - Q'M + KQm(E -Q )- 'Rl(xo)],

(k2-kt)(kA+k O-'[d-(A+C)xo]I ,

3t)2 Qm+I(EQ)_1M s"` = (

+ (!E)2 KQm(E_ Q) sup X l(k2-k1)(k1A+k2C)-'[d-(A+C)xo]l, 3 0 r= L^

(5.23)

Chapter 1

Numerical-Analytic Method of Successive Approximations

60

It is known [VoK, p. 100] that if II H II < 1, then the matrix (E + H) is nondegenerate and

II (E+H)-'

II < 1-IIHII

Consequently , taking (5.13), (5.14), and (5 .21) into account, we get

C

E - (aA p(xo'

X

(

+ 'r(Xo - Xom))J

axo

aA

J

(x0 + ti(x0 - xOm )) _ (xOm + t[(x0m axo axo

1 I

<

1 l-q2'

Therefore, it follows from (5.21)-(5 .23) that

xo

- X0m II <-

1

1- 42

I Kl II II E(A(x0 ), A . ( xo )) II,

where i (A(xo), Am (x0)) is calculated according to (5.18). By virtue of the relations obtained, there exists a matrix Al with positive elements such that

E

C

- (aAp(xo)) -' (a axo ))

p(xo) - aA m (Xo) axo axo J)

1-1

<- Al (5.24)

for all xo e D p. Then , taking (5.20) and (5.21) into account , we establish that, indeed, I xo - xo I <- A1K1£(A(xo ), Am(xo )). Since limm - Qm = 0, we get E(A(X0 ), Am(X0)) ) 0, m-oo

which proves the validity of relations (5.17). To prove (5.19), we consider the inequality

Ix*(t, x0 )- Xm(t,xom )I <_

I x*(t,x0 )- x*(t,xom )I + I x*(t,xom )- xm(t,XOm )I.

(5.25)

By using relation (4.2) with xp = xo and xo = xom, we obtain the following estimate for the first term in (5.25):

Section 5

Error of Calculation of the Initial Value of a Solution

61

I x*(t,xo)-x*(t,xom)I <- [E+&1(t)K(E-Q)-'(R,+R2)Ixo-xomlj In view of the second inequality in (5.16), we get

Ix*(t,xo)-x*(t,xom)l S [E+&1(t)K(E-Q)-1(R1+R2)A1K1£(i (xo),Am(xo))]. (5.26) The combination of inequality (2.13) and relation (5.26) for xo = x o m immediately yields the required estimate (5.19). The uniform convergence of the approximate solution xm (t, xom) to the exact solution x*(t, xo) follows from the fact that the right-hand side of inequality (5.19) tends to zero as m oo. Theorem 5.1 is proved. Below, we present a statement that enables one to make a conclusion concerning the solvability of the approximate determining equation (3.1) under certain smoothness conditions. Theorem 5.2. Suppose that the conditions of Lemma 5.2 are satisfied for the boundary-value problem (2.1), (2.2). Also assume that a mapping A.: 1 -4 R" of the form (3.1) is strongly differentiable in a ball S(xo°), r) c Dp centered at the point x6(°) with radius r and the following inequality holds in this ball:

1 T [af(t, xm( to, xo )) axm(t, x6) - af( t, xm(to, xo)) axm(t , xo)^l dt

LL

T

ax

ax

axo

axo

J

<

11 1 xo

- xff 1 , (5.27)

xO, xg e

where of (

S(xO(°), r),

t, xm) and axm (

t, xo) are the corresponding Jacobi matrices of the vector ax axo

functions f (t, x) and xm (t, x0). Furthermore, we assume that, at the point x o = xp°), there exists the inverse matrix

r 1

1

T

af(t, xm(t, x0°) axm(ax t, x(00)) 1 c°) dt = W (x°) (5.28)

T(k2 -kl)H(A+C) Tf ax 0

such that

°

62

Numerical-Analytic Method of Successive Approximations m,/>,/[ = Hx

Chapter 1

1

<

(5.29)

4'

where m, =

mm

P\ =

Wx\r(oy

1 (k2-k,)Hd(x^)T

1

T

nt,xm(t,40)))dt o

Then an approximate determining equation Am (x0) = 0 of the form (3.1) has the unique solution x0 = x0m in the ball S(XQ ', r^ centered at the point XQ with radius r j, where rx - P\y\ and yx is the least root of the quadratic equation Hly2-y + 1 = 0, and the sequence of approximations calculated by the modified Newton method

x

o

-

T

1 o ~ "n o > (kz-kOHdixW) T

x

1 f(t,xm(t,x{0k)))dt T

x

0

k = 0, 1,2,...,

(5.30)

converges to this solution so that

x

0m

S

<

x

0

\-q

(5.31)

Pi.

where <7 =

l-Vl-4//,

<

2

1 2'

Pro©/. Condition (5.27) guarantees that the Fr6chet derivative, which coincides with the Jacobi matrix —mS5Ql of the operator Am(jc0) in the ball S(x§\ r), satisfies the OXQ

Lipschitz condition

aAw(t,x&) 3^0

3A„('.*S) s
•*0» * 0

e

(0) "^ J t , r )•

Error of Calculation of the Initial Value of a Solution

Section 5

63

It can easily be verified that if relations (5.28) and (5.29) are true , then all conditions of Theorem 1 in [KoF, p . 470] on the convergence of the modified Newton method (5.30) for the solution of equation (3.1) are satisfied. Consequently, by virtue of this theorem, the equation Am (xo ) = 0 has a unique solution in the ball S(x(0), r1) and estimate (5.31) is true for the error of the approximate solution (5.30). Theorem 5.2 is proved. Note that, as an approximation of the exact initial value of a solution of the boundaryvalue problem (2.1), (2.2) zo(xo ) = xo+k1Hd(xo*),

d(xo) = d-(A+C)xo*,

(5.32)

where xo is a solution of the determining equation (2.26), one can take zc(xoki) = xoki+k1Hd( xoki), d (xoki) = d- (A+C)xoki,

(5.33)

and, as an approximation of the exact solution x*( t, xo*), one can take the function xm(t, xok) ) calculated according to formula (2.8). By using the results obtained and inequalities (5.31), (4.2), and (2.13 ), one can easily establish estimates for the deviation of zo( xok)) from zo(xo* ) and, correspondingly, for the deviation of xm (t, xOk)) from x* (t, xo). By virtue of (5.16) and (5.31), the relation xo - $k) = ( XO -X0m ) + ( X0m -XOk))

yields

IIXo-xok

Since c (xo, xoki )

'II

< 1 IIK1IIIIE(A(x0),Am(X0))II +

1-q2

-! k

1-q

pt = £(xo*,xpk)). (5.34)

m k 0, we have II xo - xoki II m k „-^ 0. In view of rela-

tion (5.34), it follows from (5 .32) and (5. 33) that

IIz0(x0)-

zo(Xpk

')II

£(x0'

xoki) +II k1 ( k1A +k2c)- 1(A

+c)II £(XO'xo i).

Further, be virtue of inequalities (4.2), (5.34), and (2.13), the relation X*(t,Xo* )- Xm(t,Xpk) ) = (x*(t,4 )- X*(t,4) )) + (X*(t,X(ok) )- Xm(t,xak)))

yields

(5.35)

64

I

Numerical-Analytic Method of Successive Approximations

X. ( t

, xo) - xm( t,

xok')

Chapter 1

I

e [E+&t(t)K(E-Q)-' ](RI +R2) 11 I II K1IIII £(A(xo), Am(XO))II+ gkPi1e l-q -q2

I

+ 61(t)[Qm(E-Q)-'M+KQm-1(E-Q)-' (k2-kl)(kiA+k2C)-1f d-(A+C)4i] I ,

]

(5.36) where e = (1, 1, ... , 1). Since the right-hand sides of inequalities (5.35) and (5.36) tend to zero as m, k -- > 00, we have

IIzO(x0)-z0 ))II (x0k

) m,k->-

0, IX*(t,XO)-xm(t,xak))I

m,k ^^

> 0.

6. Special Types of Successive Approximations and Estimates

If the boundary conditions (2.2) for the differential equation (2.1) are such that det C # 0, then it is clear that successive approximations of the form (2.8) can be constructed for the values k 1= 0 and k2 = 1. In this case,

X. (t, x0 ) = x0 +

J

1f(t

,x.-I(t,xo))

- T f f(s, Xm- i(s, x0))d s] dt

+ t [C-'d-(C-'A+E)xo],

m=1,2,..., (6.1)

x0(t, x0) = x0 = 20(x0), and xm (t, x0) coincides with the sequence constructed in [SaR4] for the boundary-value problem (0.2), (0.3). Obviously, in this case, Dp is the set of points x0 a R" contained in the domain D together with their (3-neighbourhoods, where

R = 3M+R1(xo), R1(x0) = [CC-'d-(C-'A+E)xo], and

(6.2)

Section 6

Special Types of Successive Approximations and Estimates H = C- ', d(x0) = d - (A + C)x0.

65 (6.3)

For the boundary-value problem (0.2), (0.3) satisfying conditions (2.4)-(2.6) and (6.2) in domain (2.3), the statements and estimates concerning the convergence of the sequence of functions xm (t, x0) of the form (6.1) and its connection with the solution coincide with those obtained for the iterative process (2.8). In this case, we must assume everywhere that R 1(x0), H, and d(x0) are defined according to (6.2) and (6.3) and estimate (2.13) has the same type beginning with m = 1, unlike the corresponding expression in [SaR4], where it has the same type beginning with m = 4. In this case, we have the function al(t) instead of a 1(t) in this estimate, i.e.,

I x*(t, x0 ) - xm (t, x0) I <- 6C1(t) W(xo), W(x0) = Q -(E-Q)-'M+KQm -'(E-Q)-'[C-'d-(C-'A+E)x0].

(6.14)

In Theorem 2.2, the parameter of control has the form T T[C-'d-(C-'A+E)xo] - T f f(t,x* (t,xo))dt

0 and the initial value of a solution is given in the form z0(x0) = x0. In Theorems 2.3 and 3.1, the exact and approximate determining equations have the form T A(xo) = T[C-'d -(C-'A+E)xo] - T jf (t, x*(t, xo ) )dt = 0,

0

(6.5) T A,(xo) = T[C-'d-(C-'A+E)xo] - 7f f(t,xm( t,xo))dt = 0.

0 Note that, by virtue of the equality z 0 (x0) = x0, the roots of the determining equations (6.5) immediately determine the exact and approximate initial values of a solution of the boundary-value problem (2.1), (0.3). In the estimates related to the determination of the subdomains D p that may include the initial values of the required solutions, the matrices in (4.2) have the form R 1 = E and R2 =I C-'A + E I = R. Therefore, according to (4.2) and (4.7), we have

Ix*(t,xo)-x*(t,xo)I <- [E+a1(t)K(E-Q)-'(E+R )Ixo-x0"I],

66

Numerical-Analytic Method of Successive Approximations

A(xo)-A(xo)I <

[R+K(E+!K(E_Qy1(E+R)]

Chapter 1

Ix' -xpl,

xa, xo E Dp, and the necessary condition (4.10) is written as follows:

+ (!)2Qw(i0), IOm00I <_ soup TR+K(E+ 9 K(E - Q)-1 (E + R)] x e k^ Ix -x0I

L

where W(x0) = Qm(E-Q)-1M+KQm-1(E-Q)-I Rl(x0),

R1(xo) = IC-Id-(C-1A+E)xoI, (6.6) and em(x0) is defined by (6.5). Inequality (4.14) in Lemma 4.2 takes the form

I x*(t, x0)-xm(t,

xo)I

m-1

< sup a1tW+ ( E+aiKQi J E+Rjlx_I X0EM„ 1=0

where W(x0) is defined according to (6.6), and the matrices PP(x) = (E+6cl(t)K(E-Q)-1)(E+R),

RP = (E+E-T K(E-Q)-' (E+R), 6 R3 = KQP-1K(E-Q)-1(E+R), are used in the inequalities of Lemma 5. 2. Estimates (5.16) of Theorem 5.1 are true if £(A(x0), Om(x0))

= (!)2 Qm+1(E - Q)-1 M + ()2 KQm(E - Q )-1 Xssup p I C-td - (C-1A + E) xo 1,

and inequality (5.19) takes the form

Special Types of Successive Approximations and Estimates 67

Section 6

Ix* (t, XO)- Xm(t, X0m)I

<- al(t)[Qm (E-Q)_1M+KQm-1(E-Q)-tI C-'d-(C- 'A+E)x°ml,

+ [E+a1(t)K(E-Q)-

'](E +R)A1K1E(A(x0), O m (XO)),

E(X*(t, XO), X,(t, xom))•

In Theorem 5.2, the matrix W (xo°)) is equal to

r 1 T-(C-' A+E)

1

L

( af(t, Xm(t, X0(0))) ax,,, (t, xo°)) dt - T ax

_1

ax°

0

the number pl is equal to

T W(xo°))^T (C-td-(C_'A+E)xo°)) - T T f(t,xm(t,xo°)))dt] 0

,

and the iterative process (5.30) of finding the required initial value of a solution of the boundary-value problem (0.2), (0.3) is constructed as follows:

X0(k+1) =

xok)

- W1(x(()°))I I(C-ld-(C_'A+ E)xok )) - _IT 'f(t,Xm(t,Xpk )))dt ,

L

0

J

k=0,1,2,.... For the deviation of the exact initial value xo of the solution x*(t, xp ) from the approximate initial value xok ), relation (5.35) implies the estimate

IIXO-XOk)II `-

11 1

42

II Kt II IISmI + gntj 1

which coincides with (5.34) because z0(xo) = xO* and z0 (xxk)) = xok). By analogy with estimate (5.36), we get

68

Chapter 1

Numerical-Analytic Method of Successive Approximations I x*(t, x0) -xm(t,

xj)

)I

[E+ 6 K(E-Q)-1 ](E

+ R)I

x [_i___ 1 g2IIK.IIIIE(N(xo),om(xo))II+1g-P1]e

I

+ al(t) [Qm (E - Q)-1 M + KQm-1(E - Q)-1 I C7-' d - (C-'A + E) x(0k) ]. If the matrices A and C and the vector d in the boundary conditions (0.3) or (2.2) are such that A = E, C = -E,

d = 0, (6.7)

then these conditions turn into the periodic boundary conditions (0.2) and problem (2.1), (2.2) reduces to the periodic boundary -value problem (2.1), (0.2). If we assume that the right-hand side of equation (2.1) is periodic in t with period T, then, under conditions (6.7), sequence (6.1) takes the form

xm(t,x0 ) = x + J [f(t Xm_i(t , x0)) -

m=1,2,.. ,

iT f f(s x ,

]

m-1(s ,xo))ds dt

x0(t, x0) = x0.

Functions (6.8) coincide with the sequence considered in [SaRI ] in the course of investigation of the periodic boundary-value problem (2.1), (0.2). As for the deviation of the limit function x*(t, x0) of sequence (6.8) from xm (t, x0), it follows from (6.4) that the following relation is true:

I x*(t,x0) - xm (t,x0)I <-

a1(t)

W( x o ),

W(x0) = Qm(E-Q)-1M.

m=1,2,..., (6.9)

It can easily be seen that, in this case , Dp is the set of points x0 E R" contained in the domain D together with their TM-neighbourhoods. The parameter of control for the periodic boundary- value problem is written as follows:

Section 6

Special Types of Successive Approximations and Estimates

69

T µ=- T f f (t, x*(t, xo))dt,

where x * (t, x0) is the limit function of sequence (6.8). The determining equations whose roots determine the exact and approximate initial values of periodic solutions take the form T

A(x0) = T J.f (t, x*(t, xo))dt = 0, 0 (6.10) T Am

(x0) =

j.f(t,xm(t,x0))dt.

T0

For sequence (6.8), its limit function x*(t, x0), the exact initial value xo* of the periodic solution, and its approximation xO m, the corresponding relations and inequalities used in the considered numerical-analytic method take the form

Ix*(t,xo )- x*(t,xo )I <_ [E+&1(t)K(E- Q)-1,Ixo-x0"1, xo, x0 E Ds,

IA(xo)- A(xo)l !5 1K(E+ 9 K(E-Q)-') lxo-X" l,

]

3 Am(x0) xs

E DZ[

E-T K(E-Q)-ll Ix-x01 + \3l QW(x0), \E+ 9

where W(10) = Qm(E-Q)-1M, m-1

Ix *(t,xo)- xm(t,xo)I <- a1(t)Qm(E -Q)-1M + sup CE + 61(t)KI Q` Ixo -xoI, x0EMi i=O

^x *(t, x0 )- x m( t,X Om)I a1(t)Qm (E-Q)-1M + [E+a1 (t)K(E-Q)-1]AiKi£(A (xo),Am(xo))

E(x*(t, xp ), xm(t, x0m)),

70 Numerical-Analytic Method of Successive Approximations

Chapter 1

(A(XO),Am (XO)) = (!!) 2 Qm+1(E -Q)- 'M,

Op(t) = E+&1 (t)K(E-Q)-1,

RP = E+ 6 K(E-Q)-

R3 = KQP-1(E-Q)-1.

The iterative process (5.30) for the periodic boundary- value problem is constructed as follows:

X(0k+1) = X(0k) _ W

I (X00))[T J

J

f (t, Xm( t, Xpk)))dt ,

k = 0, 1, 2, ... ,

where

w,

T (°) 1 of (t, Xm(t, x00))) aXm (t, x°°))dt1 ( x ) - [T o ax ax°

]

Pi =

T W(°))T J f (t, xm(t, 4°)))dt 0

We have

IIXp -

xok) I

I KIIIII£(A(xo),Am(xo)) I + qPI

<

1-IIK1K1411 1-q

I X (t,XO )- xm(t,XOk))I

IIKI IIIIZ(A(xo),Am(x0))II + ---p1 e 1- IIKIK4II 1-q

[ E+ ,T K(E_Q )- ' ]

6

+ &I(t)[Qm(E-Q)-IM], 1- 1- 4mip111 q= 2

Section 6

Special Types of Successive Approximations and Estimates 71

We now consider the case of the integrable differential equation (2.1) with separable two-point boundary conditions x.(0) = d,, i= 1,2,...,q, xj(T) = d;,

j = q+1, q+2,..., n.

(6.11)

The boundary conditions (6.9) can be represented in the form (2.2). Consequently, the squarematrices A and C have the block structure, namely, A= I E"

0], C r0 0 L 0 0' L 0 Eq

J

where Eq and Eq+ t are the unitary matrices of dimension (q x q) and (n - q) x (n q), respectively. The separable boundary conditions (6.11) are equivalent to the following:

L x(0) + x(T) = d. 0 0 00 [0 E„-q rEq 0

(6.13)

Clearly, matrices (6.12) are degenerate and, therefore , one cannot apply the numerical-analytic method based on the method of successive approximations (6.1) to the boundary-value problem (2.1), (6.13). This problem can be investigated with the use of approximations of the form (2.8). In this case, one must choose kl and k2 so that det(k,A+k2C) * 0

(6.14)

and a sequence xm (t, x0) of the form (2.8) depends on the vector parameter x0 = (x01, ... , xp, ). For example , for k = 1 and k2 = 1, it is obvious that condition (6.14) is satisfied because A + C is equal to the identity matrix . In this case, the sequence of functions (2.8) satisfying the boundary conditions (6.11) is written in the form

x m(t,x0 )

= it + J

[f(t Xm_i(t, x0 )) - - f f(s, xm - t(s,xo ))ds

]

dt, (6.15)

m = 1,2,...,

x0(t, xo) = d.

However, the successive approximations (6.15) do not depend on the free parameter x0. Consider (2.8) for the values k 1 = 1 and k 2 = 2. For the boundary-value problem (2.1), (6.13), it follows from (2.8) that

72 Numerical-Analytic Method of Successive Approximations

Chapter 1

xm (t, x0) = [E-(A+ 2C)-1 ]xo +(A+2C)-'d

+ J

[

i(t.

xm-i(t, x0))- T J f(s, xm-i (s, xo))ds dt

]

+ T(A+2Q-1(d -x0), m = 1, 2, ... ,

(6.16)

x0(t,x0) = [E-(A+2C)-1x0+(A+2C)- ld = z0(xo)• If x*(t, x0) is the limit function of sequence (6.16), then, according to (2.26), the determining equation for this problem has the form

0(x0) = T(A+2Q-1(d -xo) - T J f(s, x*(t, xo)) dt = 0. 0

(6.17)

With the use of the equality zo(xo ) = [E-(A+2C)-lxo] + (A+2C)-ld the solutions x0= xo of equation (6.17) determine the initial values (for t = 0) of the solutions of the boundary -value problem (2.1), (6.13).

7. Numerical-Analytic Method in the Case of Nonlinear Two-Point Boundary Conditions

Let us study problems in which the differential equation and boundary conditions are nonlinear. The numerical-analytic method can also be applied to these problems [SaL2]. Consider the following system of nonlinear differential equations with nonlinear twopoint boundary conditions: dz

dt = Pt' Z),

(7.1)

g(z(0), z (T)) = 0,

(7.2)

where z, f, and g are points of the n- dimensional Euclidean space R."

Section 7

The Case of Nonlinear Two-Point Boundary Conditions 73

We assume that the right-hand side of equation (7.1) satisfies conditions of the type (2.4), namely, the function f (t, z) is defined and continuous in the domain (t, z) E [0, T] x 0, (7.3) where cZ is a bounded domain in R ", is bounded by a vector M, and satisfies the Lipschitz condition with matrix K, i.e., If (t, z) I <_ M, (7.4)

If (t,z')- f(r,z")I

<

KIz' -z"I,

z,z"E Q. (7.5)

We also assume that the function g(u, v) is defined and continuous in the domain (u, u) E S2 X 92. (7.6) Let A and C be constant (n x n) matrices such that det C # 0. Instead of problem (7.1), (7.2), we consider a boundary-value problem of the form

dz = f(t, z) , dr

(7 . 7)

Az(0)+ Cz(T) = cp(z(O), z(T)),

(7.8)

cp(u,v) = Au+Cv+g(u,v).

(7.9)

where

We change the variables as follows: z(t) = x(t)+h(t,yo),

(7.10)

where the function h (t, yo) is defined, continuous in the domain (t,yo) E [0,T]xS21, 01 cR,,,

(7.11)

continuously differentiable with respect to t, t E [ 0, TI, and taking values in R "; furthermore, x(t)= h(t, yo) belongs to the domain Sl and yo is an n -dimensional parameter. Let D be the domain of variation of x. Then the change of variables (7.10) transforms problem (7.7), (7.8) into the boundary- value problem

74 Numerical-Analytic Method of Successive Approximations

dx

= f(t,x(t) + h(t , Yo))

Chapter 1

- ah(atyo) (7.12)

Ax(O) + Cx(T) = (p(x(O ) + h(O, yo), x(T) + h(t, yo)) -Ah(O,yo) - Ch(T,yo),

(7.13)

considered in the domain (t,x)e [ 0, T]x D. (7.14) Let us apply the numerical -analytic method presented for the boundary-value problem (2.1), (2.2) to problem (7.12), (7. 13). Since det C # 0, we can construct this method on the basis of the sequence of functions of the form (6.1). For the applicability of the given numerical-analytic scheme, we assume that the following conditions are satisfied for problem (7.12), (7 .13) in domain (7.14): f(t, x+h( t, y0))-ah(t,Yo)

at

<-

MI,

(7.15)

the set Dp of points x0E R " lying in the domain D together with their p-neighbourhoods is nonempty , and the matrix K satisfies condition (2.6), i.e., Dp#O, TMI+I(C-'A+E)xol, X(Q)<1. (7.16) For the boundary -value problem (7.12), (7. 13), assuming that yo satisfies the equation F(xo,y0) = T(xo+h (O,Yo),x(T)+h(T,Yo)) - Ah(O,Yo) - Ch(T,Yo) = 0,

(7.17)

we construct the following sequence of functions of the form (6.1) depending on x0 and y0 as parameters:

xm+t(t xo,y0) = xo + l[f(txm(txOYO)+h(tYO)) 0 _ ah(t,Yo )

at

T ah(s,Yo)1 as ds dt (f (sxm(sxo Yo )+ h(s Yo )) T 10

Section 7

The Case of Nonlinear Two-Point Boundary Conditions 75 - T(C-IA+E)xo,

m = 0, 1, 2, ... , xo(t, xo, yo) = xo. (7.18)

Since, by virtue of (7.17), the right-hand side of (7.13) vanishes , we have x (T) _ -C-1Ax(0). Therefore, equation (7.17) can be rewritten in the form F(xo, yo ) = cp(xo + h (O, yo)' -C-1Axo+ h(T, Yo)) - Ah(0, yo) - Ch(T, yo)

= 0,

(7.19)

If conditions (7.15) and (7.16) are satisfied, then, by virtue of Theorem 2.1, the sequence of functions (7.18) uniformly converges to the limit function x' (t, x 0, y 0), which is the solution of the boundary-value problem ah(t, dx = f(t,x(t) + h(t, Yo)) yo) + A(xo, Yo), (7.20) dt at Ax(0) + Cx(T) = 0,

(7.21)

where E(xo,Yo ) = (C-IA+E)xo

- T j[.f(t,x'(t,xo,Yo)+h(t,Yo ))0

ah(atyo Idt. (7.22)

Let us now establish that if (xo, yo) = (xo, yo) is a root of the system of algebraic equations F(xo, yo) = 0, A(xo , Yo) = 0,

(7.23)

z*(t, xo, yo ) = x*(t, xo, yo ) + h(t, yo )

(7.24)

then the function

is a solution of the boundary -value problem (7.1), (7.2). Indeed, in view of (7.23) and (7.20), the function x*(t, xp, yp) satisfies the equation --=

ah(t, Y o)

y* )) f(t, x(t) + h(t,at dt

76 Numerical-Analytic Method of Successive Approximations

Chapter 1

This implies that the function z*(t, xo, yo) determined according to (7.24) is a solution of the equation dz

dt

= f(t, z) .

Furthermore,

z*(0) = x*(0,xo,y*) + h(O,yo) = xo + h(O,Yo), z*(T) = x*(T, xo, Yo) + h(T, yo) = x*(T) + h(T, yo), and, consequently, z * (0), z * (T) a 92. In view of (7.9) and (7.17), it readily follows from the last equalities that a function z*(t, x*, yo) of the form (7.24) satisfies the boundary condition (7.2). It follows from inequality (6.4) that the deviation of an exact solution z*(t, x0, Yo) of the form (7.24) from the approximate solution zm(t, x*, Yo) = xm(t, xo , yo) + h(t, yo) (7.25) satisfies the estimate I z*(t, xo,Yo)-zn(t, xo, y0* )I :- at(t)[Qm(E-Q)-'M+KQm-tI (C-'A+E)xo I] . Thus, the following statement is true: Theorem 7.1. Let a function f (t, z) be defined in domain (7.3) and satisfy conditions (7.4) and (7.5) and let a function g (u, v) be defined and continuous in domain (7.6). Also assume that there exist constant matrices A and C such that inequalities (7.16) are true. Then, in order that a solution z = z * (t) of equation (7.1) passing through the point zo = xo+h(O,y0) E 92

(7.26)

at t = 0, i.e., the solution of the Cauchy problem (7.1), (7.26), be also a solution of the boundary-value problem (7.1), (7.2), it is sufficient that the parameters x0 and y o satisfy the equations F(xo, yo) = 0,

(7.27)

A(xo, Yo) = 0 (7.28)

Section 7

The Case of Nonlinear Two-Point Boundary Conditions 77

given according to (7.19), (7.22), where x*(t, xo, yo) is the limit of the uniformly convergent sequence of functions (7.18) satisfying the boundary conditions (7.21). In this case, z*(t) = x*(t, xo, yo) + h(t, yo) and estimate (7.25) is true for the deviation of the exact solution z * (t) from the approximate solution zm (t, xo, yo) for all m >_ 1. The statement formulated above is analogous to the sufficient condition given in Theorem 2.3. Note that we can also establish analogs of Theorems 2.2 and 2.3. Moreover, in the general case, one should consider the sequence of functions constructed for the boundary -value problem in the form (2.8) instead of sequence (7.18) obtained on the basis of (6.1). Thus, as in the case of boundary -value problems with the linear boundary conditions (2.2), (0.2), (0.3), the problem of existence and approximate solution of the nonlinear differential equation (7.1) satisfying the nonlinear boundary conditions (7.2) reduces to the problem of finding the roots of the determining equations (7.27), (7 .28) and construction of successive approximations with the use of the recurrence formula (7.18). Along with equations (7.27), (7. 28), we consider the system of approximate determining equations F(xo,Yo) = 0, A. (xo , Yo) = 0, (7.29) where the function A. (xo, Yo)

(C~lA+E)x0

1T f(t,xm(t,x0,y0) T

J

+ h(t,y0))-

dh(t,y0j dt dt

(7.30)

0

differs from (7.22) by the fact that xm(t, x0 , yo) is used instead of the limit function. The following statement on sufficient conditions for the solvability of the nonlinear boundary-value problem (7.1), (7.2) is true: Theorem 7.2. Suppose that that the right-hand side of the system of differential equations (7.1) and the boundary conditions (7.2) are such that the functions f ( t, z) and g (u, v) satisfy the conditions of Theorem 7.1. Moreover, assume that, in the domain Di = Dp xQ there exists a convex closed domain

(7.31)

78 Numerical-Analytic Method of Successive Approximations

Chapter 1

D3 = L x O 'c Dp x 521, (7.32) such that, for a certain m ? 1, the approximate determining equations (7.29) have a unique solution wo = wom, w0 = (x0, Yo) ' wom = (xom , yon ) of nonzero index in D2 . If the inequality ()2[Qrn+1(E_Qy. »inf l&m( xo ,Yo) I > 1M^ KI(C'A+E)xoll ES2

(7.33)

holds for the vector function (7.30) on the boundary S2 of the domain D2, then the nonlinear boundary-value problem (7.1), (7.2 ) has a solution z = z *(t) for which the initial value z *(0) = z0 determined by a formula of the form (7.27) is given by a parameter x0 = xa belonging to the set and a parameter Y0 = yp contained in the domain Q'I . This statement can be proved by analogy with the proof of Theorem 3.1 by showing the homotopy of the vector fields generated by mappings (7.27)-(7.29) on the boundary of domain (7.32) in the case where condition (7.33) is satisfied. For the nonlinear boundary-value problem (7.1), (7.2), we can obtain an analog of Theorems 4.2 and 5. 2, on the basis of which we can give an algorithm for finding approximations of the values of the parameters x0 and yo determining the required initial value of the solution according to (7.26). Let us now clarify conditions that guarantee the solvability of the approximate system of determining equations (7.29). Assume that the functions f (t, z) and g (u, v) satisfy certain additional conditions , namely, let the vector function 'bm (x0, yo ) = (F(xo, y0), Am (x0 , y0)) , (7.34) which generates the mapping (D: R2n -* R2", be differentiable with respect to w0 = (xo, Yo) in domain (7.31). Assume that a point w o. = x'om = (xOm' Yom) is an approximate solution of the system of approximate determining equations (7.29) which gives the residual II m(xOm . YO m )

I

<_ F--,

(7.35)

where 11 • 11 is the norm in R2n and Em is a sufficiently small positive number. Let us prove that , under certain conditions, on the basis of the properties of the vector function 0m (x0 , y0), its Jacobi matrix, and the value F_., one can conclude about the existence (in a certain ball centered at the point w 0 = wom ) of a point w0 = wo = (xo, yo ) at which the system of exact determining equations (7.27), (7.28) vanishes.

Section 7

The Case of Nonlinear Two-Point Boundary Conditions 79

Assume that , in the nonempty domain Dr = D ^ x S2i = two = (x0, Y0) : II WO - x'om II < r} c DDxf2 1 = D1, (7.36)

the Jacobi matrix

Ca4)w'

o

equality

YO)J ) of the vector function is nondegenerate , and the in-

)

(a 'm(iom' . om) l aw0

- sm (7.37)

is true at the point w0 =wom. Furthermore, let us indicate the number k < 1 such that

ra'm(x0, YO)1 - raom(x0m' %m) I aw0 ) I` ax0 /I

< Sm, $mE„i (1-k)-1 <- r. (7.38)

One can verify that, under assumptions (7.35)-(7. 38), the equation (Dm (x0, Y0 )

=

0

(7.39)

satisfies all conditions of the proposition in [Ural] , which implies that equality (7.39) has a unique solution w0 = wom = (x%, yon) of nonzero index in a ball Dr of the form (7.36). Further, by analogy with Theorem 7. 2, if the estimate

xinf IAm(xO,Yo )l >

(it)2 Q[Qm

(E-Q)-1 M+KQm -1I (C-'A +E)x0I ] (7.40)

is true on the boundary of the ball Dr , then there exists a point in this ball at which the vector function Z(x0, y0) = (F(x0, y0), A(x0, y0)) vanishes , i.e., (D(xo , yo) = 0. Then 11 wo

-wom 11 <- r

(7.41)

and, moreover, if £m is small , then r is also small. Let us estimate the deviation of an exact solution z*(t, xp, yp ) of the form (7.24) of the nonlinear boundary-value problem (7.1), (7.2) from the approximate solution Zm(t,xOr+YOm ) = xm(t,XOm,SOm )+ h(t,YOm )' (7.42)

80

Numerical-Analytic Method of Successive Approximations

Chapter 1

assuming that the function h (t, y0) satisfies the Lipschitz condition I h(t,Yo) - h(t,Yo)I <- K4IYo -Y6I'

Yo, yo e 0 1,

(7.43)

in domain (7.11). Then

Z*(t, xu, YO) - Zm(t, xom, YOm)

= x*(t,x*,y*)-xm (t,XOm,YOm ) + h(t,yo)+ h(t,YOm)

= (x*(t, xo , yO) - x*(t, xOm, YOm )) + (x*(t, xOm , Yom) - xm (t, xOm, YOm))

+ (h(t, yo) + h(t, Yom )) . (7.44) According to estimate (4.2) in Lemma 4.1, for the limit functions of sequence (7.18), we have ( E+al(t)K(E-Q)-1) (E+ R)Ixo

Ix*(t,xo , Yo)-x*(t,x om,Yom )I

-xOm

1, (745)

where R is the matrix whose elements are the absolute values of the elements of the matrix C-IA+E. In view of (6.4) and (7.42), we get

I x*(t, x0m, Yom) - xm (t, XOm, YOm)

<- al(t)[Qm(E -Q)-IM1 +KQm-I(E-Q)-II(C-'A+E)xOm I,,

Ih(t,yo)+ h(t,YOm )I <- KhIyo - YOm (7.46) Since, in view of (7.41), we have

Ix0

-X0m1 <-

ell wO*

-wOmII < re'

(7.47) IYo

-YOmI

<- ellwO*-wOm

I

<- re,

where e = (1, 1, ... , 1), the combination of (7.45)-(7.47) with the use of (7.44) yields

Section 8

Boundary-Value Problems with Small Parameter

81

Z*(t, x0, YO) - Zm(t, xom, YOm)

al(t)[Qm(E _

Q)-' MI + KQm-I (E - Q)-

'I (C- 'A+ E)xo m l

+ [Kh+(E+&t(t)K(E-Q)-t)(E+R)]re. (7.48) Thus, we have established the validity of the following statement: Theorem 7.3. Suppose that the following conditions are satisfied for the differential equation (7.1) and the boundary conditions (7.2): (i) the functions f (t, z), g ( u, v), and h ( t, yo) are defined and continuous in domains (7.3), (7.6), and (7. 11), respectively, and conditions (7.4), (7.5), (7.15), (7. 16), and ( 7.42) are satisfied; (ii) for a vector function matrix

CaOaw'

(Dm (x0,

yo) of the form (7.34), there exists the Jacobi

YO)1 in domain (7.31), and relations (7.35) and (7.37) are

0

true;

(iii) inequality ( 7.40) holds on the boundary Sr of the ball (7.36).

Dr of the form

Then the boundary -value problem (7.1), (7.2 ) has a solution z (t) = z*(t, x*, y*) of the form (7.24) for which the point wo = (xo , yo) belongs to the ball Dr and the deviation of solution (7.24) from an approximate solution zm(t,.rOm +Yom) of the form ( 7.42) is estimated by inequality (7.48). Note that, for a solution of the determining equation (7.39), one can use the iterative scheme of the type (5.30) and obtain an analog of Theorem 5.2.

8. Boundary-Value Problems with Small Parameter

It is convenient to apply the numerical-analytic method presented in Section 7 to the investigation of the problem of existence and approximate solution of one class of boundary-value problems considered below. This is a rather broad class of problems containing

82 Numerical-Analytic Method of Successive Approximations

Chapter 1

a small parameter a on the right -hand side of the differential equation and in nonlinear boundary conditions: dz = ef(t,z), dt

(8.1)

Az(0) + Cz(T) = ecp(z(0), z(T)),

(8.2)

Assume that inequalities (7.4) and (7. 5) are true for the function f (t, z) in domain (7.3). Let us prove that, for sufficiently small values of e, problem (8.1), (8.2) satisfies the requirements that allow one to apply the method under consideration. Let us choose the function h(t, yo) = tyo in relation (7.10). In this case, the boundary-value problem (8.1), (8.2) turns into problem (7.12), (7.13 ), i.e., dx = ef( t,x +tyo) - Yo, dt Ax(0)+ Cx( T) = ecp (x(0), x(T)+Tyo) - TCyo. Functions (7.19) and (7.30) are rewritten in the form F(xo,Yo) = ecp(xo, -C-'Axo+Tyo) - TCyo, (8.3) T A,.(xo'Yo) = (C-'A +E)xo + yo - TJ.f( t,x,^(t,xo,Yo )+tyo)dt.

0

Consequently, in the zero approximation where xo ( t, xo, yo ) = xo for m = 0, it follows from (8.3), (7.39) that bo(xo, yo) = 0, i.e., e(p(xo,-C-IAxo+Tyo) - TCyo = 0, (8.4) T (C-1A +E)xo + Yo - Tf.f(t,xo+tyo)dt = 0. 0 It can easily be seen that a solution of this system of equations is proportional to the small parameter e, i.e., xo =

exo ,

Yo

= £YO.

Substituting (8.5) in equation (8.4), we get (Do(xo , yo) = 0, i.e.,

(8.5)

Section 8

Boundary- Value Problems with Small Parameter

83

q (ex0 ,-C-lAxO +eTyo) - TCyo = 0, (8.6) T (C-IA+E)xo + xo - T f f(t,exo+etyo)dt = 0.

0 For e = 0, the root of the system of equations (8.6) takes the form

^ YO = T- C- 1 (P(0, 0), xo =

T

(C-1A+E)

-1

T

f f(t,0)dt- C-' (p(0,0) . 0

The Jacobi matrix of the vector function 0(x0, y0 ) is equal to

ac 0( x0, y0) .

-

C cDol 2

0

awo

X21

0

X22 J

'

(8.7)

where '00 = Ea(p (xO,-C-lAxO+Tyo) - eC tAa1P(xo,-C-1Ax0+Ty0) , 11 au 01V

(,o = ETatP (xo,-C-'Ax0+Tyo) - TC, 12

aU

(D o = (C-1A+E) 2t

E

af(t

,xo+tyo)dt^ T 0 az

1

(Do = E - e 1 af(t,xo+tyo)tdt, 22 T 0 az a(P au and a9 are ( n x n) Jacobi matrices with respect to the variables u and v, respectively. The Jacobi matrix of the vector function t0(xp , yo) with respect to the variable wo = (zo, yo ) has the form ajDo(xo, yo )) = I X01 21, (8.8) ' Xo J awo 21 jD2o 2

84 Numerical-Analytic Method of Successive Approximations

Chapter 1

where

*?l

£ acp (xo, - EC-1Axo + ETyo) - EC_1A acp (Exo, - EC-'Axo + ETyo )

au

av

= acp (exo, - eC-lAxo + El YO) _o E - TC, t2 au aF(t, "6 +Etyo) dt, ^o2t = (C-'A + E) - £ J T o az

4D0E - e f af(t,Exo+etyo)dt. 22 = T JO az

Comparing (8.7) and (8.8), for e = 0 we get 0 -TC a'Do(xo , Yo) _ ADO(xo, Yo) awo ( awo )E=0 = ( ) 1 =0 = [C-'A+E E This implies that if det(C-'A+E) # 0,

(8.9)

then the inequality det (aIDO(xO, Yo)0 awo is true for sufficiently small a in the ball wo = (x0, y ), II wo II <- r = Er'.

Moreover, since the right-hand side of equation (8.1) and the boundary conditions (8.2) depend on the small parameter E, we can always choose a sufficiently small positive number k < I such that conditions (7.38) are satisfied and, for m = 0, condition (7.40) is given by the inequality

inf o (xo,Yo)> ( )2 c2KM ,

wpeS,

Section 8

Boundary- Value Problems with Small Parameter

85

which follows from relation (8.4). Thus, the following statement is true:

Theorem 8.4. Suppose that the function f (t, z) is continuous and differentiable with respect to z and satisfies conditions (7.4) and (7.5) in domain (7.3), and the function cp (u, v) is defined, continuous, and differentiable in domain (7.6). Also assume that condition (8.9) is satisfied. Then, for sufficiently small values of the positive parameter E, the boundary-value problem (8.1), (8.2) has a solution z(t, xo, yo) that tends to zero as a -* 0.

Note that the considered numerical -analytic method for the nonlinear boundary-value problem (7.1), (7.2) can also be applied to the case of nonlinear problems with multipoint boundary conditions , namely,

dz = f(t, z), dt

I A1z(ti)

i=o

= W(z(to), z(t1), ... , z(tm )),

(8.10)

(8.11)

0=to
where A i are given constant ( n x n) matrices. The change of variables (7.10) reduces problem (8.10), (8 . 11) to the multipoint boundary-value problem with linear boundary conditions

dx = f(t, x + h(t, Yo )) dt

ah(t, Yo

r Aix (ti) = 0 at 1==o

(8.12)

with the additional condition

F(xo,yo) = W(xo+h(O,Yo),x(tl,xo,yo)+h(t1,Yo),...,x(T,xo,yo+h(T,yo)) M

-

E Aih(ti, yo) = 0. i=o

(8.13)

In the case of multipoint linear boundary conditions (8.12), we can construct [SaR3, p. 74] the following sequence of functions satisfying (8.12) for arbitrary xo and yo:

Chapter 1

Numerical-Analytic Method of Successive Approximations

86

xm+1(t, x0, Y0)

I f (t, xm (t, x0, yo) + h(t , yo )) L 0

= xo + f

ah(tt,tYo )

+

Am(xo,

yo ) dt,

J

(8.14)

J

where

A. (xo, Yo) r _ -

ll

I ItkAk l k=0

t

r ah(t, Yo) IZAkf f(t,xm( t,xo,Yo )+ h(t,Yo)) dt. at 0 1 IAkxOk=1 + L k=0 (8.15)

If the boundary-value problem (8.10), (8. 11) is such that the sequence of functions (8.14) uniformly converges to the limit function x' (t, x0, Y0) , then the investigation of the existence of its solution reduces to the problem of existence of a solution of the system of determining equations ID(xo,Yo ) = (F(xo,Yo),A(xo,yo )) = 0, (8.16) where

A(xo, Yo) l 1

ah( t, yo) Jr( f(t, xo,Yo)+h( t,Yo)^ - dt x Ak^ k=1 k=0 0 at

1 I AkxO + J _ -l k=0 ^tkAk

F(xO,y0) = tp(xo+h(O,Yo),x*(tl,xo,yo)+h(t1,y0),...,x*(T,x0,y0+h(T,y0)) M

- Y A.h(t,, yo). =o In turn, we can make a conclusion concerning the existence of a solution of system (8.16) with the use of the roots of the approximate system of equations I'm (xo, yo ) = ( Fm (xo, yo), Am (xo, yo)) = 0,

Section 8

Boundary-Value Problems with Small Parameter 87

where Fm(xo,Yo) = (P(xo +h(O,Yo), xm(ti, xo ,Yo)+h(ti,Yo),...,xm(T,xo,Yo+h(T,Yo))

I Ah(ti, yo), i=o and A. (xo , yo) is obtained according to (8.15). The problems considered can be investigated by the methods applied in the case of nonlinear two-point boundary conditions.

2. If1ODIFICHTIOII OF THE AUmERICHL -RnRLYTIC METHOD FOR TWO-POIIIT BOUADAAY-VALUE PROBLEMS The results presented in this chapter indicate one possible approach to the combination of different numerical-analytic methods, in particular, the method of successive approximations studied in the previous chapter and the collocation method . The combination of these methods gives a possibility to improve the convergence of successive approximations to the solutions of the nonlinear differential equations (2.1) with periodic or two-point boundary conditions (0.2), (0.3), (2.2). The idea of combination of indicated two methods is based on the use of the approximate solution obtained by the collocation method as the initial approximation in the numerical-analytic method of successive approximations . First, we study periodic boundary-value problems, for which, in Section 9, we investigate the convergence of the constructed sequence of functions , analyze the properties of the determining function, and prove theorems on the solvability of a periodic boundary -value problem on the basis of the modified method of successive approximations . In Section 10, we formulate theorems that characterize the properties of the determining function and establish necessary conditions for the existence of a point in a given domain that determines the initial value of a periodic solution . In Section 11, we establish estimates of the error of an approximate solution and the error of determination of the initial value of the required periodic solution. In Section 12, the principal ideas of the suggested modified method are described for two-point boundary -value problems . In Section 13, on the basis of the properties of the constructed sequence of functions and the approximate determining equation, we find sufficient conditions for the existence of a solution and prove a theorem that gives necessary conditions for the solvability of the boundary -value problem under consideration. In Section 14, we determine the error of determination of a solution and the initial value . In Section 15, we consider a specific periodic boundary -value problem.

9. Periodic Boundary -Value Problem

Assume that the right-hand side f ( t, x) of the system of differential equations (2.1) is T-periodic in t, defined and continuous in domain (2.3), and satisfies conditions (2.4)89

90

Modification of the Numerical-Analytic Method

Chapter 2

(2.6), where Dp is the set of points xo E R " contained in the domain D together with their 0-neighbourhoods and

p

T M. 2

(9.1)

For the periodic boundary -value problem (2.1), (0.2), the classic scheme of the numerical-analytic method of successive T-periodic approximations is based on the iterative formula (6.8). If conditions (2.4)-(2.6) are satisfied , then, according to Theorem 2.1, the sequence of functions xm (t, x0) of the form (6.8), which are T-periodic functions for all m and arbitrary x0, converges uniformly as m-*cc with respect to the set

(t,x) E (-oo,00)x Dp (9.2) to a T-periodic solution x * (t, x O ) of the integral equation

]

x(t) = xo + J [f(tx(t)_Jf(sx(s)ds dt.

(9.3)

Moreover, for t = 0, the function x*(t, x0) passes through the point x0, i.e ., x * (0, x0) = x0, and the rate of convergence is estimated by inequality (6.9) for all m >- 1. For the periodic boundary-value problem (2.1), (0.2), one can improve estimate (6.9) by proper modification of the iterative process (6.8). This modification is based on the combination of the numerical-analytic method of successive approximations with one of the most efficient (in the sense of calculations) projective methods, namely, the method of trigonometric collocation, which is also a numerical -analytic method according to our classification. It is known [SaRl ] that if there exists a T-periodic solution x*(t) of problem (2.1), (0.2), then, according to the method of trigonometric collocation , it can be found in the form of a vector trigonometric polynomial of degree k k

lajcos jou + bj sinjwt),

xk(t) = ao +

T=2.

(9.4)

j=1

In this case, the unknown coefficients (parameters) a j = (aj1, ... , a j") and bj = (bj 1, ... , bj") are determined from the system of Nn, N = 2k + 1, determining equations

xk(ti) = f(t1,Xk(t, )),

t; = iN, i = 0, 1, ... N- 1, (9.5)

and, as k-co, the approximate solutions Xk(t) uniformly converge to the exact solution x*(t), and ik(t) converge to x*(t) in mean square.

91

Periodic Boundary-Value Problem

Section 9

For the rate of convergence, the following estimates are true: IIxk(t)- x *(t)IIC <

C1Ek (v *),

II

- k(t)

- x*(t)

11 2 <

C2Ek(v*),

(9.6)

where C1 and C2 are constants independent of k, Ek(v*) is the best uniform approximation of the function v * (t) = x* (t) + x*(t) by trigonometric polynomials of degree k, and the norms are understood as follows: 1 1/2 rT I^x(t)IIC = maxmax lx(t)I, IIx(t)112 = max[ jlxi(t)I2dt

0

]

Assume that, for some fixed k, we have already determined the coefficients ao, aj, bb, j = 1, 2, ... , k, from (9.5), and, consequently, we have found an approximate solution of the form (9.4), which gives error (9.6). As already noted, in the abstract scheme of the numerical-analytic method, there is a certain arbitrariness in the choice of the zero approximation z 0 in the iterative formula (1.11) for solving equation (1.10). Taking this into account, for the solution of the integral equation (9.3), instead of the iterative process (6.8) we consider the following successive approximations of T-periodic functions passing through the point X0 +xk(0) at t=0: .i.(

xm(t, xo) =

x0

+ Xk(O) + J [f(t' m = 1, 2, ... ,

t'XO))_ T

f f(s, xm- 1(s, xO ))ds

J dt, (9.7)

xo(t, xo) = xk(t)•

It is easy to see that the zero approximation in (9.7) is chosen according to the k th approximation Xk(t) obtained by the method of trigonometric collocation, and xo is, in fact, a correction parameter for the initial value obtained with the use of the approximate solution xk(t).

The convergence of the successive approximations (9.7) is guaranteed by the following statement: Theorem 9.1. Suppose that the right-hand side of the system of differential equations (2.1) is T-periodic in t and satisfies conditions (2.4)-(2.6) and (9.1) in domain (2.3). Also assume that the set 4 of points x o E R" such that the points x o + xk(0) = zo(0) lie in the domain D together with their TM-neighbourhoods is not empty, i.e.,

fy

* 0,

(9.8)

92 Modification of the Numerical-Analytic Method

Chapter 2

and a function Xk(O) of the form (9.4) obtained by the method of trigonometric collocation (9.4), (9.5) belongs to the domain D. Then the sequence of T -periodic functions xm(t, x0 ) of the form (9.7) uniformly converges as m -> oo with respect to the set

(t, x0) a (-co, 00)x (9.9) to the limit T-periodic function x*(t, x0). In this case, z*(t, x0) is a T -periodic solution of the integral equation

x(t) = xO + Xk (0) +

f

[

]

f(txt)_f(s xs)ds dt

(9.10)

that passes through the point x*(0, x0) = x0 + Xk(O) at t = 0, i.e., it is a T -periodic solution of the initial-value problem x = f(t,x)+ A(x0),

x (0) = XO+ Xk(O),

(9.11)

where

A(x0)

T f f (t, z*(t, xo))dt 0

and the deviation of the approximation Xm(t, x0) from i*(t, x0) satisfies the following estimate:

&1(t)KQm _1(E-Q)-'lik(XO) = &1(t)W(x0), (9.12) lx-(t' xo ) - xm(t, xo ) I <m = 1,2,..., where

Ii k(XO) = IXOI +Sk,

Sk(t) = Xk (O)'+'

f

[

8k

=

max te[0,T]

ISk(t)I,

f(t,xk(t))_jf(s,xk(s)) 0 0

d s]dt- Xk(t),

W(XO)= KQni-1(E-Q)-'I3k(XO).

(9.13)

Periodic Boundary-Value Problem

Section 9

93

Proof. By analogy with Theorem 2.1, we prove that sequence (9.7) is fundamental. The direct substitution of t = 0 and t = T in (9.7) shows that the functions zm (t, x0 ) are T-periodic for all m = 1, 2, .... Moreover, if x k (t) e D, then all functions zm(t, x0) are also contained in the domain D. For m = 1, in view of Lemma 2.1, it follows from (9.7) that

Ixi(t,xo)-zo(xo)I T it, xk (t)) f(s, Xk ( s))ds dt [ J 0 0

a I (t)M <_ T M 2

whence, according to (9.8), we get 11(t, x0) E D. For all m = 1 , 2, ... and x0E DD, we have

I xm(t, x0 ) -z0(xO) I

J

[f(t,

xm-1(t, x0))

-

T f f(s, xm-1(s, xo))ds

]

dt

ct 1 (t)M <- T M 2

which directly implies that xm(t, x0) a D. Further, it follows from (9.7) that

XI (t, x0) - xo(t, x0)

= x0 + Xk (O) +

J

[ f(txk(r))_ff(sxk(

s))ds]dt - Xk

( t) = XO +

where Sk(t) is a function of the form (9.13). If we denote rl (t) = 111 Q, x0) - x0(t, x0) I , then the last equality yields

rl(t) 5

I X OI +Sk = R k(XO)•

(9.14)

For the iterative process (9.7), we can prove (in the same way as in Theorem 2.1 for the recurrence relation (2.8)) that inequality (2.15) is true for the difference rm+l(t) = Ixm +l(t,xO)-

i.e.,

xm(t,xO)I,

Modification of the Numerical -Analytic Method

94

Chapter 2

I 1-1 J rm (s)ds + t Jrm (s)ds . rm+1(t) 5 K

L

)J

T f T1

]

In view of (9.14), the last inequality implies that

r2(t) <- Kat(t)Rk(xo), and, by induction, we get rm+1(t) <- Kmam(t)(3k(xo ),

m=0, 1,...,

(9.15)

where the functions am (t) are defined according to (2.9). If we take (2.10) into account, then it follows from (9.15) that

rm +1(t) <_ al(t)KQm-1Rk(x0)• Since

xm+j (t, x0 ) - xm (t, x0 )

= (xm+j(t,x0 )- xm+j-1(t , x0)) +...+ (xm+1(t,x0)-xm (t,x0)), according to (9.15) we obtain

I xm+j(t, x0)-xm(t, x0)I

i Qm+i -2 <- j rm+i(t) S &1(t)K Rk(xo) <_ a1(t)KQm-1 Q` Pk (xo). (9.16) i=1 i=1 i=0

Condition (2.6) guarantees that lim m -, - Qm = 0 and, moreover, j-1

IQ' < (E-Q)-'.

i=0

Therefore, (9.16) implies that the sequence of functions Xm (t, x0 ) of the form (9.7) is fundamental in the space of continuous vector functions and, consequently, it is uniformly convergent in domain (9.9), i.e.,

Section 9

Periodic Boundary-Value Problem lim im (t, xo) = i*(t, xo).

95 (9.17)

Since all functions im(t, x0) are T-periodic, the limit function i*(t, x0) is also T-periodic. Estimate (9.12) follows immediately from (9.16) as j-+°°. Passing to the limit as m -4 00 in equality (9.7) and taking (9.17) into account, one can easily see that the limit function i*(t, x0) is a solution of the integral equation (9.1) satisfying the initial condition i*(0,xo) = XO + xk(0)

(9.18)

at t = 0 and the periodic boundary conditions (0.2), i.e., it is a solution of problem (9.11). Theorem 9.1 is proved. Note that, according to (9.6), by choosing an approximation xk(t) of proper order by the method of trigonometric collocation, we can make the value of Sk in (9.13) sufficiently small. In this case, the vector xo also lies in the neighbourhood of the zero vector and, consequently, the components 3k(x0) are sufficiently small. Comparing (6.9) with (9.12), we see that the improvement of the rate of convergence of the iterative process (9.7) as compared with the successive approximations (6.8) is attained due to the fact that, in (9.12), we have the vector Q k (x0) < M instead of the vector M appearing in (6.9). For the differential equation (2.1) with right-hand side T-periodic in t, the following statement concerning the parameter of control is true: Theorem 9.2. Suppose that the conditions of Theorem 9.1 are satisfied. Then, for any point x0 a D^, there exists a unique value of the vector parameter µ = (91, ... , 9n), given by the limit function of sequence (9.7) in the form

µ=

1 T

T

J f (t, I * (t, xo))dt , 0

(9.19)

such that a solution x = x(t) = i*(t, x0) of the system of differential equations obtained from (2.1) by adding the parameter µ to the right-hand side, i.e., x = f(t,x )+µ,

(9.20)

that passes through point (9.18) at t = 0 is, at the same time, a T-periodic solution of (9.20).

96

Modification of the Numerical -Analytic Method

Chapter 2

Proof. In problem (9.11) in Theorem 9.1, we have determined the value of a parameter .t of the form (9.19) for which a solution x(t) = x*(t, x0 ) of equation (9.20) satisfies the formulated conditions. It remains to prove that the value of parameter (9.19) is unique. Assume the contrary, i.e., let there exist two values of the parameter µ' and µ", µ ' * µ", for which the solutions x (t, x0, µ') and x (t, x0 , µ") of the Cauchy problems x = f(t,x)

+ l.t,

It = µ', µ = p.", x(0) = x 0 +

xk ( 0)

(9.21)

also satisfy the periodic boundary conditions (0.2). For the difference of these functions, by using equality (9.10), we obtain the following identity: x(t,xo,µ" )- x(t,xo,µ')

= J I (f(t, x(t, x0, W' ))-.f(t, x(t, x0, µ'))) 0

T - TJ(.f(s,x(s, x0,p."))-.f(s,x(s,x0,µ' )) ds dt. (9.22) 0 Relation (9.22) can be estimated with the use of formulas (2.23) and (2.24) as in Theorem 2.2, which completes the proof. Let us now formulate necessary and sufficient conditions for the limit function of the modified sequence (9.7) to be a periodic solution of the boundary -value problem (2.1), (0.2). Theorem 9.3. Suppose that the conditions of Theorem 9.1 are satisfied. Then a solution x(t) = x* (t) of the Cauchy problem for equation (2.1) with the initial conditions x(0) = x*(0) = x0 + Xk( O),

xoe D1 ,

(9.23)

is simultaneously a T-periodic solution of the system of differential equations (2.1) if and only if the determining function

A(xo) _ -T f f (t, i*(t, xo))dt (9.24) 0

97

Periodic Boundary- Value Problem

Section 9

constructed with the use of the limit function 1* (t, x0 ) of sequence (9.7) vanishes at the point x0, i.e., (9.25)

A(x0) = 0 . Moreover, in this case, x*(t) = i*(t, x0)

(9.26)

and, for the estimation of the error of an approximate T-periodic solution im(t, x0) of the form ( 9.7), the following inequality is true:

1x*(t)-im (t,xO)I <- &t(t)W(x0). (9.27) Proof. The sufficiency of the condition obviously follows from the fact that if the function x(t) = x*(t, xo) is a solution of problem (9.11), then, under condition (2.5), it is a solution of the problem x = f(t,x),

x(0) = x0 + xk ( 0),

x(0) = x(T).

The necessity of condition (9.25) is proved as follows : If x = x*(t) is a T-periodic solution of equation (2.1) that passes through point (9.23) at t = 0, then the solution x = x (t, x0, A) of the system of equations (9.20) with µ = A(x0) with the same initial condition x (0, x0, A) = x0 + xk( 0) satisfies (0.2) for µ = 0(x0 ) = 0, (9.28) because, in this case , x (t, x0 , A = 0) = x*(t ). But, according to Theorem 9.2, there exists a unique value of µ for which the solution of the Cauchy problem (9.21) is also T-periodic , and this value of the parameter has already been determined in (9.28). It is clear that, in this case , by virtue of the uniqueness of a solution of the Cauchy problem, equality (9.26) is true , which yields inequality ( 9.27) by virtue of (9.12 ). Theorem 9.3 is proved. Thus, we have the following numerical -analytic scheme for determining a T-periodic solution of the system of equations (2.1): (a) for fixed k, calculate an approximate solution xk (t) by the method of trigonometric collocation (9.4), (9.5); (b) construct a sequence of functions

im(t, x0) of the form (9.7) that depends on

x0 E D^ as a parameter and determine the limit function i*(t, x0);

98

Modification of the Numerical-Analytic Method

Chapter 2

(c) find a solution x0 = xo of the determining system of equations (9.25); (d) construct an approximate solution xm(t, xo) of the periodic boundary-value problem (2.1), (0.2) by using the recurrence relation (9.7).

10. Theorems on the Properties of Determining Functions of a Periodic Boundary-Value Problem By using Theorem 9.3, one can draw a conclusion about the solvability of the periodic boundary-value problem (2.1), (0.2) from the properties of the limit function i*(t, x0) of sequence (9.7). Let us show how this problem can be solved on the basis of the properties of a successive periodic approximation zm(t, x0) of order m. We introduce the approximate determining function T

Am (xo ) _

T

f f(t, Xm(t, xo))dt

(10.1)

0

and, on its basis, the approximate determining equation A. (x0) = 0.

(10.2)

Then the following statement is true: Theorem 10.1. Suppose that, for equation (2.1), the conditions of Theorem 9.1 and the following conditions are satisfied: (i) one can indicate a convex closed domain D 1 c bo such that, for a certain fixed order of approximation Xm(t, x0 ) of the form (9.7), the approximate determining equation (10.2) has a unique solution x0 = xOm of nonzero index in the domain D1; (ii) on the boundary S1 of the domain D1, the following inequality is true: T

inf X0 e

1 f (t, zm(t, xo))dt

s, T 0

> ()2 Q* xo) ,

where Q = T K and the vector W(x0) is defined by (9.13). R

(10.3)

Section 10

Theorems on the Properties of Determining Functions

99

Then the periodic boundary-value problem (2.1), (0.2 ) has a T-periodic solution x = x * (t) such that its initial value is determined by the point x = x* lying in the domain D1: x*(0) = XO +xk ( 0), x* E D1. (10.4) Proof. Let us prove that, for the determining functions (9.24) and (10.1), the following inequality is true:

^0(XO)-

(XO)I < ()2 Q*(xo).

(10.3)

Indeed, since Jal(t)dt = 3 J 2t(1-T)dt = 0 0

7tT2 9,

(10.6)

in view of (9.12) we have

0(X0)-Om (XO)I <-

1K

Ix

*( t,XO)-xm(t,XO)Idt

T Jal(t) dtKW(xo) _ ()2Q*(xo). 0 The subsequent proof is analogous to the proof of Theorem 3.1. By virtue of condition (i) and Lemma 3 . 3, the rotation of the vector field 1,,, (xo) generated by (10 . 1) is not equal to zero on S1. If we consider a collection of vector fields P(9, xo ) of the form (3.4), continuous with respect to 0 and xo on the boundary S1, that connect fields ( 10.1) and (9.24), then, taking (3.6), (10.3), and (10. 5) into account, one can easily see that P(9, xo) does not vanish on S1 . Consequently , the vector fields (9.24) and (10.1) are homotopic and, therefore , according to Lemma 3.2, the rotation of the vector field (9.24) does not vanish on S1. Hence, in view of Lemma 3 . 1, the exact determining equation (9.25) has at least one solution x 0 = x* E D1 in the domain D1 that determines (according to formula (9.23) in Theorem 9. 3) a T-periodic solution x *( t) of problem (2.1), (0.2) with the initial value ( 10.4). Theorem 10.1 is proved. For the approximate determination of the initial values of periodic solutions of equation ( 2.1), we need several estimates characterizing the closeness of the limit functions of sequence (9.7) with respect to the points xo and the condition of continuity of function (9.24).

100

Modification of the Numerical-Analytic Method

Chapter 2

Lemma 10.1. If the conditions of Theorem 9.1 are satisfied for the periodic boundary-value problem (2.1), (0.2), then, for arbitrary points

xo , xo a D^, the limit func-

tions x*(t, xp) and i*(t, xo) of sequences xm(t, xp) and xm(t, xp) of the form (9.7) satisfy the relation

Ix *(t ,xo )- x *(t,xo )I <- [E+a1(t)K(E-Q)-1]Ix0' -x0tlI.

(10.7)

Proof. For m = 1 , it follows from (9.7) that Ixl(t,xo)-xl(t,xo )I < Ixo xo For m = 2, in view of the last inequality and Lemmas 2 . 1 and 2.2, relation (9.7) yields I x2(t, xo ) - x2(t, xo) I

C

<_ 1x0'-x0-1 + K

t t

1-T) f Jij(s,xo)-xl(s,xo)Ids

0

T

t + Tj I xt(s,xo )-xl(s,xo)Ids t

[E+al(t)K]lxo-xoI• By induction, we get xm(t,x0 )- xm(t,x0)I

)fI

t t

xo xoI + K (1- 7.

')xm- 1(s,x0

0

t

T

+t I t

xm_ 1(s,x0 )- xm-

")Ids xm - 1(s,x0

1(s,x 0)Ids

<- [E+al(t)K+ a2(t)K2+ ...+ am_1(t)Km-1]Ixo-xo

(10.8)

Passing to the limit as m-oo in ( 10.8) and taking (2.10) and (2.6) into account, we establish that the limit functions satisfy the inequality

101

Theorems on the Properties of Determining Functions

Section 10

Ix *(t,x0)- x(t,x0)1

< E +a (t)jK( -T K)

x0-x01

1 =0

[E+a1(t)K(E-Q)-1 ]I X01_X011 11 which completes the proof. Theorem 10.2. If the conditions of Theorem 9.1 are satisfied , then a determining function A(x0) of the form (9.24) for the periodic boundary-value problem (2.1), (0.2) is defined, continuous, and bounded in the domain D^ and, for all x0, x0 E 41 the following inequality is true:

1A(x0)- A(xo)1 - K E + 9 K(E-Q)-1]1x0-x01 (10.9)

[

Proof. Since, for any point x0E 4, there exists the limit of sequence (9.7), which is continuous , we can conclude that, for xo varying in the domain D^, a determining function A(xo) of the form (9.24) is also continuous and bounded and I A(xo) 1!5 M. For xo , xo e

r,

relation (9.24) yields

1A(x0)-A(x0)1

T TJIx*(t,x0)-x*(t,x0)Idt.

( 10.10)

0

Substituting ( 10.7) into ( 10.10) and taking ( 10.6) into account, we get

A(xoA(x0)1 < TKJ[ E+a1(t)K(E-Q)-1]1x0-x01dt 0 nT < K1 x0-x01 +KZ(E-Q)-1 9 1x0-x01,

which directly implies inequality ( 10.9). Let us now formulate a statement that gives (on the basis of the properties of the approximate determining function (10.1)) a necessary condition for a subdomain of D^ to contain a point xo that determines the initial value x* ( 0) of the required periodic solution of the system of equations (2.1) according to formula ( 10.4).

102

Modification of the Numerical -Analytic Method

Chapter 2

Theorem 10.3. Suppose that the periodic boundary -value problem (2.1), (0.2) satisfies the conditions of Theorem 9.1. Then, in order that a subdomain D e e Do contain a point x o = that x0 determines the initial value of a T-periodic solution x = x*(t) of the boundary-value problem (2.1), (0.2) at t = 0 according to formula (10.4), it is necessary that, for all m and any point xo E D 2, the following inequality be true:

1T _ f (t, xm(t, x0))dt To

<_ sup K IE +'TK(E-Q) xo a DZ 9

-']I

-(4), (10.11) xo-xoI + (E), QW 3

where Q = T K,

1'(1 0) = KQm(E-Q)Qk(xO),

and (3k (xo) is according to (9.13). Proof. According to the assumption , x = x*(t) is a T-periodic solution of equation (2.1) with the initial value ( 10.4). Consequently , according to Theorem 9.3, the determining function (9.24) is equal to zero at the point xo = xo, i.e.,

A(xo) = 0. (10.12) Let us write inequality ( 10.9) for the point xo = xo, where xo is an arbitrary point in a domain D 2 e D p, and for the point xo = X* O. Then, in view of ( 10.12), we get

IA(xo)l <- K[E+ 9 K(E-Q)-t]lxo-x*l.

( 10.13)

Moreover, by virtue of ( 10.5), we have 2 ID m(x0 )I = IA(x0)I{ (,,) QW(xo). 3

Substituting (10.13) in (10. 14), we get

(10.14)

Section 10

Theorems on the Properties of Determining Functions

103

1Tr T•,) f(t, xm(t , X0))dt 0

r <- K E+ 9 K(E-Q)_ 1]Ixo-x0l

[

<- sup X0EDZ

( 7[ ) 2 +

3QW(XO)

E+K( E-Q)1]Ixo - xoI+() 2 Q(xo) , K[

which completes the proof of Theorem 10.3. Theorem 10. 3 gives a practical method for the approximate determination of the initial value of the required periodic solution of equation (2.1) on the basis of the successive approximation zm(t, x0). Assume that the set DD satisfying (9.8) is the union of subsets Di, i.e., N

U Di.

L =

(10.15)

i=1

In each subset D i , we choose an arbitrary point xp E D i and, for a certain successive approximation xm(t, xo ) of order m of the form (9.7), we calculate the vector

Am(XO) _ -T f f(t, Xm ( t,x 0 )) dt.

(10.16)

0 The subsets Di from ( 10.15 ) for arbitrary points xp E Di of which the inequality

Tr T ,1 .f (t , Xm(t, xo))dt 0

sup KrE+ XO

EDP

9

1

K(E-Q)- ]Ixo -

xoI +312QW (x

o),

(10.17)

holds according to the results of calculation of (10.16) forma certain set Mm, which, by virtue of Theorem 10.3, is a domain that may contain a point xO = xa that determines the initial value x (0) of a T-periodic solution of problem (2.1), (0.2) according to (10.4). Clearly , by increasing the partition number N for the domain D^ according to

Modification of the Numerical-Analytic Method

104

Chapter 2

the results of verification of conditions (10.16), (10.17), one can determine the set M,„ more precisely because, as i, m -* co, this set tends to the set M(xo) that determines the required initial values according to formula (10.4). If we choose an arbitrary point zo E M,,, as an approximate value of xp, then, obviously, 1zo-xo1 <- sup. lxo-xo1 (10.18) xoeM.

and it is clear that we can take the mth approximation xm(t, zo) of the form (9.7) as an approximate T-periodic solution of problem (2.1), (0.2). Let us estimate the error of Xm(t, io). Theorem 10.4. Suppose that, for the periodic boundary-value problem (2.1), (0.2), xo is a solution of the exact determining

the conditions of Theorem 9.1 are satisfied, equation (9.24), and

Xo E Mm is an arbitrary point in Mm satisfying condition

(10.18). Then, for the deviation of the exact T -periodic solution

i*(t, x*0) of equation

(2.1) from the approximate solution zm(t, x0)

=

x0

+ xk(0 ) + J

[

At' xm-1(t, io )) - I

0

T f(S> xm- 1(s, xo )) ds l dt ,

( 10.19)

0

m = 1, 2, ... , zm(t, x0) = xk(t), where Xk(t) is the kth approximation obtained by the method of trigonometric collocation (9.4), (9.5), the following inequality is true:

I x* (t, x0) - xm (t, i0 )1

m-1

sup &1(t)W(xo)+ E +a1 (t)KYQ xoeMm '

|xo-*o|>

(10.20)

i=0

where W(xo) is defined according to (9.13). Proof. For sequence (9.7), we consider the following analog of inequality (4.15):

Section 11

Solvability of the Approximate Determining Equation

105

Ix*(t, xO)-xm ( t,XO)I <- Ix*(t,xo )-xm(t,xo)I +Izm(t,xo)-zm(t,xo )I. ( 10.21) Estimating the first term on the right-hand side of (10.21) according to formula (9.12) and the second term according to (10.8), we get

Ix*(t,XO )- Xm(t,x0) m-1

&1(t)*(xp)+ [ E +^i(t)KQ' i =o

I <- sup

}x

_ioI

m-1

&1(t)1'(xo)+ E+&1(r)KjQ'

xp E M;,,

Ixo-xOI,

i=0

which completes the proof of Theorem 10.4.

11. Solvability of the Approximate Determining Equation and the Error of the Initial Value of a Periodic Solution

Denote by XOmE Mm the exact solution of the approximate equation (10.2), i.e., T J1 f (t, XOm + Xk (O) 0

- J 1 f(t'-i._ 1(t'xO. ))- -1ff(s,xm -t(s,XOm )

)ds dt Idt = 0.

]

(11.1)

It is obvious that relations ( 10.19) and (10.20) can also be applied in the case zo = xom. In this case , for the error of the approximate T-periodic solution zm (t, xom) constructed with the use of the root xOm of the approximate determining equation, we get

X*(t,x0 )- xm(t,XOm)I

sup [^i (t)i'(xo) + I E + &1(t)KQ' ]IxO-xOml. (11.2) J l i=O

X0 EM ,,

Modification of the Numerical-Analytic Method

106

Chapter 2

If the right-hand side of equation (2.1) is continuous together with the partial derivatives with respect to x, then estimate ( 11.2) can be made more constructive. Lemma 11.1. Suppose that, for the periodic system of differential equations (2.1), the conditions of Theorem 9.1 are satisfied and f (t, x) is a function continuously differentiable with respect to x and satisfying inequalities (5.3) in domain (2.3). If x0 belongs to a domain 4 of the form (9.8), then the Jacobi matrices aim (t,

xo)

axo

axp(t, xo) ao m(xo) a0p(xo) ' axo ' axo ' axo

for the vector functions (9.7), (10.1) satisfy the following relations for any natural p < m:

a xp(t, x0)

< Pp(t) < Pp,

(11.3)

axo

axm(t, x0) axp(t, x0) axo - axo a0p(xo)I axo

<- Sa(t) <- S4,

(11.5)

< K(3p,

aA m(xo) - aop (x0) I axo axo

(11.4)

< KS4,

(11.6)

where S4(t) = al(t)S3,

S3 = KQp-'(E-Q)-',

Rp(t) = E+a1 (t)K(E-Q)-I,

S4(t) < 6 S3 = S41

Qp = E + 6 K(E-Q)-'.

Proof. In view of Lemma 2.1 and conditions (5.3), it follows from (9.7) that

axl(t, xo) = E, axo

C

ai2(t , x0) = E + j af (t, X1(t, xo )) ax l(t, x0 ) - 1 J of (s, xl(s, xo)) ax1 (s, x0) ds I dt ail axo J axo 0 ax1 axo T0

Section 11

Solvability of the Approximate Determining Equation

107

xl(t, xo )) azl(t, xo) ' af (t, ail axo

E + (i_)j 0

tTr af(t, xi(t,xo )) aii(t,xo)dr - T J ax, axo ax2(t, xo) SE+K I1-!Ij axo /

L``

5

)0

ail(t, xo) dt + T fT l ax (t, axo

axo

E+aI(t)K.

1

dt]

(11.7)

By induction, according to (11 .7), we obtain axp(r, xo)

E+aI(t)K +...+ap_I (t)Kp- ,

axo

whence, in view of estimate (2.10) in Lemma 2.2 and (5. 10), it follows that ( 11.3) is satisfied. By analogy with the derivation of relations (11.7) from (9.7), we get

ail(t, xo)

- aio(t, xo) = ai,(t, xo) _ axk(t) = E,

axo axo axo axo

(11.8)

ai2(t, xo) - axi (t, xo) = f af(t, -ii (t, x0)) _ af(t, io(t, xo)) axo axo o axo axo 1 r I af(S, x1(S, xo)) - af( s, xo(S, xo ))

T ,0 I\

axo

Ids I dt.

axo J J

(11.9)

In view of the second inequality, it follows from (5.3), (11.8), and (11 .9) that

ai2(t, xo) axi(t, xo) axo axo

<_ a1(t)K,

and, by induction, az j+j (t, xo) ax j (t, x0 ) <- aj(t)KK, axo axo Further, we have

j=0,1,2,.... (11.10)

Chapter 2

Modification of the Numerical-Analytic Method

108

axm(t, x0 ) _ axp(t, x0) _ I atm( t, x0) _ axm -1(t' x0) axo axo Il axO axo

(axmaxt,x0) a rn_2 t,x0)'

0

0

+...+(ai

+I( t. x0) _axPa(t,x0)). (11.11)

Pax o

)

o

Jf

Therefore, according to (11.10) and Lemma 2.2, relation (11.11) yields

aim(t, x0)

axP(t, x0)

m-1

<_ Ea,(t)K` i=p

al(t)KQP_'(E

S4(t)5

-Q)-' = al(t)S3 = S4(t),

6 S3=

S4

i.e., the validity of inequality (11.4) is proved. To prove inequalities (11.5) and (11.6), we use relations (10.1), (5.3), (11.3), and (11.4). Then

aAp(xo) axo

1 Tr

af(t, i (t, x0))

ai (t, x0)

axo

T 0 aip

aAm (xo) - aA p (xo )

1J

< K

K(3P,

(11.12)

af(t, Xr(t, x0)) _ af(t' iP(t' xx0))1 dt

0 axo T[

axo axo

dt

axm(t, X0) ___ xo)

axo

J

<- KS4. (11.13)

axo - axo

The obtained estimates (11.12) and (11.13) coincide with (11.5) and (11.6), respectively. Lemma 11.1 is proved. Theorem 11.1. If, for the periodic boundary-value problem (2.1), (0.2), the conditions of Lemma 11.1 are satisfied and, moreover, for a certain p < m and all x 0 E D^, there exists a constant matrix K1 with positive elements such that

a_P(xo) I axo )

I
(11.14)

Section 11

Solvability of the Approximate Determining Equation

109

and the norm of the matrix K2 = K1KS4 is less than 1, i.e.,

IIK211=q2<1,

(11.15)

where K is the matrix appearing in condition (2.4), then the following assertions are true: (i) for the deviation of the solution x*0 of the exact determining equation (9.25) from the solution x Om of the approximate determining equation (10.2) of the periodic boundary-value problem (2.1), (0.2), we have

Ilx*O-xOm11 <

1 1-q2

IIKIII IISmlI, xp

- xo m

l

<-

(11.16)

A1K 6Sm,

where S. = ()2 KQm(E_Q)_1

(11.17)

SuP (I xOI+sk) • X0 Et

Sk is calculated according to (9.13), and A, is defined in (11.24); (ii) the convergence

1 xp - xO 1

0,

m

l xp - xO I m

(11.1 8)

0,

holds; (iii) the approximate T-periodic solution

xm(t, xOm ) = xOm + Xk(O)

[ f(tI xm-1(t, xOm ) )-7. j.f (s, m-1 (s, xom ))ds + f 0 0

l dt,

J

xO(t, xom ) = Xk(t),

of the periodic boundary-value problem (2.1), (0.2) converges uniformly as m -+ co to its exact T -periodic solution x * (t) = i*(t, x* ), which is the limit function of the sequence zm(t, xp ) , and, furthermore,

Chapter 2

Modification of the Numerical -Analytic Method

110

I x'(t)-

al (t)KQm_

<-

xm(t , xom)I

t(E-Q)-t (IxOm I +6k)

+ [E+at(t)K(E-Q)-1]A

=

^ Kt6m

E(x'(t, xp), xm(t, x0m )) ,

( 11.19)

where 6m is calculated according to (11.17) and S k is calculated according to (9.13). Proof. By analogy with the proof of Theorem 5.1, for solutions xp and xOm equations (9.25) and ( 10.2), we can obtain an analog of formulas (5.20)-(5.22):

XO XOm -

[

m( XOm + 'C(x

Am(x0 ),

axo

of

( 11.20)

aAm(xom+T (xp-xOm )) -t = [E_[P 0m+'C ( x0-x0m)) E axo ] axo )

x

x

C

aAp(xom +T(4O - xom)) - aAm (xOm +T ( XO -XOm)) axo axo

raAp(xom + T WO - xom ))1

L

raAm (xOm + ,c (x0 - X0m ))^

IIx4-xo ll :5 L

(11.21)

J

axo

m

1-1

-IIe(A(xo),Am(x0))II, (11.22)

axo

where

E(A(xo ), Am(xo )) =

(

)2Q*(x) =

3

(

- )2KQm (E_Q)_ 1k(X),

3

(11.23)

Ik(xo ) = I xo I + 6k and 6k is obtained by using (9.13). In view of (11.6), (11.14), and ( 11.15), relation (11.21) implies an inequality of the form (5.24). Since E (A(xo), Am(xp)) < 6m in (11.23 ), relations (11.22) and (5.24) yield the first inequality of (11.16).

Section 11

Solvability of the Approximate Determining Equation

111

The boundedness of the norm in (5.24) guarantees the existence of a matrix A 1 with positive elements such that

E-

C

aAp(xo)_l

(aAp(x0)- aom (xo)1

5 A 1 (11.24)

axo axo ))]

axo

for all x0 E D. Consequently, relations (11.11), (11.20), (11.21), and (11.24 ) indeed yield the second componentwise estimate in (11.16). The validity of the limit relations (11.18) follows from the fact that (11.25)

lim Qm = 0, m-_ which, in turn , yields the limit relation lim Sm = 0. m^-

To obtain inequality (11.19), we use the obvious relation

x*(t, x(, ) - im(t, iom) I

x*(t,xO)- x*(t,xOm ) I + I x*(t,x0m )- xm(t,xOm)I.

(11.26)

In view of the second inequality in (11.16), it follows from estimate (10.7) in Lemma 10.1 with xo = xO and xo = xOm that

Ii*(t,x*)- i*(t,x0m ) I

[E+al(t)K( E- Q)-']I xo

-xoml

5 [E+&1(t)K(E-Q)-']A,KiSm .

( 11.27)

Further, relation (9.12) with x0 = xOm yields

I i *(t,xOm )- im(t,xom ) I <- a'(t)1i(x0), 1'(xo) = KQm-'( E-Q)-'Rk (XOm).

(11.28)

The substitution of (11.27) and (11.28 ) into the right-hand side of ( 11.26) gives the required estimate (11.19). The fact that lim E(i*(t, XO), Xm ( t, xOm)) = 0 M-4-

112

Chapter 2

Modification of the Numerical-Analytic Method

uniformly with respect to t implies the uniform convergence of an approximate T-periodic solution Xm(t, xam) of the form (9.7) constructed for the root x0 = xam of the determining equation (10.2) to the exact T-periodic solution z* (t, x*) of equation (2.1). Theorem 11.1 is proved. Thus, if we find a solution x0 = xom of the approximate determining system (10.2) for sufficiently large m >: m0, then, according to Theorem 11.1, we may expect that the function xm(t, xom) obtained with the use of the recurrence formula (9.7) would be a sufficiently good approximation to the exact T-periodic solution of problem (2.1), (0.2). Let us prove the following statement, according to which one can investigate the solvability of the approximate determining equation (10.2). Theorem 11 .2. Let, for the periodic boundary-value problem (2.1), (0.2), the conditions of Lemma 11.1 be satisfied, let, in a certain ball (11.29)

S(x00), r) c f

with radius r centered at a point x00, a mapping An : Rn-* Rn of the form (10.1) be strongly differentiable, and let the inequality

, xb)) ax

m (t, moo) j raf (txm( t , xm (t, x6) _ ax 1 dt arm ax0 azm axo J TO

of (t

< It 11 xo - xo 11 ,

(11.30)

x6, X" E S(x00^, r), be true in ball (11.29), where

af (t, xm(t, x07)

a

aCm

' ax0

,

xm( t x0)

are the corresponding Jacobi matrices. Also assume that, at the point x0 = xp0), there exists the inverse matrix

[_

af (t, x0 0))) axm(t, x00)) 1c dtl = ax0 T azm xm(t,

Wi(xp0i)

(11.31)

0

and the following inequality holds: 1 m1plll = N1 <

4,

(11.32)

Section 11

Solvability of the Approximate Determining Equation

113

where m l = II W (x°°)II'

PI =

W (xo0)) -T f f (t, xm(t, xO0))) dt 0

(11.33)

Then the approximate determining equation ( 10.2) has a unique solution x0 = xOm in the ball S(x0(0 r1) with radius r1 centered at the point xo°); here , r1 = pIy1 and y 1 is the least root of the quadratic equation H I y 2 - y + 1 = 0. In addition, apxOk+I) proximations obtained by the modified Newton method

x(Ok+1) = x(k) - W (40)) I - T J f (t. xm(t, xpk)))dt I1 J L 0

(11.34)

converge in norm to a solution x0 = x0m and, for the rate of convergence, the following inequality holds: k

II xOm _ xOk)

II <_ ql PI 1 - q1

(11.35)

where 1- 1-4H1 < 1 ql 2 2 The theorem can be proved by analogy with Theorem 5.2. Assumptions (11.30)(11.34) guarantee the applicability of Theorem 1 [KoF, p.470], according to which the iterative process (11.34) converges to a solution xOm of the determining equation (11.34), which is unique in the ball S(xo0), r1 ), and the rate of convergence is estimated by inequality (11.35). In Theorem 9.3, it was proved that if x0 = x* is a solution of the exact determining equation (9.24), then the initial value x*(0) of the exact T-periodic solution of problem (2.1), (0.2) is determined by the formula x*(0) = xo +xk(0).

(11.36)

Let us take the point xok) calculated according to the iteration formula ( 11.34) as an approximation to xo, and the function xm(t, xok )) obtained with the use of the recur-

Chapter 2

Modification of the Numerical-Analytic Method

114

rence formula (9.7) as an approximation to the exact solution z*(t, x0). Let us estimate possible errors. Since X* 0 - X6 k) _ (x0 -xOm )+(xOm -XOk) ), (11.37)

in view of the first inequality in (11.16) and (11.35), estimating the right-hand side of (11.37), we get 1

(

II 4 - X0k) 11

1- q2 K,

k

II II Sm II +

1 ql p^ = E (x0, xpkl ). ( 11.38)

Taking into account that 8(x0, x0k1) m k- 0, we obtain from (11.38) that

Ilx* -XO lII m^ 0,

i.e., XOP + Xk (0) tends to the exact initial value (11.36): xok) +Xk(0)

.,k-4- ) XO +Xk(0).

Let us now estimate the deviation of x*(t, x0) from zm(t, xok)). Obviously,

x*(t, x0) - xm(t, X0 = (x*(t,x0)-x*(t, x(()k)))+(. * (t, xjkl) -xm(t,x(ok))).

(11.39)

Estimating the absolute value of (11.39) and using inequalities (10.7), (11.38), and (9.12), we get

Ix*(t' xo)-xm(t, xp >)I k

<- [E +a,(t)K(E -Q)-'] [..... 1 L.. 11K1 IIIIsmII +lq,glp, + a,(t)KQm-'( E-Q)-'[I Xpk) I+Sk]

e(x*(t, 4O), xm(t, xvk))),

where e = (1, 1, ... , 1).

(11.40)

Section 12

Modification of the Method for Two-Point Problems

115

If we take into account that the right-hand side of inequality (11.40) tends to zero as m, k -* co, then, clearly, lim

Xm (t, x((') = x'(t, xi).

12. Modification of the Method for Two-Point Problems

We can also construct an analog of the modified iteration scheme (9.7) in the case of more general boundary conditions. We consider the differential equation (2.1) with nonseparable boundary conditions (0.2), i.e., we study the boundary-value problem (12.1)

x = f(t,x), Ax(0)+Cx(T) = d,

detC * 0, det (A+C) * 0. (12.2)

Denote by Xk (t) an approximate solution of the boundary-value problem (12.1), (12.2) obtained by the method of algebraic collocation in the form [SaR4]

Xk(t)

= (A+C)- l d + (A+C)

-1

k+1 k+1

1: TjCQj +

lQjtj.

(12.3)

j=1 j=1

In (12.3), the unknown vector coefficients Qj = (Qj 1, ... , Qj„), j = 1, 2, ... , k + 1, are obtained from the following n (k + 1) determining equations: k

k+1

k+1

jQjt' -1 = f tj,(A+C)-1d + (A+C)- 1 j=1

j=1

ITjCQj + 1 Qjt,

, (12.4)

j=1

where t1, t2, ... , tk+1 are points specially chosen in the interval (0, T). For example, t,, i = 1, 2, ... , k + 1, are the roots of a polynomial 'Fk+1(t) of degree k + 1 belonging to the system of polynomials {Tm (t)} orthogonal on the segment [0, T] with weight p (t), where p (t) is a nonnegative function such that T

Jp1(t)dt < oo. 0

Chapter 2

Modification of the Numerical -Analytic Method

116

Under the condition of existence of a solution of the boundary -value problem (12.1), (12.2), it is established that the function x k (t) obtained according to (12.3), (12.4) converges uniformly to the exact solution x *( t) of this problem as k -+ 00 and the derivative ik (t) converges in the mean square to the derivative i *(t) . Furthermore, the following equations are true for their deviations:

IIXk(t)- X*(t )IIC

<-

C1Ek( x *(t)), (12.5)

Il Xk(t)-x*(t)II2 <- C2Ek(x*(t)),

rr

T

[ IIx(t)IIc= mxmx I xi(t)I, II X(t) II2 = m x J P(t) I Xi(t) I2dt

lo

1

1/2

J

Since det C *0, the classical scheme of the numerical -analytic method of successive approximations for problem (12.1), (12. 2) can be constructed on the basis of the recurrence relation (6.1) beginning with zero approximation x0(t, x0) = x0. If the set D0 is determined according to (6.2) and problem (12.1), (12 .2) satisfies conditions (2.4)-(2.6) and (6.2), then , in view of Theorem 2.1, a sequence of functions xm (t, x0) of the form (6.1) converges uniformly as m -* oc with respect to the set (12.6)

(t, x0)E [ 0, T] x DP to the solution x * (t, x O) of the integral equation

[ ftxtdt+(c-'d+c-'A+Exo) - TJ f( s,x(s))ds dt (12.7) x(t) = X0 + x(0 J 0 0

]

satisfying the two-point boundary conditions ( 12.2). Moreover, all functions of sequence (6.1) satisfy the same boundary conditions for all x0 E DP and estimate (6.4) holds for the deviation of xm (t, x0) from x* ( t, x0). As in the case of periodic boundary-value problems, one can improve the rate of convergence of the approximate solution to the exact one on the basis of the combined use of approximations obtained by the application of the collocation method and numerical-analytic method of successive approximations . Therefore, instead of the iterative process (6.1), we construct successive approximations according to the following formula: X(t, Xo)

= XO +

Xk(0)

[ + J

f(t, zm-1(t, x0)) -

0

T J f( s , Xm -,(s, xo ))ds l dt

0

Section 12

Modification of the Method for Two-Point Problems

117

+ T[C-td-(C-to+E)(xo+xk (0))], m = 1, 2, ... ,

( 12.8)

where -m ( t, x0) =xk ( t) is the zero approximation obtained by the method of algebraic collocation (12.3), (12.4). It can easily be verified that all functions of sequence ( 12.8) satisfy the boundary conditions ( 12.2) for all x0E Dp and pass through the point xm(0, x0) = XO + xk(0), m = 1, 2, ... , (12.9) at t = 0. It follows from (12.9) that xo is a vector parameter, i.e., a correction to the initial value determined by the approximate solution xk(t). The following statement enables one to conclude about the rate of convergence of successive approximations: Theorem 12.1. Suppose that the right-hand side f (t, x) of the system of differential equations (12.1) is defined, continuous in domain (2.3), and satisfies conditions (2.4) and (2.6). Also assume that the set 4 of points io E R° such that the points x0 + Xk(O) = Z0(xo)

(12.10)

are contained in the domain D together with their J3-neighbourhoods,

p = 2M+R1(ZO), (i1(z0) = IC-'d-(C-to+E)(xo+xk(0))I, (12.11) D^

(12.12)

^ 0,

and the values of a function xk (t) of the form (12.3), (12.4) belong to the domain D. Then a sequence of functions zm(t, x0) satisfying the boundary conditions (12.2) and passing through point (12.9) at t = 0 converges uniformly as m -* oo with respect to the set (t, xo)E [ 0, T]x Dj (12.13) to the limit function z*(t, x0). Moreover, i*(t, x0) is a solution of the integral equation

x(t) =

xo

+ x k (0) + J [ f(tx(t)) - TJf(s, x(s))ds 0

0

J

+ T (C-'d - (C- 'A + E)(xo + xk (0))) dt

(12.14)

Chapter 2

Modification of the Numerical -Analytic Method

118

that passes through point (12.9) at t = 0, i.e., a solution of the initial-value problem x = f(t,x)+A(x0 ),

x(0) = xO+xk(0),

(12.15) T T5 f(t,x•(t,xo))dt A(xo) = 1 [C-1d-(C-1A+E)( xo+xk (0))] - 1

0

satisfying the boundary conditions (12.2). In this case, the following componentwise estimate holds for the deviation of zm (t, x0) from the limit function:

II x* (t, x O ) - xm(t, xO) II <- a1(t)KQm-

1(E-Q)-tµk(xO) = a1(t)Y(x0 ),

m=1,2,...,

(12.16)

where µk(XO) = Yk+IXOI,

Yk(t) = Xk(O) +

J[

Yk =

f(txk(t)) - T f 0 0

max IYk(t)I+ tE[0,T]

f(s,xk(s))d s (12.17)

+ T(C-1 d - (C- 1A+ E) (x0 + xk (0)))]dt - xk(t),

Y(xo) = KQm-1(E-Q)-11kk(XO)• Proof. According to Lemma 2 . 1, it follows from ( 12.8) that

I xm(t,xO ) - zo(xO)I

J [ft, xm -1(t, x0)) - T J .f (s, xm-1 (s, x0))ds I dt

TM+Ri(zo), m=1,2,... 2

+

R 1(zo) (12.18)

By virtue of conditions (12.10)-( 12.12), it follows from relation (12.18) that all functions Xm (t, x0 ) of the form ( 12.8) belong to the domain D for all m = 1, 2, ... , t E [ 0, T] and any x0E 4.

Section 12

Modification of the Method for Two-Point Problems

119

Since ( 12.8) yields

x1(t,x0 )-x0(t,x0 ) = x0 + Xk(O) t

IT

+ J Pt, xk(t)) - T f f(s, xk(s))ds

0

0

+ I(C-1d-(C- 1A+ E) (x0 + xk(0))) dt - xk (t) = x0 + Yk(t),

J

(12.19)

where Yk ( t) is calculated according to formula (12.17), for the difference

rm+I (t) =

I xm + 1(t, x0)-xm (t, x0)I

(12.20)

it follows from (12.19) with m = 0 that r1(t) = Yk+IxOI

= 9k(x0)

(12.21)

In the case of successive approximations for difference (12.20), relation (2.15) is also true, whence, by induction and in view of (12.21), we get rm+1(t ) = Kmam (t)µk(xo ),

m=0, 1, 2 , ... , ( 12.22)

where the functions am (t) are defined by (2.9). Applying inequality (2.10) to the estimation of the functions am (t) in (12.22), we obtain rm+1(t) - 61(t)KQm_ .tk(xO). (12.23) Therefore , for a sequence of functions Xm(t, x0 ) of the form ( 12.8), by analogy with (9.16) and according to (12 .23), we get I xm+j(t , x0)-xm ( t,x0)

rm+i(t)

<_ 61(t)K I

<- Li(t)KQmm -1

I Qm+i-z 11tk(xO)

j-1 YQ`Pk(x0) <_ a1(t)KQm-1(E-Q)-1µk(xo ). i=0

(12.24)

Modification of the Numerical-Analytic Method

120

Chapter 2

Similarly to formula (9.16) in the periodic case , inequality (12.24), by virtue of the limit relation (12.25), also implies the uniform convergence of sequence ( 12.8) to the limit function lim im (t, Xo) = X*(t, Xo) m-*-

(12.25)

in domain ( 12.13). Taking into account that all functions of sequence (12.8) satisfy the boundary conditions ( 12.2), we can conclude that the limit function (12.25) also satisfies the given boundary conditions. In this case , as j -* oo, relations ( 12.24) and (12.17) immediately yield estimate ( 12.16). Finally, passing to the limit in ( 12.8) as m-4 00 and taking ( 12.25 ) into account, we establish that the limit function (12.25) is a solution of the integral equation ( 12.14) satisfying the initial condition ( 12.9) for t = 0, and it indeed can be obtained as a solution of the Cauchy problem ( 12.15). Theorem 12.1 is proved. Clearly, by the proper choice of zero approximation io(t, x0 ) = Xk(t), one can always satisfy the inequality µk (x0 ) < M. Therefore, comparing (6.4) and (12 . 16), we can conclude that the iterative process ( 12.18) converges faster than (6.1). Using the modified iterative process ( 12.8) for the differential equation (12.1), we can establish the following statement on the parameter of control: Theorem 12.2. If all conditions of Theorem 12.1 are satisfied, then, for any point xo from the domain D^ defined according to (12.10)-(12.12), there exists a unique value of the vector parameter of control µ = (µ I, ... , µ„ ),

^ =

T[C

-'d -(C-t A+E) (xo +xk (0))] - T J f(t, i*(t, xo )) dt,

(12.26)

0

(here, i*(t, x0) is function (12.25)) such that a solution x = x(t) of the Cauchy problem for the system of differential equations obtained from (12.1) by addition of the vector parameter (12.26) to the right-hand side, i.e., z = Pt, x)+ µ,

x(0) = x0 + Xk(O),

(12.27)

is, at the same time, a solution of the boundary- value problem x = f(t, x) + µ, Ax(0) + Cx(T) = d.

(12.28)

Proof. It is shown in Theorem 12.1 that a function x(t) = i*(t, x0) that is a solution of the Cauchy problem (12.15) satisfies the boundary conditions ( 12.2). Comparing

Section 12

Modification of the Method for Two-Point Problems

121

problems ( 12.15) and (12.27), we conclude that the proper value of the vector parameter µ in (12.27) is given by expression ( 12.26), i.e., in the case where µ coincides with the value of A(x0) in (12.15). The uniqueness of a value of the parameter t of the form ( 12.26) can be proved by contradiction as in the proof of Theorem 2.2 on the basis of relations (2.23) and (2.24). Theorem 12 .2 is proved. Let us now give necessary and sufficient conditions that enable one to conclude about the solvability of the boundary-value problem ( 12.1), (12.2) on the basis of the properties of the limit function of sequence ( 12.8) if x0 is regarded as a parameter. Theorem 123. Suppose that the conditions of Theorem 12.1 are satisfied. Then, in order that a solution x = x * (t) of the Cauchy problem x = f(t,x ),

x(0) = x0 + Xk ( O), (12.29)

where Xk ( t) is the approximate solution obtained by the method of algebraic collocation (12.3), (12. 4), coincide with the exact solution of the boundary- value problem (12.1), (12. 2), it is necessary and sufficient that the determining function

A(xo) = T[C-'d -(C-'A+E) (x0 +xk(0))] - T j.f(t, x*(t, xo ))dt 0

(12.30)

vanish at the point x0 appearing in the initial conditions (12.29), i.e., 0(x0) = 0. (12.31) Here, i*(t, x0 ) is the limitfunction of sequence (12.8). Furthermore, under these conditions, we have x*(t) = z*(t,x0)

(12.32)

and the following estimate for the deviation of an approximate solution xm (t, x0) of the form (12.8) from the exact solution (12.32) is true: I x*(t)- xm (t,xo) 1 <- aI(t)KQm-'(E-Q)-'ltk(xO )•

(12.33)

Proof. It is established in Theorem 12.1 that a function i*(t, x0) of the form (12.25) is a solution of the Cauchy problem (12.15) that satisfies the boundary conditions (12.2). Therefore, if condition (12.31) is satisfied, then problem (12.15) turns into

122

Modification of the Numerical -Analytic Method

Chapter 2

(12.29) and, consequently , this is sufficient for x = x * ( t) = X* (t, x0 ) to be a solution of the boundary-value problem ( 12.1), (12.2). To prove the necessity of (12 . 31), one must take into account that if x = x * (t) is a solution of the boundary -value problem ( 12.1), (12.2) with the initial value ( 12.29), then a solution x = x(t, x0, A) of the Cauchy problem ( 12.15) satisfies the boundary conditions (12.2) only if condition ( 12.31 ) is satisfied. This follows from the fact that, in this case, x(t,x0, A=0)=x*(t). We take into account that, according to Theorem 12.2, there exists a unique value of the parameter µ of the form ( 12.26) for which a solution of the initial-value problem (12.27) is simultaneously a solution of the boundary-value problem ( 12.28 ). Note that such a value has already been indicated , namely,

µ=A(x0)=0, (12.34) and, consequently , value (12.34) is unique. Since equality (12.32) is true in this case, estimate (12.33) follows directly from (12.16). Theorem 12.3 is proved. Thus, on the basis of the modified sequence ( 12.8), the following numerical-analytic scheme for the construction of a solution of the boundary -value problem (12.1), (12.2) can be proposed: (a) For a certain value k, calculate an approximate solution xk(t) by the method of algebraic collocation (12.3), (12.4); (b) regarding x0 as a parameter in the domain L , construct a sequence of functions Xm(t, x0 ) according to formula ( 12.8); (c) find the limit function (12.25) and the roots x0 = xo of the exact determining equation (12.31); (d) the solution of problem ( 12.29) for x0 = xp* is the exact solution of the boundary-value problem ( 12.1), (12.2) coinciding with the function

i(t, 4),

and

zm(t, xo) is an approximate solution. In the next section , we show how to prove the existence of a solution of problem (12.1), (12.2) without finding the limit function (12.25).

Section 13

123

Relationship between Exact and Approximate Determining Equations

13. Relationship between Exact and Approximate Determining Equations The exact determining equation (12.31) is composed on the basis of the limit function x* (t, x0 ) of sequence ( 12.8). An approximation xm (t, x0 ) of the form ( 12.8) generates the approximate determining functions

A (xo ) = T[C-1 d -(C-1 A+ E) (xo +xk(0))] - T J.f(t, xm(t, x0 ))dt 0

(13.1)

and the m th-order equation Am(xo)

=0.

(13.2)

It turns out that we can conclude about the existence of a solution of the boundary-value problem (12.1), (12. 2) not only on the basis of zeros of the exact determining function (12.30), but also on the basis of zeros of the approximate determining function (13.1). Theorem 13.1. Suppose that the conditions of Theorem 12.1 are satisfied and, in addition, the following two conditions are satisfied:

(i) there exists a convex closed domain D 1 c D^ such that, for a certain order m of an approximation xm(t, x0 ) of the form ( 12.8), the approximate determining equation (13.2) has a unique solution x 0 = xOm with nonzero index in the domain D1 ; (ii) on the boundary S1 of the domain D1, the following inequality holds:

inf xOES,

T [C-1d - (C- 1 A + E) (x0 + xk

(0))]

T 7 f f(t, xm(t, x0))dt 0

=

()2 QY(xo) ,

(13.3)

where Q = T K and the vector Y(x0) is determined by formula (12.17). It Then the nonlinear two-point boundary-value problem (12.1), (12.2) has a solution x=x*(t) such that its initial value at t = 0 is determined by a certain point x0 = xo lying in the domain D1, i.e.,

124 Modification of the Numerical -Analytic Method x*(0) = xo +Xk (O), x* a DI,

Chapter 2 (13.4)

where Xk ( O) is the initial value of the approximate solution xk(t) obtained by the method of algebraic collocation (12.3), (12.4). Proof. We establish that the deviation of an exact determining function A(x0) of the form (12.30) for the boundary-value problem (12.1), (12.2), where i* (t, x0) is the limit function of sequence (12.8), from the approximate determining function (13.1) can be estimated by the inequality

I A(xo) - Am (xo) I = () 2 QY(xo

)3

= c(A(x0 ), Om (x()),

(13.5)

where Y(x0) is determined by (12.17). According to (12.30) and (13 . 1), we have

A(xo)-Am(xo) _ -Tj[f(t,x*(t,x0))-f(t,xm(t,xo))]dt. 0 In view of the Lipschitz condition (2.4), inequality ( 12.16), and expression ( 10.6), the last inequality yields

IA(xo)-Om(xo)I = TJat(t)dtKY(xo) = ()2QY(xo), 0 i.e., (13. 5) is true. The proof is completed by analogy with Theorems 3.1 and 10.1. It can easily be verified that, according to condition (i) and Lemma 3 .3, the rotation of the vector field A. (x0) generated by mapping (13.1) is not equal to zero on St. Considering on Sl the set of vector fields P(O, x0) of the form (3.4) continuous in the collection of variables and continuous everywhere on St that connects fields (13.1) and (12.30) according to relations (3.6), (13.3), and (13. 5), we can easily conclude that P(O, x0) is not equal to zero anywhere on St . And this is sufficient for the conclusion that the vector fields (13.1) and (12.30) are homotopic. Consequently, according to Lemma 3.2, the rotation of the vector field generated by the exact determining function (12.30) is also not equal to zero on St. Then, by virtue of Lemma 3 . 1, the exact determining equation (12.31) of the nonlinear boundary-value problem (12.1), (12. 2) has at least one solution x0 = x* a Dt in the domain D1. According to Theorem 12.3, the latter value of x0 determines a solution x*(t) of the boundary-value problem (12.1), (12.2) that passes through the initial point x* (0) at t = 0 and is determined by (13.4). Theorem 13.1 is proved.

Section 13

Relationship between Exact and Approximate Determining Equations 125

Let us now obtain estimates for the differences between the limit functions of the se quences xm(t, x'0) and im(t, X'Q) calculated according to (12.8) for x'0, X'Q e Dp, which are necessary in what follows. Moreover, we establish the continuity of an exact determining function A{x0) of the form (12.30) with respect to points x 0 e Ct. Lemma 13.1. Suppose that the conditions of Theorem 12.1 are satisfied for the nonlinear two-point boundary-value problem (12.1), (12.2). Then, for any points x'0, X'Q e Da, the limit functions x*(t,x0) and x*(t,x0) of the sequences xm(t,x0) and xm(t, x0), respectively, calculated according to (12.8) satisfy the inequality |i'(f,jc'o)-i*(*,xo)| <• [£ + a,(OA:(£-0- l ](£ +

tf)|*'o-*o|,

(13.6)

where a^t) is the function appearing in (2.10). Proof. It follows directly from (12.8) for m = 1 that *i('.*o)--*i('.*o) = (■*'<)-*o) -

J.(C~1A

+

E)(X0-X'O),

(13.7) | i , ( / , X o ) - i i ( ? , X o ) | ^ (E +

R)\X'0-X'Q\,

where R = |C-U + £ |

(13.8)

is the matrix whose elements are the absolute values of the elements of the matrix C~lA+E. For m = 2, the recurrence relation (12.8) yields x2(t, x0) - x2(/, X'Q) = (x0 -

X'Q)

- ^ ( r ' A + E)(x0 - X'Q) tr + J (/(/,x,(»,xb))-/(r,x,(r.jtb))) oL - ^ J ( / ^ i , ( 5 , x ; ) ) - / ( 5 , x 1 ( 5 , x ; ' ) ) ) ^ dt. 1

0

In view of formula (4.3), the Lipschitz condition (2.4), estimate (13.7), and notation (13.8), the last equality yields

126

Modification of the Numerical-Analytic Method

I x2(t, x o)- x2(t,x 0)I

Chapter 2

<_ (E +R)I x 0 - x0I t + K (1-t)J Ixl(s,xo)-xl (s,xo)Ids T0 T +

t J l xl(t, xo )- zl(s,xU7lds Tt

<_ (E+R)Ix0-x0"l+K(E+R)Ix0-x0Ia1(t) (E +al(t)K)(E+ R)I xo - xo 1.

(13.9)

Analogously, by induction, in view of (13.7) and (13.9), we obtain zm(t,xo

xm(t,xo )

<- (E+R) x'0 - X0 + K[ (i_

.)5 lm- i(s, x)x

m- 1(s,x)ds

T + T. JI xm-I (s,x0)-xm-1(s,x0)Ids

m-2 <_ (E+R)Ixo-xol + K a1(t)K` (E+R)Ix0-x0I i=1

=

[

E+ Eai(t)K`(E+R)Ixo I

-xxI.

(13.10)

i=1

If m -+oo in (13.10), then, by virtue of relations (2.10) and (2.6), we obtain the following estimate for the limit functions: x*(t, xo) -i*(t, xo) I ll; <- E+al(t)jK(TKI (E+R)Ixo-x" // =o 7E

I

<- [E+al(t)K(E-Q)- 1](E +R) Ix0' -x"I, where the function al(t) is determined in (2.10). Lemma 13.1 is proved.

Section 13

Relationship between Exact and Approximate Determining Equations 127

Theorem 13.2. If the conditions of Theorem 12.1 are satisfied, then an exact determining function A (x0) of the form (12.30) for the nonlinear two-point boundary-value problem (12.1), (12.2) is defined, continuous, and bounded in the domain 4 defined by (12.10)-(12.12). In this case, for arbitrary points x0, x'0, xp E 4, the following inequalities hold: 1A(xo)I <- T

IA(xo)-A(xo)I <-

I[C- 'd-(C_1A+E)(xo+xt (0))]I+M, (13.11)

L T E+ ! K(E-Q)-t (E+R)IIxo-xoI, (13.12)

where R is a matrix of the form (13.8). Proof. It is proved in Theorem 12.1 that, for any points xo E DD, there exists the limit of the sequence of functions (12.8). Since the limit function X*(t, x0) is continuous, we establish that, for xoe 4, the exact determining function (12.30) is also continuous and bounded, and estimate (13.11) is true. For points xo, xo a 4, relation (12.30) yields T

A(xo) -A(xo) = -1 -(C to+E)(xo-xo) - T

5[f(t,x* (t,4)) - f(t,x'(t,xo')) ] dt.

In view of inequality (13.6) and value (10.6), the last expression yields the required estimate (13.12):

A(%) - A(x0) I T

< TRIxo

-xoI + TKJ I x*(t,xo )-x'(t,xo)Idt 0

1 Rlxo-x"I + K(E+R)Ixo-x"I T T + K(E-Q)-1(E+R )Ixo-x0I T Jat(t)dt

= [R+K(E^

KE_Q_1)E^R ]Ix _ x.

128

Modification of the Numerical-Analytic Method

Chapter 2

Below, we present a result that allows one to establish necessary conditions for the solvability of the nonlinear boundary-value problem (12.1), (12.2) on the basis of specific properties of the approximate determining function (13.1). Theorem 13.3. Suppose that the nonlinear two-point boundary-value problem (12.1), (12. 2) satisfies all conditions of Theorem 12.1. Then a certain subdomain D z of the domain Dzc contains a point xo = x* determining the initial value of the solution x = x*(t) of the nonlinear boundary-value problem (12.1), (12. 2.) at t = 0 according to formula (13.4) only if, for all integer positive m and an arbitrary point Xo E D2, the following inequality holds: T T[C-'d-(C-'A+E)(X0+xk(0))] - T f f(t,X m(t,xo))dt

0

< xsup o Dz

1 R+K T

E E-T

K(E-QV

(E + R)\

XQ-XQ

I + ()2 QY(io), (13.13)

where Q = TK, Y(xo) = KQm_I(E-Q)-'µk(xo), and µk(xo) is calculated according to (12.17). Proof. To prove the necessity of condition (13.13) in the case where there exists a solution x = x * (t) of the boundary-value problem (12.1), (12.2) with the initial condition (13.4), we take into account the fact that, according to Theorem 12.3, the determining function (12.30) vanishes at the point xo = xo. We use inequality (13.12) with xo = xo and xo = xo, where xo is an arbitrary point lying in the domain Dzc DR. Since A(xo) = 0, we have

IA(xo)I = 1-L TR+KE+ 9 ET- K(E-Q)- '(E

+R)]Ix

o

-xo (13.14)

Estimate (13.5) for the point xo = xo immediately yields the inequality

IA (zO)I - IA(zo) + m

( ,,)2 QY( 10). 3

If we substitute (13.14) in the right-hand side of (13.15), then we get

(13.15)

Section 13

Relationship between Exact and Approximate Determining Equations 129

U(rld-(C-lA '

+ E)(xo + xkm]

~ ' o

±]f{t,imit,x0))dt

< [l^ + ^ £ + ^^(£-e)-'j(£+i!)]|jr-x5| + g j QY(i0), <

sup [iR+KfE

+ ^KlE-Qr^lE+Rillx-xol

+ fc) QY(x0),

which which completes completes the the proof. proof. This theorem can actually be used for the approximate determination of the initial value of a solution of the boundary-value problem (12.1), (12.2) at t = 0 . For this purpose, we proceed by analogy with Section 10. Assume that, for the boundary-value problem (12.1), (12.2), the domain Dp determined by formulas (12.10)—(12.12) can be represented as a union of subsets DL in the form (10.15). In order to clarify whether some subset D contains the point x0 = JCQ determining the initial value of a solution of the boundary-value problem (12.1), (12.2) according to formula (13.4), we proceed as follows: For a certain successive approximation of order m of the form (12.8), we calculate the vector

A m ( 4 ) = ±[
i

+ E){xii +

'

T

xtm]--!zfAt.Xmfrxo))'l'< ' 0

where x'0 is an arbitrary point of the set Dt. On the basis of Theorem 13.3, we choose subsets D ( , is {1, 2

N], whose ar-

bitrary points X'Q e D( satisfy the relations T

I

Urld - (C'A+E) (4 + *t(0))] - i J /(», *„(/, xi))dt\ ' 'o I <

sup [ I * + K{E + ^ K ( E - Q T ^ ( £ + / ? ) ] \xo-4\

+ (j) gy^).

(13.16)

Denote by M'm the union of such sets D-t. Obviously, M'„ includes a point x0 = x^ that determines the initial value of a solution of the boundary-value problem (12.1), (12.2) according to (13.4).

130

Modification of the Numerical-Analytic Method

Chapter 2

Note that the more the number of subdomains verified by inequality (13.16), the more exactly we can determine M,n because the set

M,n tends as i, m - 00

to the set

M(xo) whose points determine the initial value (13.4). We can take an arbitrary point x0 a Mm as an approximate value xo in (13.4). In this case , it is clear that lx0-x* 1 <- sup li0-x01, (13.17) xpeM;,

and it is natural to take the mth approximation calculated according to formula ( 12.8) for x0 = zo as an approximate solution of the boundary -value problem (12.1), (12.2). Let us estimate the deviation of . m (t, x0) from the exact solution z *(t, x* ). Theorem 13.4. Suppose that, for the nonlinear two-point boundary-value problem (12.1), (12. 2), the conditions of Theorem 12.1 are satisfied, xo is a solution of the exact determining equation (12.31), and x0 is an arbitrary point from the set Min that satisfies the condition

1 [C-id-(C-'A+ E)(xo+xk(0))] - TJ.f(t, im(t,x0))dt T 0

<- supr[TR+KI E+ - K(E-Q)-1 I(E+R)] Ixo - ioI + ()2QY(io). 9 3 X0 a Mm Then the error of the approximate solution im (t, z0) of the nonlinear boundaryvalue problem (12.1), (12.2) calculated according to the formula xm(t, x0) = x0 + xk(0)

]

+ J I f(t, Xm-1(t, i0)) - T J f(s, xm-i(s, xo))ds dt

+ [C-'d - (C- 1A+ E) (xo + xk(0))],

-'0(t,'0) = Xk(t),

(13.18)

where Xk ( t) is the approximate solution calculated by the method of algebraic collocation (12.3), (12.4), satisfies the inequality

Section 14

Determination of Initial Values of Solutions

i*(t, x o) - im(t, xo)

131

I

sup [a, (t)Y(xo)+(E+at(t)K(E-Q)- ')(E +R)Ixo -io l], (13.19) xpEM,,

where Y(xo) is the vector appearing in (12.17). Proof. It is obvious that

Ii*(t,4)-im(t,io)I

i'(t,xo)-im(t,xo)I +Iim(t,xo)-zm(t,io)I.

Estimating the first term on the right-hand side of the last inequality according to (12.16) and the second term according to (13.10) and using (13.17) and (13.18), we arrive at estimate (13.19). Theorem 13.4 is proved.

14. Determination of Initial Values of Solutions of Two-Point Boundary-Value Problems

Assume that xOm is an exact solution of the approximate determining equation (13.2) of the two-point boundary-value problem (12.1), (12.2). In view of (13.5) and (13.16), we can conclude that xom n M.

If we take the function im(t, xOm) obtained according to (12.8) as an approximate solution, then, taking (13.19) into account, we get

x*(t, x0) - xm(t, xom) I

<- sup

[a,(t)Y(xo)+(E+a,(t)K(E-Q)-' )(E +R)Ixo -xom Ii • (14.1)

xp a M,,,

Under certain conditions of smoothness of the right-hand side of the system of differential equations (12.1), estimate (14.1) can be made more convenient for practical verification. First, we prove several auxiliary statements. Lemma 14.1. Suppose that, for the nonlinear two-point boundary-value problem (12.1), (12.2), the conditions of Theorem 12.1 are satisfied, the function f (t, x) in (2.3) is continuously differentiable with respect to x, and the corresponding Jacobi matrices of the vector function f (t, x) satisfy conditions (5.3).

Modification of the Numerical-Analytic Method

132

If a point x0 belongs to a domain

Chapter 2

Di of the form (12.10) -(12.12), then the Jac-

obi matrices axm(t, xo)

axp(t, xo) aAm(t, xo)

aAP(t, xo)

axo ' axo ' axo ' axo of the vector functions (12.8) and (13.1) satisfy the following estimates for any natural p<m: azp(t, x 0)

axo

< 7P(t) < 7p, (14.2)

axm(t, xo) _ axp(t, xo) axo

< H4(t) <- H41

(14.3)

axo

aA P(t, xo)

axo

(14.4)

< 1R+KYp, T

aA m(t,xo) aAp(t, xo) axp axp

(14.5)

<- KH4,

where H4(t) = &1(t)H3, H3 = KQp-1(E-Q)-'(E+R),

H4(t) <- T 7cH3 = H4, yp(t) = [E+ a1(t)K(E-Q)-'] (E+R),

yp =

(14.6)

[E+K(E_Q-']E+R.

Proof. By virtue of Lemma 2.1 and conditions (5.3), the iteration formula (12.8) for m =1, 2 yields

o ail(to o) = t i laxt(t,xo)I ax o

E - T(C A+E), ax

<_ E+R,

(14.7)

Determination of Initial Values of Solutions

Section 14 ai2(t,

x 0)

133

_ E + Tr I af (t, xl(t, x0)) axl(t, x0)

axo

O

ail

axo

xo) I t t 1 f of (s, i1(s, xo)) ait(s, ds dt - T(c- A+E) ax o ait To t

E + 1-1 ) J

T

o

I

af (t, xl(t, xo )) a xl (t, xo) dt ail axo

t faf(t , it(t, xo)) ait(t, xo) dt I t (c- t A + E), - T

ail axo

Tt ai2(t, xo) axo

E+K(1

-T)J

a il(t, x 0) dt

0

+ TtJ

t T ail(t, x0)

axo

dt] + R <- [E + Ka I ( t)] (E + R).

axo

Taking ( 14.7) into account, by induction we obtain the following inequality:

aip(t, x0) axo

p-t <- LE + I ai (t)K' (E+ R), L i=l

where ai ( t) is a function of the form (2.9). In view of (2.10) and (5 . 10), inequality ( 14.8) yields

aip(t, x0)

<- [E+ al(t)K(E-Q)-l] (E+R),

axo

i.e., relation ( 14.2) is true. To verify ( 14.3), we consider the following differences obtained from (12.8):

ail(t, x0)

ax 0

ai0(t, xo) = ail(t, xo)

- ax 0

ax 0

-

aXk(t)E= t (C- 1 A + E), - T ax 0

(14.8)

Chapter 2

Modification of the Numerical -Analytic Method

134 a-' 201 x0)

ai 1(t, x0) ( [af(ti1(t, xp )) - of (t, (t, x0 ))

axo

0

axo

_

i j 1af

s, a , x0)) - af(s, a(s, xo) }ds dt

]

T0

_ (1- t

T/) J0

[af(t' -'I(t'xO )) - af(t, x0( t, xo))ldt

axo axo

J

_ t f raf(t, xl(t, x0)) _ af( t, 4(t, x0))^dt . T f L axo axo

( 14.9)

By virtue of the second relation in (5.3) and (14.9), the last equality yields ai2(t, x0)

a il(t, x0)

axo axo

< K (1-T ) $(E+R)dt + T J(E+R) dt] <_ al(t )K(E+R).

[

o

r

By induction, one can easily obtain

aij +1(1, x0) _ aij(t,x0)

< _ al(t)KK(E+R),

j=0, 1,2, ... ,

axo axo

where a j ( t) is a function of the form (2.9). Estimating the terms on the right-hand side of the equality axm(t, x0 ) axp(t, x0) axo - axo

raxm(t, x0) - axm-1(t, x0 )1 raxp+l(t, x0 ) - axp(t, x0)1 Il Jl+...+ll J axo axo

by using formula (14.10), we get

135

Determination of Initial Values of Solutions

Section 14

a

M_ I

a xp(t, XO)

xm(t, x0)

(14.11)

jai(t)K'(E+R). i=p

axo axo

By virtue of Lemma 2.2, relation (14.11) yields axm(t, x0) axp(t, x0) ax0 ax0

i-1 m-1 ( 7 < ^K(KI (E+R)6c1(t), Zl i = p 1t

<- a1(t)KQp-1(E-Q)-1(E+R) = a1(t)H3 = H4(01 i.e., inequality (14.3) is true. Proceeding to the proof of inequalities (14.4) and (14.5), we obtain from (13.1) the following relation:

a0p(x0 ) = _ 1

_ 1 J of (t, xp(t, xo)) aip(t, x0)

' T (- A+ ^ T 00 azp

axo

axo dt

.

(14.12)

According to the first inequality in (5.3) and (14.2), this yields aAp(xo)

<

TR

axo

+K ' P.

If follows from (13.1) that aAm(xo ) xo

a

_ aep(xo ) = _ 1 f raft, xm(t,

axo

T O

)) - af(t, xp(t, xo))l dt,

I axp axo J

whence, in view of (5.3) and (14. 3), we get

a 0m(xo) a0p(xo) axo axo

T axm(t, x0)

< IKJ O

axp(t, x0)

dt <- KH4,

axo axo

which completes the proof of Lemma 14.1. Let us obtain an estimate for the deviation of solutions of the exact and approximate determining equations (12.31) and (13.2). Theorem 14.1. Suppose that, for the two-point nonlinear boundary-value problem (12.1), (12.2), the conditions of Lemma 14.1 are satisfied and, for certain p < m and

136

Modification of the Numerical-Analytic Method

all x0 e

Chapter 2

one can indicate a constant matrix K3 with positive elements that sat-

isfies the inequality

C

a,P(xo)1 -1 axO

<- K3 (14.13)

)

and, furthermore, the norm of the matrix K4 = K3KH4 is less than 1, i.e., II K411 = q4 < 1, (14.14) where K is the matrix appearing in the Lipschitz condition (2.4). Then the following assertions are true: (i) for the difference between the solutions xo and xom of the exact and approximate determining equations (12.31) and (13.2), respectively, of the nonlinear boundary-value problem (12.1), (12.2), the following estimates are true:

Il xo - x 0, 11

<-

1 1g4IIK3IIII0m

II,

1 xp

-xOml

<- A2K3Am ,

(14.15)

where 'cQm(E_Q)_ 1 Om = ()2

SUP (IxOI +Yk), (14.16) zOEDo

Yk is determined according to formula (12.17), and the matrix A 2 is determined according to (11.23); (ii) the following convergence takes place: 11 x0 - xom 11 M-4- ) 0, I x0 - xom l 0; (14.17) (iii) the approximate solution zm(t, xom ) of the nonlinear boundary-value problem (12.1), (12.2) obtained by the recurrence formula (12.8), namely, xm(t, x0 ) = xOm + Xk(O)

f(t' .iM _j(t'XO. )) + f

I

T J f(s, xm -1(S, xOm))dS1dt

137

Determination of Initial Values of Solutions

Section 14

+ T[C'd -(C- 'A+E) (xOm +Xk(0))] uniformly converges as m - 00 to its exact solution x * (t) = z*(t, x*), which is the limit function of the sequence zm(t, x* ) constructed on the basis of xO* ( a solution of the exact determining equation (12.31 )); moreover, the following estimate is true: x*(t)-

Xm (t,

I

XO)

:5 &i(t)KQin-1( E-Q)-'(I XOmI +1k) + [E+a1 (t)K(E-Q)- '](E+R)AZK39m ( 14.18)

= E(i*(t,x*O), xm(t, x0m )),

where Om and Yk are determined by formulas (14.16) and (12.17), respectively.

Proof. We use the scheme of the proof of Theorems 5.1 and 11.1. In the case of the nonlinear boundary-value problem (12.1), (12.2), the following relations of the form (11.20), (11.21) are true for the determining functions (12.30) and (13.1): _1

xp -xom

[

- [m(xom + 'Cx* - XOm ) I

ax Am(XO), ) 0

1

t m(XOm +,r(x0 -x0m))

)

(14.19)

-1

- aAm(X0m +'C(x* -XOm )

_ E

x

1

-ax0 axo

aAp(X0m +'C(XO -XOm)) aAm (XOm +'t(x0 - X0m

a x0

ax0

[

))/

(XOm t(xp - xOm))1 0 (14.20)

J

In view of estimate (13.5) and the fact that A (x*) = 0, relation (14.19) yields

C

_1

az m (xOm + ,C ( XO - XOm ))

I I Xo - Xom I I - ax0

l

IIE(A(x0 ), Am(XO ))II' (14.21)

Modification of the Numerical -Analytic Method

138

Chapter 2

where £(A(xo), Am (xo)) = () 3 QY(x),

Y(xo) = KQ'"-'(E-Q)-tµk(x0), ltk(xo) = Yk + 1x0* 1, (14.22)

and Yk is calculated according to (12.17). By virtue of conditions (14.5), (14.13), and (14. 14), relation (14.20) yields the following inequality of the form (5.24):

r E

-

))l tl a0p(XOm +T(x0-XOm)) X0 m axo l aXO la a^m(XOm+ t(xo - XOm )] axo

<- 1 . (14.23) 1-q4

If we estimate (14.22) with the use of a vector Om of the form (14.16), then relations (14.21) and (14.23) indeed yield the first inequality in (14.15). It follows from the boundedness of the norm of the matrix in (14.23) that there exists a matrix A 2 with positive elements such that

aA m (xo) X O) aAP( aXO aXO axo /

E- (aA'(XO) ) _' (

< A2-

(14.24)

for all x 0 e 4 . Therefore , according to (13.5), (14.13 ), ( 14.19), (14.20), and (14.24), we can conclude that the second inequality in (14. 15) (which estimates the deviation of xo from x0m componentwise) is true. Formulas ( 14.17) follow immediately from ( 14.15) in view of (11.25). To prove inequality ( 14.18), we use relation ( 11.26). Estimating the first term on the right-hand side of (11.26 ) with the use of formula ( 13.6) and taking the second inequality in (14. 15) into account, we get I x*(t, xo)-z*(t, xom)I 5 [E+ at(t)K(E-Q)-'] (E+R) Ixo -xom I <- [E+ai(t)K(E-Q)-1](E+R)A2K3Om. It follows from (12.16) for xo = xom that

(14.25)

Section 14

Determination of Initial Values of Solutions

I x*(t, x0m )-Xm(t, xom )I :5 a1(t)KQm-'(E-Q)-'(Yk +I xOm D .

139 (14.26)

Combining (14.25) and (14.26), we obviously obtain estimate ( 14.18).

By virtue of (11.25), we have lim 0 m = 0

m->-

and lii1 E(x*(t, x0),

xm(t, XOm))

=

0

on the right-hand side of inequality ( 14.18). This implies that the approximate solution .xm(t, x0m ) constructed on the basis of the solution x0 = xom of the determining equation (13.2) converges uniformly to the exact solution of the nonlinear boundary-value problem (12.1), (12.2). Theorem 14.1 is proved. For the practical construction of xm(t, xom), it is necessary to know a solution x0 = x0m of the approximate determining equation (13.2). We investigate the solvability of this equation by analogy with Theorem 11.2. Theorem 14.2. Suppose that all conditions of Lemma 14.1 are satisfied for the nonlinear two-point boundary-value problem (12.1), (12.2). Assume that, in a certain ball of radius r2 centered at the point xoo), (14.27)

S(x00j , r2) c t,

the mapping Am : D^ -4 R" defined according to (13.2) is strongly differentiable and the inequality

1 ( raf t, xm(t, xo)) axm(t, xo) xm(t, xo')) az To L

aim ax°

axm

l dt

ax0 J

<_ 12I^ x0 - xO II, x0, x0 E S(x0(°)> rz),

(14.28)

holds in ball (14.27). Also assume that, at the point x0 = x0(°), there exists the inverse matrix

r of(t, x (t, t0i) ax (t, t0^) 1 1 log -(C_1A +E) = W2(x °) T'° aim ax° T

(14.29)

140

Modification of the Numerical-Analytic Method

Chapter 2

and the following inequality is true:

m2P212 = H2 <

1 4

(14.30)

where m2 = II W2 ( x0(e))H and

P2 =

[

T

W2(xo°^) IC_id -'(C-to + E) (xo) + xk(0)) - T J f (t, zm(t, x(O)))dt 0

Then, in the ball S^x00) , r3), where x(O is its center, r3 = p2y2 is its radius, and y2 is the least root of the quadratic equation H2y2 - y + 1 = 0, the approximate determining equation (13.2) of the nonlinear two-point boundary-value problem (12.1), (12.2) has a unique solution x0 = xOm . In this case, the approximations xok+1) constructed by the modified Newton method, x(0k+t) xoki _ W2(xoo

x ITC-td-T(C-&A+E)( x^i+xk (0)) - T

L

J f(t,xm(t,xpk)))dt] ,

(14.31)

0

converge in norm to the solution x0 = x0m and the rate of convergence is estimated by the inequality

IIrOr-xk,II q2 1-q2

(14.32)

where q

=

1- 1-4H2 < 2 2

Proof. By analogy with Theorem 5.2, one can prove that conditions (11.28)-(11.30) guarantee the applicability of Theorem 1 from [KoF, p.470] to equation ( 13.2), on the basis of which all assertions of Theorem 14.2 are, in fact, established. For the nonlinear two-point boundary -value problem (12.1), (12.2), it is proved in Theorem 12.3 that the initial value x*(0) of the exact solution of this problem can be determined according to the formula x*(0) = xp +xk(0),

(14.33)

Section 14

Determination of Initial Values of Solutions

141

where xo is a solution of the determining equation (12.31) and xk(k ) is the approximate solution obtained by the method of algebraic collocation (12.3), (12.4). In this case, the deviation of the approximate solution xm(t, x* ) from the exact one is estimated by inequality (12.33). If we construct an approximate solution on the basis of the exact solution x0 = xOm of the approximate determining equation ( 13.1) in the form of the function xm(t, xom), then the error satisfies estimate ( 14.18). In practice , as an approximate solution of the nonlinear boundary -value problem (12.1), (12.2), it is natural to take a function xm(t, xok)) of the form ( 12.8) constructed on the basis of xok^-approximation (14.31) of the exact solution x0 = xQm of the approximate determining equation (13.2). Let us estimate the error in this case. If we estimate the first term on the right-hand side of the obvious equality (11.37) on the basis of the first inequality in (14.15) and the second term on the basis of (14.32), then we obtain

I^xo- x(k' I

I K31111 © . II+ 42 P2• 1-q4 1-q2

(14.34)

It follows from the statements of Theorems 14.1 and 14.2 that lira E(xok), xp) = 0 m,k- -

and, therefore, lim (xok) + xk(O)) = xp+xk(O). m,k-3-

To estimate the deviation of xm(t, xok) ) from z*(t, x* ), we consider an equality of the form ( 11.14). According to inequality (13.6) for xo = x*0 and xo = xok) (14.34) and (12.16), for the boundary -value problem ( 12.1), (12 .2), we have

x *(t,4 )- zm(r,xoki )I <_

and

[E+a^(t)K(E-Q)- '](E +R) k

IIIImII x 1+IIK3 O + 1q2 44 - q2P2 , t, xOk))), + &I(t)KQm - 1(E-Q)-' (x0kl +7 k) = E(x*(t x0), xm ( where e = ( 1, 1, ... , 1) and yk is a vector of the form (12.17).

(14.35)

142 Modification of the Numerical-Analytic Method

Chapter 2

Since the right-hand side of inequality (14.35) tends to zero as m, k -4 oo, we have lim

m,k-^_

Xm (t, XOk)) =

X'^ (t,

xo).

Note that the modification of the numerical-analytic method of successive approximations for the two-point boundary-value problem (12.1), (12.2) considered in the present chapter admits an extension to the case of the boundary-value problem (2.1), (2.2). In this case, an approximate solution obtained by the method of algebraic collocation is used together with the successive approximations (2.8). Instead of (12.8), we can consider the recurrence relation

xm(t,XO) = xO + Xk(O) + k1Hd(xo+xk(0))

+ J [i(t, m_i(t, xo))_ T f .f(S+ xm- 1 (s,xo ))dtl dt

+ T(k2-k1)Hd(xO+xk(0)),

zO(t,xo ) = Xk(t), m= 1,2,...,

(14.36)

where d(xo+xk (0)) = d-(A + C )(xo+xk(0)). By direct verification, we can conclude that the successive approximations (14.36) satisfy the boundary conditions (2.2) for arbitrary xO:

A(xo+xk (0)) + Ak1Hd(xo+xk ( 0)) + C(xO+xk(0)) + Ck1Hd(x0+xk ( 0)) + C(k2 - k1)Hd(x0+xk(0)) _ (A+C)(xO+xk(0)) + d - (A+C)( xO+xk ( 0)) = d.

15. Realization of the Method for Systems of Two Equations We now consider the possibilities of the modified numerical-analytic method of successive approximation for a specific example of a periodic boundary-value problem.

Section 15

Realization of the Method for Systems of Two Equations

143

Assume that it is necessary to investigate a 27t-periodic solution of the following nonlinear periodic boundary-value problem: i, = x2,

x2 = -0.05JC 2 - 0.02sinrjc,2 + »P(r),

(15.1)

¥ ( / ) = 0.00005sintcos2/ - 0.05cosf - 0.0025sinf, x(0) = x(2n).

(15.2)

Assume that theright-handside of the system of equations (15.1) is defined and continuous in the domain (U)6[0,2it]xD, (15.3) D: {x = (xi,x2)eR2,

|JC,|<

1.7, | * 2 | ^ 0 . 5 l } .

Clearly, D is a closed and bounded domain in R2. Moreover, the right-hand side of (15.1) is bounded by a vector M and satisfies the Lipschitz condition with matrix K, where M =

r 0.511 0.14J

and

K =

ro LO-07

li 0.051.

Then the vector (3 in (9.1) for the problem (15.1), (15.2) is determined in the form P = (0.51 Jt, 0.14TI). Consequently, the set of points D» contained in domain (15.3) together with their P-neighbourhoods has the from D p : {JCeZ?2: |jct| < 1.7-0.51it, \x^\ <. 0.51-0.147c} and, moreover, it is clear that Do * 0 . Calculating the maximum eigenvalue of the matrix

r Q = LK

n

ro

2i

= 2K =

|_0.14 °-102J

we get X(j2) = 0.051 + V 0.2826 < 1. Consequently, one can apply the classical scheme of the numerical-analytic method of periodic successive approximations based on the iteration formula (6.8) to the periodic boundary-value problem (15.1), (15.2) considered in domain (15.3).

144

Modification of the Numerical-Analytic Method

Chapter 2

To obtain an approximate 216-periodic solution of the system of differential equations (15.1), we use the algorithm of the modified method of successive approximations in the form (9.7). Let us first calculate an approximate solution Xk ( t) by the method of trigonometric collocation (9.4), (9.5). Assume that k = 1, i.e., xk(t)=(x11(t),x12(t)),

x11(t) = ao1+a11cost + bllsint

x12(t) = a02+a12cost+b12sint.

If we take three equidistant points

t0 = 0, tl = 3 16, t2 = 376

on the segment [ 0, 2n] as the points of trigonometric collocation , then the system of determining algebraic equations (9.5) for (15.1) with respect to the coefficients a01, a 11, b 11, a02, a 12, and b 12 takes the form b11-a02-a12 = 0, 'V3-a11+b11+2a02-a12+-F3b12 = 0,

V_3a11-b11-2ao2+a12--F3b12 = 0, b22+0.05a02+0.05a12+0.05 = 0, -v 3 a12+b12- 0.02 ,[3 a01 -O.005V ail -0.O15 F3 b11 + 0.02-ao1a11-0. 06aonbll +0 . 03a11b11 -0.1ao2 + 0.05a12-0.05Jb12+0.05-0.0024875 = 0, _^3 a12 -b12 - 0.020 41-0.015Vb11 -0.0050_ai1 + 0.02Vaoiall +0.06a01b11 -0.03a11b11 + 0.1a02 - 0.05 a12 - 0.05 [ b12 - 0.05 - 0.0024875 , = 0. It is easy to reduce these equations to the following system of four linear and two nonlinear equations:

Section 15

Realization of the Method for Systems of Two Equations a02 = 0,

a12 = bll = -20b12-1,

all = -b12,

145 (15.4)

6.005 biz + (20.65 + 0.02a01)b12 + 0.02a01 + 1.0174875 = 0, 20b12+(1+40a01)b12+20a01 = 0. (15.5) Solving the nonlinear system (15.5) by the Newton method, we obtain a01 = 0.02456 and b12 = -0.04913. Then, by using (15.4), we get a02 = 0, a 11 = 0.4913, and a 12 = b11 = -0.0174. We choose x 1(t) obtained by the method of trigonometric collocation as the zero approximation of the iteration process (9.7), i.e.,

x01(t,x0 ) = x11(t) = 0.02456+0.04913 cost - 0.0174 sin t, (15.6) x02(t,x0) = x12(t) = -0.4913 sin t - 0.0174 cost. The approximate solution (15.6) determines the initial values xk(0) = x1(0) = (0.07369, -0.0174) at t = 0. Therefore, a domain 4 of the form (9.8) is as follows:

4: {x a R2, 1 xi 1 <- 1.62631- 0.5n, 1x2 < 0.4926 - 0.141t} ,

AJ

*0-

Thus, all conditions of Theorem 9.1 are satisfied and, according to it, the successive approximations (9.7) for problem (15.1), (15.2) are uniformly convergent. In this case, the first three iterations have the following form: .j, 1(t, x0) = x 0l + 0.2456 + 0.04913 cos t - 0.0174 sin t, z 12 (t, X O) = X02 - 0.01746 + 0.04913 sin t + 0.00005 cost + 0.00001 sin3 t, x21 (t, x0) = x01 + 0.049125 cost + 0.00005 sin t + 0.02457, x22(t,x0) = x02-0.02x01+0.00147x01-0.0176 -0.05 sint-0.00005 cos t + 0.0000 1 cos 2 t + 0.0000 1 sin3 t + 0.02 x01 cost + 0.00098x01 cost - 0.0017x01 sin 2t +0.00049x01 cos 2 t,

146

Modification of the Numerical-Analytic Method

Chapter 2

x31(t, x0 ) = 0.99915x01 + 0.02369 + 0.05 cost

+ 0.00005 sin t + 0.02x01 sin t + 0.00098x01 sin t + 0.00085x01 cos2t+0 .00025x01 sin2t , (15.7) x32(t,x0 ) = x02-0.01741 - 0.02x01 - 0 .05sint + 0.00001 cost + 0.00001 cos 2t + 0.0242 1 cos t + 0.00098x01 cos3t - 0.00147x01 +0.00049x01 cos2t, By using the second approximation i2(t, x0 ) = (x21(t, x0 ), x22(t , x0)) obtained, we construct the following approximate determining equation of the form (10.2): 2n

A21 (x0 ) = ? n f x02 -0 .02x01- 0.01746+0 .00147x01 0 - 0.05 sin t - 0.00005 cos t + 0.00001 cos 2t - 0.00001 sin3 t + 0.02 x01 cost + 0.00098 x01 cost -0.0017x01 sin2t + 0.00049x01 cos2t]dt = 0,

A22(xo ) =

21c0f [

( 15.8)

-0.05i22 (t, xo)-0.02 sin txi1 (t, x0)+`1(t)]dt = 0.

Performing necessary calculations, we obtain from ( 15.8) the system of equations x01 = 0.02x01 + 0.00147x01 + 0 .01746,

x02 = 0.02x01 - 0.00147x01 + 0.01746 -10-6x01- 2.457.10- 8, whence x01 = -0.02457, x02 = 0 .01744.

( 15.9)

If we substitute ( 15.9), e.g., in (15 .7), then we obtain the following approximate 2,n-periodic solution of the system of equations ( 15.1):

Section 15

Realization of the Method for Systems of Two Equations 147 i 31(t, x o ) = 0.05 cost + 0.00004 sin t +0.00086 - 0.000006 sin 2 t - 0.00002 cos2 t, i32(t, xo) = -0.05 sint + 0.00005 .

( 15.10)

One can verify that the exact solution of the boundary -value problem (15.1), (15.2) has the form xi (t) = 0.05 cost,

x2 (t) = -0.05 sin t. (15.11)

Comparing the exact and approximate solutions ( 15.10) and (15 . 11), we get

Ii31(t,x0)-xi(t)I :5 0.001,

I x32(t,x0)-x*2(t)I 5 0.00005.

The exact solution (15.11) and approximate solutions (15.10), (15.6) take the following values at the corresponding points:

t xi (t)

0

n/6

is/4

n/3

tc/2

0.05000 0.043300 0.035355 0.025000 0

131(t, x0) 0.04912 0.042470 0.034530 0.024160 -0.00080 i01(t• x0)

0.07369 0.058408 0.046965 0.034056 0.00716

X2* (t) 0 -0.025000 -0.035355 -0.043300 -0.05000 i32(t, x0) 0.00005 -0.024950 -0.035310 -0.043250 -0.04995 i020, x0) -0.01740 -0.260719 -0.359705 -0.434178 -0.49130 The results obtained show that the third approximation determined according to the iteration formula (9.7) is quite close to the required solution of the periodic boundaryvalue problem (15.1), (15.2).

3. AUmERICIIL-RnRLYTIC METHOD FOR BOUADHRY -VALUE PROBLEMS WITH PRHR ETERS IA BOUADRRY COIIDITIOIIS

In the present chapter, the scheme of the numerical-analytic method of successive approximations is generalized to problems containing one or several controlling parameters in boundary conditions. We study problems related to the investigation of the existence and finding the values of parameters providing the solvability of the boundary-value problems under consideration. We also present an algorithm for the direct construction of an approximate solution. Note that the problems analyzed are, in a certain sense, related to the eigenvalue problems in which unknown parameters are regarded as eigenvalues and the required solutions of the boundary-value problems are the corresponding eigenfunctions. In Section 16, we study properties of a certain sequence of functions satisfying given boundary conditions for arbitrary values of a controlling parameter A, that appears in the boundary conditions and for arbitrary values of a free parameter x0 having the dimension of the system of differential equations under consideration. We prove the uniform convergence of this sequence and establish the relationship between the limit function and a solution of the original boundary-value problem. In Section 17, we obtain sufficient conditions for the solvability of the boundary-value problem under consideration with a parameter in the boundary conditions. Moreover, we justify an algorithm for the approximate calculation of the initial values of solutions of boundary-value problems for t = 0 in the case where the parameter is linearly involved in the boundary conditions. In Section 18, we study a problem with nonfixed right boundary point as a boundary-value problem with a parameter nonlinearly involved in the boundary conditions. We investigate the relationship between the limit function and its m th approximation for the constructed sequence of functions and the exact and approximate solutions of the problem considered. In Section 19, we prove theorems on the properties of exact and approximate determining functions, estimate the error of the approximate solution and the values of the parameter that determines the right boundary point of the interval of integration. In Section 20, the basic ideas of the numerical-analytic method are extended to the case of several controlling parameters for a problem with free right boundary. In Section 21, we study the case of linear dependence of boundary conditions on parameters and illustrate the scheme of the method by an example. 149

150

Numerical-Analytic Method for Problems with Parameters

Chapter 3

16. Successive Approximations for Problems with One Parameter in Linear Boundary Conditions

Most investigations of boundary-value problems with parameter refer to the case where a parameter is contained only in the differential equation. Clearly, from the practical point of view, it is interesting to investigate problems in which a parameter appears in the given boundary conditions. It turns out that the numerical-analytic method described in the previous chapters can also be efficiently used for establishing the existence of solutions of boundary-value problems with parameters and their construction. For convenience, we first consider the simplest problem. Consider a system of differential equations x = f(t,x),

x,fE R ',

(16.1)

on the segment tE [0, T] with two-point linear boundary conditions with an unknown scalar parameter ?. E R Ax(0)+XCx(T) = d,

detC*0, (16.2)

under the condition that a certain fixed coordinate xi of the vector x = (x1, ... , x takes a value xoi for t = 0. Without loss of generality , we can assume that j = 1, i.e., x1(0)

=

x01.

(16.3)

Assume that the right-hand side of equation (16.1) is defined and continuous in t and x in the domain (t, x)E [ 0, T]x D,

(16.4)

where D is a closed bounded domain in the space R" and A and C are constant (n x n) matrices. We consider the following problem: Find a continuously differentiable solution x = x* (t) of equation (16.1) and a value of the parameter . = X*, i.e., a pair (X*, x* (t)), for which conditions (16.2) and (16.3) are satisfied. Among the boundary-value problems (16.1)-(16.3), one selects the class of problems for which the function f (t, x) is bounded by a vector M in domain (16.4) and satisfies the Lipschitz condition with a matrix K, i.e., conditions (2.4) are satisfied. Denote by Do the set of points x0 = (x01, x02, ... , x0ri) that lies in the domain D for all values of 2 from a certain domain IX = [? 1, a' 211 A. 1 > 0, together with its J3neighbourhood, where

Section 16

Successive Approximations for Problems with One Parameter

151

R = 2M + R 1(xo, A), 131(x0, ? ) = I -1 [C-d-(C- 'A+XE)x,)jj.

(16.5)

Denote by G the set of (n -1) -dimensional vectors YO = (x02, ... , xOn) such that x0 = col (x01, yo) belongs to the domain Do. We also assume that, as in Section 2, D0#0

(16.6)

A.(Q) < 1,

(16.7)

and

where ? (Q) is the maximum eigenvalue of the matrix Q = T K . n Consider the sequence of functions

x,n (t, Y0 , )L) = x0 + J I .f(t, x.-1(t, Yo' ^)) 0 T - T,I .f(s, xm-1(s, Y0,)L))ds dt + 0 m=1,2,...,

7, [C-ld - (C-'A +XE)x0j,

(16.8)

x0(t, Y0, X) = x0 E DP,

where the first component of the vector x0 is defined by (16.3). It can be directly verified that all functions xm (t, y0, X) depending on the parameter that appears in the boundary conditions (16.2) and on an (n - 1)-dimensional vector y0 (or an n-dimensional vector x0 with fixed first coordinate) satisfy the boundary conditions (16.2) and (16.3) for arbitrary values of y0E G and XE I X. The following statement concerning the convergence of successive approximations (16.8) is true: Theorem 16.1. If the right-hand side f (t, x) of the system of differential equations (16.1) is defined and continuous in domain (16.4) and conditions (16.5)-(16.7) are satisfied for the boundary-value problem (16.1)-(16. 3), then a sequence of functions xm (t, y0, ?) of the form (16.8) satisfying the boundary conditions (16.2), (16.3) for arbitrary y 0 E G and A. E I X converges uniformly as m - co with respect to the domain

152

Numerical-Analytic Method for Problems with Parameters (t, y0, A.) E [0, T] x G x I.

Chapter 3 (16.9)

to the limit function x*(t, yo, A.). Furthermore, x* (t, y0, X ) is a solution of the integral equation

x(t) =

T

x0

+ J { f(tx(r)) - 1 J f(s, x(s))ds TO

+ A1T [C-'d - (C-'A +AE)x0] }dt,

( 16.10)

which passes through the point x*(t, y0, A.) = x0 = (x01, y0) for t = 0. Moreover, x*(t, y0, A.) satisfies the boundary conditions (16.2), (16. 3), i.e., is a solution of the boundary-value problem x = f(t,x)+A(y0,A.),

Ax(0)+A,Cx(T) = d,

x1( 0) = x01, (16.11)

where

A(Y0,.) = AL [C-Id - (C-1A+AE)x0] - T Jf(t, x(t))dt. T 0 In this case, the deviation of the limit function x* (t, y0, A.) from xm (t, y0, A,) for all m = 1, 2, ... is estimated coordinatewise by the inequality

x* (t, Y0, %) - X. (t, YO, X ) I <_ a1(t)W(Y0,?) = £(x*(t,y0,A.)-xm(t,y0,A.)), (16.12) where W(Y0, ), ) = Qm(E-Q)-1M + KQm_1(E- Q)-' (x0, A,)

(16.13)

and a1 is defined by (2.10). Proof. We prove this theorem by analogy with Theorem 2.1. According to Lemma 2.1, it is easy to show that if (y0, A.)E Gx I)L, then all successive approximations (16.8) belong to the domain D. Estimating the difference xm+1(t, y0, A,) - xm (t, y0, A.), we get

Successive Approximations for Problems with One Parameter

Section 16

xm+l(t,y0,x )- xm(t,YO ,X) =

JIf(t

153

+xm( t,YO,X))

0

T - f (t, xm-1(t, YO, ?)) - 17 f f (s, xm(s, YO, a')) 0

J

- f (s, xm - 1(s, yo, 71,))ds dt,

or

1

[(

)j YO,^')) f sIx.(s' xm +1(t, y0,?)-xm(t , YO,?) = (l-- ,

-

0

.f(s• xm - 1(s , Y0, %))

J

ds

T - T J [.f (s, xm (s, Yo' ? )) - .f (s, xm - 1(s, Yo' ? ))]d s . 0

(16.14)

If we denote rm+1(t,y0,%)= Ixm+t(t, y0,?)-xm(t,y0,)) I =rm+1(t), then, by virtue of the Lipschitz condition (2.4), it follows from (16.14) that

rm

+1(t,y0 , a)

K((

1-T)Jrm ( s,Yo'X ) ds + TJrm ( s,Yo'?)ds . 0

t

]

(16.15)

For m = 0, in view of Lemma 2 . 1, relation ( 16.8) immediately yields

r1 (t, Y0, ?) <- I xl (t, Y0, A) -x0(t, Y0, A)

JI f(t, xo ) - T J.f(s, xo)ds dt + PI (X0, )-)

]

< a1(t)M+ (31 (x0, A,).

(16.16)

Consequently , in view of ( 16.16) and (2. 9), it can easily be obtained by induction from (16.15) that

154

Numerical-Analytic Method for Problems with Parameters

r2(t) :5 (1_

Chapter 3

)j[a,(t)M+ii(xo,)]dt 0

T + T J[a, (t)M+[i1(xO,?)]dt = K[a2( t)M+a,(t)R,(xo,k)], r

rm+i(t) <- Km[am + l(t)M +am(t)p1(xo,

? )],

(16.17)

m = 0, 1, 2, ... .

We estimate the functions am (t) and am+1(t) by using (2.10). Then ( 16.17) yields

rm+I(t,yO,?,) <- &i(t)[QmM+KQm-' 31(x0,?)]•

(16.18)

Since

xm+j(t, YO, ?)-xm(t, YO,

( xm+j(t,YO,xm+j-i (t, YO, X )) + ... + (xm+i(t,yO, ?1 )-xm(t,YO,

(16.19) in view of ( 16.18) equality ( 16.19) yields J I xm+j(t,Yo , X)-xm(t ,YO,?)I

rm+i(t,YO,?) i=1

<- a1(t)

I i=O

(Qm+iM+ KQm+ i-' R1(xo, a ))

J 1

j-1

_ & i(t)

Qm^`

Q`M +Qm-'

i=o i=1

Q`Rt(xo ,

(16.20)

X)

Since lim Qm = 0, m-4-

according to (16. 7) one can easily conclude from ( 16.20) that the sequence xm (t, yo, X) is fundamental in the space of continuous vector functions . Consequently , by virtue of

Section 16

Successive Approximations for Problems with One Parameter

155

the Cauchy criterion, the sequence x (/, y 0 , X) converges uniformly in domain (16.9), i.e., lim xm(t,y0,X)

= x*(t,y0,X).

(16.21)

Sequence (16.8) is constructed so that all functions xm (t, y0, X) satisfy the boundary conditions (16.2), (16.3). Therefore, the limit function x*(t, y0, X) also satisfies these conditions for arbitrary Xe 1 x and y 0 e G. For ,/-»<», by virtue of (16.21), the re quired estimate (16.12), (16.13) can easily be obtained from (16.20). Finally, if m—>■» in (16.8) and we take the limit relation (16.21) into account, then it is obvious that the limit function x'(t, y0, X) satisfies the integral equation (16.10) and passes through the point x*(0,y 0 , X) = x0 = (*01, y 0 ) for / = 0, i.e., it is indeed a solu tion of the boundary-value problem (16.11). Theorem 16.1 is proved. Let us show that one can always introduce an additional parameter of control on the right-hand side of the system of differential equations (16.1) so that the solution of the given Cauchy problem for the system obtained also satisfies the original boundary con ditions. Theorem 16.2. Under the conditions of Theorem 16.1, for any pair (yQ, X) y^ e G, X € lx, there exists a unique value of the additional parameter of control u, = (H

H„). T

u = Mc-id-lC-^ Kl

+ lEixo] - ±]f(t,x'(t,y0,X))dt, I

(16.22)

0

where x*(t, yQ, X) is the limit function of sequence (16.8), such that a solution x = x(t) of the Cauchy problem x =/(*.*)+ji.

(16.23)

x(0) = * 0 = ( x O I . y 0 )

(16.24)

is also a solution of the boundary-value problem (16.23), (16.2), (16.3) and x(t) = x'(t,y0,X).

(16.25)

Proof. It has already been proved in Theorem 2.1 that function (16.25) is a solution of the Cauchy problem

156

Numerical-Analytic Method for Problems with Parameters

T J f(t,x

Chapter 3

T

x =.f(t,x)+ I[C-'d-(C-'A+Xpxo] -

0

*(t, Yo,k))dt, (16.26)

X(O) = xo = ( x01, y0) and, moreover, it satisfies the boundary conditions (16.2), (16.3). Thus, for equation ( 16.23), we have already found one required value of the additional controlling parameter µ equal to (16 .22). We now prove that this value of µ is unique. Assume that there exist two different values µ = µ' and µ = µ*, µ' # µ", such that the solutions x(t, y0, A., µ') and x ( t, y0, A,, µ") of the Cauchy problem (16.23), (16.24) satisfy the boundary conditions ( 16.2), (16.3). Then we obtain from (16 .26) the following equality for the difference of these functions:

x(r YO A µ")- x(t,YO,A.,.L') t

J [(f(ts x(t, Yo'

f(t, x(t, y0, A., R')))

0

T 1 T J (f(s, x(s, Yo, A., µ")) - f(s, x(s, Yo' A,, µ')))ds dt.

]

0

(16.27)

As in the case of Lemma 2. 1, by virtue of the Lipschitz condition, it follows from (16.27) that

r(t) <- K '1-T I J r(s)ds + T Jr(s)ds ,

1

]

(16.28)

where

r(t) = Ix(t,YO,A.,µ")-x(t,yo,),,µ')I. If we denote

I r(t)l0 = (suPIri (t)I...suPI1(t)I), t

r

(16.29)

then, in view of ( 16.9), inequality ( 16.28) yields

r(t) <- al(t ) KIr(t)IO , <-

r(t) <- a2(t)K21r(t)10...,

am +1(t)Km

+II

r (t)I0.

(16.30)

Section 16

Successive Approximations for Problems with One Parameter

157

According to inequality (2.10), it follows from (16.30) that

Ir(t)l0 s ai(t)KQmIr(t)lo s

KQmjr(t)Io.

(16.31)

G But, under condition (16.7), inequality (16.31) can be satisfied only if (16.29) is identically equal to zero, i.e., for We arrive at a contradiction. This completes the proof of Theorem 16.2. The statement below gives necessary and sufficient conditions for the limit function of sequence (16.8) to be a solution of problem (16.1)-(16.3). Theorem 16.3. Suppose that all conditions of Theorem 16.1 are satisfied for the boundary-value problem (16.1)-(16.3) with a parameter in the boundary conditions. Then a solution x = x* (t) of the Cauchy problem

x = .f (t, x), x( 0) = x0 = (x01 , yo)

(16.32)

is simultaneously a solution of the boundary- value problem with parameter (16.1)-(16.3) if and only if the pair ( yo, ?,) is a solution of the determining equation T

0(y0,

_T[C-td -( C-'A+Wxo] - T .f(t, x*(t, yo, ^,))dt = 0, ( 16.33)

f0

where x* (t, yo, ?) is the limit function of sequence (16.8). Moreover, x* (t) = x* (t, y 0, ?) and the deviation of an approximate solution x m (t, y0, X) of the form (16.8) from the exact solution x*(t) = x*(t, y0, %) of the boundaryvalue problem (16.1)-(16.3) is estimated by inequality (16.13). Proof. The sufficiency of condition (16.33) follows from the fact that, according to Theorem 16.1, the function x* (t, y0, ?) with the initial value x* (0, y0, ?) = x0 is a solution of problem (16.11). If A (yo, X), then a solution of problem (16.11) coincides with a solution of the original boundary-value problem (16.1)-(16.3) and the Cauchy problem (16.32). The necessity of the fact that the pair (y0, ?) satisfies equation (16.33) follows from Theorem 16.2. Indeed, if x = x* (t), as a solution of the Cauchy problem (16.32), is a solution of the boundary-value problem (16.1)-(16.3), then a solution x = x(t, y0, ?, .t) of the initial-value problem (16.22)-(16.24) with additional controlling parameter µ satisfies the boundary conditions (16.2)-(16.3) only for one value, namely, for µ=t(yo,X)=0,

158

Numerical-Analytic Method for Problems with Parameters

Chapter 3

because, in this case, x(t, yo, X , .t) = x*(t) = x(t, yo, X). This completes the proof. Thus, according to the numerical-analytic scheme presented above, the procedure of finding a solution of the boundary-value problem with parameter (16.1)-(16.3), i.e., the pair (X*, x*(t)), is as follows: (i) Construct a sequence xm (t, yo, A,) depending on yo and ? as parameters according to the recurrence relation (16.8); (ii) determine the limit function x*(t, yo, ?) of the sequence xm (t, yo, X) ; (iii) fmd a solution yo = yo, ? = X* of the determining equation (16.33); (iv) find a solution x =x* (t) of the Cauchy problem x =f (t, x), x (0) = (x01, Yo) or, in other words, calculate the limit function (x*(t, yo, X,*)) of the sequence xm t, Yo* , Al In what follows, we show how to investigate the problem of solvability of the boundary-value problem (16.1)-(16.3) without finding the limit function of sequence (16.8).

17. Sufficient Solvability Conditions and Determination of the Initial Value of a Solution of the Boundary -Value Problem with Parameter

Denote by Am (yo, A,) the following approximate determining function of the boundary-value problem (16.1)-(16.3):

A.(Yo'X ) =

7,[C-'d-(C-l A+A,E)xoI - T j.f(t,xm ( t,Yo,A,))dt, 0

(17.1)

where xm (t, yo, A,) is calculated according to (16.8). Along with (16.33 ), we consider the approximate determining equation Am(YO,A.) = 0, (17.2) where Am (yo, A,) is defined according to (17.1).

Section 17

159

Sufficient Solvability Conditions and Determination of Initial Value

Theorem 17. 1. Suppose that the conditions of Theorem 16.1 are satisfied. Also assume that there exists a convex closed domain A' = G' X Ij, c Gx IX (17.3) such that, for a certain m > 1, the approximate determining equation (17.2) has a unique solution (y0, ?) = (yam , km) in (17. 3) with index different from zero and the following inequality holds on the boundary S 1 of the domain A':

(Yo in S'

(!) 3 Qw(yo,&), IA.(YO,X)I >

(17.4)

where W(yo, ?) is determined according to (1613) and (16.5) and Q = T K. tc Then, for problem (16.1)-(16.3) with a parameter ? in the boundary conditions, there exists a solution, i.e., a pair (x* (t), X* ), and the initial value of this solution is x* (0) = col (X01, yo ),

(17.5)

where yo a G' and X* E I^. Proof. Let us estimate the difference between the exact and approximate determining functions A(y0, ?) and A. (y0, X ). It follows from ( 16.33) and (17.1) that

A(Y0, ^) - A. (Y0, X)

J 0

[f(t, x*(t, Yo, ?))-.f(t, xm(t, Yo, k))]dt.

(17.6)

According to the Lipschitz condition (2.4) and relations ( 16.12) and (10.6), equality (17.6) yields T TKW(Yo,?) Ja1(t)dt A(Yo,X) - Am(Y0,a')I <

0

= 9 KW(y0, .) = E(A(yo k), Om (Y0, X))

(17.7)

The subsequent proof is analogous to that of Theorem 3.1. Since ( y0m, Xm ) is a unique solution of equation ( 17.2) in domain ( 17.3), the rotation of the vector field A. (y0, ?.) is different from zero on the boundary S 1 of the domain A'.

160

Numerical Analytic Method for Problems with Parameters

Chapter 3

On S', we consider the set of vector fields P(O,YO,A.) = Am(Yo,A.)+9[A(yo,7.) - Am(YO, X)],

0<-0<-1,

continuous everywhere with respect to all variables . Then, in view of inequalities (17.4) and (17.7), the relation

IP(O,YO,X)I

>

IA.(YO,^)I- IA(YO,a )

- A M (YO,)L)I > 0,

implies that P(O, yo, A,) is not equal to zero on S. This guarantees the homotopy of the vector fields A. (Yo, A,) and A(yo, ).). Consequently , by virtue of Lemma 3.2, the rotation of the vector field A (yo, 7.) is not equal to zero on S. Therefore, according to Lemma 3.1, the exact determining equation ( 16.33) of the boundary-value problem (16.1)(16.3) has at least one solution A(Yo, X) = (Yo, r) e A' (17.8) in the domain bounded by S' (i.e., in the domain A'). By virtue of Theorem 16.3, values (17.8) determine a solution x = x* (t) of the Cauchy problem x = .f (t, x),

x (0) = (x01, Yo), (17.9)

which is simultaneously a solution of the boundary -value problem ( 16.1)-(16 .3) for A. _ A.*. Theorem 17. 1 is proved. Prior to studying problems connected with the calculation of initial values of solutions of the boundary-value problems ( 16.1)-(16.3), we prove the following statement: Lemma 17.1. If the conditions of Theorem 16.1 are satisfied, then, for arbitrary pairs (Yo' X), (Yo X") E G'x IX,

(17.10)

the limit functions x* (t, y;, .') and x * (t, ya , A") of sequences x m (t, yo , .') and x,,, (t, yo , a.") of the form (16.8) satisfy the inequality

I x* (t, Yo , A.) - x* (t, Yo , A.") I <- [E+&1(t)K(E-Q)-'] [I x6-41+at (A.',A",Yo,Y6')1, (17.11)

Section 17

161

Sufficient Solvability Conditions and Determination of Initial Value

where

, „[(

al(Vd, ",YO,YO1

a,'-?b") C-'d+C-

' A(X'xo-7J "x0)]+(xg

-x')1.

(17.12)

Proof. For the difference of the functions x (t, yo , X') and x I (t, yo , ?") calculated according to formula ( 16.8), we have xI(t, Yo, ?J) - xI (t, Yo , X")

= (jtf-jtf) +

C

f ( f(s, xo)-f(s,xo))ds l dt t, 4 )- f(t, xo)) - T 1(f( o

+ {[c'-'d +(C- 'A+^,'E)xo ] -

„[C-Id-(C- 'A+^,"E) xo]

According to (4.3 ) and the Lipschitz condition, the last inequality yields

IxI(t, Yo, X') - x1 (t, Yo ,?")I r

"

Ixox0 + K

t T " ( t 1 --1 jlxo -xIdt + - J I xo -xoldt + al(x,,X" ,Yo,Yo) TJo T t I

t

(

= [E+a, (t)K] Ixo-xoI + a1(?',X" , Yo,Yo)•

( 17.13)

Analogously, for the difference x2(t,Yo,^)-x2(t,Yo,? ), in view of (17.13), we get

Ix2(t, Yo, x2(t, Yo , ?") t

< Ixo -xo + K (1 -T 1I xt( t,yo,?)- xt(t,yo,X )Idt

+ TJTr I xl(t,YO,a'')-xl(t,Y0,^1'")Idt

+ al(^1,',X",YO,YO)

t

< [E+al(t)K + a2(t)K2 ]I 4-xoI + [E+al(t)K] Ixo -x' a,(X', k",Yy,Yo)• (17.14)

162

Numerical-Analytic Method for Problems with Parameters

Chapter 3

By induction, estimates (17.13) and (17.14 ) yield the inequality

I x m (t, YO , A,')

- xm (t, YO ,

X") I

// \t Ixp-xl + K I1-T)J l xm-1(t,Yo,A')-xm-1(t,yo,X")Idt \\ o T + 7 J l xm-1(t, Yo+ a') - xm_1(t,

r

l i =O

l

1

Y0, ^1 ") dt]

+ a1(X^, ^1 ", YO y )

a1(t)K` Ilxo-xol +[lai(t)K` a1(X',?",Yo(17.15)

J

a =o

where a.(t) are functions of the form (2.9). Passing to the limit as m- oo in (17.15) and taking relations (2.10) and (16.7) into account, we obtain

Ix*(t, Yo,

x*(t, Yo E + al(t)Klj

/ \\r TK X

(

J ] I xp - xo I

lt + E+&1(t)Ki(T KI Ja1Yo^ Yo) =0n /J

<- [E+&1(t)K(E-Q)

-][I xo -xoI + a1(^',

", Yo, yo)].

Lemma 17.1 is proved. Theorem 17.2. If the conditions of Theorem 16.1 are satisfied, a function A (y0, X) of the form (16.33) for the boundary-value problem with parameter (16.1)-(16.3) is defined and continuous in the domain G x Ix defined according to (16.5) and an arbitrary pair of values (17.10) satisfies the inequality

Io(Yo,^') - YO ,^")I <- 1 a,(X',r,y ,yo)

+ K[E+ 9 E-T K(E-Q)-1]Llxo- xoI + at

Yo,yo)]. (17.16)

Section 17

Sufficient Solvability Conditions and Determination of Initial Value

163

Proof. According to Theorem 16.1, the limit of the sequence of functions (16.8) is a continuous function for all (yo, %) e G x Ix. Consequently, for (yo, X) E G x I., the function A(yo, X) is also continuous and bounded, namely, ( C - 1 A l + M.

I ^(Yo, ^) I <_ -

By the definition of function (16.33), in view of (17.11 ), we obtain the following estimate for arbitrary pairs (17.10): at( ^,^,

A (Yo , V) - A(Yo , ?") I <

Yo, Yo )

T T

'KS I x* (t, Yo, ?') - x* (t, yo> )") I d t 0 +T 1 a1(^^,

Yo, yo")

+ IKJ[E+&1(t)K(E-Q)-1] [Ix0-x0I +a1p!,?",Yo, yo') ]dt. 0 By virtue of equality ( 10.6), the last relation yields (Yo,V) - A(Yo ,?")I <- Tat(O•,?",Yo,Yo)

[l o - x0' 1 +a1(X?J', Yo,Yo')]

+ K x'

T

+ TK2(E-Q)-1[Ixo- xp1 +al(X',X",Yo,Yo)]

Ja ,(t)dt

= T a1(A,', Yo, Yo )

+ K E+ 9 K(E-Q)_1] [Ixo-xol+a1(^,^,X^^,Yo,Yoe)].

L

i.e., inequality (17.16) is satisfied. We now prove a theorem that gives necessary conditions for a certain set from the domain of definition of the boundary-value problem with parameter (16.1)-(16.3) to contain a pair (yo, A.) that reduces it to the Cauchy problem (17.9).

Numerical-Analytic Method for Problems with Parameters

164

Chapter 3

Theorem 17.3. Suppose that the boundary-value problem with parameter (16.1)(16.3) satisfies the conditions of Theorem 16.1. Then in order that a domain (17.17)

A" = G" x I'/ c G x Ix contain a pair of values

(yo, V) that, for A, = X*, reduces the boundary-value

problem (16.1)-(16.3) to the Cauchy problem (17.9), it is necessary that, for all numbers m and an arbitrary pair (yo, ?,) from set (17.17), the following inequality be true: T 71T-(C-IA+71,E)XO] - T J f(t,xm(t,YO,a,))dt - [C-'d 0

sup H 1 a1 (X1,?,YO,Yo) + K[E+ 9 K(E-Q)-1] (yo, X) E A"I

x [Ixo-xOI+a1 (i,,%,YO,Yo)]}+E(A(yo,7),Am(yo,X )),

(17.18)

where the vectors E (A(yo, ? ) , A. (YO, A)] and a1(? 1, X, Yo, y(,) are defined by the right-hand sides of (17. 7) and (17.12), respectively, and x0 = (x01, YO). Proof. By assumption, the function x = x* (t) that passes through the point x` (0) = (x01, yo) for t = 0 is a solution of the boundary-value problem with parameter (16.1)(16.3) for X = X*. Then, according to Theorem 16.3, the pair (yo, r) is a solution of the determining equation (16.33), i.e., A(yo,X ,*) = 0. (17.19) If we write inequality (17.16) for the pairs (YO, V)

and (YO, X") = (yo*, *),

where (yo, ?) are arbitrary points in the domain A", then, by virtue of (17.19), we get

^0(Yo^^)I

Tat (^,r,Yo,Yo)

+ KrE+

( E-Q)-1][lxo-x01+a1 (^,r,yo,y)]. (17.20)

9K At the same time, by virtue of (17.7), we have

Section 17

Sufficient Solvability Conditions and Determination of Initial Value 165 Iom(yo, ^)I <- JA(yo, i)I + E(0(Yo ^), om(yo^))

(17.21)

for (yo, X ) = ( yo, ?,). Combining inequalities ( 17.20) and (17.21), we obtain the required relation ( 17.18), i.e.,

IDm(yo^^)I < Tai(X,V,YO,yo)

+ KI E+LT9 K(E- Q)-1][Ixo-xo + a, (1, V, Yo, YO*)]

+ E(A(Yo, ^), om(Yo. ).)) sup J a,Yo,yo ) + K[E+ET K( E-Q)-1] (yo,A..)ET 9

x [Ixo-xoI+a, (7, ,Yo,yo)]}+E(0(yo,4Am(Yo,'))The last theorem enables one to propose the following algorithm for the approximate calculation of the pair (yo, r) that determines the equivalent Cauchy problem ( 17.9) for the original boundary-value problem ( 16.1)-(16.3) with parameter: We represent the set Dp defined according to (16 .5) and (16 .6) and the segment I. _ [ 1, X 2 ] as a union of the finite number of subsets: N

Dp =

U Di,

N1

I,. =

U Ij•

(17.22)

?, E Ii,

(17.23)

j=1

i =1

In all subsets Di and Ii , we choose arbitrary points xo = (xol,Yo)E Di, and calculate the vector T

_

Om(Y0+ -j)

_

[C-'d-(C-A- ^1,jE)i

1T i

]

-

Jf(t,xm(t,y0+ )j )) dt

T o

(17.24)

for a certain successive approximation xm(t, yo , iI) of the form ( 16.8). Denote by the union of the subsets Di and Ij from (17.22) for which

M`J,

166

Numerical-Analytic Method for Problems with Parameters

Chapter 3

A. (Yo, Xj) <- sup {1-ai (i,,yt,yo) + K [ E + ET K(E - Q)-' J T 9 1 (YO, X) E A;{

x [Ixo - xoI + at(^,X,yo,Yo)]}+E(A(yo,i),Am(Yo,^j) ), (17.25) where Am ( yo, X. j) and (yo, ^ j) are defined by (17.24 ) and (17.23), respectively, A,` = G` x Ij, and G` is the set of (n -1 )-dimensional vectors y0 =(x02,... , xa) such that xo = (x01, YO) 1= Di for the indices i that determine M. According to Theorem 17.3, Mm'J is precisely the set that contains the pair ( yo, r) that determines the initial-value problem ( 17.9) equivalent to (16. 1)-(16.3). Obviously , as i, j, m-oo, Am tends to the set G* X I* that determines the exact initial value yo E G * of a solution and the value of the parameter X* E I* for the boundary-value problem ( 16.1)-(16.3). We can choose an arbitrary point yo E G' as an approximate value of yo, and an arbitrary ? E Ii as an approximate value of the parameter X*. It is clear that the deviation of the exact initial value xo* = (x01 , yo) from the approximate one i0 (x0j, Yo) and the deviation of X* from ? are estimated as follows:

1 4-41 = sup Po - xo 1, yo EG'

x 0 = (xor YO),

IA,*-ij <_ sup Ii-7'I. ;L e 1i

(17.26)

Obviously, the subsets Di and Ij in (17.22) whose points do not satisfy inequality (17.25) cannot contain the required pair (yo, r). Thus, the function xm(t, yo, A,) calculated by the recurrence relation ( 16.8) can be regarded as an approximate solution of the boundary -value problem (16.1)-(16.3) for

x=x. Let us estimate the error of the approximate solution: Theorem 17 .4. Suppose that, for problem (16.1)-(16. 3) with a parameter in the boundary conditions, the conditions of Theorem 16.1 are satisfied, the pair (yo, a.*) is a solution of the exact determining equation (16.33), and is an arbitrary pair from the set A`m .

Section 17

Sufficient Solvability Conditions and Determination of Initial Value

167

Then, for the deviation of the exact solution x*(t, yo* , r) of the boundary-value problem (16.1)-(16.3) for = 7^* from its approximate solution xm (t, yo , ?) for ?= ? of the form

xm(t, yo, A) _

1i(t,

xo +

xm_1(t, Y0, ?)) -

T f f( s , xm- 1(s, 3'0, ? ))ds

]

dt

1 [C-'A+(C-'A+iE)xo],

xo(t, Yo , i) = xo, the following estimate is true:

I x*(t,Yo, *)- xm(t,3'0, -) I S sup tE (X * (t, Yo, yoeG',?€I

xm(t, Yo,

m-1

1

+ E+ al(t )K I Q' [ Ixo- xol +a1(X,i,Yo,Yo)] =o

J

(17.27)

where E(x* (t, yo, ? ), xm(t, yo, ? )) is calculated according to (16.12), (16.13), and (16.5) and a1(?, i, Yo, yo) is calculated according to (17.12). Proof. We write the obvious inequality x*(t, Yo, - xm(t, Yo, i) <_ I x*(t,y(*,,k)-xm(t,yo,V)I + I xm(t,yo,?*)- xm(t,yo,X)I.

(17.28)

Let us estimate the first term on the right-hand side of (17 .28) by using (16.12), and the second term by using (17.15 ) as follows:

Ix*(t,Yo,?*)-xm(t,Yo,?)I !5 FE (x* (t, YO*' r), X. (t, YO*' m-1

+IQ' [Ixo-XOI+a1 (V,i,yo,Y0)]. E+dcl(t)K i=O I

168

Numerical-Analytic Methodfor Problems with Parameters

Chapter 3

In view of (17.26), the last relation takes the form of inequality (17.27), which completes the proof of Theorem 17.4.

Assume that (yo, i) = (yon, km)' where (y, ,'m) is the exact solution of the approximate determining equation (17.2), (17.1). We take a function x m (t, y om , Xm) calculated according to (16.8) as an approximate solution of the boundary-value problem with parameter (16.1)-(16.3). We can estimate the error of the approximate solution xm (t, y(rn, a m) by using formula (17.27). In this case, we get

1X* (t, y0, )*) - xm (t, Yom, XM) I

<- c(x'(t,Yo, ),xm(t,yo,?`)) -1

+

[ E+al(t)K " Q i] =o

5 sup Yo

E

[Ix-

xOm

l +at

(^1,',?1.m'YO*'

yom)

Jc(x* (t, Y0, X), xm(t, Yo' A)^

G',A. E/i

m-I

1

+ CE+at(t)KyQ`l[jxo-xomj+ai(X,Am,Yo,Yom)] • (17.29) i =o J Estimate (17.29) can be made more constructive if we impose certain additional conditions of the form (5.3) on the right-hand side of the system of differential equations (16.1). Moreover, for the boundary-value problem (16.1)-(16.3), one can obtain analogs of the statements of Sections 5, 11, and 14, which were established for problems without parameters.

18. Boundary-Value Problems with Nonfixed Right Boundary

In Sections 16 and 17, we have studied problems with two-point boundary conditions linearly depending on a parameter. Let us now investigate a boundary-value problem with boundary conditions nonlinearly depending on a parameter. Consider a two-point boundary-value problem with nonfixed right boundary point. Assuming that this point is a parameter X, we can regard this problem as a boundaryvalue problem with boundary conditions nonlinearly depending on a parameter.

Section 18

Boundary- Value Problems with Nonfrxed Right Boundary

169

Consider the boundary-value problem (18.1)

x = f(t,x), x,fE R" Ax(0)+?Cx(T) = d,

(18.2)

detC# 0,

x1(0)= x01 .

( 18.3)

Assume that, as in Section 16, the right-hand side of equation ( 18.1) is defined and continuous in t and x in domain (16.4). E ( 0, T], i.e., the right boundary of the We seek the values of the parameter interval of integration, and a solution x = x* (t) of equation ( 18.1) continuously differentiable on the segment tE [ 0, ),*1 for which conditions ( 18.2) and ( 18.3) are satisfied. It is easy to see that the problem posed is analogous in a certain sense to the periodic boundary-value problem [SaR4] for autonomous systems of differential equations, where the value of the parameter ? in (18 .2) plays the role of the unknown period of the required periodic solution. For establishing the existence of solutions of the nonlinear boundary -value problem (18.1)-(18 . 3), i.e., the pair (X*, x* (t)), and finding them, we use the numerical-analytic method described in Sections 16 and 17. Assume that a function f (t, x) satisfies the boundedness condition and Lipschitz condition (2.4) with a vector M and matrix K, respectively, in the domain of definition (16.4). Let Dp be the set of points x0 = (x01, ... , x0i) contained in the domain D for all fixed ? E I X = [ X 1, T], ? 1 > 0, together with their (3-neighbourhoods, where

(3 = 2M+1j ( xo), 31(xo ) = I C-1 d-(C-'A+E) x0I ,

(18.4)

and the first component of the vector x0 is defined in (18.3). Denote by G the set of (n- I)-dimensional vectors y0=(x02..... xOn) such that the vectors x0 = (x01, yo) belong to Da for x01 defined by (18.3). Furthermore, we make certain assumptions typical of the numerical-analytic method under consideration , namely, we assume that the set Dp is nonempty and the maximum eigenvalue of the matrix Q = T K is less than 1, i.e., n Dp # 0,

?.(Q) < 1. (18.5)

Consider the sequence of functions xm (t, y0, ? ,) depending on t, y0, and ? as follows:

170

Numerical-Analytic Method for Problems with Parameters

xm

(t, YO, ?U)

= x0 +

f

Chapter 3

f(t'x'._I(t' yO'x)) - x f .f (s, xm-1(S, y0, ? ))ds I dt 1

+[C-ld-(C- A+E)xo], m=1,2,...,

(18.6)

x0(t,Yo, ?) = x0 = (x01,Y0)•

Substituting ( 18.6) in (18.2) and ( 18.3), one can easily see that , for all m = 1, 2, ... , the functions xm (t, yo , ?.) satisfy the given boundary conditions for any X E [X 1, T] and YoE G. Let us prove that a sequence of functions x m (t, yo, ?,) of the form ( 18.6) is uniformly convergent. Theorem 18.1. Suppose that conditions (2.4), (18.4), and (18.5) are satisfied in domain (16.4) for the boundary-value problem (18.1) -(18.3). Then a sequence of functions xm (t, y0, ?.) of the form (18.6) satisfies the boundary conditions (18.2) and (18.3) for all m = 1, 2, ... and arbitrary y o E G and X e [%j, T] and converges uniformly to the limit function x * (t, y o, A.) as m -+ oo with respect to the domain

(t, y0,X)E [ 0,X]x Gx[ X1, T].

(18.7)

Furthermore, f (t, yo , ?) is a solution of the integral equation

x(t) = xo + j {f(tx(t)) - f f(s,x(s))ds + [C-1d-(C-IA+E) xo]}dt

(18.8)

that passes through the point x* (0, yo, ? ,) = xo = (x01, yo) for t = 0, and x*(t, y0, X) satisfies the boundary conditions (18.2) and (18.3), i.e., x * (t, yo, X) is a solution of the boundary-value problem

x = f(t,x)+A(yo, &), Ax(0)+Cx(A) = d,

x1(0)= x01, (18.9)

where X A(Yo,k) = [C-'d-(C-IA+E)xo] - -f f(t,x(t))dt.

0

For the deviation of xm (t, yo, 7.) from x*(t, yo, X) in domain (18.7), the following coordinatewise estimate is true:

Section 18

171

Boundary- Value Problems with Nonfrxed Right Boundary

x*(t,Yo, X. (t, Yo' ') I a(t,A)W1(Yo,?) = E(x*(t,Yo,?), xm(t,Yo,) ,)),

( 18.10)

where W1(Y0,?) = Qm(E-Q)-1M+KQm- 1(E-Q)-1N1(xo),

Q = ^ K, ot(t, ^) = 3 tttl 1 - I, and ^i1(x0) is a vector of the form (18.4). Proof. We have already verified that all functions xm (t, yo, a,) satisfy the boundary conditions ( 18.2) and (18.3) for arbitrary A,E [? 1, T] and yor G. The inequality

Ix1(t,Yo,),)- xoI <_

j

[

it xO)

<- 2t(1- T

-

J f(s,xp)ds dt + Pl(*b)

]

+ 31(x0) = 2 M + 81(x0), (18.11)

which is obtained from ( 18.6) for m = 1 , shows that x 1 ( t, yo, A,) E D whenever xp E Dp. By analogy, we can establish that the functions xm (t, yo, A,) are contained in the domain D for all m = 1, 2, ... , t(=- [ 0, A.], and arbitrary x O E Dp. Let us estimate the difference xm+1(t, Yo' A,) -xm (t, y0, A.) . Relation ( 18.6) yields xm+l(t, YO, A.) -xm (t, Yo, X)

=

J j [f (t,

xm (t,

-

C

l-

T) J0

Yo, A.)) - f (t, xm - 1( t, Yo, A.))]

J[f(s,xm(s,YO,A.))-f(s,xm-

1(S ,Yo ,.))]ds}dt

[f (s, xm( s, yo, a.)) - f (s, xm- 1(s, yo, A.))]ds

f

- t t

[f (s, Xm(S, YO, a')) - f (s, Xm-1(S, Yo,

.))] ds.

172 Numerical-Analytic Method for Problems with Parameters

Chapter 3

This inequality can be rewritten in the form

rm +t(t,

Yo , X) K[(1

J

J rm(s, yo, A,)ds + f rm(s, yo , A)ds , (18.12) r

0

where rm+l (t,YO, A. ) _ (x m +1(t,

yO, A-) - xm (t ,YO, A.) I.

Since, by virtue of (18.11),

r1(t,yo,A.) <- a1(t,A)+81( x0), a1 (t , A.) = 2t(1-X), (18.13) by induction, in view of (18.13) and (2.9), relation (18.12) yields r2(t, Yo' A.) <- K[a2(t, ),)M+ a l (t, .)01(x0)]; (18.14) rm+1(t,yo, A) <- Km [am+1(t, A,)M+ am(t, A.)(31(x0)] , where

(

am +1(t,

1

-) J

a m(S, ), )ds

+ J

a0(t, A,) = 1.

a m(s , A-) ds, (18.15)

For functions (18.15), it follows from estimate (2.10) with T=X that

am+1(t, A,) =

Am gma1(t,

A,) <

7cm

(18.16)

a1(t, A.) = 3 a1(t, %) = 23 t (1- T) In view of inequality ( 18.16), relations (18.14) yield

rm+1(t,Yo,A.) <_

x a1(t,A.)[( -K)

m

A.

M+K( -K)

m -1

(31(xo) ,

tE [ 0, A,], m = 1.2, .... (18.17)

Section 18

Boundary- Value Problems with Nonfixed Right Boundary

173

- 1, the relation Further, by virtue of (18.17), for all j > j-1 xm+/t, YO , x ) - xm (t,

Y O, x )

=

0 ( Xm+j - k(t, YO , Y ) k=0

- xm+j-k - 1(t, YO

, a.))

yields

^xm+J{t,YO,?)-xm(t,YO,? )I <_

i 1 rm+i(t,YO,?) i=0

\m+i / lm+i-1 l a1(t,jM+ K I KI L\- KI , 0l(xo)] n // `n / % k^

_

[( a

ma1(t,))IUiIM+IKQm-' I a 0 1 (xO) , i= 0 ') i=0

(18.18)

where Q = X K. X In view of the second relation in (18.5), it follows from (18.18) that the sequence of functions xm (t, yo, X) is fundamental. Therefore, according to the Cauchy convergence criterion in the space of continuous vector functions, the sequence xm (t, yo, )L) is uniformly convergent in domain (18.7), i.e., lim xm (t, YO, A) = x*(t,Yo, ^ ).

(18.19)

In addition, the limit function x * (t, y o, ?) also satisfies the boundary conditions (18.2) and (18.3) because sequence (18.6) is constructed so that every function from it satisfies these conditions for arbitrary )LE [X 1, T] and yoe G. For the deviation of x * (t, y 0, X ) from x m (t, y 0, ?) in domain (18.7), relation (18.18) indeed yields estimate (18.10) as j-4 oo for all m > 1. Taking (18.19) into account and passing to the limit as m - - in (18.6), one can easily establish that the function x*(t, yo, )') is a solution of the integral equation (18.8). Moreover, similarly to the functions x m (t, yo, X), the limit function x * (t, yo, X) also passes through the point x0 = (x01, yo) for t = 0. Thus, the function x*(t, yo, ?) is a solution of problem (18.9). Theorem 18.1 is proved. By analogy with the case of the boundary-value problem (16.1)-(16.3), the righthand side of the system of differential equations (18.1) can be transformed with the use of a certain vector parameter so that a solution x(t) of the equation obtained with the initial

174 Numerical-Analytic Method for Problems with Parameters

Chapter 3

conditions x(O) = xo coincides with its solutions for the boundary conditions (18.2) and (18.3). Theorem 18.2. Suppose that all conditions of Theorem 18.1 are satisfied. Then, for an arbitrary pair (yo, A.), yo e G, A. E [A.1, T], there exists a unique value of the controlling vector parameter

n =

1 [C-1d - (C-1A + E)xo] -

X

XI0f

f(t, x *(t, yo, A))dt

(18.20)

such that a solution x = x(t) of the initial-value problem x = f(t,x)+µ,

(18.21)

x(0) = (xo1, yo)

(18.22)

is simultaneously a solution of the differential equation (18.21) with the boundary conditions (18.2) and (18.3) and x (t) = x*(t, yo, A), where x*(t, yo, A,) is the limit function of sequence (18.6). Proof. It follows from Theorem 18.1 that the limit function x* (t, y o, A.) is a solution of the differential equation

X

=.f(t,x)+ [C-'d-(C- 'A +E)xo] - -f 0

f(t,x *(t,yo,A,))dt

(18.23)

with the initial conditions (18.22) and, furthermore, it satisfies the boundary conditions (18.2) and ( 18.3). Thus, we already have one required value of the parameter µ equal to (18.20) in equation (18.21 ). It remains to prove that this value of µ is unique . Assume the contrary. Then there exist two values µ = µ' and µ = µ", µ' * µ", such that the solutions x(t, yo, 7l.,µ') and x(t, yo, A,,µ") of the initial-value problem ( 18.2), (18 .22) also satisfy the boundary conditions (18.2) and ( 18.3). In this case , for the difference of these two functions, it follows from ( 18.23) that

x(t, Yo, 7L,µ") - x(t, yo, A.,µ') r

=J 0

P(t, x(t, Yo, 71, p.")) - At' x(t, Yo' A., µ')))

- f (.f (s, x(s, y0, 7L, µ")) - .f (s, x(s, yo, X, µ'))) ds dt. 0

(18.24)

175

Boundary- Value Problems with Nonfixed Right Boundary

Section 18

If, as in Theorem 16.2, we assume that r(t) = I x(t, y0, ?, µ")-x(t, y0, ?, µ') I ,

Ir(t)Io = (suPIri (t)I...suPIi(t)I)> t

t

then, taking the Lipschitz condition and Lemma 2 . 1 into account , we can obtain from (18.24) that (18.25)

r(t) <_ KI (1- t) J r(s)ds + t J r(s)ds .

L o t

]

By iterating inequality (18.25), we get

r(t)

5

at(t,)L)Klr( t)lo, r(t ) 5 a2(t,?) K2Ir(t)lo, ... , r(t) 5 am+1(t,?)Km

+

tlr(t)1o.

In view of ( 18.16), we have Ir(t)Io <- &1(t, ?.)KQmI r ( t)Io <5 KQmI r ( t)Io. 6

(18.26)

However, under the second condition of (18 .5), relation ( 18.26) holds only in the case where I r(t)Io= 0, which implies that µ' = µ" . The contradiction obtained completes the proof of Theorem 18.2. The following statement on necessary and sufficient conditions of the solvability of the boundary-value problem under consideration is true: Theorem 18.3. Suppose that the conditions of Theorem 18.1 are satisfied for the boundary-value problem (18.1)-(18.3) with a nonfixed right boundary point. Then, in order that a solution x = x*(t) of the Cauchy problem for the given differential equation x = f (t, x),

x(0) = xo = (x01, Y0)

( 18.27)

be simultaneously a solution of the boundary- value problem (18.1)-(18. 3), it is necessary and sufficient that the initial value in (18.27) and the value of the parameter in the boundary conditions (18.2), i.e., the pair (yo, A,), satisfy the determining equation

176

Numerical-Analytic Method for Problems with Parameters

A(yo,% ) = ! [C-ld-(C-I A+E)xo ) -

Chapter 3

5f(t,x*( t,Yo,X )) dt = 0, (18.28) 0

where x*(t, y0, ?) is the limit function of sequence (18.6). In this case, a solution x*(t) of the Cauchy problem (18.27) and a solution x*(t, y0, ?) of the boundaryvalue problem (18.1)-(18.3) coincide, i.e., x*(t ) = x*(t, yo, X), and we can take a function xm (t, y0, X) of the form (18.6), whose error is estimated by inequality (18.10), as an approximate solution. Proof. The sufficiency follows form the fact that, according to Theorem 18.1, the limit function x*(t, yo, ?) that takes the value x * (0, yo, X) = (x01, y0) for t = 0 is a solution of the boundary -value problem ( 18.9). In this case, the pairs (Y0, X) for which A(Y0, ?) is equal to zero (i.e., solutions of equation ( 18.28)) reduce the boundary-value problem ( 18.9) to the original problem ( 18.1) -(18 . 3). Then it is obvious that a solution of the Cauchy problem (18.27) with the obtained initial value yo coincides with a solution of the boundary-value problem ( 18.1)-(18 .3) for the calculated value of X. The necessity of the condition that the pair (y0, ?,) is a solution of equation (18.28) follows from Theorem 18 .2. The function x =x*(t), as a solution of the initial-value problem (18.27), is simultaneously a solution of the posed boundary-value problem (18.1)-(18 . 3). Therefore , for a solution x =x(t, Y0, ?, µ) of the Cauchy problem (18.20)-( 18.22) with an additional controlling parameter t, the boundary conditions (18.2) and ( 18.3) are satisfied for only one value of µ, namely , .t = A(y0, A,) = 0, because, in this case, x(t,yo,X, t=0) =x*(t )=x*(t, yo, X). Theorem 18.3 is proved. Thus, it follows from the reasoning presented above that , to obtain a pair (X*, x* (t)) that is a solution of the boundary-value problem (18.1)-(18 .3) with nonfixed right boundary, one can use the following numerical -analytic scheme: (a) Beginning with x0(t, yo, X ) = x0 = (x01, yo), construct (by using the iteration formula ( 18.6)) a sequence of functions xm (t, y0, ?) depending on the (n - 1)-dimensional vector yo and on the scalar ? as a parameter; (b) construct the determining equation (18.28) by using the limit function x* (t, yo, X) of the sequence xm (t, y0, X); (c) obtain a solution y0 = Y0, ? = X* of the determining equation (18.28) by using any numerical method; (d) on the basis of the pair (yo*, a.*) obtained, construct a sequence x m (t, y0, X*) according to (18.6);

Solvability Theorems for Problems with Nonfixed Right Boundary 177

Section 19

(e) the limit function x* (t, yo, A*) determines (for A. = A.*) an exact solution of the boundary-value problem (18.1)-(18 .3) that coincides with a solution of a Cauchy problem of the form x =f (t, x), x(O) = (x01, Yo )

19. Solvability Theorems for Problems with Nonfixed Right Boundary

Let us show how one can investigate the solvability of the boundary-value problem (18.1)-(18.3) without calculation of the limit function of sequence (18.6) on the basis of analysis of the properties of the mth approximation xm (t, yo, A,). Parallel with the exact determining function (18.28), we introduce an mth approximation of this function of the form

0(Yo>A.) _

[C-'d-(C- 'A+E)xo ] -

f f(t,xm(t,Yo'.))dt, 0

(19.1)

where xm (t, yo, A,) is calculated by using (18.6). It is natural to define the approximate determining equation for the boundary-value problem ( 18.1)-(18 .3) with the use of function ( 19.1) in the form Am(YO,A) = 0.

(19.2)

Theorem 19.1. Suppose that all conditions of Theorem 18.1 are satisfied for the boundary-value problem with nonfixed right boundary (18.1)-(18.3). Furthermore, assume that one can indicate a convex closed domain A' = G'x [.i,?4] c Gx[ T] ,

(19.3)

such that, for a certain m >- 1, the approximate determining equation (19.2) has a unique solution (yo, %) = (yom, Am) in (19.3). Also assume that the index of this solution is not equal to zero and the following inequality holds on the boundary S' of the domain A':

(YOinfE S,

2 3) IAm(Y0,a')I > (,, QW(YO,A.),

where Wt (yo, % ) is defined according to (18 . 10) and (18.4), and a = K.

(19.4)

178

Numerical-Analytic Method for Problems with Parameters

Chapter 3

Then the boundary-value problem (18.1)-(18. 3) has a solution, i.e., the pair (x*(t), X*), and the initial value of this solution x*(0) = (x01, Yo) (19.5) is such that the ( n - 1)-dimensional vector yo belongs to G' and the corresponding parameter A* lies in the segment [a,,, ?,2]. Proof. We use the scheme of the proof of Theorem 17.1. Estimating the difference between the exact and approximate determining functions, we can obtain from (18.28) and (19.1) that

A(yo,^)- A .(Yo,?)

j[f(t,x*(t,y0,?))-f(t,xm(t,Yo,??))}dt.

(19.6)

0

In view of the Lipschitz condition (2.4), relation (18.10), and the equality

f a(t,X)dt = 1 l- !)dt = 9 231t ( 0 obtained from (10 .6) for T = ?,, equality ( 19.6) yields a componentwise estimate of the form

IA(YN,^)- A m(YO,^ )) <

JKjx *(t,Yo ,;L)-xm(t,yo,;L)jdt 0

2 <- KW(Yo,?)jal(t,?)dt =

3)

QW(Yo,X)

= c(A(YO, k), Am (Y0, A.)) • (19.7) Since, by assumption, (yom, am ) is the unique solution of equation ( 19.2) inside domain (19.3), the rotation of the vector field A. (Yo, ?) is not equal to zero on the boundary S' of the domain A'. As in Theorem 17.1, we introduce the family of vector fields -1, (19.8) P(9,Yo,)-) = Am(YO,?)+O[A(Yo,,X)-Am(YO,X )], 0<-9<

Section 19

Solvability Theorems for Problems with Nonfixed Right Boundary

179

continuous with respect to all variables everywhere on S', which connects the vector fields A. (yo, ?,) and A(yo, .). In view of (19.4) and (19.7), the inequality IP(9,Y o,?) I ?

- I A(Yo,X )- A m(VO,X)I > 0

I Am(VO,) ') I

(19.9)

implies that the vector field (19.8) is not equal to zero on S. Therefore, condition (19.9) enables one to establish the homotopy of the vector fields A. (yo, %) and A (yo, %). Then, according to Lemma 3.2, the rotation of the vector field A (yo, ?') is not equal to zero on S. In this connection, according to Lemma 3.1, the exact determining equation (18.28) of the boundary-value problem (18.1)-(18.3) with nonfixed right boundary has at least one solution (yo, ') = (Y0, A.*)E A', A(Yo, X*) = 0, (19.10) inside the domain A'. By virtue of Theorem 18.3, the values of (yo*,, %*) in (19.10) determine a solution x = x * (t) of the Cauchy problem x = f(t,x),

x(0) = (x01, YO ),

( 19.11)

which coincides with a solution of the boundary-value problem ( 18.1)-(18 .3) in the case where the right boundary of the interval of integration is A. = A*. Theorem 19.1 is proved. Let us now investigate the problem of calculation of the initial values of problem (18.1)-(18.3) and the values of the parameter a,. Lemma 19.1. If the conditions of Theorem 18.1 are satisfied for the boundary-value problem (18.1)-(18.3), then, for arbitrary pairs (yo, A.'), (Yo, A.") c Gx [ 7. 1, T],

(19.12)

the limit functions x * (t, yo , A,') and x * (t, y;, A.") of sequences x m (t, yp , A,') a n d xm (t, yo, A.") of the form (18.6) satisfy the inequality

Ix*(t, Yo, A')-x*(t, YO," )")I < [E+al(t,y2 )K(E-Q)-'] [Ixo-xo'I + a2 (A.',.", y6,Yp7], (19.13) where y1 = min (X" X") 1 y 2 = max (X', (18.16), and

,

al(t, 72) is a function of the form

180

Numerical-Analytic Method for Problems with Parameters

Chapter 3

a2O', ?", Y4, Yo')

= I '7^"^^^„-^')C-1d+(C-1A+E)(^'x4'- "x6)]I

Q = 1 2-

+ 2I^'-^"IM,

Y1

(19.14)

it

Proof. Let us estimate the difference x m (t, yo x m (t, yo , k") . In this case, for m = 1, assuming that X">V , we obtain from ( 18.6) that x1(t,Yy,?')-xI(t,y ,A.") = (x o' - xo") r

1

X,

J (f(t, x4 )- f(t,x4^) o

X.

J f(s,xo)ds

o

+ 1, f

1 f(s, x4')ds dt

]

o

+[C-1d+(C-1A+E)x4] -

[C-'d +(C- 1A+E)x4']

r

7^' 1 (f(t,x6)-f(t,xo')dt - i,5 f(s,xo) - f(s,x4')ds dt (x4-xo) + j o ^o

]

a + J f(s, x4)ds + „1, f(s, x4')ds XIXII 0 + ^X„[(;."-W)C-ld+(C-1A+E)(X 4 -X"x4)].

( 19.15)

By virtue of (4.3 ) and the Lipschitz condition, it follows from ( 19.15) that

Ixt(t, Y4,A.')-xI(t, Yo,&")I ( lt <_ Ixo -x4'1 + K I1x l\ o

, ')5Ix4-

1

'Idt +51x4- x 'Idtl r J

+ 21 XX" M + I^'^"[(^"-^'

)C-1d+(C IA+E)(^'x4'-"^r4)]I

_ [E+a1(t,X')K] ^Ix4-41 + a3 (?',?",Y4>Y4)j +

2 V- X" M

,

( 19.16)

Section 19

Solvability Theorems for Problems with Nonfixed Right Boundary

181

where

a3(?', A", A, Yo') =

_ ^"[(^,,- ?')C ld+(C-'A +E)(?'xo'-

X"x )]

,

and a,(t, ?.') is a function of the form (18.15). If ?. > X", then (18.6) with m = 1 yields xl(t, Yo, .')-xt(t, Yo 'r )

_ (xo-x4') = t [(At' xo)-f(t,xo^) - f (f(s,xo)-. f(s,x4'))ds dt (A.' - A.") '° X. + VV, $ f(s,xo)ds - , $ f(s,xo)ds 0 A" + 7^"[(%"-.')C-d+(C-'A+E)( ^,xg_ A."x )]. (19.17) By analogy with ( 19.16), equality (19.17) yields

I xI(t, Yo, A.')-x1(t, Yo , A.") I

<_

Ixo

-41 + K

t

xo- 41dt + „ $ I „)$I0 0

a,'-X"I M + a2(X',X",y'0,yS) + 2 1 A ) + 2 Vx,,X„ M, (19.18) <_ [E+al(t, 7.")K]Ixo-xol + a3(A,',)L",Yo,Yo'

It follows from ( 19.16) and ( 19.18) that the inequality IxI(t, Yo, 7.')-xi(t, Yo, A.") I <- [E+a,(t,Y2)KIIxo-4I + a2(A,, A.",yo,yo'). (19.19) holds independently of the relationship between A ,' and X". Similarly , in view of (19.19), for the difference x2(t, yo , A.')-x2(t, Yo , A,") we have

182

Numerical-Analytic Method for Problems with Parameters

Chapter 3

for X" > X', X2(t, YO , )) -X2(t, Yo, ?")

_

(4 - xo)

+ J (f(t,X1(t, Yo, ?'))-f(t, X1(t, Yo,k"))) 0

I X, - f (f (s, x1( s, yo, ?')) - f (s, X1(s, yo, X "))) ds dt 0

+

) ,

J f(s, X1(s,

"

0

yo, )."))ds

+ f

X,

f(s, X1(s, yo, A"))ds

+ X„[(A."-X')C 1d+(C'A+E)(?'4-%"X;)]

X2(t,

A, ?') -X2(t, Yo , ?") I t I4-4I + K 1- )JIX \\ 0

1 (t, yo,))-Xl(t,yg X")Idt

,l'

+ -' f I X1(t, Yo, )) - X1(t Yoe 7") I dt

+ a3() , ?" , V., V.") +

<- [E +al (t, X')K +a2(t,Y2)K2 ]

2 1 k' - X"

A

M

I4 -41

+ [E+a1(t, X')K]a2O,', X", yo, Yo') ; for X, >)," , X2(t, Yo, ))-X2(t, Y9, ?")

_ (XO - 4) + J I Wt, X10, YO" X')) - f(t, X1(t, Yo, "))) 0

( 19.20)

183

Solvability Theorems for Problems with Nonfixed Right Boundary

Section 19

1V - ^„ J (f(s, xl(s, yo, ?'))-f(s, xi(s, Yo, X")))ds dt 0

+ ^,x1; f f(s, x1(s, y;, X')) ds - ), f f(s, xi(s, Yo, %')) ds 0 X" + - ,,[(?"-^')C-ld+(C- IA +E)(7'xo- ? "xo)],

1 x2(t, Yo,?')-x2(t, YO ,?")I

<

Ixo

-4

xl(t, yo, ?') - x1(t, yo", X") Idt 0

+ ^., t xl(t Yo, X x t,

+ a3(?',)", Yo, Yo) + 2

[E +al(t,? ") K+a2(t , Y2)K2 ] + [E+al(t,

1 W- ,^,, M

IX6-4I

?")K]a2()V, ?.", y6, Y6)

(19.21)

It can easily be seen from (19.20) and (19.21 ) that, for any pairs ( 19.12), we have

I x2(t, Yo ,?')-x2(t, Yo , ?")I <- [E+al(t,

Y2 )K +a2(t , 7 2)

K2] x6 I -4

I

+ [E +al(t, Y2)K] a2(V, ?", yo, Yo) . (19.22) By induction, it follows from inequalities (19.19) and (19.22) that for k" > X',

Ixm(t, Yo >W) - xm( t, Y011, X") t, ' ^')-x t 0+ y"")Idt J Ix Ixo-xo'I + K 11- -L ^,) m-1( Y0+ m-1(+ 0

184

Numerical-Analytic Method for Problems with Parameters

Chapter 3

2.' + Xt .11 xm-1(t, y0, A.') - xm-1(t, Y0, a.") I d t t

+ ()L', V Yo, Yoh +

[

<-

21 V- A"

M

A,

E+.,(t,X')K+ j ai(t,Y2)K` IIxo-41 i =2

J

m-1 1

+ E+a1(t, A')K+ I ai(t, Y2)K` I a2(X', A.", Yo, Yo") ; (19.23) i=2 I for A,' >

A. ",

I xm(t, Y0 , 71.')-xm(t, Yo, a'")I

t

/

Ix'p-xp'I + K I1--tir)f Ixm-1(t,yo,xm-1(t,yo,A")Idt /1 0

+ X"" f I xm-

1(t, yo, )1) - xm-1 (t, y0 Al ) I d t

J

+ a3 W, A.", Yo, Yo") + 2

1

. X.. M

m

[ ^ a i tt, A.")K+ m ai(t, Y2 )K` I xo -xo I i=2

+ E+a1(t ,

J

m-1 11 A")K+ I ai(t,'Y2)K' i=2

J

a2 (.', A, ", YO, YO) •

(19.24)

Consequently, by virtue of (19.23) and (19.24), for arbitrary pairs (19.12) we have Ixm (t, YO ,

<

xm(t,

YO,A.V ')I

,Y 2)K ` 1x6-41 + ^E ^ Li=0 1 i =0 ai(t

where ai(t, y2) are functions ( 18.15).

ai(t ,Y 2)K`

J

a-(A.',A. ",y

,yo'),

(19.25)

Solvability Theorems for Problems with Nonfixed Right Boundary

Section 19

185

Passing to the limit as m -4 °° in (19.25) and taking ( 18.16) and (18.5 ) into account, we get

I x*(t, yo, X

')

- x *(t, YO"' X-) I [ E +ai(tT2 Kj (xzK)i lxo-xo'I i =o n

1

+ E+b.x(t,y2)K

y, u71 A:

,'

02(v,r,^.3«)

i=0

[E+&t(t,72) K(E-Q)-1j [Ixo - 41 + a2 (X', ) ", yo, yo')], i.e., estimate (19.13) is proved. Theorem 19.2. For the boundary-value problem (18.1)-(18.3) with nonfixed right boundary that satisfies the conditions of Lemma 19.1, a function A (yo, ?) of the form (18.28) is defined and continuous in the domain G x [ a.1, T ] and, for arbitrary pairs (19.12), the following estimate is true:

I A(y6, V) - A(yo. k") I

< as(k',X", yo, yo) + K

z

L

E +^( E -Q)-1 97,

[I xo -41 + a2(1',% yo,yo^] ",

(19.26) where a5(.' ?" yo, yo) = I^'"(.,_^')C-1d+(C-1A+E)(^'xo-"xo)]I +21 w-X" M. Y1 Proof. According to Theorem 18.1, the limit of the sequence of functions (18.6) is a continuous function in domain (18.7) and, therefore, the function A (yo, ?,) is also continuous and bounded for (yo, ?.) e G x [ a,1, T], i.e.,

A(yo,A )I !5 1IC_1d-(C-1A+E)xol+M.

186

Numerical-Analytic Method for Problems with Parameters

Chapter 3

By the(yo, definition of the function "" y X) for O^ , > X', relation ( 18.28) immediately yields

0(Y0I X") - A(Yo, k")

J [C_id+(C-'A+E)xo] - ^, f f(s,x* (s,y6,?'))dt 0

- .""[C-'d-(C-'A+E)xo' ] + J", jf(s,x*(s,Yo,X""))dt

e

1 C'dtf." - V) + (CT]A + E){\%- \'x%) XT' -, j [f (t, x*(t, Yo, ?") - f (t, x*(t, yo X"))] dt 0 %, +

x°

r

f(t, x*(t,Yo,? "))dt + f f( t, x*(t, yo, ).")) dt. 0 x

(19.27)

By virtue of estimate (19.13), it follows from (19.27) that

IA(YoI V)-o(Yo,A."')I = aa( A,' ^", YO' , yo) + IX" K

<- aa(X", A." , Yo, Yo") +

21

10 I x* (t, y;,", x* (t, y;, k"") I dt + 2 ^X„ ", M

^.. V M

+ X'K [E+al(t,Y2)K (E_th '] [!xo-4 + a2 (A,',A.", yo,Yo")]dt,

(19.28) where

aa(A",A.", yo,Yo") = 1 Vlk"^(A."-^')C td + (C-^A+E)(^'xo'- "xo)]I

Section 19

Solvability Theorems for Problems with Nonfixed Right Boundary

187

Since X,

2

f at(t,Y2 )dt = 92 0 relation (19.28) yields

A(Yo, ^') - A(Yo, ") I = a4(?', %", y , Yo') + 2 X.. ^.. M

r

z

[(xo - xo l

+ K E+ 9 K(E-Q)-^

[

+ a2 (X',X",Yo>Y0J]• (19.29)

For X' > X", it follows from (18 .28) that

A(Yo, X') - L(Yo, A")

_ X IX"

[(?L" - ,')C-'d+(C- 'A+E)(X )q - X"xo)]

-„ J [f(s,x*(s,Yo,A'))-f(t,x*(s,yo,?."))]dt 0 X.

X^

+ , x* (s, yo, .')) dt + ^'::. J f(t^

Jf( t, x* (t, y,^,')) dt.

( 19.30 )

The last equation yields

I A(Yo, X') - A(Yo, V) I <- aa(A',A",Yo,Y0') + 2I^'% ^M

+ X„ K J [E + a i ( t, r2) K ( E -

z ][l o' - xo" f + a2 ( A,', V > Yo, yo")] dt

0

aa(X',X",YoIYo) + 2IX- IM 2

+

KIE +99 K(E-Q)- 1

[I xo-xol+a2 (% )'" Yo.Yo)]-

188

Numerical-Analytic Method for Problems with Parameters

Chapter 3

Thus, relations (19.29) and (19.31) yield the following estimate for arbitrary pairs (19.12):

1 A(Yo, ?') - A(Yo' X") I <- a4(A', A ",Y6,Y9) + 2

1?"-A.' I M Y1

z + K E+2K (E-Q)-1J[lxo - xol+a2 (A',A.", yo,yo)], 9Y1 where Q = 12 K and a2 (A,', A.", yo, yo) is calculated according to (19.14). Theorem R 19.2 is proved. Theorem 19.3. Suppose that all conditions of Theorem 18.1 are satisfied for the boundary-value problem (18.1)-(18.3) with nonfixed right boundary. Then, in order that the domain A" = G"x [Ai,A']c Gx[

T], (19.32)

contain the pair of values (ye, A.) that determines the Cauchy problem (19.11) equivalent to the boundary-value problem (18.1)-(18.3) for A. = A,", it is necessary that the following relation hold for all numbers m and an arbitrary pair (yo, A,) from domain (19.32):

[C-1d-(C-1A+E)x0] - J f (t,xm( t,YO,7.))dt 0

SUP

2 p,,^ as(^,^,yo,Yo) + K E + 2(E-Q)-'

(YO,X)EA" 9il J

x [IX0-xoI+a2(^,X,Yo,Yo)] r + E(A(Yo,X),Am(Yo,X)), (19.33)

where 10

= ( x01, Y0 ) ,

E(A( YO, 1), Am( YO, A )),

are calculated according to (19.7) and (19.14).

and

a2(A.,A.•70,Yo)

Section 19

Solvability Theorems for Problems with Nonfixed Right Boundary

189

Proof. By assumption, the function x = x*(t) with the initial value x* ( O) = (x01, yo) is a solution of the boundary -value problem (18.1)-(18 .3) for the right boundary point ? = X*. Therefore , according to Theorem 18.3, the pair (yo*, ?,*) is a solution of the determining equation ( 18.28). Choosing (Yo, ?') = (yo, ?) and (Yo, A.") = (yo, ?* ) as pairs (19.12), we write inequality (19.26). Since ° (yo, ?*) = 0, we have r

A(5o,X) :5 a5(X' V" Yo, yo) + K E +

L

2

(E-Q)-^ 971

x [Ixo-x*ol +a2 (^, *,yo, Yo)]. (19.34) According to formula ( 19.7), we obtain

°m(yo, ?) l :5 J °(Yo, A) l + E(°(Yo, ?), °m(YO, X)) (19.35) for (yo , A.)= (yo, 7S). Substituting ( 19.34) in ( 19.35), we get

I°m(yo, X) I <- a5(X, *, yo, Yo) r

2

+ KI E+ Y (E-Q)-1 [ xo-.ol+ai (A,r,5o,Yo)I

+ E(°(Yo, C), °m(Yo, ^)) 2

a5 sup Yo,Yo) + K E+^(E-Q)-' (YO, ^,) a A„{ 97,

+ [Ixo-xoj +a2(X,X,yo,Y0)] + e(A(yo,1),°m(Yo,X))-

Theorem 19 . 3 is proved. This statement generates the following scheme for the approximate calculation of the pair ( yo, ?,*) with the use of which the boundary -value problem (18.1)-(18 . 3) is transformed into the equivalent Cauchy problem ( 19.11): Assume that a set Do of the form ( 18.4) and the segment [X I, T] are unions of finitely many subsets, i.e.,

190

Numerical-Analytic Method for Problems with Parameters N

Chapter 3

N,

(19.36)

DR = UDi, [X1,T] = U I;.

i=t

i=i

We choose arbitrary points xo = (x01, Yo) E Di, X, E I;, (19.37) in subset (19.36) and, for a certain fixed order of approximation xm(t, YO', ?) of the form ( 18.6), calculate the value of the approximate determining function

Om(Yo' X j) _ [C-td -(C-'A+E)xo] - j f(t, xm(t, Yo,

a,;))dt.

(19.38)

.i o

Denote by Mm the union of the subsets Dj and Ij in (19 .36) for which 2

Om(Yo,Xj) sup .. as(xj,X,Yo,yo) + K E+9 2K(E-Q)-1 (yo,X)E Am

L

ll

7

J

x [Ixo-xol+a2(I;,2L,A,yo)] + E(A(Yo,Xj),Am(Y0,Xj)), (19.39)

where L m (yo, ? ) and (yp, ^j) are determined according to (19.38) and (19 .37), respectively , yI = min (A. , ?), Y2 = max (Xj, A,),

2 K, A;; = G` x IJ, and G' is

the set of ( n - 1)-dimensional vectors y0 = (x02, ... , xo„) for which xo = (x01, yo) E D i M. for the numbers i that define the set Mimi According to Theorem 19.3, this set is the domain that contains the pair (yo, A,*) that appears in the initial conditions of the Cauchy problem (19.11 ), and its solution coincides with a solution of the boundary-value problem (18.1)-(18 .3) under consideration. It is clear that A; tends as i, j, m -* co to the set G * x I", which determines the exact values of the pairs (yo*, V). As approximate values of yo and r, we choose arbitrary yo E G` and ? E Ii, respectively . In this case , it is obvious that

Ixo-4I = sup ixo-xoI,

4 = (xol, y0),

yo EG'

I2-?I. Ii -A''I = sup x .=- ii

(19.40)

Section 19

Solvability Theorems for Problems with Nonfixed Right Boundary

191

It is easy to see that the subsets D; and Ij in (19 . 36) whose arbitrary points do not satisfy relation (19.39) do not contain the required pair of the initial value and parameter (Yo, A*). Let us estimate the error of an approximate solution of the boundary -value problem (18.1)-(18 . 3) chosen in the form of the function xm(t, yo, ?) calculated according to ( 18.6) for an arbitrary point yo e G' and X e F. Theorem 19.4. Suppose that, for the boundary-value problem (18.1)-(18.3) with a nonfixed right boundary point, the pair (yo, ?,*) is a solution of the exact determining equation (18.28) and (yo, ?,) is an arbitrary pair from the set A;; . If the conditions of Theorem 18.1 are satisfied, then the error of an approximate solution of the form x,n(t, Yo' i)

zo + J [(t Xm_i(t, 'o, ?)) - J f(s, xn_1(s,yo ^))ds dt 0 0 + [C-1d-(C_ 1A+E)XoJ,

xo ( t,Yo,^) _

(x01, S'0),

(19.41)

is estimated as follows:

I x*(t, Yo , *)- xo(t, yo, i) sup

E(x*(t, Yo, ?), xm(t, Y0, ,)

Yo eG ', ke /'

+ [ E ^^i(t ?2)K+,Q` =0

[jXo- xol + a2 (?,'> Yo >yo)1 ,

(19.42)

where E(x*(t, YO, X), x,,,(t, yo, ?.) ) is obtained according to (18.10) and (18.4), a2(?*, ?, Yo, Yo) is obtained according to (19.14), Y2 = Max (A,, ?),

Q = 72 R',

a,etj

2

and x*(t, yo, r) is the limitfunction of sequence (18.6).

192

Numerical-Analytic Method for Problems with Parameters

Chapter 3

Proof. For the difference x*(t, yo, A.*) - xo(t, yo , ?) of the functions calculated according to ( 18.6), an inequality of the form ( 17.28) is obvious . If the first and second terms in (17.28) are estimated according to (18 .10) and (19.25), respectively , then we get

I x*(t, Yo, r) - Xm (t, 5o• ?) I : E(x*(t, Yo, A ), xm(t, YO*' ?*)) m _

+ [E+^i(t 72)KQ` [ Ixo-xoI+a2W,i,Yo,Yo)J, (19.43) i=o where

Y2 =

Max

Q = Y2 K.

(X\5L).

X

In view of (19. 40), inequality ( 19.43) can be rewritten in the form ( 19.42), which completes the proof of Theorem 19.4. By using inequality ( 19.42), one can easily estimate the error of the approximate solution xm (t, yom, A.) obtained in the case where (yo, i) = (yom, 'm ) is an exact solution of the approximate determining equation ( 19.2), (19 . 1). Here , as a particular case, we obtain the inequality

I x*(t,YO+a'*)-xm(t,Yom, Am)I

E(x*(t,Yo+a'*),xm(t, YO*' A!))

m +

5 sup Yo EG',?.EN

[E ^^i(t, Y 2)K 1: Q' i =o

4 -

x0ml +a2

*

Y0 +Y0m

E (x*(t, yo, A), xm(t, Yo,.))

+ E+&I(t,Y2)K^Q` xo- xomI+a2 (k,Xm,Yo,Yom)] i=o where 72 = max

y2 = max. (1%, gym), 7lEV

)]

Section 20

The Case of Several Parameters in Boundary Conditions

193

a1(t, Y2) and a1(0 ¶'2 ) are functions of the form (18.16), and Q =) K, Q=Y2K. n It

20. The Case of Several Parameters in Boundary Conditions The numerical-analytic method under consideration can be extended to the case of problems with several parameters of control in the boundary conditions. Consider the boundary-value problem x = f(t,x),

x,fe R", n>3,

(20.1)

X1Ax(0)+Cx(A.2) = d, detC#0, (20.2) x1(0) =x01,

x2 (0) = x02.

(20.3)

One can see that the boundary conditions (20.2) contain two scalar parameters a' 1 and X 2 and, moreover, a' 1 appears linearly and ? 2 nonlinearly. As above, we assume that the function f (t, x) is defined and continuous with respect to t and x in domain (16.4). It is necessary to determine a solution x =x*(t) of equation (20.1) continuously differentiable on the segment [0, X2] and the values of parameters 1 = Ai and X2 = 2 for which the solution x*(t) satisfies conditions (20.2) and (20.3). It is obvious that problem (20.1)-(20.3) is, in fact, a boundary-value problem with a nonfixed right boundary point, which is analogous to the eigenvalue problem. We also investigate the boundary-value problem (20.1)-(20.3) with the use of the numerical-analytic method. We select the class of problems for which the right-hand side of the system of differential equations is continuous and satisfies the boundedness conditions and Lipschitz conditions of the form (2.4) in domain (16.4). Denote by Dp the set of points x0 = (x01, x02, ... , xo,) contained in the domain D for any values of A. from a certain domain

A. _

(a,1,

A, 2) E

[Ai, Ai] X [A2, T

^

= A, AZ > 0, ( 20.4)

together with its 0-neighbourhood, where fi = 22M + R1(xo,X1 ),

131(x0,A.1 ) = IC-(.1C- IA+E)xol. (20.5)

194 Numerical-Analytic Method for Problems with Parameters

Chapter 3

In this case, the set G is understood as the set of (n - 2) -dimensional vectors yo = (x03, ... , x0:) such that the vectors x0 = (x01, x02, yo) belong to Dp for x01 and x02 defined according to (20.3). In addition, we assume that the set Dp is nonempty and the maximum eigenvalue of the matrix Q = T K is less than 1, i.e., inequalities (18.5) are satisfied. X We construct a sequence of functions xm (t, yo, ?) depending on t, y0, and ? _ ()L1, X 2) according to the recurrence relation t

xm(t,y0,^) = xo + J

0

+

1

XZ

f(t,xm- 1 (t, Yo,X)) - f f(s,xm- 1(s,y0,?))ds dt 20

2[C-'d- (?.1C- 1A+E)xo], m=1 ,2,...,

xo(t,Y0,X) = x0, x0 = (x01,xo2,Yo)•

(20.6)

Direct verification shows that, for all m = 1, 2, ... , the functions xm (t, yo, ?) satisfy the boundary conditions (20.2) and (20.3) for arbitrary values of ? from domain (20.4) and Yoe G. By analogy with Sections 18 and 19, we can prove that all functions xm (t, y0, X) of the form (20.6) belong to the domain D for X0 E Dp. For the boundary-value problem (20.1)-(20.3) with two controlling parameters X 1 and X2, according to the properties of the sequence of functions (20.6) and its limit function, we can obtain analogs of all theorems proved in Sections 18 and 19 for the boundary-value problem (18.1)-(18.3) with one controlling parameter. We do not prove some of these theorems but only formulate the corresponding statements. Theorem 20.1. Suppose that the boundary-value problem (20.1)-(20.3) satisfies conditions (2.4), (20.5), and (18.5) in the domain of definition (16.4) of the right-hand side of the system of differential equations (20.1). Then a sequence of functions xm (t, y0, ?) of the form (20.6) satisfies the boundary conditions (20.2) and (20.3) for all m = 1, 2, ... and arbitrary Y0 e G and ? e A, and, as m-*oo, it converges uniformly with respect to the domain (t,y0,A.)e [ 0,X2]x GxA

(20.7)

to the limit function x*(t, yo, ). Moreover, x*(t, yo, X) is a solution of the integral equation

Section 20

195

The Case of Several Parameters in Boundary Conditions

x(t) = xo + J jf(t,x(t)-- f(s,x(s))ds + 1 [C-'d-(^,1C-IA+E)xo]}dt J o T2 o X2 JJJ that passes through the point x*(0, yo, ?) = xo = (x01, x02, Yo) for t = 0. In this case, the limit function x * (t, y o, A.) satisfies the boundary conditions (20.2) and (20.3), i.e., x * (t, yo, A.) is a solution of the boundary- value problem with modified right-hand side

x = f(t,x)+A(yo,.), x1(0)= x01,

.1Ax(0)+Cx(.2) = d, x2(0)= x02,

where

A(Yo,A.) _

)L2 2 [C-'d-(A.1C-'A+E)xo] - 2 Jf(t,x(t))dt.

0 For the deviation of the m th approximation x m (t, y o, A.) from x * (t, yo, A,) i n domain (20.7), the following coordinatewise inequality is true: Ix*(t,Yo, A.)-xm(t,y0,X)I <_ a1(t)W2(Yo,A.) = e(x*(t,Yo , A.),xm(t ,Yo,A.)), (20.8) where

W2(yo,A.)= Qm(E-Q)-1M+KQm-1(E-Q)-131(xo,Al), Q = K, &1(t) is a function of the form (2.10), and I3 1 (x0, A.1) is a vector defined in (20.5). For the boundary -value problem (20.1)-(20 .3), the theorem on the controlling parameter is also true . According to this theorem , equation (20.1) can be modified so that its solution satisfying the boundary conditions (20.2) and ( 20.3) passes through a certain point. Theorem 20.2. If all conditions of Theorem 20.1 are satisfied, then, for an arbitrary pair (yo, A,), yOE G, AE A, there exists a unique value of the additional controlling n -dimensional parameter

196

Numerical-Analytic Method for Problems with Parameters

Chapter 3

^z

n=

Z [C-'d-(o1C"1A+E) xo] -

Yo '2 Jf(t,x*(t, 0

,%))dt

for which a solution x =x(t) of the Cauchy problem

x = Pt, x)+ µ,

x(0) = (xo1, xo2, Yo)

coincides with a solution of the boundary- value problem x = f(t,x)+µ, ?,1Ax(0)+Cx(.2) = d, x1(0) = x01, x2(0) = x02• In this case, x(t) = x*(t, yo, X), where x*(t, yo, functions (20.6).

is the limit of the sequence of

The statement below gives a necessary and sufficient condition for the existence of a solution of the boundary-value problem (20.1)-(20.3). Theorem 20.3. Suppose that the conditions of Theorem 20.1 are satisfied for the boundary-value problem (20.1)-(20.3) with two parameters X t and ' 2 and a nonfixed right boundary point. Then, in order that, for the given differential equation (20.1), a solution x = x*(t) of the Cauchy problem

x = f(t,x),

x(0) = x0 = (xot,x02,Yo)

(20.9)

coincide with a solution of the boundary-value problem (20.1)-(20.3), it is necessary and sufficient that the initial value in (20.9) and the values of the parameters X1 and X2 in the boundary conditions, i.e., the pair (yo, A,), satisfy the determining equation X2

A(y A(Yo,X) 0 .*) = _ Z[C-' d- (?,1C-'A+E) xo] -

= 0, - Jf(t,x*(t,Yo,A))dt 0

where x*(t, yo, .) is the limit of the sequence of functions (20.6). Moreover, for these values of y o, ? 1, and ' 2, a solution of problem (20.9) is x * (t) = x*(t, yo, ?) and we can take a function xm (t, Y0, A) of the form (20.6) as an approximate solution with error estimated by (20.8). Let us investigate the problem of solvability of the boundary -value problem (20.1)(20.3) without determining the limit function x*(t, Yo, ?,).

197

The Case of Several Parameters in Boundary Conditions

Section 20

Theorem 20.4. Suppose that, for the boundary- value problem (20.1)-(20.3), the conditions of Theorem 20.1 are satisfied and there exists a convex closed domain A' X A'1+ A" ]x I> A' = G' x A 1] A" L x c 2T ] [ 1+ 2G2] such that, for certain m >- 1, the approximate determining equation X2

A.(Yo,X) _

2 [C-'d-(7.1C-1A+E)xo] -

X2

J f(t,xm(t,Yo,?)) dt = 0

(20.10)

0

has a unique solution (yo, X) = (yon , X. ), X m = (a'm 1, X m 2), in the domain A'. Also assume that the index of the solution of equation (20.10) is not equal to zero and the following inequality holds on the boundary S' of the domain A':

(yo,

Es' IAm(Yo,X)1 > ( ")' Qw2(Yo, X), 3

where W2(yo, X ) is obtained by using (20.8) and (20.5), and L2 K. X Then the boundary- value problem with two parameters X 1 and X 2 and a nonfixed right boundary point (20.1)-(20.3) is solvable. In this case, the pair (x*(t), ?*) that is a solution is such that the initial values of this solution x*(0) = (x01, x02, Yo) are determined by an ( n - 1)-dimensional vector yo that belongs to the domain G' f o r ? 1=

A1.1E

[A j ,A ; ] and

?2= A,Z E

[A2 2 ].

The determination of subdomains of the domain of definition of the boundary-value problem (20.1)- (20. 3) that can contain the values yo, )^,, and ^z determining the required solution is based on the following auxiliary statement: Lemma 20.1. Under the conditions of Theorem 20.1, for arbitrary pairs (Y' o+ V' 1+ X' 2)> (yo,? ,?2) E Gx [A',Ai]x [A' ,T],

the limit functions x*(t, yo, &') and x*(t, yo, ") of sequences xm(t, xm(t, yo, a.") of the form (20.6) satisfy the inequality

(20.11)

yo, ?')

and

x* (t, Yo, - x* (t, Yo, X")

< [ E + a1(t, 74)K(E xo 41 + a6(?", A. ", Y6, y6')] where y4 = max (?4,.2),

&1(t, Y4), is a function of the form ( 18.16), and

(20.12)

198

Numerical-Analytic Method for Problems with Parameters

Chapter 3

a6(A',A.", Yo, Yo) = a7(A.',A.", A, YD + 21 X2-"zIM, (20.13) 73 a7( ,y ,yo)

2 -A2)C-d+C- A(A 1 ?62x0-A. A. x0 )+

i2, Xz

12x0- AZxO]

,

73 = max (?4, A2) •

Proof. To obtain inequality (20.12), we estimate the deviation of the functions xm(t, yo', a,') and xm(t, Y0, A.").

If we set m = 1 in ( 20.6) and assume that A2 > '2, then we get x1(t, Yo, A.') - x1(t, Yo, A.")

1 + f (f(t,x)_f(t,4)) f (f(s, 4 )- f(s,xo)) dsldt 2 r

_

(xo-xo')

1

X,

X.

(.f( s, 4)) ds +2 (f(s, 4)) ds ;Lj

+

11 11 1 t , [(A2 -A.2)C- d+C- A(^11 A62x0 -a,1 ,2x01+' )+ CA2x0 -A,2x0]. 2 X0

21 2

(20.14) By analogy with (19.16), relation (20.14) yields

I xt (r, yo A,') -x1(t Yo„, A.")I t

f

7l'

l

" " ) f I xo-x4 Idt +i I x0-xO dt

i

i -x0l +t K 1--

2

0

2

t

2 A2 - A.2

+ ^„ M+a7 (^ ,)L ", YO,Yo) X2

_ [E+a1(t,A.2)K] x0-xoI+ a7(A.,^,",y0,YO) + 2 X ^..^2IM. (20.15) 2

Section 20

The Case of Several Parameters in Boundary Conditions

199

If ?2 > AZ , then relation (20.6) with m = 1 yields x1(t,Yy,? )-x1(t, Yo,^ )I 21 X'2 - Xt2'1 <_ [E+al(t,Xz)K]lxo-xol+ a7(k,% ,Yo,Yo) + ^, M. (20.16) z It follows from (20.15) and (20.16) that x1(t, yo, A.') - x1(t, Yo, x") I <_ [E+c1(t,Y4) K]Ixo-xol + a6 (?",)"', Yo, Yo) (20.17) for any ?2 and ?'2 . Similarly, if A2 > Az, then, in view of (20.17), it follows from (20.6) with m = 2 that

J X20, Yo, A.') - x

2(t,

yo, A.")

"

I

<_ Ixo- xo + K

C

t lr

IJ Ix1(t,Yo,7.')-x1(t,Yo,X")Idt 1- X2 0

I

+ 7,2 t

jxl(t,yo ,X')-xl(t,Yo,X")jdt

+ X2' I

1 11 M+a7()L',X",YO,Yo)

X2

[E+al(t,A.i)K+al(t,Ya)K 2] Ixo-xol + [E +al(t,X2)K]a6 (),i,A",yo,yo )• Assuming that A,2 >

A.'2 , we get

x2(t, yo, A.') - x2(t, Yo, A.")

„

5 Ixo - xol + K

C

t 1--, JIx1(t,yo•7-)-x1(t,Yo,A )Idt

X2 0

(20.18)

Numerical Analytic Method for Problems with Parameters

200

Chapter 3

A.z

+ 2 J Ixt(t,Yo,A.')- x1(t,Yo,A.")Idt

, + a7 A' A„ + 21 X z Az M YO, YO ) X 2 ^" K+a t, -xo z ". [E+a 1(, 1)t 1( 74)K Ixo + [E +a1(t, Az)K]ab(%', A", YO YO ). (20.19) Thus, it follows from (20.18) and (20.19) that

Ixz(t,YO,A.')- xz(t,Yo , A,")I [E+a1(t,Y4 ) K+a2 (t,74 )K2Ixo-xoI " (20.20) + [E+a1(t,74 ) K]Ixo-xola6 (A,1,A,",Yo,Yo) for arbitrary pairs (20.11). By induction, inequalities (20.17) and (20.20) yield the following relations: for X2 > X2,

I xm (t,

Y0, A') - xm (t , y0, A") I

^ 1*0-*o I + KR11ltt ) p I xm-t(t , Yo, Yo, A.") Idt L,

?Lz t „ )L2 j 1X, _1(t,yo,A )- xm_t(t,Yo,71 )^dt

2I Az, -Az M

+ a7(A,', X ", yo,Y0) + ^„

X2

5 [E+a1(t,A2 )K+a2(t,74)K2+...+ am(t,7a)KmIIxo-xo1 + [E+a1(t, ?4)K+a2(t,74)K2+...+ am_1(t ,74)Km-1 ]a6(A,',)'",Y61yo); (20.21)

The Case of Several Parameters in Boundary Conditions

Section 20

201

f o r X2 > ?2, Xm(t, Y0 , ?') - Xm(t, Yo,

<

I xp

1- Tr

-I Xp + K

lXm -1(t,y,X)- xm-1(t,YO,a

)Idt

a2 0

t

+ ^, $ X2 0

I

Xm - 1(t,Yo,a.')-Xm-t(t,YOIdt

r

+ a7(^' ,

&",

YO,YO)+

2I A,2 -? 2

M

m

-- E +a1(t,,%2)K +1a1(t,Y4)K` Ixo -xoI i=2 m-1

+ [E+al(t, X2")K+ ai(t,74)K` ]a6O'. 1=2

Yo, Yo). (20.22)

Inequalities (20.21 ) and (20.22) imply that, for arbitrary pairs (20.11 ), the following relation holds: X. (t, Y0,

Xm (t, Y0, k")

m

lai(t, Y4)K` =0

i

r

"

Xp x0 I + ^ ai(t, 74)K i=o

r

"

a6(k, X , Y0,Y0 ), (20.23)

where a.(t, 74) are functions of the form (18.15). Passing to the limit as m -^ oo in (20.23 ) and taking ( 18.16) into account, we get

I x* (t, Y0, A') - X. (t, Y0, A") I [E+cxi(t14 )Ki (Y4 K'l Ixo -xoI )j i =0 It

(

K)`] ab(A., A", YO, YO) + E +a1(t, 74)Ki )

i

=0

202 Numerical-Analytic Method for Problems with Parameters

Chapter 3

<- [E+al(t,Y4) K(E-Q)-1]I xo-xoI + as (X',X",Yo,Yo), where a6 (?.', k", yo,yo) is a vector of the form (20.13). Lemma 20.1 is proved. Theorem 20.5. If the boundary-value problem (20.1)-(20.3) with two parameters satisfies the conditions of Lemma 20.1, then its determining function X2

AN, X) = 12 [C-ld -(X C- l A +E)xo] - _2 f

0

f(t, x*(t, yo, ? ))dt,

(20.24)

where x*(t, y0, ),) is the limit of sequence (20.6), is continuous in domain (20.7) and, for arbitrary pairs (20.11), the following inequality holds: A(yo' x) - A(Yo, x ) I = a9(X , X

YO,Yo )

[

l

+ K E+ 1.Ed K( E_ y' [x -x1 + a6(k',X",yo,y )] , (20.25) 973 where

a9(X ,X ,Yo'Yo) = a8(). ,X ,Yo,Yo)+ M, Y3

a8(X , X , Yo,Yo) _

1

^-2xtt 2

n

r

1

1

" ,

[(X2-^.2)C-d + C-A(a1

n , n X2xo -Al^2xo) + X '4

n,

- a2X0]

Q = 1± K %

,

(20.26)

Proof. Theorem 20. 1 guarantees the continuity of the limit function of sequence (20.6) in domain (20.7). Consequently, for (yo, X) E G x A, the determining function (20.24) is also continuous and bounded, namely,

JA(Yo, )I <_ 2IC-1d-(?,1C-1A+E )xoj+M.

By definition, for arbitrary pairs (20.11) and X2 > i4, relation (20.24) yields

The Case of Several Parameters in Boundary Conditions

Section 20

203

A(YO, ).') - A(YO, ?")

2[C-'d-(X'C-' A+E)xo]

1 j.f(t,x* (t,y ,?'))dt - Z

0

X2

- 2[C-'d-(?L1C-'A+E)xo] + 42 5f( t,x'(t,Yo,^,"))dt

0 n r n r u, n ,,,

I A(711 A.zxo - 7v11L2xo) +?z xo -? 2xo] d+C-7V2) C1 [(^ 2n

1

2X

z

22

j [f(t,x' (t,Yo, ))-f(t,x'(r, yo, A. ))Idt

72

••

2

^2

X2

A"))dt.

(20.27)

+ -ZK j [E +&1(t,Y4)K(E-Q)-'][Ixo-41 + ab(A.',A",Yo,Yo)]dt.

(20.28)

+ A.2,

J f(t, x'(t, Yo, A"))dt +

J f(t, x'(t, Yo,

2^2 0 A.2 22

In view of estimate (20.12), relation (20.27) yields .. .. , „ X'2' 1 M X11

I A(Yo, ^) - A(YO, X) a8 (,% 'I, A , Yo,YO) +

2

X2

By analogy with (19.29), it follows from (20.28) that

..

..

.

..

,

11

21

X2

-

A2

1 A(YO, A) - YO, ) I = as(%1, , Yo, Yo) + X,, M z

+ K1E+9K(E-Q)-1 [Ix, -x0I + a6(?V>^ Y,, )], z For arbitrary pairs (20.11) and 42' > A2 , relation (20.24) yields

(20.29)

Numerical-Analytic Method for Problems with Parameters

204

Chapter 3

0(Yo> X) _'&( Yo, )

22 l(X2 - a,2) C- d + C- A (a1 ?2xo - 2t 1,2x0) + 1,2x0 - A 2x01

X2 + 2 f [f(t,x*(t,Yo 0

f(t,x*(t,Yo• "))^dr

„ a,2 X2

+ X2^- 7^2

r f(t, x*(t, Yo, &'))dt + 1 J f(t, x*(t, Yo, X'))dt. a 2a 2 0 a,2 ;L2

(20.30)

Hence,

(Yoe ) (Yo, ) a( i, , Yo^Yo) X, 2

)L;Z + .. K [E+al(t,74)K(E-Q)-1^ ^Ixo -xoI + a6 y0,y0dt f20

, < 11, 2" , „) + 2X2 -X2 M - as I, , Yo,yo 2.2

2

+ K E +9 44 K(E -Q)-' 2

„ „ „ [I x0- xoI+ab (X,2. ,Yo,Yo )J,

(20.31)

It is obvious that, for arbitrary pairs (20.11), independently of the relationship between A,2 and 22 , inequalities (20.29) and (20.31) indeed yield estimate (20.25), which completes the proof of Theorem 20.5. Theorem 20.6. Suppose that all conditions of Theorem 20.1 are satisfied for the boundary-value problem (20.1)-(20.3) with two parameters and a nonfixed right boundary point. Then, in order that a certain subdomain of the domain of definition of the given problem A" = G"x [Ai, Ai ] x [A2, A2 ] c G x [Ai, Ai ] x [A2, T^ (20.32)

The Case of Several Parameters in Boundary Conditions

Section 20

205

contain the pair (yo,X*) that determines a solution x = x*(f) of the boundary-value problem (20. l)-(20.3) with X=X* = (X\,X*2) that passes through the point x'(0) = (x0l,x02,yt),

X

= X* = <*;,Ar2).

(20.33)

it is necessary that the following inequality hold for all numbers m and an arbitrary pair (y0, X) from domain (20.32):

-i-fC-'d-CMrU+ £)*,] -±\f(t,xm(y0,X))dt K2

K2 o

<

sup \a9(X,X,y0,y0)+

K E + ^f-K(E-Q)~l

x [l^-xol + a6(X,X,y0,y0)]\ + e(A(y0,X),Am(y0,X)), (20.34) wAere *b = (*oi.*02.?b).

e(A(y0,^,Am(>oJ)) = ( f f f ^ ^ J ^ ^ o A ) -

and a6(X,X,y0,y0), a9(X,X,y0,y0), formulas (20.13), (20.26), and (20.8).

and W2(y0,X) are calculated according to

Proof. By definition, in view of the Lipschitz condition and relation (20.8), it follows from (20.24) that \A(y0,X)-AJy0,X)\

< - M K\x*(t,y0,X)-xm(t,y0,X)\dt K

2

<

^-KW2(y0,X)jai(t,X2)dt K

0

2

= (^JAT^OtoA) = (f)(^K"jW2(y0,X).

o

(20.35)

Taking (20.35) into account, we prove this theorem by analogy with Theorem 19.3.

206

Numerical-Analytic Method for Problems with Parameters

Chapter 3

By virtue of Theorem 20.6, we can formulate an algorithm for the approximate calculation of a pair (yo, ),') of the form (20.33) with the use of which the problem of construction of a solution of the boundary-value problem z = f(t,x),

) Ax(0 )+Cx(? ) = d,

x1(0) = x01 ,

(20.36)

x2(0) = x02

reduces to the problem of construction of a solution of the Cauchy problem x = f(t,x),

x(0) = (x01, x02, Yo). (20.37)

Assume that sets Do and A of the form (20.5) and (20.4) are represented as a union of finitely many subsets, i.e., N

Do =

UDi,

N,

A = [A', Ai ] x [A2, Az ] = U 1j.

i=1

(20.38)

j=1

In subsets (20.38), we choose arbitrary points

xp = (x01,x02,YO)E Di,

Xi = (xj1,xj 2)E

Ii

.

(20.39)

Then, for an approximation of a certain order xm (t, yo, A.j) calculated by the recurrence relation (20.6), we find the following value of an approximate determining function of the form (20.10):

Am(yo"%j) = -j2 [C -1d -(^,J1C-1A+

E)xo]

IIj2

-

^,2

f f(t,xm(t,yo,?j ))dt.

(20.40)

0

Denote by M; the union of subsets Di and Ij appearing in (20 .38) for which inequality (20.34) is satisfied, i.e.,

Am(yo,'j) <

ag (^^^ ^^ yon Yo)

sup (YO.%)aAO

2

E

+

I

+ qA;,yo,Yo)]

9Y3

+ £(A(o, ^j), Am(yo ,

Xj)),

(20.41)

Section 20

The Case of Several Parameters in Boundary Conditions 207

where Am(yo, ?) is calculated by (20.40), (yo, ij) are arbitrary pairs from (20.39), 73 =min (X , X), y4 = max (Xj , k,), Q = 4 K, Am = G` X II, and G' is the set of (n - 2)-dimensional vectors y0 = (x03, ... , xOn) for which x0 = (xo1, x02, yo) a Di for M. the numbers i that define the set M`m According to Theorem 20.6, the set A; is the domain that contains the pair (yo, r) from (20.33) for which the problem of calculation of a solution of the boundary-value problem (20.36) is equivalent to the problem of calculation of a solution of the Cauchy problem (20.37). It is clear that, as i, j, m-aoo, A;; defines the set G* X I* that determines the exact values of the pairs (yo, A!). If we take arbitrary y0 E G' and ?e Ii as approximate values of yo and r, then the following estimates are obviously true: xo - xoI = sup Rio - xoI,

xo=(xo 1,x02, Y0 ),

I^-^*I = sup bEN

yo r= G'

I i -?I. (20.42)

Note that the subsets Di and Ii from (20.38) for arbitrary points (20.39) of which inequality (20.41) is not satisfied cannot contain the pair (yo, r). As an approximate solution of the boundary-value problem (20.1)-(20.3), we take a function xm(t, y0, Xj) of the form (20.6) calculated for arbitrary points y0 E G' and i e P. The error of this solution is estimated as follows: Theorem 20.7. Suppose that, for the boundary-value problem (20.1)-(20.3), all conditions of Theorem 20.1 are satisfied, the pair (yo, )*) is a solution of the exact determining equation r2

A(yo, *) _ 2 1C-td -(rC-IA+E)xo] -

2 J f (t, x`(t, yon * ))dt = 0,

and (y0, A.) is an arbitrary pair from the set A. Then, as an approximate solution of the problem considered, one can take the function %2

xm(t,

xo + J [f (tXm_i(t OX)) - 1 J.f (s, xm-1(s, yo' ^))ds dt o i2 2 + [C-1d - (i,C-1A+ E).i(,

xo(t, yo , A.) = (xo1, x02, yo),

208

Numerical-Analytic Methodfor Problems with Parameters

Chapter 3

with the error

I x*(t, Yo, *)-xo(t, Yo' X)I <- sup

YO EG',AEN

+

1

E (x *(t,

[

Y0, X), xm (t, Y0+

E+&I(t , y4)Kii Q

'][IXO-

iol+a6O',', YO,Y0),

1

>

(20.43)

where E(x* (t, YO, A,), xm(t, yo, X)) is calculated according to (20. 8) and (20.5), a6(A,, yo, Y0) is calculated according to (20.13), y4 = max(,,, ^), Q = Y4 K, and x * (t, XE fi 7t

yo, A.) is the limit function of sequence (20.6). As in the case of Theorem 19.4, the proof of Theorem 20.7 is based on the use of inequality ( 17.28). Estimating the first term of it according to (20 .8) and the second term according to (20 .23) and taking (20.42) into account, we arrive at estimate (20.43). It is natural to choose the exact solution

(yo., A 'm) of the approximate determining

equation (20.10) as (yo , A,). In this case , we have x* (t, Y0, A,*) - Xm(t, YOm+ )1.)

E(x*(t, Yo, V), xm(t, Yo, X*)) _ 1

l

+ E+ ai(t, 7a)Kia i][Ixo XOm +a6(A,*,A,m,YoYom)] J i=0 sup Yo

E

G', ;L E P

I

E ( x *(t ,Yo, A.), xm(t,Yo,

A.))

1

+ [ E+&,Q, y4)K j Q' i =o

] [I xo - xo

m

l +a6 (A.+A'm, Yo, YOm)J

where 74 = max (A.*, A.m), T4 = max (A,, .m), 71.E P

Section 21

209

Linear Dependence of Boundary Conditions on Two Parameters

a1(t, y4) and &1(t, y4) are functions of the form (18.16), and

Q = 'Y4 K, l4K. it It

21. Linear Dependence of Boundary Conditions on Two Parameters Note that a specific form of the dependence of boundary conditions on controlling parameters for the numerical-analytic method under consideration essentially determines the form of successive approximations xm (t, yo, A.) on the basis of which we construct approximate solutions and investigate the solvability of boundary-value problems with parameters. The parameters may enter given boundary conditions linearly (as, e.g., in problem (16.1), (16.2)) and the successive approximations (16.8) nonlinearly. Consider the following two-point boundary-value problem with two parameters that enter the boundary conditions linearly: x = f(t,x),

(21.1)

x,fE R",

A.1Ax(0 )+7.2Cx( T) = d, detC# 0 ,

d#0, (21.2)

x1(0) = x01, x2(0) = x02•

(21.3)

We pose the problem of finding a solution x = x* (t) of equation (21.1) continuously differentiable on the segment [ 0, T] and the values of the parameters A.1 = Ai and (21.2) and (21.3). A.2 = A:2 for which the required solution satisfies conditions Under conditions typical of the numerical- analytic method under consideration, we assume that the function f (t, x) is continuous in domain (16.4) and satisfies conditions of the form (2.4). Let D0 denote the set of points x0 = (x01, x02, y0), y0 = ( x02, ••• x0r, ) contained in the domain D for any values of A. from a certain domain A, (.1,.2)E [A1', A"] X [A',A"] = A, A2' >0, (21.4) together with its

a-neighbourhood, where R = 2111 + P1(x0, X),

R t (x0, 7^ ) =

I-L[2 C-ld -

(21.5) (X,C-'A + '%2E) XOIII

Numerical-Analytic Method for Problems with Parameters

210

Chapter 3

and let G be the set of ( n - 2)-dimensional vectors yo such that xo = (x01, x02, y0) e D0. We also assume that inequalities of the form (18.5) are true for the set D5 defined by vector (21.5) and the matrix K appearing in (2.4). We choose a sequence of functions x, (t, yo , X) satisfying the boundary conditions (21.2) for arbitrary values of X = (X 1, X 2) from domain (21.4). For this purpose, we consider the sequence of functions

xm

t (t, Yo, ^) = x 0 + f f(t, xm -1(t, yo, X)) dt + at,

0

(21.6) m = 1, 2, ... , xo (t, Y0, ?) = x0, where the vector parameter a = (a I,... , a„) is chosen so that all functions (21.6) satisfy the boundary conditions (21.2), (21.3) for arbitrary x0 and Xe A. Substituting (21.6) in (21.2), we obtain the following system of algebraic equations for the calculation of a: rr T ^.lAxo + ^,2C xo + f f(t, xm -1(t, yo, % )) dt + aT = d,

L

0

whence a = -ZT[C-'d -(O1C-'A+X 2E)xo] - T f At' xm-1(t, Yo, X)) dt.

(21.7)

Thus, relations (21.6) and (21.7) enable one to construct the sequence of functions

X.

(t, Y0, ^) = X() +

f [f(t,

X xm -1(t ,

f

Yo, ))dt - T f (s> xm-1(S, Yo, A.))ds1 dt

+ _,2T1C-ld-(XIC-lA+), 2E)xo1 ,

m = 1, 2, ... , (21.8)

x0(t, Y0, ?.) = xo,

which satisfy the given boundary conditions for arbitrary y0 and Xe A. For the boundary-value problem (21.2)-(21.3), we can also formulate analogs of all statements of the previous section concerning problem (20.1)-(20.3).

Section 21

211

Linear Dependence of Boundary Conditions on Two Parameters

Theorem 21.1. Suppose that, for the boundary -value problem (21.1)-(21.3) with two parameters entering linearly, the values of the vectors M, K, and d, the square matrices A and C, the scalar T, and the domain of definition (16.4), (21.4) are such that this problem satisfies conditions (2.4), (18.5), and (21.5). Then a sequence of functions xm (t, y0, A,) of the form (21.8) satisfies the boundary conditions (21.2) and (21.3) for all m = 0, 1, 2, ..., y0 a G, and A, C= A. As m 00, it converges uniformly with respect to the domain (t,yO,A.)E [0,T]xGxA (21.9) to the limit function x*(t, y0, A,). The function x* (t, y0, X) passes through the point x* (0, y0, %) = x0 = (x01, x02, y0) for t = 0, satisfies the boundary conditions (21.2) and (21.3), and is a solution of the integral equation

- 1-Jf(s, x( s)) ds x(t) = xo + J{f(tx(t)) To 0 + -ZT[C-'d-(A,IC-'A+A, 2E)xo]}dt,

i.e., x *(t, y0, A.) is a solution of a "perturbed" boundary- value problem of the form x = f(t,x)+ A(yO,A ), x1(0) = x01,

.1Ax(0 )+ A.2Cx(T) = d, x2(0) = x02,

where T

A(YO,X) =

--L[CX2T

1d-(.1C-1A+A. 2E)x0] - TJf(t ,x(t))dt.

Moreover, in domain (21.9), the error of the m th approximation satisfies the inequality

Ix*(t,yo,A,) - xm(t,Y0,A)I

<-

a1(t)W3(yO,A.)

= € (x*(t,y0, A. )- xm(t,y0,A )),

where W3(Y0, A.) = Qm(E-Q )- IM+KQm- '(E-Q)-1 131(xo, A.),

( 21.10)

212 Numerical-Analytic Method for Problems with Parameters

Chapter 3

Q = T K, a1( t) are functions of the form (2.10), and 131(x0, ?,) is vector (21.5).

Theorem 21.2. If all conditions of Theorem 21.1 are satisfied, then, for an arbitrary pair (yo, X), yo E G, X = (a,1, a.2) a A, there exists the unique value of the additional controlling parameter T

n=

-L[C-1d- (?1C- 1A+%2E)xo] X2T

T

f .f(t,x*(t,yo, ))dt

for which a solution x =x(t) of the Cauchy problem

x=

f(t,x) +µ,

x(0) =

(xol,xo2,y0)

for the "perturbed " equation coincides with a solution of the boundary-value problem x = f(t,x)+µ, X1Ax(0)+? 2Cx(T) = d, x1(0) = x01,

x2(0) = x02.

Moreover, in this case, x(t) = x*(t, yo, X), where x*(t, yo, X) is the limit of the sequence of functions (21.8). Theorem 21.3. Suppose that the conditions of Theorem 21.1 are satisfied for the boundary-value problem (21.1)-(21.3). Then, in order that a solution x = x* (t) of the Cauchy problem x = f(t,x),

x(0) = (xo1,xo2,yo)

( 21.11)

be simultaneously a solution of problem (21.1)-(21.3), it is necessary and sufficient that the initial value yo in (21.11) and the values of the parameters a'1 and ) 2 in the boundary conditions, i.e., the pair (yo, A), be a solution of the determining equation

T e(yo,X) = ^ZT[C-1d-( A.1C 1A+%2E)xoI - TJf(t,x* (t, Yo, ?))dt = 0, (21.12) where x*(t, yo, ?) is the limit function of sequence (21.8). In this case, the equality x*(t) = x*(t, yo, ?,) holds for the indicated pair (yo, ).) and if the function of the m th approximation xm (t, yo, ?) is taken as an approximate solution, then its error is estimated by inequality (21.10).

Section 21

213

Linear Dependence of Boundary Conditions on Two Parameters

Theorem 21.4. Suppose that, for the boundary-value problem (21.1)-(21.3), all conditions of Theorem 21.1 are satisfied and there exists a convex closed domain A' = G' x [ Ai, ] x [

A2,

AZ ] c G X A (21.13)

such that, for a certain m >- 1, the approximate determining equation T Am(Y0,X ) _ ^ZT[C-'d-(.1C-'A+. 2E)xo] - Tj.f(t,xm (t,yo, A.))dt

(21.14)

has a unique solution (yo, A') = (Yom, X. I A. m = ()Lm 1, A. m 2) in domain (21.13). Also assume that the index of this solution of equation (21.14) is not equal to zero and the following inequality holds on the boundary S 1 of the domain A': 2 inf IAm(yo,A.)I > (!) QW3(y0,A.),

(YO,2,)es' 3

where Am(yo, A,) is calculated according to (21.14), W3(yo, A,) is the vector appearing in inequality (21.10), and Q = T K. tc Then the boundary-value problem (21.1)-(21. 3) has a solution (x*(t), A.*) such that, for t = 0, the initial value of this solution x * (0) = (x01, x02, yo) is determined by an ( n - 2) -dimensional vector yo lying in the domain G' and the parameters Ai and A2 in (21.2) are such that A,; E and A! 2,E [ A2, AZ ^. Lemma 21 .1. If the conditions of Theorem 21.1 are satisfied, then, for arbitrary pairs (yo, A.'), (yo,A,")E Gx [A',A'1] x[A',A2(21.15) the following inequality holds for the difference between the limit functions x*(t, yp, A,') and x*(t, yo, A.") of sequences of the form (21.8):

I x*(t, yo, 7.') - x*(t, yo 7.") I <- [E+ a1(t)K(E- Q)-1][1x' -xoI+b(A.', A.", yo, yo (21.16) where

b(A.' A." yo, yo)

= I^

,z[( 7^;- ' C-1d+C-1A(^Z^ix4 '-A.zAixo)+4-xo^ 2)

214 Numerical-Analytic Method for Problems with Parameters

Chapter 3

Proof. It follows directly from (21.8) that

xt(t,y',?')-xt(t,Yo )L")

= xo - xo + f

[(f(t, xo) - fit, xoll - T J (f(s, xo) - f(s, xo))ds

[

(

)l- [1

+ c- 1+E xo d_ c- ' A 2 2 2

]

dt

r i

1

dc- ' A + E ) xo

I

2

The last equality yields

I xl (t, A, A.') -

xt (t, Yo, )"" ) t

T

Ji

I xo - x4'I + K C1- t f I xo - x '1dt + xp Tr /0

- x'1dt

11

]

+ b(.', A", Y', yo )

_ [E+at(t)K]Ix6-xI+b(.' A."' Y6,Yo)• By analogy, we get

X2 (t, Yo,

x2(t, Yo, A.")

t Ix' -x4I + K r1- t f l xt(t,Y',),')- xt(t,Yo,X

\ o

")Idt

T + T ft 1 X'(t'

Yo,A.')-x,(t, Y, A")Idt + b(A, A ,y0,y )

<- [E+at (t)K + a2(t)K2 I 1 -xoI + [ E+at(t)K]b(),',A.", yo, Yo )• By induction, one can easily obtain

I xm(t, y0, A.')-xm(t, YOX")

r <_ l xo -xoI + K 1 T) J l xm-1(t, y , A.') - xm-t (t, yo", A") I dr 0

Section 21

215

Linear Dependence of Boundary Conditions on Two Parameters T

+ T J I xm -1 ( r, y ' , > k')- xm - 1 (t + y0", r')I dt + b(%', ?," , Yo, Yo ) t

C

M ai( t)K` lxo-xol+

J

i =o

I

ai(r)K` +b(X',7^",Yo, Yo),

J

LLi=o

(21.17)

where a ; ( t) are functions of the form (2.9). Passing to the limit as m-+oo in (21.17) and taking (2.10) into account, we get

I x'(t, Yo, A,') - x*(t, Yoe A") I [ + 1

K

Q tK ` x - x f + E +& 1O 1=o =o

EQ ]I

L

b( A', A" ' Y) ,Y0

<- [E+al(t)K(E-Q)-I][I xo-xo"l +b(A',?" Yo, Yo )], i.e., inequality (21.16) is true. Theorem 21.5. If the conditions of Lemma 21.1 are satisfied, then a determining function A(y0, 2,) of the form (21.12) is continuous in the domain G x A and, for arbitrary pairs (21.15), it satisfies the inequality |AO>6,X')

A(yo, ),") I < T b (,', ,", Yo, Yo )

<- K E + 9 K E - 1 x' x" + b ^

,', a,,, ' " Yo A 1, (21.18)

Proof. According to Theorem 21.1, the limit of the sequence of functions (21.8) is a continuous function . Therefore , for (yo, A ) E GxA , the function A(yo, A ) is continuous and bounded, namely,

IA(Yo,).)I

I

d- (AiC- 'A+AZE)xx + M. = 1IC-' T Xa

According to the definition of A(yo , A), in view of Lemma 21.1 and by analogy with Theorem 17 . 2, for arbitrary pairs (21.15) we get

216

Numerical-Analytic Method for Problems with Parameters

Chapter 3

A(Y6, X') - A(YY' A.") I T Tb(X',A.", yo,Yo) + TKf I x*(t,A,A.')-x*(t,Yo, ;,")Idt 0

T .,] b(A.', X" -5 Tb(X,,?",Yo,YY )+ K+T K2( E-Q)_h f at(t) dt [L xo -x0 , Yo' ,yo ] I+

0

< Tb(X',X",Yo , yo) + KI E+ 9 K(E-Q)][I xo-xoI + b ( X',A,",yo,yo], i.e., inequality (21.18) is true. Theorem 21.6. Suppose that the conditions of Theorem 21.1 are satisfied. Then, in order that a certain subdomain contained in the domain of definition of the boundary-value problem (21.1)-(21.3) A" = G"x [Ai,Ai']x [A2', A2" ] c Gx [ AI,A]"]x[A2,A2'] (21.19) contain a pair (yo*, A.*) that determines the solution x = x* (t) of this problem with the initial value x* (0) = (xo1, x02, yo) for A. = .* = (MI, r2), it is necessary that, for all m and an arbitrary pair (Yo, ?) from domain (21.19), the following inequality be satisfied: T _ ^ZT[C-td-( ^1C to+_ ^ZE)xo] - T f f( t,xm(t,Yo, ^ ))dtl

0

5 sup { 1 b (A., A., Yo, Yo) + K E + ET K(E - Q)-11 (yo, X) E A'^ T 9 1

L

1

x [ 4-41 + b(X, X, Yo, Yo }} +

e (A(Yo, 1), Am(Yo, ^)),

where

)2 xo = (X 01, X021 YO ),

E(A(Yo' i), AM(Yo. ^)) = ( l3

C 1 K) W3(Yo,

and the vectors W3(Yo , 7.) and b (., A,, Yo , Yo) were used in (21.10) and (21.16).

Section 21

Linear Dependence of Boundary Conditions on Two Parameters 217

Consider an example of the application of the given numerical -analytic scheme of investigation and calculation of solutions of boundary -value problems with controlling parameters in boundary conditions. Consider the following two-point boundary -value problem with one parameter in boundary conditions on the segment t e [ 0, 1 ] : 11 =x2,

X2 = 19

(21.20)

x(0)-7x(1) = d,

(21.21)

x1(0) = x01 = 0.1, d=(0,-0.1). (21.22) It is obvious that this boundary-value problem is a particular case of (16.1)-(16.3) for

f(t,x) = (f1( t,x),,f2 ( t,x))

_(x2 ' 10 x1 4x2+100)$ (21.23)

A = E, C = -E,

d= (0,-0.1).

Assume that the right-hand side of (21.23) is defined and continuous in the domain

(t,x)E [0, 1]x D, D = {x=(xl,x2) ER2; Ix1I:5 1, Ix21-1 . (21.24) 2 2 In the closed bounded domain (21.24), the function f (t, x) is bounded by a vector M and satisfies the Lipschitz condition with respect to the variable x with matrix K, where

M=

0.5 0 1 and K= 1 1 . 0.31] 10 4

It is easy to see that, for problem (21.20), the vector R in (16.5) is defined as follows: R(1>

R

0.16

1(x0,

+ 81(x0, ?) =

R (2 )

k) = 1 I ... I + I -1 Ix0 I I. (21.25)

Numerical-Analytic Method for Problems with Parameters

218

Chapter 3

Therefore, the set of points Dp contained in the domain D together with their (3-neighbourhoods for all X from the domain 1X = [ 0.4, 0.9 ] has the form

Dp

=

{xER2;

Ix1152 -R(1 ), Ix2I52- 13`2' },

(21.26)

The direct verification shows that there exist points x0 = (x01, x02 ) = ( 0.1, x02) for which the values calculated according to (21 .25) satisfy the inequalities (3(1) < 1/ 2 and p(2) < 1 /2 . Consequently , a set Dp of the form (21.26) is not empty . Calculating the maximum eigenvalue .(Q) of the matrix Q = 1 K, we establish that it is less than 1. 71 Thus, we can apply the numerical-analytic method described in Sections 16 and 17 to the two-point boundary-value problem with a parameter in boundary conditions considered in domain (21.24). For problem (21.20)-(21.22), the successive approximations ( 16.8) take the form X. 1(t, y0, A.)

= 0.1 + j Cf (t, xm-1(t, yo , A.)) - f f (s, xm-1(s, yo, ),)) ds { dt + t ( -110.1

(21.27) xm2(t,YO, )')

x02 + j [h (t, xm-1(t,Yo , 7.))- j f2 (s, xm- 1(s, yo, A)) dsl dt

+t

L

1+1

-1 x02

where m = 1,2,..., x0 (t,y0,A. ) = (x01, x02 )= ( 0.1,x02), xm(t,y0,A.) = (xm1(t,y0, )L ),xm2(t, y0,)L)),

and f, (t, x) and f2(t, x) are defined by (21.23). According to Theorem 16.1, the successive approximations (21.27) converge uniformly to the limit function x*(t, y0, A,) as m -* co. For m = 1, by using (21.27), we obtain the first approximation

Section 21

219

Linear Dependence of Boundary Conditions on Two Parameters x11(t,YO,A.) = 0.1+c11(A.,x02)t,

(21.28) x12(t,Yo,X) = xo2+dii(X,xo2)t, where c11(?,x02) = 0.1( 1 -1),

[+(_1)xo2] , Yo = x02• dll(',xo2) 2.-'-+

In view of (21.28 ), relation (21.27) with m = 2 gives the second approximation x21(t, Y0, a,) = 0.1 + c21(?, x02) t + c22 ( %, x02)t2, (21.29) x22(t,YO,?) = x02+d21(?,x02)t+d22(?,xo2)t2+d23(?,x02)t3,

where

c 21( A.,x o2) = x 02 - d 11( ?1, ,x 02) +c 1l( X,x 02 ),

cz2(a+xo2)

=

dll(? +x02)

2

d2 ^,, x d21(a', xo2) = d11(X, xo2) + 11(4 02) + I X02d11(a', x02)-0.1c11(XI x02)+

2

d22(A', xo2) = 21 c11(? ,xo2) - I xo2d11(),, x02),

d1(?, xo2) d23(A' ,x02) = 12

Let us calculate the solution of the approximate determining equation (17.2) for m = 0. This equation has the form Aol(Yo,X) = (X-1)0. 1-x02 = 0,

A 02(Y0,^ ) +(X-l )x02-5+4x02 = 0. Solving the last system of equations, we get

220 Numerical-Analytic Method for Problems with Parameters

Chapter 3

0.0975 X2 +0.105,-0.1025 = 0, whence X(0)

= 0.62,

x02) = 0.06.

(21.30)

We can take values (21.30) as an approximate initial value x02 of the solution of the boundary-value problem (21.20), (21 .21) and an approximate value of the parameter A. . Note that the exact solution of this boundary-value problem is

xi(t) = i t , ).* = 1/2 xi (t) = 1 + 1 t2, 10 10 5

(21.31)

and, consequently , x*2(0) = 0. Substituting the obtained values (21.30) in the first and second approximations (21.28) and (21.29), we get x1 t(t, x(62),

%(0))

- 0.1 + 0.6 t,

x21(t,x00,A,(O))

2)

x1 2 (t, x1 , X(°)) -• 0.06 + 0.2 t, - 0.1 - 0.07 t + 0. 1 t 2,

x22(t, xo2i,X,iOi)= 0.06+0 .2t+10-5t2+3 . 10-3t3. Comparing these expressions with (21.31 ), we conclude that the values obtained with the use of the zero approximation fairly well characterize the required values that enter into a solution of the boundary -value problem (21.20)-(21.22) both from the quantitative and qualitative point of view.

4. COLLOCRTIOII METHOD FOR BOUIIDRRY-VALUE PROBLEMS WITH IMPULSES This chapter is devoted to the investigation of some classes of boundary-value problems for ordinary differential equations with impulse influence. We consider the problem of existence and construction of Green functions for linear homogeneous systems of impulsive differential equations. For the collocation method, a rather popular and efficient approximate method, we justify its applicability to the construction of solutions of twopoint and multipoint boundary-value problems with finitely many impulses. The collocation method was first proposed by Kantorovich [Kan] in 1934 for the solution of linear boundary-value problems; three years later was proposed again for the same purposes by Frazer, Jones, and Skan (Frazer R. A., Jones W. P., and Skan S. W., Approximation to functions and to the solutions of differential equations, Rep. Mem. Acr. Res. Com., 1799 (1937)). The main idea of the method is to seek a solution of the problem in the form of a linear combination of coordinate functions with unknown coefficients. The latter are determined from the condition that the approximate solution must satisfy the given differential equation at certain fixed points (collocation nodes). This allows one to interpret the collocation method as a version of the general Galerkin method [KVZRS] and, hence, to regard it as a numerical-analytic method. We distinguish between the trigonometric and algebraic collocation methods in which the coordinate functions are trigonometric and algebraic polynomials, respectively. The algebraic collocation method for the solution of nonlinear boundary-value problems for ordinary differential equations was first developed by Vainikko [Vail]. The justification of the trigonometric collocation method for the investigation of periodic solutions of nonlinear systems of differential equations can be found in the book of Samoilenko and Ronto [SaR2]. In the substantiation of the algebraic collocation method for nonlinear impulsive boundary-value problems, the concept of Green function is used. The existence of these functions for homogeneous impulsive systems is established in Section 22. Since the properties of Green functions for impulsive boundary-value problems are of independent interest, in Section 23 we describe the application of these functions to inhomogeneous linear boundary-value problems with impulses. In Section 24, assuming the existence of the exact solution of a two-point nonlinear impulsive boundary-value problem, we prove the existence of approximate solutions determined by the algebraic collocation method. We establish the convergence of the ap221

222 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

proximate solutions to the exact solution and give an error estimate. The method of trigonometric collocation for finding periodic solutions of nonlinear impulsive systems is justified in Section 25. In Section 26, we give illustrative examples of application of the theoretical results presented. The existence of the Green function for a homogeneous linear three-point boundaryvalue problem with single impulse influence is proved in Section 27. In Section 28, we construct the Green function for a two-point m-impulse boundary-value problem and, in Section 29, we show how to use it for the solution of inhomogeneous linear two-point mimpulse problems. For an m-impulse (m + 3)-point homogeneous boundary-value problem, the Green function is constructed in Section 30. In Section 31, we justify the trigonometric collocation method for the construction of periodic solutions of nonlinear differential equations with piecewise-continuous righthand sides in the case of finitely many impulses. Finally, in Section 32, we give several examples illustrating the theoretical results obtained.

22. Green Function of a Homogeneous Two-Point Boundary-Value Problem

Let C,I[ a, b] denote the space of vector functions x (t) piecewise-continuous on the segment [a, b I, x(t) _

1x1(t),

tE[a,ti1],

l x2 (t), t e [tit, b],

with discontinuity of the first kind at the point t ='C 1 E ( a, b): x('r1+0)-x(ti1-0) = x2(tit)-xl(i1), where x1(t)=(xtt(t),...,x,,(t))E C[a,ti1], x2(t)=(x21(t),...,x2,(t)) E C['c1,b], and C [a, ti i ] and C [ti 1, b ] are the spaces of continuous vector functions on the segments [a, ti 1 ] and ['t1, b ] , respectively, equipped with the norm

IIxIIC[a,T,] = IIXIIC[ti,,b] =

max max i=1,n tE[a,ti]

I xi (t) I ,

max max I x2i(t)I. i=1,n tE[t1,b]

Section 22

223

Green Function of a Homogeneous Two-Point Problem

The space CT i [ a, b ] is the direct sum of the spaces C [ a, T 1 ] and C [T 1, b ] : CC,[a,b] = {x(t): x(t)=x1( t)S('C1-t )+x2(t)S( t-'C1), tE [a, b)},

where I,

t>-0,

{ 0,

t < 0.

( x1(t)E C[a,T1], X2(t)E C[ti1,b],

S(t)E

The norm in CT [a, b] is defined as follows: = max {11 X1

IIXIICTI[a,b]

11C[a, 1,11 II X2 11C[T,,b1}.

In what follows , a function x(t)E C. [a, b] will also be regarded as an element of with the space 2[a, b] of vector functions square summable on the segment [ a, b ] weight p ( t), where b

fp1(t) dt < Co. a

The norm II X II I[a, b] i n 4[a, b] is defined as follows:

11X114 [a,bl = 11 X1 II4,[a,T,] + 11x2114, [Tl,bl, where T

11X111, ,[a,1,1

maxllx1i

1/2

M41 [JPISIXISl2ds]

l tp, [a,T,l

i =1,n

a 1/2

b

I1 x2114, [T,, b]

I

= max X2i II4Z[[,,b] =

M41 (f P2 (s)1 X2i (s)12 dsl i=1,n 'Cl

r -b pi(t)=Pla tit + b a t, TI -a T1-a

PM =

b-a P2(t) = P(b a b --TT1 1

+ b' - Tlt

t E [a,'Ci],

t E [T1, b].

224 Collocation Method for Boundary - Value Problems with Impulses

Chapter 4

Let {''k (t)} -= o , where

!k (t) is a polynomial of degree k, be a system of functions orthogonal with weight p (t) on the segment [a, b ]. Let to, t 1, ... , t m E (a, b) be the roots of T. + 1 ( t) . The change of variables

,(1)

t(2 ) = T1 -bt + a-'tl b T1 -at + b-Tl a, b-a b-a a-b a-b

=

maps the points to , t 1, ... , tm e (a, b) into the following points of the segments [ a, ti 1 ] and [tit, b], respectively:

, tml ) E (a, til ), to(t), t12), ... , tm2) E ('tl, b).

to('),

(22.1)

Let C, , denote the space whose elements are T-periodic functions (xt(t), x x2(t),

tE[a,t1], tE[,tl,b], x1

( 0) = x2(T),

with a finite discontinuity of the form x (,r 1 + 0) - x (,r 1 - 0) = q at the point t = T l E (0, T). For xE C,, we define the norm

IIxl1C, I

Fm^

max{IIx1IICEO , 1,3'11x2IIc[T,,T]

}•

We also use functions x(t) from the space Lit of functions square summable on [0, T] with the norm 1/2

IIxII4^ =

I x1IIL2[o,til] + 11X2llL2[T^.T]

5lxl(s)I2ds

1

+

T

1 1/2

5 1 x2(s)I2 ds el

Let us construct the Green function. Consider the linear homogeneous boundaryvalue problem with impulse influence dx = A(t)x, dt

t*'tl, tE [ a, b], xE R°,

't1e (a, b ), (22.2)

S1x(ti1+0)+S2x(T1-0) = 0,

(22.3)

Blx(a)+B2x (b) = 0,

(22.4)

Section 22

Green Function of a Homogeneous Two-Point Problem 225

where A (t) is an (n x n ) matrix continuous on [a, b] , S 1, S2, B 1, and B2 are constant (n x n) matrices , and, furthermore , det S 1 * 0 and det S2 * 0. Assume that the boundary-value problem (22.2)-(22.4) has only the trivial solution x(t) - (x1(t)=0,

I

x2(t)

=0,

tE[a,rI], tE[C1,b],

in the space C ,, [ a, b ] . Then it can be shown that the matrices S 1, S 2, B 1, and B 2 are such that detD # 0, D = Bl - BicI(b)I-1(til)Si 1S2c' (t1), (22.5) where cb(t) is the fundamental matrix, normed at t = a, I (a) = E, of the homogeneous system (22.6)

dx = A(t)x, tE [ a, b],

dt

without impulses. It is well known that the solution of system (22.6) that passes through the point x = x(a) at t = a is determined by the relation

x(t) = (D(t)(-1(a)x(a) = D(t)x(a),

tE [ a,,ct].

Taking into account the jump condition (22.3) for t =,r 1 and the last relation, we get x('t1+0) = -S1i1S2x('Cl-0) = -S1'S2')(21)x(a). Therefore, the solution of equation (22.6) for tE [ ti 1, T] takes the form

x(t) = -fi(t)(D-1(til)Sl 1S21'(,cl)x(a), tE [ 'cl, b]. Then it follows from the boundary conditions (22.4) that

[B1- B2(D(b)O-1(cl)Si 1S2cb(til)1x(a) = 0. Since the boundary-value problem (22.2)-(22.4) has only the trivial solution, we conclude that relation (22.5) is valid. The following statement is true:

226 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

Lemma 22.1. If the impulsive boundary-value problem (22.2)-(22.4) has only the trivial solution, then the matrices S1, S2, B1, and B2 and the fundamental matrix cb(t) of the homogeneous system (22.6) without impulses normed at t = a are such that relation (22.5) is true. Let us introduce the concept of Green function for a boundary -value problem with impulse influence. Definition. A matrix function G (t, s) of two variables t and s defined for all t,sE [a,b], t, S#'r1, t#S,

Gi(t, s), t E [a, s], (22.7)

G(t, s) = G1(t, s), t E [s, b], where

Gl(t,s) =

G2 (t, s) =

Gll )(t, s),

t E [a, till,

s E [t, til],

G12)( t,s),

tE[a,t11,

s E[t1,b],

G13)(t, S ),

t E [t1, b],

s e [t, b],

GZ')(t,s),

tE[a, ,cil,

sE[a,t],

G22)(t, s),

t E [Tl, b],

s E [a, 'C1 ],

Gi3)(t, s),

t E [til, b],

s E [a, ti11,

(22.8)

(22.9)

is called the Green function of the homogeneous impulsive boundary -value problem (22.2)-(22.4) if it satisfies the following conditions: (i) For every fixed s E [ a, b], s # 'C 1 , G (t, s), as a function of t, satisfies the homogeneous differential equation (22.6) for all t E [ a, b], t # s *,r 1: dG(t, s) = dt - A(t) G(t, s), s e [ a, b],

t # s # ti 1 ;

(22.10)

(ii) for t = a, t = b, and all fixed s E [a, b], s # T 1, the function G (t, s) satisfies the boundary conditions (22.4): B1G(a,s )+ B2G(b, s) = 0,

sn[a,b], s#t1; (22.11)

Section 22

Green Function of a Homogeneous Two-Point Problem 227

(iii) for t = s *ti 1, the function G (t, s) has a discontinuity of the first kind: G(s+O,s) - G(s-O,s) = E, SE [a,b], s*ti1;

(22.12)

(iv) for t = ti 1 and all fixed s e [ a, b 1, s # ti 1, the function G (t, s) satisfies the jump condition (22.3): S1G(ti1+O,s) + S2G(tt1-O,s) = 0, se [a,b], s* t1. (22.13) One can see that the Green function (22.7) for equation (22.2) with single impulse influence at the point t =,r 1 consists of six functions (22.8) and (22.9), each of which is defined in a separate interval of variation of the variables t and s . [This is schematically illustrated in Fig. 1.] It is obvious that if there is no impulse influence , then G (t, s) coincides with the Green function for the homogeneous boundary-value problem (22.6), (22.4). S

b G(3) (t, s) G(t, I2) S) G23) (t, S) ti

1--(2) (t, s)

a a

'tl

0. t

Fig. 1 Theorem 22.1. If the homogeneous boundary-value problem (22.2)-(22.4) has only the trivial solution, then there exists a unique Green function G (t, s) of the impulsive boundary-value problem that satisfies conditions (22.10)-(22.13). Proof. Since function (22.7) satisfies (22.10) on the intervals t E [a, 'r 1 ] , t E ['r 1, b] , sE [ a, c 1 ] , se [ 'C 1, b] , the functions G, (t, s) and G2(t, s) in (22.8) and (22.9) must have the following representation:

228 Collocation Method for Boundary-Value Problems with Impulses

ID(t)C1)(s),

t E [a,

t1],

Chapter 4

SE [wrl],

G, (t, s) _ c(t)C 2)(s), t E [a,'r1 ], s E [t1, b], (D(t)Cj3)(s),

t E [Tl, b], s E [t, b],

(22.14) (D(t)C21)(S), G2 (t, S) _

t E [a, ,rl ],

'D(t)C22)(S), t e [t1, b],

O(t)d3)(S),

SE [a, t], s E [a, 't1],

t E [r1, b], s E [,rl, t],

where C;3)(s), i = 1, 2, j = 1, 2, 3, are certain matrices dependent on s. For the determination of the six matrices C,D)(s), we obtain the following two equations from condition (22.11): B1C;1)(s) + B24D(b)Cz2)(s) = 0, (22.15) A q2) (s) + B2(D(b)C'2) (s) = 0. Condition (22.12) also gives two equations, namely, 41)(5)_

1)(5) _

'D -'(S), (22.16)

C<2 )(S) - 3)( S) _

(D-1(S),

and, finally, condition (22.13 ) yields

Sl(D(Tl)C22)(s)+52(D(til)C2)(s) = 0,

(22.17) S14)(til)1 -3)(S)+52(D( ,rl)C(2)(S) = 0.

Solving system (22 . 15)-(22. 17), we get 1-j1)(S) = C21)( S) - -1(s), C,l3)(S) = C23)(s) --1(S),

A Ci2)(s) + B4(b) C23)(s) = 0,

S2`D('rl ) c(2)(s) + S1(b('rl) 43)(s) = S1 b(il).O_l(s)• AC21)(s)+ B2(D(b)C?)(s) = B1('-1(s),

Section 23

Inhomogeneous Linear Impulsive Boundary-Value Problems 229 SA(rl)C(2)(s)+SlD('Cl)C,22)(S) = 0.

Hence, taking Lemma 22.1 into account, we uniquely determine C 2)(s) D- 1BZC)i(b)cD-1(s), C23)(s) = (D-1(s) + (D-1Si1 Sz b(til ) D-1B24)(b ),O-1(s), Cz1)(s) = D-'B1cI '(s), (22.18)

C 2(2)(s)

= - cp- 1(til)Sl 1Szc ('Cl)D-

'Bl4 -1(s),

Cl(') = D-B1(D_ (s)-(D-1(s), Li3)(s) _ -1(ti1 )S1 1Sz^(til ) D-1B24 (b)c^-1(s).

Substituting (22.18) in (22. 14), we obtain the general form of the Green function for the impulsive boundary -value problem (22.2)-(22. 4) in terms of the fundamental matrix of system (22.6) without impulse influence: (D(t)ID- 1•'1 - E]0_ 1(s), G1 (t, s) _ - (D (t)D-

1B2'(b)D -1(s),

t E [a, il ],

s E [t, till,

t E [a, till, s E [t1, b],

(D(t)(D-1(til)Si lS2(D(til)D-1Bz4'(b)('-1(s), t e [ 'cl, b],

s E [t, b], (22.14)

fi(r) D-1B1cp -1(s), G2 (t,

t E [a, it],

s E [a, t],

s) = - c^(t)c^ -1(t1)Si 1S2^(C1)D-1B1^-1(s), t E [t1, b], SE [a, till,

b(t)[E+0-1(•cl)Sl 1S2cb(,tl)D_1B2cb(b)]cb 1(s),

t e [til, bl,

s e [t1, t].

23. Inhomogeneous Linear Impulsive Boundary - Value Problems

Consider an inhomogeneous system of linear differential equations

dx = A(t)x+g(t), dt

t*Tl , tE [ a, b ], (23.1)

230

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

with the impulse influence (22.3) and homogeneous boundary conditions (22.4); here, 81(t), t E [a, ti1], g (t) =

1 g2(t),

t E

[ t1,

b],

i.e., g(t)E CTJa, b], and A(t) is a matrix continuous on [a, b]. Theorem 23.1. Suppose that the homogeneous impulsive boundary-value problem (22.2)-(22.4) has only the trivial solution. Then, for every piecewise-continuous function g (t) E C1[a, b], there exists a solution, x = x1(t) for t e [a, tit] and x= x2 (t) for t e [r1, b], of the semihomogeneous impulsive boundary-value problem (23.1), (22.3), (22.4). This solution is given by the formula b

x(t) = f G(t, s)g(s)ds

(23.2)

a

and admits the following representation:

x1(t) _ ('(t) JG21)(s)g1(s)ds+ f Ci1i(s)g1(s)ds+ f C12i(s)g2(s)ds , a

t

tE[a,T1],

Tj

(23.3) Tj

x2(t) = 4 (t) J

t

G22>(s)g1(s)ds+

a

b

J

,,t

tE [ 61, b].

G23)(s)g2(s)ds+ J C, i3) (s)g2(s)ds . t

Tj

(23.4) Proof. The direct verification shows that the functions x 1(t) and X2 (t) defined by (23.3) and (23.4) satisfy equation (23.1) and conditions (22.3) and (22.4). The function under the integral sign in (23.2) is continuously differentiable on each of the intervals [a, t], [t, i1], and [tit, b]. Therefore, differentiating (23.3) with respect to t, we obtain

dx1 = j^ aG(t,s)g1(s)ds+G(t,s)I g1(t) dt j at L t-o a

+

" aG(t, s) b aG(t, s)

J

at

g1(s)ds-G(t,s)js= +ogt (t) + J

t

t

Ti

at

g2(s)ds

Section 23

Inhomogeneous Linear Impulsive Boundary - Value Problems

231

= b ac(t ' s) g(s) d s + gl (t). At

a

Consequently, b aG(t, s) a

b g(s)ds + gl (t) - A(t)$ G(t, s)g(s)ds at

a

= J [aG(ts)

at

- A(t)G(t, s)]g(s) ds + g,(t).

Since, by virtue of (22. 10), the Green function satisfies the homogeneous equation (22.5) for t e [a, s 1, (s, ti 1 ), ( r 1, b), it follows from the last relation that, for t E [a, ti 11, a function x 1(t) of the form (23.3) satisfies the inhomogeneous equation dx = A(t)x+g1(t), dt

tE [ a,'r1}.

Similarly, one can prove that, for t E ['r 1, b ] , the function x2(t) given by (23 .4) satisfies the inhomogeneous equation dx = A(t)x+g2(t ), dt

tE [ti1, b].

Let us show that functions (23.3) and ( 23.4) also satisfy the boundary condition (22.4) and jump condition (22.3). Indeed, inserting (23.2) into (22.4) and (22.3) and taking (22.11) and (22.12) into account, we get

Bt5 G(a, s)g(s)ds + BZ J G(b, s)g(s)ds a a b

= J [B1G(a, s)+ BZG(b, s)]g(s)ds = 0, a

Sl J G(ti1 +0, s)g(s)ds + SZJ G(T1-0, s)g(s)ds a a b

=

J [S

SG('rl

a

The theorem is proved.

+ 0, s)+SZG('r1-0, s)]g(s)ds = 0-

232 Collocation Method for Boundary - Value Problems with Impulses

Chapter 4

Note that, according to (23 .3) and (23.4), the solutions of the semihomogeneous boundary-value problem (23.1), (22.3), (22.4) on the segments [ a, ti 1 ] and ['r 1, b ] can be found independently of one another. The last assertion also applies to the case of the inhomogeneous impulsive boundaryvalue problem, i .e., the case where the inhomogeneous differential equation (23.1) is considered with the inhomogeneous boundary condition (23.5)

Blx(a)+B2x(b) = d and the inhomogeneous jump condition

(23.6)

S1x('r1+0) + S2x(T1-0) = q.

Theorem 23.2. Suppose that the homogeneous impulsive boundary-value problem (22.2)-(22.4) has only the trivial solution. Then, for any piecewise-continuous function g (t) a C, I [ a, b ] , the solution x (t) of the inhomogeneous impulsive boundary-value problem (23.1), (23.5), (23.6) is given by the formula b

(23.7)

x(t) = H(t) + f G(t, s)g(s)ds, a where the integrand is determined by (23.2) and H(t) _ (H1(t)

for t e [a, ti1],

1 H2(t)

for t e [tit, b],

H1(t) =

c(t)D -1(d

(23.8)

- B2(D(b)('-1(tit)Si q),

H2(t) = (D(t)(D-1('r1)Si 1 [(E +S2cD(tit)D-1B2(D(b)(D-1('t1)St')q-S2 tit)D-1di Proof. According to Theorem 23.1, the integrand in (23 .7) satisfies the differential equation (23.1) and the homogeneous boundary conditions (22.3) and (22.4) for all t e [a, b], t #ti 1 . Let us show that a function H(t) of the form (23.8) satisfies the homogeneous condition (22.6) for t * ti 1. Indeed , substituting H(t) in (22.6), we get

Cd

O(t) - A( t)(D(t)l p dt J = 0,

where P = Pt = D-1(d-B2(D(b'- 1('r1)Si 1q)

,

tE [ a,ti1],

Section 23

Inhomogeneous Linear Impulsive Boundary - Value Problems

233

P = P2 = 4D-'('t1)Si' x [(E+S24)( ,t1)D-'B24 (b)(D-'('t1)Si')q-SS(D(tii)D-'d], to [ ' ti, b], and 4) (t) is the fundamental matrix of the homogeneous system (22.6) such that d(D(t) = A(t) c'(t), dt i.e., H(t ) satisfies (22.6). It now remains to prove that solution (23.7) satisfies the inhomogeneous condition (23.5). Since the integrand in (23 .7) satisfies the homogeneous conditions ( 22.3) and (23.7), it suffices to show that H(t) satisfies the inhomogeneous conditions (23.5) and (23.6). Substituting H(t) in (23.6) and (23.5), we obtain

S1H2('t1 )+ S2H1('r1 ) = Sl(D(ti1A-'(tii)Si' x [(E+52(D(i1)D-'B24 ( b)(D-'(ti1)Sl' )q-S2fi('ti)D-'d]

+ S2'1('t1 )D-'(d - Bz(D( b -'(tit)S,'q] = q + S 24 (,q )D-'B2(D(bA-1 (ti1 )Sj'q - S2I('t1 )D-'d + S2cb(i1 )D-'d-S2(D( ti1)D-'Bzc(b)(D-'(ti1)S-'q = q,

B1D-'(d - B2(D( b)'-'(ti1)Sj'q) + B2-D(b)('-'('t1)Si

x [(E+524)( ti1)D-'Bz(D(b),O-'('t1)Sj' )q-S24)('tt)D-'d]

B1D-'d - B1D-' Bi(D(b)-D-' ('t1)S^'q + BL-V(b)-O -'('t1)Siq + B2(D(b)4)-'(tit ) ST'S2b(ti1)D-'B2cb(b A -'('t1)St'q - B24)(bA _'('t1)Si'S24D (tit)D-'d

234 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

= [Bi - Bacp (b)(D-1(tii)Si 1S2' (t1)JD-d + B2'D(b)(D-1('t1)Si 1 q - (B1 -B2I ( b)(D-(ti1)Si 1S2(D(t1))D-1B2(D ( b)c-1(ti1)Sl 1q

DD-d + B2(b(b)(-1(tt1)Si lq - DD-1B20 ( b)(D-1('r1)Si 1q = d. The theorem is proved.

24. Convergence of the Algebraic Collocation Method for Nonlinear Systems

Extensive investigations of the collocation method proved that it is a general-purpose tool for the approximate solution of fairly broad classes of two -point, multipoint, and periodic linear and nonlinear boundary - value problems for ordinary differential equations and differential equations with delay. For a survey of works devoted to these problems, see, e.g., [SaR4]. It is interesting from the practical and theoretical point of view to study the possibility of extension of the collocation method to the case of boundary -value problems for differential equations with impulses . In [SaP3] and [SPT], it was shown that the solutions of impulsive differential equations possess certain specific properties that distinguish them from equations without impulses. In this section , assuming the existence of an exact solution of the impulsive boundary-value problem, we study the solvability of determining equations that determine an approximate solution according to the algebraic collocation method . We also estimate the rate of convergence of the approximate solution to the exact solution. Consider the following system of differential equations with impulse influence at fixed times [SaP3 ] (for simplicity, we assume that there is a single impulse): dx _ cp(t), dt = f(t,x)+

t* tit, to [ a,b], x,fe R", (24.1)

S1x('r1+0)+S2x(ti1-0 ) = q, (24.2) where T ( t) is a piecewise- continuous function with discontinuity of the first kind at t = t1,

Section 24

Convergence of the Algebraic Collocation Method

235

J (pi(t), [a, til], 1 P2(t), t E [til, b], tE

(p(t) =

and S 1 and S 2 are constant ( n x n) matrices satisfying the conditions det S 1 # det S2 # 0, q = (q 1, q2 ... , q") is a constant vector, the function f (t, x) is defined and continuous for (t, x) E [ a, b ] x D, and D is a closed bounded domain in R". Let us investigate the problem of finding a piecewise-continuously differentiable function

X =

XO(t)

xi ( t),

t E [a, t1],

= I

X2 (t), t E [t1, b],

with finite discontinuity at t = 1 that satisfies the equality dxO 0 = f (r, X (t)) + (p(t) dt for all t E [ a, ti 1) U (ti 1, b ], the jump condition (24.2) for t = ti 1, and the two-point linear boundary condition

(24.3)

B1x(a)+B2x (b) = d,

where B 1 and B 2 are constant ( n x n) matrices and d = (d1, d2, ... , d") is a constant vector. Assume that the impulsive boundary-value problem (24.1)-(24. 3) has a solution

XO(t) =

J Xl(t), t E [a, ti1], 1 x2 (t), t e [,rl, b],

in the space C, I [a, b]. Let us prove that xO ( t) is isolated and the approximate solution obtained by the algebraic collocation method converges to the exact solution . We seek an approximate solution of problem (24.1)-(24 .3) in the form of a vector polynomial of degree m + 1: m+1

*m(t) =

Q01) + Qkl)tk = xlm(t), t E [a, til], k=1

m+1 Q02) + E Qk2)tk - X2m(t), k=1

where Qk) = (Qlk), Q2k .... , Qnk ), j = 1, 2.

t E [til, b],

(24.4)

236

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

We choose the coefficients Qol), Qo) so that expression (24.4) satisfies the boundary conditions (24.3) and jump condition (24.2) for arbitrary Qk ), k = 1, ... , m + 1, j = 1, 2. Then we have

B1 1 Qo1) m+l + I Qk1)a' + B2 IQ(2 ) + Qk2)b" IL k=1 k=1

I

Sl [Q(2)

+

J

= d,

[Q(1 + M+lE Qk')T1 = q. E Qk2'T J + S2k=1 0

1 k=1

Consequently,

I

BIQO ) + BZQ02) = d - l B1akQg ) B2bkQ(k2), k=1 k=1

(24.5) SZQ0I) + S1Q02) = q -

E

E

Ti S2 Qk2) . Ti S1 Q(k2) k=1 k=1

Assume that the equation (24.6)

dx = 0, t^T1, dt

with conditions (22.3) and (22.4) has only the trivial solution. Since, for the last equation, we have (D (t) = E, by virtue of Lemma 22.1 we get det [Bl - B2Si% * 0. This implies that system (24.5) is uniquely solvable: Qo) [BI - B2ST

X

C

152] _1

d - B2S11q +

I (B2S1 1S2T1 -Blak)Qk) + IB2(Tl -bk)Qk2) k=1 k=1 11

Q0 = S1 1q-Si1S%[B1-B2Si 1S2]

X

L

d - B2Sj 1q +

(BZs, 1 S2T1 k=1

-Blak)Qki

) + I B2(Tl - bk)Qg) 1

k=1

Section 24

Convergence of the Algebraic Collocation Method 237 k T Qk2) - SI % Ir Qkl) . k=1 k=1

Therefore, if Qol) and Q(2) are chosen in the form (24.7), then the approximate solution (24.4) satisfies conditions (24.2) and (24.3) for arbitrary coefficients Qlt), Q(m)+l, Q(I2)+•••. Qm+1'

According to the algebraic collocation method, the unknown coefficients Qkl), Qk2), k = 1, m + 1, of the approximate solution (24.4) are determined by the solution of the system of 2 (m + 1) vector algebraic equations dxm(tii dt

i + (24.8) )) = f (ti (j), t) ) )+ (P(t(j))

i = O, 1, ... , m, j = 1, 2, where ti l) and ti(2) are the points from (22.1). Theorem 24.1. Let x = xO(t) = Ixj(t), tE[a,til], x2(t), t E[,rl, b],

be a solution of the boundary-value problem (24.1)-(24.3) that belongs to the space Cr i [ a, b ] . Suppose that the components of the vector function f (t, x) and the Jacobi matrix F(t x) = af(t, x) ax are defined and piecewise continuous for tE [ a, b],

llx-xcll

Ct i [a, b]

<- S, S>0. (24.9)

Assume, in addition, that the linear homogeneous variational equation corresponding to the solution x°(t) of system (24.1)

dx = F(t,xe(t))x, t#tl, dt

238

Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

with conditions (22.3) and (22.4) and the homogeneous impulsive boundary-value problem (24.6), (22.3), (22.4) have only the trivial solutions. Then one can indicate x > 0 such that the solution x = x°(t) of the impulsive boundary-value problem (24.1)-(24. 3) is unique in the ball dx dx° dt - dt

S x (24.10) L;' [a, b]

and, for sufficiently large m ? m °, the algebraic collocation method (24.4), (24.8) determines a unique polynomial approximate solution xm (t) in (24.10). Furthermore, the sequence of approximate solutions xm (t) uniformly converges to the exact solution x°(t) as m - oo, t # T1, the derivatives xm(t) converge to x°(t) in the mean square, and the following error estimates are true:

II xm(t)-x°(t )II C[a b] < C1Em(x°),

(24.11)

I

(24.12)

x'm(t)-

x°(t)II L[a,b] <

where Em(ac°) = max (Em(x°), Em(x2)),

C2 Em(x °),

Em(x^), i = 1 , 2, is the best uniform ap-

proximation of the derivative xo by a polynomial of degree not higher than m, and C1 and C2 are constants independent of m. Proof. By assumption, the Green function G (t, s) of the impulsive boundary-value problem (24.6), (22.3), (22.4) exists. It is obtained from (22.19) with (D (t) = E and D = B1- B2Sj1S2 as follows: D-'B1-E,

tE[a,T1],

G1 (t, s) _ -D-'B2, t E

[a, T l],

SE[t,T1],

s E [Tl, b],

S1'S2D-'B2, t e [Tl, b], S E [t, b],

(24.13) D-'A, G2 (t, s) _ -Si'S2D-' B1, E+S-'S2D-' B2,

tE[a,T1],

sE[a,t],

t E [T1, b], s E [a, T1], t r= [T1, b],

s E [TI, t].

We rearrange the components of G1 (t, s) and G2 (t, s) in (24.13) as follows:

Convergence of the Algebraic Collocation Method

Section 24

D-'B1= G1 1(t,

239

s), t E [a , T1], s E [a, t],

G1(t,s) = D-1B1-E=G12( t,s),

D-'B2 = (j1 3( t, s), -Si'S2D-' Bi = G21(t, s),

tE[a,T1] , sE[t,T1],

t E [a, Tl], s n [T1, b],

t E [Tl, b],

s E [a, T1],

G2(t, s) = E+ Sj'S2D-'B2 = G22(t, s), t e [tit , b], s E [T1, t],

Si'S2D-'B2 = G23( t, s), t E [Tl, b],

s r= [t, b].

By virtue of (23.7) and (23.8), the functions G1(t, s) and G2(t, s) enable one to represent the solution of the inhomogeneous differential equation { vl(t), t E [a, T1 ], (24.14) E CTI [a, b], = v(t) = u(t)

dx

D2(t), t E [Tl, b],

dt

with the inhomogeneous boundary conditions (24.3) and impulse influence (24.2) in explicit form on the intervals t r= [ a, r 1 ] and t e [ T 1, b ], respectively, as follows: b

xl(t) = JG,(t,s)v(s)ds+D-'(d-B2S1'q) a

f

f"

= t Gll(t,s)vl(s)ds + T 612(t,s)vl(s)ds 6 a t b

+ JG13(t,s)u2(s)ds+D-'(d-B2Si-'q), Tj

b

x2(t) =

fO2(t,s)v(s)ds+S- [q-S2D-' (d-BzSi-lq) '

a

f G21(t, s) vl(s)ds + f G22(t, s) v2(s)ds a

TI

b + f G23(t,s )v2(s)ds + Si-'[ q-SZD-'(d-B2Si-'q)]• t

240 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

On the basis of the functions G1(t, s) and 62(t, s), we define the following operators: b G1 : G1v = J Gi(t, s)v(s) ds, tE [ a, till, a

(24.15) b

G2: 62u = JG2(t,s)v(s)ds,

tE [ i1,b],

a

K 1 : K 1v

= f(t,

Hi +Gl v) +(p1(t),

tE [ a,

ti11, (24.16)

K2: K2v = f (t, H2 +G2u)+ p2(t), tE [ tit, b],

where v (t) E C [ a, b ] , and

H1 = D-1(d-B2Sj tq), H2 = Si '[q - S2D-(d - B2Sj lq)].

By using the properties of the functions G1(t, s) and G2(t, s), one can readily prove that operators G1 : 4[a, b] -> C[a, ti 1 ] and G2 : 4[a, b] -* C[T 1, b ] of the form (24.15) are completely continuous . Since the operators Gl and G2 are bounded, we can choose a ball < K1 z Lp[a, b]

(24.17)

with radius K 1 such that the functions x 1= G1v + H 1 and x2 = G2v + H2 satisfy condition (24.9). In ball (24.17), we consider the operators

K1 :

2 [a,-4 b] C[a, ti 1 ] and K 2:

2 [a, b] -* C [ ,r 1, b ] defined by (24.15) and (24. 16). In this case, the continuity of f (t, x) and the complete continuity of G1 and G2 guarantee the complete continuity of the operators K1 and K2. Let P1 and P2 denote completely continuous imbedding operators of C[a, ti 1 ] into 4 [a, t1] and C[ti1, b] into 2 [T1, b], respectively . Let PI,,, and Pa„ be the linear operators that associate every continuous function from C [a, ti 1 ] and C [ ti 1, b ] with its Lagrange interpolation polynomial of degree m constructed for nodes (22.1). Then the boundary-value problem (24.1)-(24.3) is equivalent to the operator equations

Section 24

Convergence of the Algebraic Collocation Method 241 vl =PKv,

U = 1u2 = P2K2U,

tE [a,ti1],

(24.18) t E [tit, b],

and the solutions of the problem under consideration are related to the solutions of the operator equations (24.18) as follows: Uj = x^, x? = H1 +Giv°, t E [ a,'C1], UU = I

u° = jO, x° = H2 +G2v°,

/ e[T,,b].

In (24.8), we set

dX

= (Ulm, tE[a,t1],

"'

= v Sl

dt m

(24.19)

t e [t1, b].

V2.,

We have Xlm =

Hi

+GIUm,

tE [ a,'C1

x2m = H 2 +G2Um,

-T

and equations (24.8) can be rewritten in the form I Pmllim = P1mK1Um,

t E [a, 'L1 ],

Pm Um b]. P2mV2m = P2mK2Vm , t E [t1,

Since v Im and V2m are polynomials of degree m, we conclude that P im Vim = u Im and P2m u2m = u 2m and, therefore, relations (24.8) are equivalent to the equations Vim = PmKillm,

t E [a, til],

U2m = P2mK2Um ,

t e [til, b].

Um

(24.20)

It is known [Gon] that, for arbitrary functions continuous on the corresponding intervals of variation of an independent variable, the Lagrange interpolation polynomials of degree m constructed for nodes (22.1) converge to the approximated function in the mean square with weight p r (t), p 2 ( t). This means that the sequences of the operators P i m and P b„ strongly converge to the operators P 1 and P2: mim P1m = Pi,

nlrm P2m = P2.

242 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

By virtue of the Banach-Steinhaus theorem [Nat], this implies that the norms of the operators indicated are uniformly bounded: II Pea II <_ M"•

II PImII <_ M',

The continuity of the derivative of the function f (t, x) in domain (24.9) implies that the operators K1 and K2 are continuously differentiable in ball (24.17) and, furthermore,

af(t,

H1 +Gtv^ K 1 = ax G1,

K2 =

tE [ tit, b],

v) af(t, H2ax+GG2, tE [ Tl, b]. 2

Since, by assumption , the homogeneous variational equation with the boundary conditions (22.3) and (22.4) has only the trivial solution, the operator equations v1 = PKf(v°) v, t E [a, TI], v =

V2 = P2K2( v°)v,

(24.21)

t E [T1, b],

have only the trivial solution v 1 = v 2 = 0. Therefore, there exist the bounded inverse operators for E - PKi(v°) and E - P2K2(v°), and, as is known [Vai2], the operators E PmKq(v°) and E-P2mK2(v°) are also invertible for sufficiently large m. Furthermore, the norms of these inverse operators are uniformly bounded , namely, 1I (E - P1mK1 (v°))

-111

S C3,

I1

(E - P2mK2 (v°)) -111 <- C4,

m = 1,2,..., and v? and v2 are isolated solutions of equations (24.18) with nonzero index. Regarding (24.18) and (24.20) as equations in the spaces ^, [a, Tl ] and L

[,r1, b],

we conclude that they satisfy all conditions of Theorem 3 in [Vai2], according to which the solution v° of equation (24.18) is unique in ball (24.10) and, for sufficiently large m, equation (24.20) has a unique solution v lm, v2m in this ball. Moreover,

Ilvlm - v0114,ja ,T,l

<_ C3IIv^ -PmP 14

II ,[a,T,l (24.22)

1 v2m 1 L [t^ , b] - v2

c4 I v2 - P2m P2 1v2 11 lpZ [tip, b]' 0

Convergence of the Algebraic Collocation Method 243

Section 24

where the notation P_1vo and PZ- 1v2 means that v° and v2 are regarded as elements of the spaces C[a, 't1] and C[ti1,b], respectively. Let 01m ( t) and 01m (t) be arbitrary vector polynomials of degree not higher than m: 0j,n(t) = (0j 1(t),0j 2(t),...,0jn(t)), j=1,2. Since , for such polynomials, one has

PmP-101m(t) = 0lm (t),

P2mP2 102m(t) = 02,n (t),

relation (24.22) yields

Il u

m-V 011 4[a,bl

<_ C3[

IIUI-

01m11

4. [ a,T^] + I1P mP- 1(Ul- 01m )III^[a,q

+ C4[IIv2-

C3 max [IIvI -01i

02 m

1142

[ TI,b)

+IIP2mPF' (V2- 02m )11 42[T, bl]

II4,[a,T,] +IIPmP

i=1,n

_1(v1 -

01i)II4'[ ,]]

°'T

+ C4mx[IIv2i-02iII42[ T,,b]+IIP2mP21 (U2i-02i )III [TI bl]

T,

1/2

1

max Iv° -Ol;(t)IJ < C3 M41 I jPt(s)ds +M' i=1,n l te[a,T,] a

b

1/2

+ C4 max fP2(s)ds +M " i =1,n

,

max Iu2; - O2i(t)I

te[T,,b]

C3Ci max Em(v? (t)) + C4C2 mxEm(v2i(t)) i=1,n t=1,n

I C2[Em(vO( t))+Em(U2 (t))] <- C2Em(vO(t)),

]

244 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

where 2 C2 = max (C3Ci, C4C2). Passing to the variables x and xm according to formulas (24.14) and (24. 19) in the last inequality , we obtain estimate (24.12). To establish estimate (24.11), we use the relation b

xm(t) -x °(t) =

J

G(t,s)( vm(s)- v °(s))ds

a b

561 (t,s)(vm(s)- v°(s))ds,

^ X'M(t) - X?w

t E[a,'t1},

a (24.23) x2m (t) - X20 (t) b f G2(t, s)( vm(s) - v° (s))ds, t E ['C1, b]. a

We denote by G1ij(t, s) and G2ej(t, s) the components of the matrices G1(t, s) and G2 (t, s) and set

1Icki(t,5)I a P(s) ds 2

Cki = max max i=1,n t

1/2

, k = 1, 2.

(24.24)

By definition, we have I xm(t) - x°(t) I cif [a b]

max max max

L

I

I

ximf (t) - x9(t) I , max max x2mi (t) - x2i (t) I . (24.25) i =1 n tE[a , Tj1 i=1,n te[Ti,b]

J

It follows from (24.23) that

max max I xt„u(t) - x° (t) j

i = 7-tE [a , 'r ]

= max max Y f Gj (t, S)(vmj (S)-v?(s))ds i=1,n te[a,T,I j=1 a

= max max

5O1 ij(t,s)

i=1,n tE[a,T1] j=1a

P(s) ,rp(s)(u mj (s)-

In view of (24.24) and the Cauchy-Buniakowski inequality , relation (24.26) yields

(24.26)

Method of Trigonometric Collocation for Periodic Systems 245

Section 25

max max i

=1,n

t e [a, r

I xlmi (t) - x° (t) 2

max max

;1 IGw(t,)

n

l1/2

ds ( J P(s)Iumj (s)-u9(s)I2ds s a J

i= 1,n t E [a, Tl

Max C" 1 II i=n 1, j =1

1/2 b

Vmj(t)-

v^(t)II tp [a,b]

max C1inC2Em(xc) = C1Em(x°). i=1,n

By analogy, one can show that

max max

Ix2, (t)- z2i(t)I

<_ ClEm(i o).

(24.27)

i =1,nIE[r1,b]

Relations (24.25)- (24.27) now imply the required estimate (24.11). The theorem is proved.

25. Method of Trigonometric Collocation for Periodic Systems

Consider a system of ordinary differential equations of the form dx = f(t, x)+cp(t ), dt

f(t,x) = f(t+T,x), (p ( t) = cp(t+T ),

(25.1)

where f (t, x) is an n-dimensional vector function defined and continuous in domain (t, x) e [ 0, T] x D and periodic in t with period T; here, D is a closed bounded domain in Rn and cp (t) is a piecewise-continuous function with discontinuity of the first kind at t =,c 1 e ( 0, T) that is periodic with the same period T. Let us find a T-periodic solution of system (25.1) satisfying the following jump condition at t =,r 1 :

x(•t1+0)-x ( ti1-0) = q, T1e (0,T). (25.2) We seek an approximate T-periodic solution of system (25.1), (25.2) in the form of a trigonometric polynomial , namely,

246 Collocation Method for Boundary-Value Problems with Impulses M xlm(t)

k

= 4 + Y, (ak coskwt+b sinkwt),

*m(0 =

Chapter 4

t e [0, -r 1], (25.3)

M

x2m(t) = ao + I (ak coskwt +

bk sin kwt), t e [i1 , T],

where w = 2n /T and ak = (aik, ... , a;,k ) and bk = (4k ,..., Q, i = 1 , 2, are unknown coefficients. We choose the coefficients ao and ai so that the approximate solution xm (t) satisfies condition (25.2) and the periodicity condition, i.e., x(0)

=

x(T).

(25.4)

Furthermore, m

ap

=

ap

+

E (ak - ak) - (1- coson,)-' k=2 x {(b k - bk) sin karcl + q - j (ak - ak )(1-cos karcl ) k=1 k=2

a 1l

4 1

al + (1- cos ko t1)-1

«/.

x Y,(bk - b2) sin kcot1 + q - j (ak - ak )(1- cos kohl ) k=1 k=2

Then the approximate solution (25.3) takes the form m

xim(t) = 2 +ai coswt+ xm(t)

=

x2m(t) = 4 +

m

m

I ak cos k wt+ Y,

k=2

bk sin kwt,

t E [a, t1],

k=2

b? Y, (ak cosk(at + cos kwt),

t e [TI, b].

k=1

(25.5) One can verify that the solution xm (t) defined by (25.5) satisfies conditions (25.2) and (25.4). The unknown coefficients ak and bk, i = 1, 2, are determined by the collocation method from the condition that the residual function must vanish at the collocation nodes (in the case considered, as collocation nodes, we take equidistant points from the segments [0,cc1] and [t1,T]):

Section 25

Method of Trigonometric Collocation for Periodic Systems dXm(tk) =

247

k=l ,I,

(25 . 6)

tj = (2j-1)T-tit +til, t i = (2i-l) til , 4m 4m

(25.7)

dt

f(tk, Xm (tk ))+(P(tk ) ,

where i, j = 1, 2, ... , 2m. Let us show that, under certain natural assumptions, the existence of an exact T -periodic solution x =x°(t) of the periodic system (25.1) that satisfies the jump condition (25.2) implies the solvability of the system of 4m determining algebraic equations (25.6) and the convergence of the approximate solutions obtained to the exact solution as m-4 oo. The following statement is true: Theorem 25.1. Let x°(t) E C 1 be a solution of the system of impulsive differential equations (25.1), (25.2), (25.4). Assume that the function f (t, x) and matrix function F(t x) = af(t, x) ax are defined and piecewise continuous for tE [ 0,

TI,

IIx-X

°IIC

< 8, S>0, (25.8)

and the function (p(t) is defined and piecewise continuous on the set [0, 'r 1 ] U ['r 1, TI. The homogeneous linear variational equation for the solution

dx = F(t, x°(t))x, t *'C1, dt

with the homogeneous jump condition x('[1+0)-x(t1-0) = 0 does not have T-periodic solutions in C 1 except the trivial one. Then, there exists ic > 0 such that, in the ball

^

I1z+x-(.0 +x)II 4 <- x, (25.9)

248 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

the solution x°(t) is unique, and, for sufficiently large m, the trigonometric collocation method (25.3), (25.6), (25.7) uniquely determines the approximate T periodic solution x = xm (t ) in domain (25.9). The sequence of approximate solutions x m (t) uniformly converges as m -* oo, t = [ 1, to the exact solution x 0 (t), and the sequence of zm(t) converges to .io (t) in the mean square. Moreover,

Il

xm(t)+x0 (t)II Ct <- C1Em(x0),

Il

xm(t) +x0 (t)

I4

<-

CZEm(x °),

where Em(X0) = max {Em(i?), Em(z2)}, C1 and C2 are constants independent of m, and Em(i9) is the best uniform approximation of the function (x°) by trigonometric polynomials of degree not higher than m. This theorem is proved by analogy with Theorem 24.1. Indeed , since the homogeneous system of differential equations

dt +x = 0, t# 'r1,

with the homogeneous boundary conditions x(0)-x(T) = 0,

x(ti1+0)-x('C1-O) = 0

has only the trivial solution, this problem has the Green function G(t, s), which is determined by (22.19) with 4D(t) = Ee-t and hasthe form

(D-1 -E)es-t, t E [O, ti1 ], s E [t, t1], G, (t, s) = D-les-t-T, t E [O, t1], D-Ies-t-T,

D_les-t, G2 (t, S) =

D-lest,

(E+ D-le T ) es-t, where D= E-e TE.

S E [t1, T],

t e [t1, T], s E [t, T],

tE [0,ti1],

SE[O,t],

t E [TI, T], S E [0, ti1],

t E [TI, T],

S E [t1, t],

Section 25

Method of Trigonometric Collocation for Periodic Systems 249

Let v (t) e L. L Then, using the Green function and applying Theorem 23.2, we establish that the solution of the semihomogeneous system

!Lx dt +x = v(t), t#ti1, x(0)-x(T) = 0,

x(tt1+0)-x(tt1-0) = q

has the form T x(t) = H(t)+ JG(t,s)v(s)ds 0

or, which is the same, T x1(t) = H1(t)+ JGj(t,s)v(s)ds

0 for to [ O, 't l ], and T x2(t) = H2(t)+ JG2(t,s)v(s)ds

0 for to [tit, T]. Here, H1(t) = eT'-t-TD-lq, H2(t) = et'-`(E+e-TD-1)q G1(t, s) = G(t, s) for to [ 0,'t1], and G2(t, s) = G(t, s) for to [ t, T]. We introduce the operators

G1 : G1v = f G1(t, s) v(s) ds, 0 (25.10) T G2 : 62v = J G2(t, s) v(s)ds,

0 K1: Klv = f(t, H1 +G1v)+G1v+ H1(t)+(p1(t),

K2: K2v =

f(t,H2 +G2v+G2v+ H2(t) +(p2(t).

250

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

It follows from the continuity of the operators G1 and G2 that one can indicate a ball in the space ` 1 with radius x 1 such that, in the ball

IIU-UOII5 K1.

the functions x1 (t) = H1(t )+ G1v and x2 ( t) = H2 (t) + G2u satisfy condition (25.8). The operators K 1: 4 1 --+ CEO, ti 1 ] and K2: Lit - C ['t 1, T] defined by (25. 10) are completely continuous in ball (25.8). Denote by P1 and P2 linear imbedding operators of the spaces C[0, 't 1 ] and C [ 21, T] into L2 [ 0,ti1] and L2[ti1 , T], respectively. Let P1m and Pa„ denote the linear operators that associate every function from C[0 , T 1 ] and C [T 1, T ] with its trigonometric interpolation polynomial constructed for the equidistant points ti and tj, respectively, defined by (25.7). According to [Bun2] and [Vai3 ], the operators Plm and PZm are uniformly bounded,

i.e.,

IIP1m1I s

M',

II P2m II <_

M„

for all m. The proof can now be completed by analogy with Theorem 24.1.

26. Practical Solution of Impulsive Problems

Example 26.1. Consider the simplest semihomogeneous boundary-value problem with impulse influence, namely,

g1(t) = t2, dx = g(t), t^ 1,

dt

t E [0, 11, (26.1)

g(t) _ 82(t)=t+1,

-8x(1 +0)+9x(1 -0) = 0,

tE[1,2],

4x(0)-x(2) = 0.

(26.2)

By virtue of relation (22.19) and the equality (D (t) = E, the Green function for the corresponding homogeneous impulsive boundary-value problem has the form

Section 26

Practical Solution of Impulsive Problems C{%) = ^ ,

r e [0,1],

5€[r,l],

G, U 5) = • Cf >(*) = A ,

t e [0,1],

^ e [1,2],

Cf 3 ) (s)=^,

te[l, 2],

se[f, 2],

C^ I ) ( 5 ) = 23'

r e [0,1],

se[0,f],

G 2 «, 5) = ■ C<2)(s) = g , C23)(') = § .

r e [1,2],

' 6 [1.21.

251

5 e[0,1],

*e[l,r].

According to (23.5) and (23.4), the solutions of the boundary-value problem (26.1), (26.2) * = *,(/) for re [ 0,1 ] and x = x2(t) for re [ 1,2] are given by the following formulas: for re [ 0 , 1 ] , /

xAt)= 1

]

2

— [s2ds +— J[s2ds + — f(5 + l)ds = -V + l; 23 J 23 , 23 J 3

for r e [ 1,2],

XM) 2

= ¥±\s2ds 23 J

+—$s + $ds + — $s + $ds = i f 2 + l . 23 J 23 J 2

Example 26.2. Consider the impulsive equation (26.1) with inhomogeneous bound ary conditions and a jump condition at t = 1: -8^(1 + 0) + 9 * ( l - 0 ) = 0.5,

4x(0)-x(2) = 1.

(26.3)

According to (23.7) and (23.8), the solution of the inhomogeneous boundary-value problem (26.1), (26.3) has the form

252 Collocation Method for Boundary- Value Problems with Impulses

x1(t) = Hl(t)+ o(t)

Chapter 4

f C21)(s) gl(s)ds

1 0'

2 + f Ci1i(s)g1(s) ds + f C(2)(s)g2(s) ds I,

tE [ 0, 11,

(26.4) x2(t) = H2 (t) + b(t)

f0

c22)(s )gl(s)ds

J C3)(s)g2(s)ds+c3) (s)g2(s)ds]

,+

te [ 1, 2],

r

where 15 8 - 1) 23 (1 16) 46'

H1(t) -

H2(t) = -1

9

231 23

According to (26. 4), the solution of the inhomogeneous problem (26.1), (26 .3) is as follows: 3

X1(t) = 3 +46, tE [ 0, 1],

X2(t) = 2 +t+23 ,

tE [ 1, 2].

Example 26.3. Let us find, by using the algebraic collocation method (24.4), (24.8), an approximate solution of the scalar second -order equation z d2 - dt +x2(

r)+g(t), r 0, (26.5)

gl (t) = -O.Olt2 - 0.1, t E [-1, 0], g(t) = g2(t) = -O.Ole2r, with the inhomogeneous boundary conditions

t E [0,1],

253

Practical Solution of Impulsive Problems

Section 26

2x(-1)+ x(- 1)+e1x(1)+x(1) = 0.1 e, (26.6) x(-1)+ 2x(-1)-x(1)+x(1) = 0.1, and the impulse influence x(0+0)-x(0-0) = 0.1, x(0+0)-x(0-0) = 0. (26.7) We seek an approximate solution in the form of a polynomial of degree 3, i.e., for tE [ -1, 0],

)t2 + Q3(I)t3 ;

(26.8)

+Q22)t2 +Q(2)t3.

(26.9)

x 1(t) = Qp) + Ql 1)t + Q2

for t E [ 0, 1 ] , x2(t) = Q(2 )+Q12)t

In order that functions (26.8) and (26.9) satisfy conditions (26.6) and (26.7) for arbitrary values of

Q2), Q31)

QZ2), and

Q32 ),

we must choose the coefficients Q(1), 0 Qo2^,

Qi1), and Q(2) according to (24.7). As a result, we obtain the following numerical values:

Q01)

= 0.06819 - 0.46609

Q2) + 0.35449 Q31) - 0.84464 Q22) -1.11159 Q32) ,

Q02) = 0.16819 - 0.46609 Q21) + 0.35449

Q31) - 0.84464 Q22) -1.11159 Q32)

Qj 1 ) = Ql2) = 0.2 + 3 Q(21) - 5Q31) - Q22) -

2Q32).

Therefore, the approximate solution (26.8), (26.9) takes the following form:

for tE [ -1,0],

x1(t) = 0.06819+0.2t+(t2+3t-0.46609)Q21) + (t3 - 5t + 0.35449) Q31) - (t + 0.84464) Q22) - (2t + 1.11159)

Q32);

(26.10)

254 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

for t e [0,1],

x2(t) = 0.16819+0.2t+(3t-0.46609)Q21) + (0.35449 - 5t) Q3t) - (t2 - t - 0.84464) Q22) + (t3 - 2t -1.11159) Q32i . (26.11) Taking the Chebyshev polynomial 'P2 (t) = 2 t2 -1 as the polynomial 'Yk (t) and using formula (22.1), we get t 1l = -0.146447, t12 = -0.853553, (26.12) t21 = 0.146447, t22 = 0.853553. 2 Substituting (26.10) and (26.11) in (26.5), we determine the unknown coefficients Q1>

Q31), Q22>,

Q32) for the collocation nodes (26.12) from (24.8) by using the Newton method. Taking Q21) = 0, Q3 1 ) = 0, Q22) = 0.05, and Q32 ) = 0.016 as the zero apand

proximation, after the second iteration we get

Q21)

= -0.00138,

Q31)

= -0.00019, (26.13)

Q22)

= 0.04344,

Q32)

= 0.02829.

Substituting (26.13) in (26.10) and (26.11 ), we obtain the required approximate solution: for tE [ -1, 0], x1(t) = 0.00063+0.09682 t- 0.00138t2-0.00019t3; for tE [ - 1, 0], X2 (t) = 0.10063+0.09682 t+ 0.04344 t 2 + 0.02829t3. One can verify that the deviation of the obtained approximate solution from the exact solution X1(t) = 0. lt,

t e [-1, 0],

x2(t)=0.le`,

tE[0,1],

x°(t) =

of problem (26.5), (26.6), (26.7) does not exceed 0.003.

Section 26

255

Practical Solution of Impulsive Problems

Example 26.4. Consider the scalar second-order equation

(

_

dx

dt2 = 4x I Ze sin4t+(p (t), t 4, (26.14) dt dt2 / where spi(t)=-1.6(sin 4t + cos4t + 0.le sin4tsin24t) , tE[0,

(p(t) =

41,

` 92(t)= 1.6es n4t(9cos24t - 10sin4t+2cos4t - 1), te[!, 2J,

Let us find an approximate 2 -periodic solution of equation (26.4) with the following jump condition at the point t =

4

x(4+0)-xl 4-0I = 0.2, * ( + O ) _ ( _ 0 ) = 0. (26.15) According to the trigonometric collocation method , we seek an approximate solution of system (26.14), (26.15) in the form of polynomials, namely, m(t) = aoli + I ( ak1) cos4kt + bk1) sin 40, K =1

(26.16)

X. (t) = I m

(t) = ao2j + I (a(k2) cos 4kt + bk2i sin 4kt). k=1

In the case under consideration, we choose functions (26.16) as follows: x12(1) = a(()l)+a(' cos4t+b1' sin 4t + b2(1) sin 8t, (26.17) 4t + b(2) sin 4t + b22) sin 8t. X 22 (1) = a0(2) +a (2) cos

In order to satisfy the jump conditions (26.15) and the periodicity condition

x(0)- x(2 'c) = 0, by virtue of (25.5) the coefficients ao1),

a12),

x(0)-xl2/ = 0, b

11>,

and b21) must be such that

256

Collocation Method for Boundary- Value Problems with Impulses aot) = a0(2) -0.1, bit) = bi2),

a^t^ = a^2^ +0.1,

Chapter 4

bet) = b22^

In this case, relation (26.17) takes the form x12(t) = a02> -0.1+(ai2) +0.1)cos4t+bi2isin4t+ b22)sin8t , tE 01 fl, (26.18) x22(t ) = ao2i+a^2icos4t + 42isin4t+b221sin8t , tE [42 As collocation nodes, we choose the following points: ti = 0.0625x, t2 = 0.1875x, t3 = 0.3125x, t4 = 0.43757n, (26.19) Substituting (26.18) in (26.14) at points (26.19), we obtain the following nonlinear determining system: 0.49307 [ao2) + 1 .4121a(2) + b22) + 0.04142

]2

+ 0.70711(a12) + b(2)) + 4b22) = 0.07318, 0.49307 [a?) + 1 .41421 bf 2) - 42) - 0.1] 2

+ 0.70711(bb2)-a(2))+4622) = 0 .07318,

2.02811 [ 2)-1.41421 ai2)+b22'l2

- 0.70711(a(, 2) +4 2)) +4 2.02811

[a(2) -

b? )

= -0.04515,

22,

= - 0.0591.

22

1.412142) - b )j 2 + 0.70711 lai2) -b12)) -46

Solving the last system by the Newton method and choosing ao2) = 0 . 1, a^2^ = 0, 02) = 0.1, and b22) = 0 as the zero approximation, on the fourth iteration we obtain the following approximate values of the unknown coefficients: apt) = 0. 10602, bt(2) = 0.08954 ,

a(2) = 0.00162, b22) = -0.00056.

Section 27

Green Function for a Three-Point Boundary-Value Problem

Substituting these values of aff\

257

a{2\ b{2\ and b^) in (26.18), we obtain the ap-

proximate — -periodic solution

jc12(t) = 0.00602 + 0.10162cos4r + 0.089541sin4r-0.00056sin8r,

re [ o , - l

= 0.106023 + 0.00162cos4r + 0.089541sin4r-0.00056sin8r,

re [ - , - ] •

JC 2 2 (0

The error of the obtained approximate solution as compared with the exact —periodic solution of problem (26.14), (26.15)

x°(0 =

x,°(r) = 0.1 (sin 4/ + cos4r),

t e [o, - 1

x°2(t) = 0.1e sin4 ',

' e [f'?}

does not exceed 0.03, which means that reasonable accuracy is attained even if we choose m = 2 in the approximate solution (26.16).

27.

Green Function for a Three-Point Boundary-Value Problem with Single Impulse Influence

Consider a three-point boundary-value problem for a linear inhomogeneous system of differential equations with impulse influence at the point t = x, of the form ^ = A(t)x + f{t), dt Cx(ix - 0 ) + Dx(x, +0) = 0 ,

r * x,,

te[a,b],

x, e (a, b),

3,x(a) + £,*(*,) + B^fc) = 0,

delD * 0,

t * x,,

(27.1) (27.2) (27.3)

where A(r) is an ( n x n ) matrix function continuous for r e [a, fc], C, D, fl0, Bx, and B 2 areconstant ( n x n ) matrices, x = (xx,x2 xn)e Rn, and f(t) is a piecewise-continuous vector function with discontinuity of the first kind at r = x,:

258 Collocation Method for Boundary-Value Problems with Impulses

f(t) _

f(t)=(f 1(t),..., fn(t)), t

1

f2(t) = (f21(t),..., f2n( t )),

Chapter 4

e [a, t1 ],

t E [ t1, b ]•

Let us find a solution of the semihomogeneous impulsive boundary-value problem (27.1)-(27.3). First, we construct the Green function for the corresponding homogeneous impulsive boundary-value problem (27.4), (27.2), (27.3): dx dt = A(t)x,

x c Rn,

t * TI,

t e [a , b]. (27.4)

The form of the Green function depends on the time of impulse influence t =,c 1 with respect to the point t = t 1 from the boundary conditions (27.3). There are two possible cases. Consider the first case where ti 1 E (t 1, b ). Denote by x1(t), t e [ a, 't1 ],

*(0 = x2 (t),

tE[t1,b],

the solution of the impulsive differential equation (27.4). We assume that x(t) belongs to the space C1 [ a, b ] of vector functions piecewise-continuous on [a, b ] with discontinuity of the first kind at t = ti 1, namely,

x1(t) = (x11(t),...,x1n(t)) E C[a,t1],

x2(t) = (x21(t),...,XN(t)) E C[til,b],

where C[ a , T1 ] and C[ i1, b ] are the spaces of vector functions continuous on the closed intervals [ a, i1 ] and [t1 , b ] , respectively. The following auxiliary statement is true: Lemma 27.1. If the homogeneous single-impulse three -point boundary-value problem (27.4), (27.2), (27.3) has only the trivial solution x (t)=_0 (x 1(t) _= 0, X2(t)=O) and ti 1 E (t 1, b), then the matrix L = Bo + B1cb(tl) - B2'D(b)L1,

I,1 = (D-1(il)D-1C(D(ri)

is nonsingular, i.e., det L * 0, (27.5)

Section 27

Green Function for a Three-Point Boundary- Value Problem

259

where fi(t) is the fundamental matrix of the homogeneous system (27.4) without impulse influence normalized at t = a ( i.e., O (a) = E). Proof. It is well known that, in terms of the fundamental matrix (D (t), a solution of the homogeneous system of differential equations (27.4) can be represented as follows:

x1(t) = (D(t)(D-'(a)x1(a) = cb(t)xl(a),

t E [a, tit ] , (27.6)

x2(t) = ci)( t)(D-'(ti1)x2 ( t1), t e [tit , b]. (27.7) By using relations (27.6) and (27.7), and the jump condition (27.2), we obtain x2(tit) = - D-' C4)(ti1)x1(a)• Substituting this relation into (27.7), we obviously get x2(t) = - 4D(t)L1x1(a ),

t e [ ti1, b ] . (27.8)

Since t 1 < ti 1 , we have

x(t1) = (D(tl )xt (a), x(b) = - cD(b)L1x1(a), and, therefore , taking the boundary conditions (27.3) into account, we get [ Bo + B14)(t1) - kb(b)L1 ]x1(a) = 0. By assumption, the homogeneous three -point boundary-value problem (27.4), (27.2), (27.3) possesses only the trivial solution . Hence, the last equation implies that condition (27.5) is satisfied. Lemma 27. 1 is proved. We introduce the matrix function of two variables G(t, s) (_ 1)'+i4(t)IL- 1[ B24(b)L1- j - B1cI'(t1)S(t1 - s )14)-,(S), t !5 s' (-1)t+jcb(t)[ L.L ' (B2c(b)L1-

(27.9) j - B1(D(tl)S(tl - s)) + L, - j ](D-' (s), t? s,

defined for all ii<-t!5tii+1 and 'rj<-s<-'cj+1 , t, s#t1, s#t1 , t#s; here , i,j=0, 1, Lo=E, do=a, tie=b, L and L1 are matrices of the form (27.5), 'c1>t1, and

260

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

J1, t1>s,

6(t1-s) = 0, t1 < s.

Let us show that a function G(t, s) of the form (27.9) can be taken as the Green function of the homogeneous single-pulse three-point boundary-value problem (27.4), (27.2), (27.3) in the case where T 1 E (t 1, b ) , i.e., a function satisfying the following conditions: (i) For every fixed s e [ a, b ], s *ti 1, s # t, G(t, s) satisfies the homogeneous differential equation (27.4) as a function of t for all t e [ a, b ], t * s, t * T 1, i.e.,

dG(t, s) =

dt A(t)G(t, s), t * T 1, t * s, t E [ a, b ] ;

(27.10)

(ii) for t = a , t = t1, and t = b, the function G(t, s) satisfies the three-point boundary conditions (27.3) for all fixed s r= [a, b ] , i.e., BOG(a, s) + B1G(t1, s) + B2G(b, s) = 0, s # t 1,

s # ti 1;

(27.11)

(iii) at t = s * 1, the function G(t, s) has a discontinuity of the first kind such that G(s+0,s ) - G(s-0,s ) = E, s # 'r1,

s * t1;

(27.12)

(iv) for t = ti 1 and all fixed SE [a, b ] , s * t 1, s * T 1, the function G(t, s) satisfies the condition of impulse influence (27.2), i.e., CG('r1-0,s) + DG('r1+0,s ) = 0,

s * 'r1,

s # t1 .

(27.13)

Theorem 27.1. Suppose that the homogeneous single-impulse three-point boundaryvalue problem (27.4), (27.2), (27.3) has only the trivial solution . Then the problem under consideration possesses a unique Green function G(t, s) of the form (27.9) satisfying conditions (27.10) -(27.13). Proof. It follows from conditions (27.10)-(27.13 ) that, in the case where three points a, t 1, and b are used in the given boundary conditions and there is an impulse influence at a single point 'r 1 > t 1, the domain of definition of the Green function a <- t, s:5 b can be decomposed into nine subdomains such that, in each of them, the function G(t, s) is given by an individual formula . Since condition (27.10) must be satisfied, we necessarily have

Section 27

Green Function for a Three-Point Boundary-Value Problem

261

SA

b

t(t)C8(s)

c(t)C7 (s) (t)C9(s) til

C(t)C4(S) tl

(D(t) C6(S)

(D(t)C5(s) <(t)C1(s)

b(t)C3(s)

c(t) C2(s) a a

tl

'Cl

b t

Fig. 2 G(t,s) = (D (t)C; (s), i = 1, 2, ... , 9. (27.14) in each of the subdomains mentioned. These subdomains and the form of the Green function are schematically depicted in Fig. 2 by layers with respect to the variable s. To determine the nine matrices C,(s), i = 1, 2, ... , 9, from condition (27.11), we consider the latter in different layers with respect to the variable s and obtain the following three equations: BOC1(s) + Bl(D(t1 )C2(s) + Bz4(b)C3(s) = 0, BOC4(s) + Bl'(t1 )C4(s) + B2c(b)C6(s) = 0,

(27.15)

BOC7(s) + B1(D(tl )C7(s) + B2(D(b)C9(s) = 0. Similarly, by using (27.12) with s varying within these three layers , we also obtain three equations, namely, C2(s) - C1( s) = 4)-1(s), C5(s) - C4( s) =(s).

C9(s) - C8(s) = (-1(S)•

Finally, condition (27.13) yields the following three equations:

(27.16)

262 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

C(D(T1)C2( s) + Dc(ti1 )C3(s) = 0, C$(til)C5( s) + D' (ti1 )C6(s) = 0, C1(til)C7( s) + D'(,r1 )C8(s) = 0. Solving the system of algebraic equations (27.15)-(27.17), we get BOCI(s) + B1b(t1)Cl( s) + B21 (b)C3(s) = -B1(D(tl)(D-I(s), BOC4( s) + B2b (tl)C4(s) + B2(D(b )C6(s) = 0,

BOC7( s) + B1(D(tl )C7(s) + B2' (b)C8(s) = - B24)(b ) (D-'(s), C(D(TOCl( s) + Dt(tl )C3(s) = -C(D( til)(D-I(S),

Cc(t1)C4( s) + D(D('C1 )C6(s) = =CO( til)4)-I(s), C(D(til)C7( s) + D' (tl ) C8(s) = 0.

By simple calculation, we determine the unknown matrices from the last system: CI (s) = L'(B2(D(b)Ll -Bl4(t1)P-1(s), C2(s) = [L'(B2I(b)L.1-B1b (t1))+E -I(s), C3(s) = -L1[L'(B2cb(b)I-I -Blcb (tl))+E]-,(s), C4(s) = L'B2-D(b)Li(D-'(s), C5 (s) = (L' B2(D(b)L1 + E)(D-' (s) , C6(s) = -LI(L'B2(D(b)Ll +E)-'(s), C7(s) = L'B2(D(b)(D-'(s), C8(s) = L1 L'B2-0(b)-D-1(s), C9(s) = (LIL'B2(D(b)+E)4)-'(s).

(27.17)

Section 27

Green Function for a Three-Point Boundary- Value Problem 263

c1(:) L^L^ B24'( b)4 ' (s)

_^(t) L' Bt4( b)q, ' (s)

m(t)(L,U' Bt4,(b)+E)(D-1(s) Tl

-o(t)L1(L 'B24(b )4 +E^4 i(s)

D(t)L Bt^(b)L14 ^(s)

^(t)C'(Br41(b)Li +E)V'(s) tl

(b(t)L'(Bi4(b)L1 Bi^( t))^ (s)

m(t)[L^(BZcZ(b)4 -Bz(ti))+E](V '(s) -Bi^(:1))+E]l (s)

a a

b t

tl

Fig. 3 Substituting these values of C; (s) into (27.14), we obtain the required Green function on all intervals of variation of the variables t and s. The form of the Green function for problem (27.4), (27.2), (27.3) in the case tit > t 1 is displayed in Fig. 3. It is not difficult to verify that the Green function obtained can be written in the form (27.9) with the use of the 8-function. The theorem is proved. Now consider the second case where ti 1 e (a, t I ). The following statement is analogous to Lemma 27.1: Lemma 27.2. If the homogeneous single-impulse three-point boundary-value prob-

264 Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

lem (27.4), (27.2), (27.3) has only the trivial solution x (t) = 0 (x 1(t) = 0, x2(t) == 0) and T 1 E ( a, t 1), then det M = det (Bo - B14)(t1)L1- B2,0(b)L1) * 0,

(27.18)

where L1 =-1(T1)D-1Ct(T1) Proof. As shown above, a solution of the homogeneous impulsive equation (27.4) on the intervals t E [a, T 1 ] and t E [ T 1, b ] is given by relations (27.6) and (27.8), respectively. Taking the inequality T 1 < t 1 into account, we get x(t1) = x2(tl) = -(D(t1)L1x1(a), x(b) = x2(b) = -(D(b)Ltx1(a)• Therefore, by virtue of the boundary conditions (27.3), we have (Bo - B1(b(t1)L1 - B2'D(b)L1)x1(a) = 0, which immediately implies condition (27.18). Lemma 27.2 is proved. Let us show that, in the case considered, the Green function G(t, s) has the following form for Ti:5 t:5 Ti+1 and Tj:s<-Tj+1, t,s#T1, t1*s:

G(t, s) (-1)i+jcI(t)L.M-

1(B1c1(t1)L1-i&(t1 -s )+ B2(D(b)L1- j)(D-1(s ),

(-1)i+jcI(t)[4M- 1(B1cb(t1 )L1- j&(t1-s)+ B2(b

t<- s,

(b)Li-j)+I.-j -1(s), t? s, (27.19)

where i,j=0, 1, to=a, T2=b, and Lo=E. Theorem 27.2. If the homogeneous single - impulse three-point boundary-value problem (27.4), (27.2), (27.3) with r 1 E (a, t 1 ) has only the trivial solution, then, for this problem, there exists a unique Green function of the form (27.19) satisfying conditions (27.10)-(27.13). The proof can be carried out by analogy with Theorem 27.1. In this case, the domain of definition a <- t, s <- b of the Green function is also decomposed into nine subdomains, in which the function G(t, s) is represented in the form (27.14), which is schematically shown in Fig. 4. By analogy with (27.15)-(27.17), we obtain the following three groups of equations for finding the unknown matrices Q i (s) :

Section 27

Green Function for a Three-Point Boundary-Value Problem

265

floCi(*) + * i * ( ' i ) C 3 (*) +B 2 «'(*)fi3( i ) = 0, * 0 e 4 < ' ) + * i * < ' i ) G 6 ( ' ) + * 2 * ( * ) G 6 < * ) = 0,

(27.20)

SoG 7 (*) + B, «&(',) G 8 (*) +B2(b)Q9(s) = 0,

e 2 (')-ei(*) = *-'(*). fi6(')-es(*) = *-'(*). (27.21)

e9(*)-e8(*) = *-'(»). C
(27.22)

C * ( t , ) C 7 ( * ) + 0 * ( T , ) j 2 8 ( 5 ) = 0.

System (27.20)-(27.22) yields B0Q](s)+Bi4>(ti)Q3(s)+B2(b)Q3(s)

= 0,

*ofi4(*) + * i * ( ' i ) G s ( 0 + « 2 * ( * ) C s ( ' ) = - ( B i * ( r i ) + * 2 * ( * ) ) * - , ( * ) . B 0 G 7 (^) + B l « ( » | ) G 8 ( * ) +»2*(*)C8(») = - * 2 * ( * ) * - I ( * ) ,

266

Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

CO(t1)Q1(s) +D4(ti1)Q3(s) = C(1( ti1)(1-1(s),

C(D(til) Q4( s) + DO('tl ) Q5(s) = 0, C()('t1) QC( s) + D(b(til ) Q8(s) = 0. Solving the last system with respect to Q1(s), we get Q1(s) = M-l (B1?( t1) + B2(D (b))L1(D-l (s), Q2(s) = [M- 1(B1I(tl) + B2 (1(b))L1 +E] (D-t(s), Q3(s) = -Ll[M- I(B1(1(t1)+ B2(D (b))L1+E](D -1(s), Q4(s) _ -M- 1 (B1(D(tl ) + B20(b ))(D-1(s), Q5(s) = L1M- 1(B1(i(tl)+B2 $(b))1-1(s), Q6(s) = [L1M- 1(B1?(tl) + B2(D (b)) +E](D-1(s),

QC(s) _ -M- IBA(b ) 0-1(s), Q8(s) = L1M- 1B2Z(b ) 1-1(s), Q9(s) = (L1M- 1B2'(b )+ E)(D-1(s). Substituting these values in (27.14), we obtain the unique Green function in the case t 1 > ti 1. The explicit form of this function in all intervals of variation of the variables t and s is presented in Fig. 5. One can readily verify that the Green function obtained can be written in the form (27.19). Theorem 27.2 is proved. By using Green functions of the form (27.9), (27. 19) constructed above , one can find a solution of the semihomogeneous boundary -value problem (27.1)-(27.3). Theorem 27.3. Suppose that the homogeneous single - impulse three-point boundary-value problem (27.4), (27.2), (27.3) has only the trivial solution. Then, for any piecewise-continuous function f (t) E CC1 [ a, b], there exists a piecewise -continuously differentiable solution of the semihomogeneous boundary -value problem (27.1)-(27.3) with discontinuity of the first kind at the point t = ti 1:

Green Function for a Three-Point Boundary-Value Problem 267

Section 27 s

b

(b ( t)L,Mt-'B2,b(b),b '(s)

-,b(t) M-'BO(b) 41-'(s)

^(t)^LiM1 -^ Bz^(b)i + E^^-^ (s) tt

^(t)IM '(B,^U1) + B O(b)) O-'(s) 0(t)[LMt-'(B1O(ti)+B2O(b))+Ellb-'(s)

-(b(t)M-'(B1(b (ti)+B2 (b )) O_'(s)

tit

0(t) Mf ' (Bi-D(t i )

+ Bz^(bU^^-^(s) -lb(t) Li [M-' (Bi-V(ty ) + B2-D(b ))+ E](b-, (s )

^(t)^M>-^ (Bi^(h ) + B20(b)) Li +E]0-'(s)

a a

tt

Tl

Fig. 5

x(t) =

xl(t),

t E [a, t1],

xi1(t),

tE[t1,T1], xil(t)=xi(tl),

x2(t),

t E [t1, b],

x1(t),

tE[a,T1],

xz(t),

t E [Ti , t1], x2(t) =x21(11),

x 1(t),

tE[t1,b],

for TIE( t 1, b) and

x(t) =

b t

268

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

for 't 1 E ( a, t I). This solution is given by the formula b

x(t) = JG(t,s)f(s)ds,

tE

[ a,b],

(27.23)

a

and admits a representation in the form r

t

t,

X11(t) = (1(t) f C2(s)fi(s)ds + JC^(s)fi(s)ds Ia

t

b + JC4(s)fl(s)ds + JC7(s)f2(s)ds , tE [ a , t1], r,

(27.24)

T,

*"(0 = fi(t)[ f C2(s)f1(s)ds + 5C5(s)fl(s)ds a

T,

t,

b

+ JC4(s)fl(s)ds + 5C7(s)f2(s)ds , t

tE [ t1,ti1],

(27.25)

tE [ T1, b] ,

(27.26)

T,

t,

T,

x2(t) = fi(t) JC3(s)fl(s)ds + JC6(s)fl(s)ds a t, t

b

+ JC9(s)f2(s)ds + JCs (s)f2(s) ds T, r 11

x1(t) = fi(t)[ JQ2(s)fi(s)ds + JQ1(s)fl(s)ds a

4

t

b

+ JQ4(s)f2(s)ds + JQ7(s)f2(s) ds , tE [ a , t1], T,

T,

xi(t)

= fi (t)I

(27.27)

t,

t

5Q 3(s)f1(s)ds + 5 Q6 (s)f2(s)ds a

t,

T,

b

+ JQ5(s).f2(s)ds + JQ8(s)f2(s)ds , t t, J

tE [ 'E1,

t1 ],

(27.28)

269

Green Function for a Three-Point Boundary- Value Problem

Section 27

xil(t) = t(t) 5Q3(s)fi(s)ds + JQ6 (s)f2(s)ds a

t,

(27.29)

+ f Q9(s)f2(s)ds + f Qs (s)f2(s) ds l , to [ t1, b]. t, t J

Proof. First, we consider the case where tit E (t 1, b ) and show that the functions 4(t), xl t(t), and x2(t) given by (27.24)-(27.26) satisfy the inhomogeneous differential equation (27.1) and the homogeneous conditions (27.2) and (27.3). In each of the intervals [ a,t], [t,t1] , [ t1,ti1], and [ti1,b] for xi (t), [a,t1], [t1,t], [t,til], and [,c1,b] for x111 (t), and [a,t1], [t1,-T 1], [til,t], and [t,b] for x2(t), the integrand in (27. 23) is continuously differentiable . Differentiating (27.23) with respect to t (regarded as a parameter), we get dxt(t) = f aG(t, s) fi(s)ds + G(t, s)Is-r_of (t) dt a at

+ f acct ' s) f (s)ds - G(t, s) I _ f (t) at

s- t+o

+ J'El aG(t, s) f (s ) ds + J aG (t, s)f2(s)ds t, at

at

b = f a^t's)f(s)ds + f (t), tE [ a,t1l, a at

f at f (s)ds

t a G(t, s) dxi t(t) = t' aG(t, s) fl (s)ds + dt a t,

f at

+ G(t, s)I _ _ f(t) + J aG(tat, s)ft(s)ds - G(t, s) I _ s- t o

+fa J,

G(t

r

, s) f2(s)ds

b

= f aG(t,s ) f(s)ds + f( t), at a

tE [ t1,T1],

s - t+0

270 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

a G(t, s) T' a G(t, s) f,(s)ds + f (s) ds

dx2(t) dt

a

q

+ f aG(t's)f2(

t,

s)ds + G (t,s)l S_,_of2(t)

+ 1 aG(t,s)f2(s)ds - G(t ,s)I s_t_of2(t)

t at

b

_

aG(t,s ) f(s)ds + f2 (t), J a.

tE [ tit, b].

a

For the expression dz - A(t) x, taking (27.23) and the last relations into account, we get

j a t , s) f(s)ds + f(s) - A(t) j G(t, s)f(s)ds at a a _ ! I aG(t's) - A(t)G(t,s)]f(s)ds + fi(t), tE [ a,t1], at J a L

tE [ t1,t1], (27.30)

b ja G(t, s) f(s) ds + f2 (s) - A(t)! G(t, s) f (s) d s a at a f r aG( t, s ) - A(t)G(t, s)]f(s)ds + f2 (t), a t at

tE [ til , b]. (27.31)

In view of (27.10), relations (27.30) and (27.31) imply that, for t E [ a, t 1 ] and t E [ t 1, T 1 ] , the functions xi (t) and xl 1(t) satisfy the inhomogeneous differential equation dx = A(t)x + f(t), dt

tE [ a,'t1],

and, for tE [ti1, b ], the function x2(t) satisfies the equation dx = A(t)x + f2(t), dt

tE [ i1, b].

Section 28

Semihomogeneous Linear Two-Point Boundary-Value Problem 271

Let us show that functions (27.24)-(27.26) and (27.27)-(27.29) also satisfy the three-point boundary conditions (27.3) and the jump condition (27.2). Indeed , substituting (27.23) into (27.3) and (27.2) and taking conditions (27.11) and (27.13) into account, we get b b b Bp JG(a,s)f(s)ds + B1 JG(t1,s)f(s)ds + B2 f G(b, s)f(s)ds a

a

b

f [BOG(a, s) + B1G(t1 , s) + B2G(b, s)] f (s) ds = 0, a

b b Cf G(ti1-0, s)f(s)ds + Df G('c1+0,s)f(s)ds a

a

b = J[CG('tl-0, s)+DG('tl+0,s)]f(s)ds = 0.

Thus, we have shown that, for 't 1 E ( t 1, b), the solution of the semihomogeneous boundary-value problem (27.1)-(27. 3) is given by relations (27.24)-(27.26). Similarly, for 't 1 E (a, t I), the solution of this boundary -value problem is given by formulas (27.27)-(27.29). The theorem is proved. It is easy to see that, with the use of formulas (27.24)-(27.26) and (27 .27)-(27.29), the solution of the semihomogeneous impulsive boundary-value problem (27.1)-(27.3) can independently be found in explicit form on the groups of intervals [a , t 1 ] , [ t 1,'r 1 ] , ['tl,b] and [a ,ti1], [til,t1]. [ t1,b], respectively.

28. Semihomogeneous Linear Two-Point Boundary-Value Problem with m-Impulse Influence

Consider a linear system of differential equations with impulse influence at m fixed points T1,T2,... ,TmE (a,b)

dx = A(t)x + f(t), XE R", t * ri, dt Cix('ri - 0) + Dix('ci + 0) = 0,

tE [a,b] ,

i=1,2,...,m ,

t = ii, i = 1, 2, ... , m,

(28.1)

(28.2)

272

Collocation Method for Boundary-Value Problems with Impulses Chapter 4

with two-point boundary conditions B]X(a,b) + B2x{b) = 0,

(28.3)

where C,, £>,-, B ) t and B2 are ( n x n ) constant matrices such that detD, = 0, i* = 1, 2,...,m. We assume that A(r) is an ( « x n ) matrix continuous for fe [a,b] and / ( / ) is a piecewise-continuous vector function with discontinuity of the first kind at t = T |, x 2 > ... , xm of the form '/,(/), f2(t),

f(t) =

te[a,xxl re[T,,T 2 ],

(28.4)

./m+iC). *e[Tm,l»]. We consider the problem of finding a piecewise-continuously differentiable function

*<0 x{t) ==

*i(0.

f€[fl,T,],

x2{t),

/e[x,,x 2 ], (28.5)

W ) ,

te[T„,b],

with discontinuity of the first kind at t = x,-, i = 1,2 m, such that the vector functions *,-(/), ' = 1, 2,..., /«, satisfy equation (28.1) and conditions (28.2) and (28.3), i.e., ^ L = A(0*i+/,
/e[x(_„x,],

C(*,(x() + D,*,+1(x() = 0 , B,jt,(a) + B2.xm + 1 (6) = 0,

/ = x,,

i= 1,2,...,m,

«=l,2,...,m+l, i = 1,2,..., m, x 0 = a,

xm + 1 = b.

Assume that the homogeneous m-impulse system of differential equations ^ = A(t)x, at

t*xt,

i=l,2,...,m,

(28.6)

with the impulse condition (28.2) and boundary condition (28.3) has only the trivial solution x(t)= 0 of the form (28.5). As above, we denote by 4>(r) the fundamental matrix of system (28.6) without impulse influence normalized at t = a and define the matrix

Section 28

Semihomogeneous Linear Two-Point Boundary-Value Problem 273 Lk = (D-'(tik)DkiCk'(tk). (28.7)

Then we can show that the following inequality is true:

det Bi + (-1)m B20(b) fl Lk # 0. k=m

Indeed, for all t E [Ti- 1, 'r ] , i = 1, 2, ... , m + 1, the solution of the homogeneous system (28.6) can be represented in the form xi(t) = (D( t)(D-1(ti-1 )xi(tii-1)•

(28.8)

Substituting (28.8) in relations (28.2) and (28.3), we get

C1c('t1)x1(a)+D1x2(ti1) = 0, C2c(,t2)C-1('t1)x2(til)+D2x2(ti2) = 0,

Cm(D(Tm)(D-1 (tim-1)xm(tim-1)+Dmxm+l(tim) = 0,

Blxl(a)+B2(D(b)4-1(tim)xm+i(tim) = 0.

Solving this system of equations from the bottom upwards with respect to x 1(a) , we obtain [B1 + (-1)m

1 B24(b) H Lk xl (a) = 0. k=m

Since, by assumption, we have x1 (a) = 0, we conclude that the condition

det [B1 + (-1)m B2cb(b) i ^ 0 k=m

is indeed satisfied. The following statement is true: Lemma 28.1. If the homogeneous m -impulse two-point boundary-value problem (28.6), (28.2), (28.3) possesses only the trivial solution, then the matrix

274 Collocation Method for Boundary-Value Problems with Impulses

K = B1 + (-1)mB2cI(b)

Chapter 4

1 T7 Lk k=m

that corresponds to this problem is nonsingular. Definition. The Green function of the homogeneous m-impulse two -point boundary-value problem (28.6), (28.2), (28.3) is defined as the following matrix function G(t, s) of two variables t and s defined for all ti i-1 <_ t!5 ti i and ti j-1 5 s <_ T j, d o= a, tim+1=b, i ,j=1,2,...,m+1:

G(t, s) 1 j (_1)m+i-j+lcI(t) fl LkK-IB2ct'(b)[I Lkcr-I(s),

s>t,

k=i-1 k=m

j=i,i+l,..., m+1; (D(t) (-l)'-f 11 Lk k=i-1 ^7 I

j

+ (-1)m+i-i +I b (t) 11 LkK-1B2fi(b)

U Lk

(D-1 (s),

k=i-I k=m

s <_ t, j =1, 2, ... , i, (28.9)

i

where I I Lk is understood as the identity matrix E if j > i. k=i

It follows immediately f r o m (28.9) that, f o r every fixed s E [ a, b ] , s * ' r , , i = 1, 2, ... , m, G(t, s), regarded as a function of the variable t for all t E [a, b] , t # s , t ^ ,ci, i = 1 , 2,... , m, satisfies the homogeneous differential equation (28.6). For t = a, t = b, and fixed s E [a, b], s # Ti, i = 1, 2,... , m, the function G (t, s) satisfies the boundary conditions (28.3): i

(- 1)m -j+2 B1K - 1 B20(b)

II LkcD

-1(s)

k=m

+ (-1)m-j+IB2ct(b)

II Lk(D -I (s)

k=m

+ (_l)2(m+1)-i B2c(b) h LkK-I B2cI(b) h Lk(D-I (s) k1=1m k=m

Section 28

Semihomogeneous Linear Two-Point Boundary-Value Problem 275 J (-1)m-J+1 B24(b ) ^ Lk(D-1(s) k=m

+ (-1)m-J+2 p_ 1 +(-1)mB2? (b) 11 Lk I K- 1B2 (b) fl Lkcb-1(s) '' k=m JJ k=m J (-1)m-J+1B2t(b) I I Lkt-1(s) k=m

J

+ (-1) m- J+2B20(b)

H LkD

-1(s) = 0.

k=m

Moreover, at t = s # i i, i = 1, 2, ... , m, the function G(t, s) has a unit jump of the form (27.12):

4)(S) (-1)I-l A Lk + (-1)m+l1 [ I LkK-1B2(b(b) lTT Lk^^-1(s) l k=1-1 k=1-1 k=m 1 J - (-1)m+1cb(s) TT LkK- 1R_ (b(b) H Lk(D-1(s) k==11-12• k=m 1

_ (b(s) 11 Lk(D -1(s),

l = 1, 2, ... , m + 1.

k=I-1

Taking into account that

h4 = E k=i

for j > i, we conclude that the last expression is indeed equal to E. Finally, for t =,r i, i = 1, 2, ... , m, and all fixed s e [ a, b ] , s # ti i, the function G(t, s) satisfies the condition of impulse influence (28.2):

(_1)m -J+l+1

1

J

II

Lk(D-1(s) Cl(b(til) [J LkK-1 B2(b(b) k=l-1 k=m J

+ (-1)m-J +1+1D1(b( ,rl)fl LkK- 1B2fi(b) 1I Lk( -1(s) k=1 k=m

276 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

= (-1)m-I+1+l(C14 (,O - Dicb(T1)LI)

x

II LkK- 'B2 b(b) II Lk(D -'(s)

k=l-1 k=m

= (-1)m-i+1 +l(C14,(Tl)- Dl(D (Ti)(D-'(Tl )D1'C1cb(Tl))

x

II LkK-' B24(b) II Lk(D-'( s) = 0,

k=1-1 k=m

I = 1,2,...,m, j = 1+1,1+2,..., m+1, for s ? t, and

Cl4)(il)j (-1)1-i 4 Lk + (-1)m-i+1+1 A LkK-' B2(D(b) i LkIc-'(s) l k=1-1 k=l-l k=m

+ D1(D(TI) (-1)1-i

1 -1h Lk + (_1)m-i+1+2 TT LkK-'BZ'(b) J7 Lk}4>-'(s) k=1

k1=1 k1=m ))J

1-i ^ _ (-1)l-j(Cib('tl)-D1' (ti1)L1) 11LkD(s) k=1-1 1 + (-1)m-i+l+1(Clb (til)-D1c'(Tl)L1) 11 LkK-1B2I(b) Lk(D-'( s) = 0, k=1-1 k=m

1 = 1,2,...,m,

j = 1,2,..., 1,

for s < _ t. The following statement is an analog of Theorems 27.2 and 27.3 for the m -impulse boundary-value problem under consideration. Theorem 28.1. Suppose that the homogeneous m-impulse two-point boundaryvalue problem (28.6), (28.2), (28.3) has only the trivial solution. Then, for this problem, one can construct a unique Green function G (t, s) of the form (28.9), and for any piecewise- continuous function f (t) of the form (28.4) solution (28.5) of the semihomogeneous m -impulse boundary-value problem (28.1)-(28.3) is given by the formula

Section 28

Semihomogeneous Linear Two-Point Boundary -Value Problem 277

xi(t) = JG(t,s)f(s) ds = a

a

JI(t)I

j=1

I

(-1)`-t I I Lk

k=i-1

1 + (-1)m+i-j+1 TT LkK-1B2I (b)h Lk ]_1(s)f(s)ds k=m

k==i-1

+

b m+1 1 1)m +i -j +1 TT

J c(t)I (-

LkK-1B2ljt(b)

t j=i k1=ii-1

j

x jjLk(D -1(s)f(s)ds, k=m

i=1,2,...,in +1. (28.10)

tE [tii_1,tii] ,

Proof. In view of Lemma 28.1, the constructive proof of the existence of the unique Green function can be performed by analogy with Theorem 27.1. By using the fundamental matrix (D (t), we can write the solution of system (28.1) on the segments t E [tii_ 1 , ti;] , i = 1, 2, ... , m + 1, as follows: I

xi(t) =

4)(t) - 1(tii- 1)xi(Ti-1) + J c(t)(D- 1(s)f(s)ds,

(28.11)

Ti-I i = 1,2,...,m+1. By virtue of (28.11), (28.2), and (28.3), after corresponding transformations, we get

*/
(-1)i-1D,_1C;_ tcI'('Ci_t) 11 Lkxl(a) k = i-2 1

2i

+ (-1)'_1D, _j1Ci-1c(ti-1) [I Lk J(D-1(s)f(s)ds k=i-2 a

2 tie +

(-D'-VJQ-^V,)

Jc-

II Lk k=i-2 T2

+ ... + (-1) -1 D;=jC-1(D( t _1) T" J 2i-2

i = 2,3,...,m+1,

1(s)f(s)ds

c -1(s) f

(s)ds,

(28.12)

278 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

where 2

xl(a) =

(-1)m+1K-

jI- '(s)f(s)ds

T1

1B2cb(b) [I Lk

k=m a

T 2 + (-I)mK-'B2cD(b) f Lk J (D-1(s)f(s)ds

k=m T1

Tm

+ K-1B2I(b)Lm

1 (D-1(s)f(s)ds

Tm-1

b - K-1B2c(b) f c-1(s)f(s)ds. Tm

In view of expression (28.12) obtained for xi (r1_ 1), relation (28.11) yields

xi(t) = f fi(t)[( -1)i-1 a

+

Lk + (-1)m+' 11 LkK-1B2(D(b)f Lk 1-1(s)f(s)ds

k=i-1

k=i-1

T2 2 c(t)I (-1)i-2 fl Lk + (-1)m+i-1 1

1

L

T1

k=i-1

k=m

]

LkK-1 B2I(b) A Lk d-1(s)f(s)ds

k= i-1 k=m

J

+ ... + j cp(t)I E+ (- 1)m+l T7 LkK_'B21(b) H Lk ^-1(s)f(s)ds Ti -1

k ==ii -1 k=m

T

]

i

+ ( (-1)m+' (D(t ) I LkK-'B2,D(b) H Lk'-1(s)f(s)ds t k==ii-1 k=m b + ... + j (-1)` c(t) fl LkK-'B2,D(b)1-'(s)f(s)ds, Tm

tE [tii-1,ti;1,

i = 1,2,...,m+1.

This relation can be rewritten in the form (28.10) with expression (28.9) as the function G(t, s). The theorem is proved.

Section 29

Inhomogeneous Linear m-Impulse Two-Point Problem 279

29. Inhomogeneous Linear m -Impulse Two-Point Boundary-Value Problem

The results of the previous section can also be extended to the case of the inhomogeneous differential equation (28.1) with inhomogeneous impulse influence Cix(,ci -0) + Dix(Ti +0) = di, i = 1, 2,... , m, die R",

(29.1)

and inhomogeneous boundary conditions Blx(a)+B2x(b) = dm+1I dm+1E R. (29.2) The solvability of the boundary-value problem (28.1), (29.1), (29.2) is established by the following statement: Theorem 29.1. Suppose that the homogeneous m-impulse two-point boundaryvalue problem (28.6), (28.2), (28.3) possesses only the trivial solution. Then, for every piecewise- continuous function f (t) of the form (28.4), the linear inhomogeneous boundary-value problem (28.1), (29.1), (29.2) admits the following solutionfor tE [Ti_1, ri], i=1, 2,...,m+1: b xi(t) = hi(t) + 5G(t,s)f(s)ds, a

(29.3) tE [Ti_1,Ti], i = 1,2,...,m+1,

where G(t, s) is the Green function given by (28.9) and hi(t) is defined as follows:

i-2 i-j

n

Lk4)-1(Ti-j-1)Dl J-1 di -j (-1)' j=0 k=i-1

hi(t) = fi(t )

Lk4-1(t)hl(t) , i = 2, 3, ..., m + 1, (29.4)

+ k=i-1

m-1 m- j+1 h1(t) =

(b(t)K- 1 dm+i - B2(D(b) (-1)j 11 Lk(D- 1(Tm -j) D;1 jdm_i

{

j=0 k=m

280

Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

Proof. By virtue of Theorem 28.1 and equality (28.10), the second term on the right-hand side of (29.3) satisfies the inhomogeneous differential equation (28.1) and the homogeneous boundary conditions (28.2) and (28.3). Therefore, to prove the validity of representation (29.3), it suffices to show that the function h i(t), t e [ i i _ 1, T j] , is a solution of the homogeneous system of differential equations (28.6) with the additional inhomogeneous conditions (29.1) and (29.2). To satisfy the conditions indicated, we seek the vector function h i (t), t e ['r ti in expression (29.3) in the form hi(t) = c(t)li,

tE [tii-1,'ril,

i = 1,2,...,m+1, (29.5)

where li E R" are some constant vectors to be determined and ( (t) is the fundamental matrix of the homogeneous system (28.6) without impulse influence normalized at t = a. Substituting (29.5) in (29.1), (29 .2), we get Ctc(tii)li + Dicb('r1)li+1 = di, i = 1, 2, ... , m,

(29.6) Bill+B2,D(b ) lm+I = dm+i.

Solving system (29.6) with respect to li, i = 1, 2, ... , m + 1, we obtain i-2

I

li = (-1)) 11 Lk1-I(tii-j-I)D-i-ldI_J_I + (-1)r-1 fl Lkll, j=0 k=i-1 k=i-I

i = 2,3,...,m+1, where 1+l m-I l1 = K-1 dm+1 - B2^(b) 1 (-1)J m--jT+I^ 1 1 Lk -1(tm- j) Dml j dm-j). j=0 k=m

Thus, the required function hi (t) indeed has the form (29.4). The theorem is proved.

30. Multipoint m-Impulse Boundary-Value Problem

Let us now investigate homogeneous and inhomogeneous multipoint m-impulse linear boundary-value problems. Consider a homogeneous m-impulse system of differential equations

Section 30

281

Multipoint m-Impulse Boundary- Value Problem dx = A(t)x, t # 'C,, i = 1 , 2, ... , m, dt

(30.1)

C,x(t1- 0) + D,x(•t, + 0) = 0, i = 1, 2,... , m ,

(30.2)

with (m + 3) -point homogeneous boundary conditions of the form m+2

I Akx(tk) = 0, tk * -Ti,

k = 0, 1,... , m + 2,

k=0

(30.3) i = 0, 1,...,m,

t0 = a, tm+2 = b,

where A k are constant (n x n) matrices and the times of impulse influence are located with respect to the points t k from the boundary conditions as follows: t k < ti k < t k+ 1, k = 1, 2,... , m. Assume that the boundary-value problem (30.1)-(30.3) has only the trivial solution. Then r

m+1

1

1

II Lj + (-1)mA,,,+24)(b) A Lj j=k-1 j=m

# 0,

det Q = det I Ao + (-1)k-lAkD(tk) l

k=1

(30.4) where the matrices L j are defined by (28.7). Indeed, substituting solution (28.8) of the homogeneous system (30.1) into (30.2) and (30.3), we obtain the equations Ckb(tk)xk(tik-1)+ Dkxk+l(Tk) = 0,

k= 1, 2,... , m,

m+1

A0x1(a)

+

E Akcb(tk)c-1(tk-1)xk(tik-1) + Am +2' (b)fi- '(tim)xm

+1(ti,n)

= 0.

k=1

Solving the last system with respect to x 1(a), we get

[Ao [Ao + I (-1) k-lAk(D (tk) ft Lj + (- 1)mAm+2t(b) n Li xl(a) = 0. k=1

j=k-1

j=m

]

Since x 1(a) = 0, inequality (30.4) is obviously valid, i.e., the following statement is true: Lemma 30.1. Suppose that the homogeneous (m + 3) point m -impulse boundary-value problem (30.1)-(30.3) has only the trivial solution . Then the matrices A k,

282 Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

Ci, and Di in the boundary conditions and conditions of impulse influence and the points tk and ii are such that the matrix Q of the form (30.4) is nonsingular. Definition. The matrix function G(t, s) of two variables t and s defined for all T i - 1 : 5 and tij-t<-s<-'tj, s* tk, k=1,2,...,m+1 , i,j=1,2,..., m+1, T o = ti m + 1= b, and given by the expression

cb(t) Lt + (-1)i-1 [J Lt Q-1 l t=i-1 t=t-t m+1

x

E

(_1)k-j-lAk(D(tk)

k = j+1

j

X

II

Lq +(- 1)m- lAm+2cb(b)

q = k-1

^-j7 l X 11 Lq-AjI(tj)S(tj-s) I (D'1(s), G(t, s) =

q=m

(30.5) s:5 t, j=1, 2,.. . , i,

i m+I (-1)i-1 fi(t ) Lq Q-1 (_l)k-j-1 q=i-1 k=j+11 j

X Ak(D(tk) I I Lq + (- 1)m- j Am+2D(b) q = k-I j x I I Lq -Ajc(tj)S(tj -s) (D-1 (s), q=m

s> _t, j=i+1,..., m+1, where j (1 for t >- 0, HLq = E for j> i, S(t)= {l q=i 0 for t<0; and Lq are matrices of the form (28.7), is called the Green function for the homogeneous (m + 3)-point m-pulse linear boundary-value problem (30.1)-(30.3). One can directly verify that the function G (t, s) thus defined possesses all properties specific for Green functions. Note that, for every fixed s E [ a, b], s # 't i, i = 1, 2,... , m, s * tk, k = 1, 2, ... , m + 1, function (30.5) is the product of the fundamental matrix and a constant matrix. In view of this fact, it is clear that function (30.5), regarded as a function of the variable t, satisfies the homogeneous equation (30.1) for all to [ a, b], t ^ s, t * tit, i = 1, 2, ... , m. Further, by direct calculation, one can establish that, for t = a, t = b, t = tk, k = 1, 2,...,m+1, and any f i x e d se[a,b], s#Ti, i=1,2,...,m, s# tk, k=1,2,...,m+

Section 30

Multipoint m-Impulse Boundary-Value Problem

283

1, a function G(t, s) of the form (30.5) satisfies the (m + 3)-point homogeneous boundary conditions (30.3). It can also be proved that, for t = s, s # T t, s = tk, i = 1, 2,... k = 1, 2, ... , m + 1, a function G(t, s) of the form (30.5) has a discontinuity of the first kind, namely, a unit jump. One can easily verify that, for t #,r i, i = 1, 2, ... , m, and fixed s e [ a, b 1, s # T s # T k, function (30.5) satisfies the homogeneous condition of impulse influence (30.2). The Green function (30.5) also enables one to investigate solutions of inhomogeneous multiimpulse boundary-value problems. Instead of (30.1)-(30.3), we now consider an inhomogeneous m-impulse system of differential equations

dx =A(t)x+f(t), t#T;, dt

i=1,2,...,m,

(30.6)

C;x(Ti-0)+Dix(T;+ 0) = d;, i = 1 ,2,...,m,

(30.7)

with inhomogeneous conditions of impulse influence

and inhomogeneous ( m + 3) -point boundary conditions m+2 IAkx( tk) = dm+i, tk# Ti, i=1 , 2,...,m, k=0

(30.8) tk < Tk < tk+1, k = 1, 2,... , m.

The following statement is true: Theorem 30.1. Suppose that the conditions of Lemma 30.1 are satisfied. Then the homogeneous m -impulse (m + 3) -point boundary-value problem (30.1)-(30.3) has a unique Green function of the form (30.5) and, for every piecewise-continuous function (28.4), a solution x (t) of the form (28.5) of the corresponding inhomogeneous boundary-value problem (30.6)-(30.8) is given by the formula b Xi(t) = Hi ( t) + f G(t,s) f(s) ds, a

where

to [ Ti _ 1, T;],

i = 1, 2,..., m+ 1 , (30.9)

284 Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

i-2 i Hi(t) = c (t) (-1)a fj -q-1)D q_ldi-q-1 q=0 k=i-1 LkI-1(Ti 1

+

11 LkZ-1(t)Hl(t) , i = 2, 3, ..., m + 1, k = i-1 m+1 k-2 k-

H1(t) = O(t)

Q- 1d m+1 -Q -1

I Ak^(tk) Y. (

-1) q k=2 q=0 j=k-1

LjI -1

m ^-`1

_

x (Tk-q-1)Dk 'q-ldk-q-1 -Q-1``1m+2 D(b) G (-1)4 q=0 m-q+1

1

x it Li0-1(tim-q )D;' qdm-q Jl' j1=1m

Proof. With the use of Lemma 30.1, the existence of the Green function (30.5) can be established, as in Theorem 27.1, by the direct construction of it. The validity of relation (30.9) can be proved by analogy with (29.3) in Theorem 29.1. For t r= [ Ti-1, T i ] , i = 1, 2, ... , m + 1, we seek the vector functions Hi(t) in the form Hi(t) = (D(t)li,

tE[Ti-1,Ti ],

i = 1,2,...,m+1 , (30.10)

and choose li c- R ' so that function ( 30.10) satisfies the homogeneous equation (30.1), the inhomogeneous conditions of impulse influence (30.7), and the inhomogeneous boundary conditions (30.8). Substituting (30.10) in (30.7) and (30. 8) and solving the system obtained with respect to l i , we get i-2 i7-q 1 7 71 q li = (-1)q (-1)`-1 H Ljl1, 11 Lj(D-1(Ti -q-1)Di - ldi-q-1 + q=0 j=i -1 j=i-1

i = 2,3,...,m+1, where m+1

l1

= Q-1dm +1 - Q-1

Ak(D(tk)

k-2 k-2 (_ 1 )q 11

LjI-1

k=2 q=0 j=k-1 d

m-

x (Tk-q-1)Dklq-ldk-q-1 -Q-1Am+z((b) "A Lj(D-1(Tm-q)Dml qdm-q• j=m

Section 30

Multipoint m-Impulse Boundary- Value Problem

36

37

38

31

32

33

26

27

21

22

285

39 A 40

t4

34

35

ti3

28

t3

/ 29

23

30 25

24

T2

17

16

18

19

20

14

15

8

9

10

3

4

5

t2

11

12

13

tit 6

7

tt 1 2 a a tt 'C1

t2

t2

t3

ti3

t4

b t

Fig. 6 Substituting the calculated values of I ; in (30. 10) and taking into account the fact that the second term on the right-hand side of (30.9) is a solution of the semihomogeneous m-impulse ( m + 3) -point boundary -value problem (30.6), (30.2), (30.3 ), we establish that expression (30.9) is indeed a solution of the inhomogeneous boundary-value problem (30.6)-(30. 8). The theorem is proved. It follows from formulas (27.9), (27.19), (28.9), and (30.5) for the Green functions of the three-, two-, and (m + 3) -point single -impulse and m-impulse boundary-value problems that the domain of definition a <- t, s <- b of these functions consists of several subdomains in each of which these functions are given by individual explicit analytic formulas. For the single-impulse two-point boundary-value problem, these subdomains are shown in Figs. 2-5. Note that, for the m-impulse (p + 1)-point boundary-value problem dx = A(t)x, dt

t*Ti,

i=1,2,...,m,

Cix(tii-0)+Dix(ri+0) = 0, P I Aix(ti) = 0, i=O

i = 1,2,..., m,

(30.11)

286

Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

the domain of definition a <_ t, s <_ b of the Green function is decomposed into m + p layers with respect to the variable s. Each layer, in turn, contains m + 2 subdomains with respect to the variable t. Consequently, with respect to both variables t and s, the domain of definition of the Green function for problem (30.11) is decomposed into N subdomains, where N = (m+p)(m+2).

(30.12)

For example, in the case of problem (30.11) with p = 5, m = 3, ti 1 E (t 1, t2), ti 2 E (t2, t3), and 't3 E (t3, t4), we have N = 40; the domain of definition of the Green function in this case is schematically shown in Fig. 6. In this case, we have G (t, s) = {Gk (t, s), k = 1, 2, ... , 40} and the solution of the semihomogeneous boundary-value problem corresponding to (30.11) can be obtained in explicit form in each of the intervals [a, t 1], [t1, til], [t1, t2], [t2,'r2], [tit, t3], [t3, t3], 1r3, t4], and [t4, b] with the use of formula (27.23) and integration with respect to s over the given time interval.

For example, in the case t E [t2, 't 2 ] , we get q Tj t2 x(t) = f Q, (t, s) f(s) ds + f G9 (t, s) f(s) ds + f G14 (t, s) f(s)ds a

tj

Tj

T2

t

+ f G19(t, s) f(s)ds + t2

f G2

1(t,

s) f(s)ds +

T2

T3

f G23(t, s) f(s)ds t b

t4

+ f G28(t, s) f(s)ds + f G33(t, s) f(s)ds + f G38(t, s) f(s)ds. t3

T3

t4

31. Equations with Piecewise -Continuous Right -Hand Side

In this section, for simplicity, we study a single-impulse T-periodic boundary-value problem. Under the assumption of the existence of a periodic solution, we establish the solvability of the equations that determine the coefficients of the approximate solution. We also determine the rate of convergence of the approximate solution. Note that analogous results are also true in the case of finitely many pulses. Consider the system

dx f (t, X), dt -.f(t'X) = 1 f2(t,x),

t E [0, t), (31.1) t E ('t, T],

Section 31

Equations with Piecewise - Continuous Right-Hand Side 287

with piecewise-continuous right-hand side under the conditions S1x('t+0)+S2x(t-0) = g, detS1,S2 * 0, (31.2) x(O) = x(T). (31.3) This boundary-value problem is a reduction of the problem on solutions T-periodic in t E R for problem (31.1)-(31. 2) with impulse influence at t = 'c + kT (k = 0 ± 1, ± 2,...). Let vector functions J (t, x) = (f,1 (t, x), ... , f,n(t, x )) and the Jacobian matrices

A, (t, x) =

(aJ (tx))' i = 1, 2,

be defined and continuous in the domains f1 : [0,,c]x D-* R', (31.4) f2: [,r,T]X D-4 R where D is a bounded closed set in the n-dimensional Euclidean space Rn and the right-hand side of (31.1) is a function periodic in t with period T, i.e.,

fl (0, x) = f2 (TI X) Let C, [0, T] = C, denote the space of piecewise-continuous vector functions x(t) = (x1 (t),

tE[O,t),

j x2(t),

tE ('c,T],

defined on [ 0, T] and having a discontinuity of the first kind at t =,r:

x('c+0)-x(ti-0) = x2('c)-xl(ti), where x1(t) = (x11( t),...,xln(t ))E C[0,'t],

x2(t) = (X21 (t), ... , x2, (t)) E C[ti, T].

(31.5)

288

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

We define the following uniform norms in C[0, t[] and C[ti, TI:

I x1 I c[ o, tl

=

Ix2IRt, T] = IxICt[O,T] =

IxIC,

i max, n r mo ] I xi, (t) I,

max Max i=l,...,n te[t,T]

I x2i(t) I ,

= max0 x1IC[O,t],Ix2IC[t,T]).

Consider the vector functions x(t) (31. 5) in the space of square -summable vector functions LT[0, T] = Li defined on the interval [0, T] with the norm

Ix1910,T]

=

I xI4 = 1 X1 IL2[ 0,t)

+ Ix2IL2[t,TI 1

t

l1

f Ixi(t)I2dt

1

r l 2 + f I x2(t)I2dt 2.

] Lt

1

We shall also need the Green functions in order to prove the convergence of the trigonometric collocation method applied to the approximate solution of problem (31.1)(31.3). Green Function for Systems with Impulse Effect and Piecewise- Continuous Coefficients. Consider the homogeneous single -pulse two-point boundary-value problem dx = A(t) x, t #T, dt

to [ a, b ],

XE Rn,

(31.6)

S1X(T+0)+S2x(t -0) = 0,

Te ( a,b),

detS1,S2 * 0

(31.7)

Blx(a)+B2x (b) = 0,

(31.8)

where A (t) is a piecewise -continuous ( n x n) matrix on [ a, b ] the elements of which may have a discontinuity of the first kind at t =,r, A (t) = I Ai(t),

1 A2(t),

t e [a, ti), t e ( i, b].

and B 1 and B2 are constant ( n x n) matrices. Let b 1 be the fundamental matrix of the following system of homogeneous differential equations without impulse influence normalized at t = a:

Section 31

Equations with Piecewise-Continuous Right-Hand Side dx = A1(t)x, dt

289 (31.9)

tE [ a,T), 11(a) = E;

also let (D2(t) be the fundamental matrix of the system dx = A2(t)x, tE (T, b]. dt

(31.10)

Lemma 31 .1. Suppose that the homogeneous boundary-value problem (31.6)(31.8) has only the trivial solution in the space CT [a, b] , i.e., x(t) = xi(t)=-O, X1(t) = 0,

tE[a,T),

(31.11)

t E (ti, b]

[in particular, the continuous solution x ( t) _- 0 ]. Then the matrices S 1, S 2, B 1, B 2, and the fundamental matrices (D 1, (D 2 of the homogeneous system (31.9), (31. 10) satisfy the inequality det D * 0,

(31.12)

D = B1- B2(D2(b)4'2'(ti)52(D1(i)•

Proof. It is known that the solution x(t) of system (31.9) that passes through the point x = x (a) at t = a can be written in the form x(t) = (D1(t)Jt(a)x(a) = cb1(t)x(a),

tE [ a,T). (31.13)

It follows from (31.7) and (31.13) that x(T+0) = -S^ 1S2x(T-0) = -ST 'S2(b1(T)x(a). (31.14) Hence, the solution of (31.10) with the initial condition (31.14) has the form

x(t) =

(D2(t)(D2'(T)x(T)

= -(D2(t)4)2' (ti)Si-'Sz(D1(ti)x(a),

t E (T,b].

In view of this relation and the boundary condition (8), we obtain [BI - B2 c2(b)C2'(T)S2(D1(T)]x(a) = 0.

Since the boundary-value problem (31.6)-(31.8) has only trivial solution (31.11), hence x(a)=-O and relation (31.12) is valid. This completes the proof. Corollary 31.1. If relation (31.12) holds for the linear single-impulse boundaryvalue problem (31.6)-(31. 8), then this problem possesses only the trivial solution (31.11).

290

Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

Definition 31.1. Consider a matrix function G(t, s) of two variables t and s defined for all t, s E [ a, b ], t, s * ti, t * s, as follows: 1 G1(t, s),

t E [a, s),

(31.15)

G(t, s) _

G2(t, s), t r= (s, b], where

G11)(t, s), G1(t,s) =

t e [a, T), s E (t, t),

G12i(t,s), tE[a,ti), sE('t , t),

(31.16)

Gf3i(t, s), t e (ti, b], s E (t, b],

G2 (t, s) =

G(21)(t, s),

t r= [a, T), s e (a, t),

G22)(t, s),

t E (rr, b] , se [a, ti),

G23k(t, s),

t e (ti, b], s e (ti, b].

(31.17)

The function G (t, s) is called the Green function of the homogeneous boundaryvalue problem (31.6)-(31.8) if it satisfies the following conditions: (i) For all fixed s E [ a, b], s # ti, as a function of the variable t, G (t, s) satisfies the homogeneous differential equation (31.6) for all t E [ a, b], t * s # ti, namely, dG(t, s) =A(t)G(t ,s), sE[a ,b], t* s#ti, dt

(31.18)

i.e., dG(t, s) dt = A1(t) G(t,s),

SE [ a,b],

sit,

tE[

a ,'r),

t* s,

(31.19) 4G(LL) _ dt -AZ(t)G(t,s),

sE[a,b] ,

s*T,

tE(ti,b), t# s;

(ii) for t = a, t = b and all fixed s E [ a, b], s # t, G(t, s) satisfies the boundary condition (31.8), i.e., B1G(a,s)+B2G(b, s) = 0,

se[a,b], s#i; (31.20)

Section 31

Equations with Piecewise-Continuous Right-Hand Side

291

S A,

b G, (t, s) G(2)(t, s) r2(3) (t, S) t

G(1)(t, S)

G(2) (t, s) 1

GZ (t, s)

a 4 a

T

b

0 t

Fig. 7 (iii) for t = s * ti, G (t, s) has a discontinuity of the first kind, namely, G(s + 0, s) - G(s - 0, s) = E, se(a,b), s#'t; (31.21) (iv) for t = ti and all fixed S e [a, b ] , s * T, G (t, s) satisfies the condition of impulse influence (31.7), i.e., S1G(tt+0,s)+S2G(ti-O,s) = 0, se [ a, b], s

(31.22)

It follows from (31.16) and (31.17) that the Green function (31.15) constructed for problem (31.6)-(31.8) is a piecewise-continuous function represented by six functions (see Fig. 7). Lemma 31 .2. Suppose that the homogeneous two-point impulsive boundary-value problem (31.6)-(31.8) has only the trivial solution (31.11). Then the Green function G(t, s) of problem (31.6)-(31.8) given by (31.15)-(31.17) and satisfying conditions (31.18)-(31.22) is uniquely defined. Proof. We can directly construct a function of the form (31.15)-(31.17) satisfying conditions (31.18)-(31.22). Since G( t, s) satisfies (31.18) and (31.19) with respect to t

292

Collocation Method for Boundary-Value Problems with ImpulsesChapter 4

for all fixed se [a, b], t*s*x, in each subdomain of [a, b]x [a, b], the functions in (31.16), (31.17) necessarily admit the following representation: <$\t, s) = QtfXfks),

te [ a, x),

G{2\t,s) =
te[a,x),

G{3\t,s) = (s),

f£ ( x, b),

se ( t, x), se(x,b], se ( t, b], (31.23)

C$\t.s) = *,(r)Cf (s),

te [ a, x),

se [ a, t),

G?\t, s) = Q2(t)C?\s),

te (x,b],

se[ a, x),

G?\t,s) = Q>2(t)C?\s),

te ( x,b],

se ( x, b],

where CJ-*\s), i = 1,2, j= 1,2,3, are unknown matrices depending on s. To find the matrices Cp ) , we substitute (31.23) into (31.20)-(31.22). As a result, we get Bi^(a)C{%)+

B^2(b)C^\s)

= 0,

B,<&,(a)Cf)(s)+ B&2{b)d?\s)

= 0,

,(*) Cf(s) - *,(»)C{l,W = E, (31.24) 3

2(s)Ci \s) = £, Sj*2(T)C$2)(*) + SptQdtHs)

= 0,

^•jWCl^W+^WCPV) =0. By using (31.24) and condition (31.12) from Lemma 31.1, we get

(<%) =

CJ2\s) =

(ET^-E^is),

-D-^iibWjis),

C f ' ( 5 ) =
(31.25)

C$\s) =

Dr-^^s),

Section 31

Equations with Piecewise-Continuous Right-Hand Side 293

CZ2'(s) = -(D21(c)Sl 'S2(D1( t)D-1Bj 1(s), C23)(s) = [E+c21(ti)S1 1S2(D1(' C)D-1B202 (b)](D21(s).

Substituting (31.25) in (31 .23), we can express the Green function of the impulsive boundary-value problem (31.6)-(31. 8) in terms of the fundamental matrices of the system (31.9), (31.10) without pulse influence . This completes the proof. Instead of (31 .6), we now consider an inhomogeneous impulsive system of differential equations with piecewise -continuous coefficients dx = A(t)x+g(t), t#tt, dt

(31.26)

where g ( t)E C,[a,b], i.e., dx __

tE [a, c),

(31.27)

dx = A2(t)x+g2 (t), t e (ti, b]. dt

(31.28)

dt A1(t) x+gl(t),

We investigate the above equations with either homogeneous conditions (31.7), (31.8) or the following inhomogeneous condition of impulse influence and two-point boundary condition: S1x(ti+0 )+ S2x(t-0 ) = q, B1x(a)+B2x (b) = d,

qE R ", de R".

(31.29) (31.30)

Lemma 31 .3. Suppose that the impulsive boundary -value problem (31.6)-(31.8) has only the trivial solution (31.11). Then, for any piecewise-continuous function g(t)r= C t [a, b] , the following assertions are true: (i) There exists a unique solution x (t) of the semihomogeneous single -impulse boundary-value problem (31.26), (31.7), (31 .8) in Cja, b], which can be represented in the form b

x(t) = j G(t,s)g(s)ds a

or, in more detail,

294 Collocation Method for Boundary- Value Problems with Impulses

x1(t) =

c1(t ) f

L

1 )(s)8t(s)ds + f C(1) (s) g1(s)ds + f Ci2)($)g2( s)ds11

a

t

Chapter 4

tE

[ a, ti),

T

(31.32) x2(t) = (D2 (t)

c2)(s)gI(s)ds + f CV) (s)g2(s )ds + f C 3i(s)g2(s)ds], T t Ca f

te(x,b], (31.33)

where G( t, s) is the Green function given by (31.15)-(31.17), (31.23), and (31.25). (ii) There exists a unique solution x (t) of the inhomogeneous single-pulse boundary-value problem (31.26), (31.29), (31.30) in CT [a, b], which can be represented in the form b

x(t) = H(t)+ f G(t, s)g(s)ds,

(31.34)

a

where H(t) E CT [a, b] and HI(t) = (D 1(t)D-1 Id- BA 2(b)

21(ti)Si Ig], t E [a, T],

H(t) =

H2 (t) = (D 2(t)4) 21(r)S1 1

(31.35)

x [ (E + Set 1(ti)D-1B2(D2(b) 21(ti)SI 1)g - S2fi1(T)D-1 d], t e (ti, b]. Proof. Functions (31.32) and (31.33) satisfy equations (31.27) and (31.28), respectively, and conditions (31.7) and ( 31.8). Since the integrand in (31 . 31) is continuously differentiable in the intervals [ a, t], [t,'r], and ['t, b] we get (t, s)g1(s)ds+G(t,s)Is=t-og1(t) drt) = Ja aG(

+ j (t' s) 9, (s) ds - GO, s) I _

t

at

= j^ aG(t, s) g(s) ds + gl (t) a at

g1(t) +

f aG(t , s) g, (s) ds

s - t+o T at

Section 31

Equations with Piecewise-Continuous Right-Hand Side

295

Thus, for every t E [a, C), we have (t )

b

a

b

dx G( t, s) - A1(t)x1 = J g(s)ds - Al (t) G(t, s)g(s) ds + J d a a

at

= b L at, s)

a at

g1(t)

_ A1(t)G(t, s)^g(s)ds + g1(t).

Similarly, one can show that function (31.33) satisfies equation (31.28) for all t E (ti, b]. The homogeneous conditions (31.7) and (31.8) are also satisfied by function (31.31):

Sl

b

b

a

a

f G(ti +0,s)g(s)ds + S2 JG( ti - 0,s)g(s)ds b

= 5 [S1G(ti + 0, s) + S2G(t - 0, s)] g(s) ds = 0, a

b b B1J G(a,s)g(s)ds + B2 JG(b,s)g(s)ds a

a b

= J [B1G(a, s) + B2G(b, s)]g(s)ds = 0. a

Since the second term on the right hand side of (31.34) satisfies the inhomogeneous equation (31.26) and the homogeneous conditions (31.7) and (31.8), the functions H1(t) and H2(t) in (31.34), as solutions of the homogeneous equations (31.9) and (31.10), must satisfy the inhomogeneous conditions (31.29) and (31.30). If we take H1(t) and H 2 (t) in the form HI(t) = c1(t )P1, H2(t) = (D2(t)P2,

then the above requirements can be satisfied by the proper choice of the constant vectors P 1 and P2- Thus, 5102('t)P2+S2(D1(ti)PI = q, B101(a)PI + B2'2 (b)P2 = d, and we obtain

296

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

Pt = D-1(d-B24)2 (b)4)2(T )Si 1q)1 P2 = (D2-1 (t)Sl 1[(E+SZ(D1 ( ti)D-1B2c2 (b)021(t)S^ 1)q-S24Dt (r)D-ld]. Consequently, H1(t) and H2 ( t) have the form given in (31.35). The uniqueness of solutions (31.31) and ( 31.34) follows from the fact that the homogeneous problem (31.6)-(31 . 8) has only the trivial solution (see [Nai]. This completes the proof. Convergence of the Trigonometric Collocation Method . According to the trigonometric collocation method , an approximate solution of the periodic impulsive boundaryvalue problem (31.1)-(31.3) in C, [0, T] is sought in the form of trigonometric polynomials of degree m. Thus, M XIm

= ao1) + I (a kl) cos kwt + bkl) sin kwt), t e [0, t), k=1

xm (t) _

m x2m = aa2) + I (ak2) cos kwt + bk2) sin kwt), t e (t, T],

(31.36)

k=1

where w = 2 and T ak`) _ (an,..., an)),

bk`)

_

i=1,2, k = 0, 1,...,m,

b;;j), i=1,2, k= 0,1,...,m,

are unknown (4m + 2) vector coefficients. It is possible to reduce the number of unknown coefficients by substituting (31.36) into conditions (31.3) and (31.2). Consequently, ail) and a o1 ) can be expressed as follows:

ail) = ail) = (1-coswt )-

1I(E +S21S1)ao2)

M

+ (E+S2S1coskwt)ak2)-S21q k=1

M m + (-l+cos kwt)'1ak1) + I [SZ 1,^bk2) + bk1)]sin kwt , k=2 k=1

Section 31

Equations with Piecewise-Continuous Right-Hand Side 297

a 01) = 4) = [E -(1-cos(ot)-1(E+Si1Sj)]ao2) M

+ [E - (1- cos wt)_1(E + S21S1 cos kutt)]ak2) k=1 M

[1 + (1- cos (ot)-1(-l + cos ko t)-l)} akl) k=2

+ (1-coswt)-1Sz]q-(1-costar)-1 M

X [SZ

l,^bk2) + Ebk1 )]sin kwt.

(31.36')

Then the approximate solution m

m

= ao1) + ail) cos wt + I akl) cos kwt + l bkl) sin kwt, t e [0, t), k=2

X. (t) =

k=1

m X2m = ao2) + I (ak2) cos kwt + bk2) sin kwt), t e (r, T], k=1

(31.37)

satisfies (31.3) and (31.2), and the following 4m vector coefficients (m is a finite number) remain unknown: ay),..., a.(), 4'),...,b„,1), (31.38) 42),ai2),... , am2), bi2),...,bm2).

The unknown coefficients in (31.38) can be determined by the trigonometric collocation method if we require that the approximate solution (31.37) satisfy (31.1) at 4m equidistant points in the intervals [ 0, t) and [t, T). Thus,

_L_E_ = f lti, Xim(ti)^, dt dx2L = .f2(t'+x2m(tj))+ dt

where

(31.39)

298

Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

ti = (2i-1 ) 4m, i=1,2 ,..., 2m, (31.40) tj = r+(2j-1) 4mti, j=1 , 2,...,2m. Let us consider the following homogeneous periodic single-impulse boundary-value problem: dx + Qx = 0, t *ti, tE [ 0, T],

S1x('[+0)+S2x (ti-0) = 0,

(31.41)

x(0)-x(T) = 0, where x ( t) is a function of the form (31.5) in CT [0, T] and Q is a certain constant (n x n) matrix chosen so that (31.41) has only the trivial solution (31.11 ). In this case, the matrix D in (31 .12) takes the form D = E-e(',-T)QSj S2e-TQ

(31.42)

and, by virtue of the choice of Q, the condition det D # 0 is also satisfied . Thus, there exists a uniquely defined Green function G(t, s) of problem (31.41 ). By using (31.15)(31.17), (31.23), (31.25), and (31.42), we obtain 'DI(t)= 12 ( t)= etQ, Bt = E, B2 = -E, and e-tQ(D-1 - E)esQ,

G1 (t, s) =

t E [0, T), s E (t, Sr),

e-tQD- le(s-T)Q, t (= [0, T), s E ('t, T], -e tQeTQSj 1S2e-TQD-1e(s-T)Q, t E (T , T], S E (T, T],

(31.43) e tQD- 1e''Q,

t E [0, T), S E [0, t),

G2 (t, s) = -e(T-t)QSi 152eTQD- 1esQ, t E ('t, T], S E [0, ti), [ e_tQ[E - etQSi 1S2e TQD-1e TQ]esQ ,

t e (t, T], s e (ti, T].

It follows from (31.32) and (31.34) that, for any piecewise -continuous function u(t)E CT[0,T],

Section 31

Equations with Piecewise-Continuous Right-Hand Side

u(t) _

I U1(t),

t E [0,T),

U2(t),

t E (ti, T],

299

(31.44)

the solution of the inhomogeneous single-impulse boundary -value problem

dx + Qx = u (t), t #,r,

S1x(t+0)+S2x('t-0) = q,

(31.45)

x(0)-x(T) = 0, in the intervals t e [ 0, ti ) and t E (ti, T j takes the form b

x1(t) = JG1(t, s)v(s)ds+ H1(t),

te [ 0,t),

H1 (t) = e `QD-1e(t-T)QSi 1q, (31.46) b

x2(t) = JG2(t,s)v(s)ds+ H2(t), tE ( T, T], a

H2(t) = e(t-t)QSj 1(E - S2e tQD-1e(t-T)QSi 1)q.

Here, G1(t, s) and G2(t, s) denote the following rearranged form of G1 ( t, s) and G2(t,s): e tQD-1esQ, t E [0, T), S E [0, t), G1(t, s) = e tQ(D-1 - E) esQ,

t E [0, T), S E (t, T),

etQD- le(s-T)Q, t E [0,'r), s E ('r, T], (31.47) -e(t-t)QSi 1S2etQD- 1esQ, t E [ti, T), s E [0, r), G2(t, s) =

e-tQ[E - etQSi 1S2e -tQD-1eTQ]D -lesQ t e (r, T], s r= ('r, t), - e-tQ etQSi 1S2e-tQD-1e(s-T) Q ,

t E ('r, T], s E (t, TI.

300

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

Theorem 31.1. Suppose that

x = xo(t) _

x°(t), t E [0, ti), 0 x2 (t),

(31.48) t E (ti, Tl,

is a piecewise -continuously differentiable solution of the T -periodic single -impulse boundary-value problem (31.1)-(31.3). The functions f; (t, x) in (31.4) and the Jacobian matrices A i (t, x) are defined and continuous in tE[0,T], tE (ti,Tl, I x-xo(t)I c S 8, 8>IHIc. (31.49)

Also assume that the impulsive variational system of equation (31.1) with respect to xo(t) dx = F(t,xo(t))x, t*'[, dt

(31.50)

where t)), t E [0, i), F(t, xo(t)) = A1(t, xo( A2(t, xO(t)), t E (t, Tl, possess only the trivial T-periodic solution (31.11 ) under condition (31.7). Then the following assertions are true: (i) There exists a > 0 such that, in the ball (x+Qx-(xo+Qxo)I4 < a,

(31.51)

the solution xo(t) of the periodic single-impulse boundary-value problem is unique; (ii) for sufficiently large m ( m >- mo), the system of equations (31.39) obtained by the trigonometric collocation method has a solution . This solution gives coefficients (31.38), which determine the unique approximate T-periodic solution xm (t) (31.37) that is valid in domain (31.51); (iii) for all t * ti, as m -*oo, the sequence of approximations xm (t) uniformly converges to xo(t) and pm( t) converges to mo( t) in the metdt dt ric of Lt [0, 71 so that

Section 31

Equations with Piecewise-Continuous Right-Hand Side

(31.52)

I xm(t) - x0 (t) I C,< < CIEm(U0),

dxm(t) dxo(t) dt dt 4

301

C2E.(u0), (31.53)

where c l and C2 are m-independent constants,

Em(UO) = max(Em(v?),Em(U2)),

and Em(u?) (i = 1, 2) are the best uniform approximations of the periodic continuation of 0 4f )+Qx°(t)

V9

(i=1,2)

among trigonometric polynomials of at most m th degree. Proof. On the basis of the matrices in (31.47), we introduce the operators G, and G2 that acting on functions v (t) of the form (31.44) as follows: T

GI : Glu(t) = f 61 (t, s) v(s) ds, 0

to [ 0, ti], (31.54)

T G2

G2v (t)

=

JGZ(t, s)v(s) ds ,

tE ('C,

T].

0

It readily follows from the properties of the matrices GI (t, s) and G2(t, s) that GI and G2 are completely continuous as operators from 4(0, T] into C J0, T ] . Since G, T] we can choose a ball centered at and G2 are bounded operators, in the space e ,[OT]

v0(t) = d0 + Qx0(t)

with radius

I - OI14 < s l U

V

such that the functions xl (t) and x2(t) (31.46) satisfy (31.49). We now define an operator x on ball (31.55) as follows:

302

Collocation Method for Boundary - Value Problems with Impulses

Chapter 4

Ku(t) A 1v(t) = f (t, Hi(t)+ylv(t)) + Q[y1v(t)+Hi(t)], t E [0, t), = x2u(t) = f2(t, H2(t)+` g2u(t )) + Q[G2u(t)+H2(t)],

t E (ti, T],

(31.56) where Ht (t) and H2( t) are the functions defined in (31.46). Since f1(t, x) and f2(t, x) are continuous and C31 and G2 are completely continuous, the operators N i and K2 are completely continuous in ball (31.55). Denote by Pi and P2 linear continuous operators that imbed the spaces C[0, ti] and C[t, T] into the spaces L2[0,'r] and L 2[T, T], respectively, and let P1m and P2m denote the linear operators that associate every continuous periodic function with its trigonometric interpolation polynomial of degree m. Let us change the variables in (31.1)-(31.3) and (31.39) as follows: dr + Qx = u(t), t* r, dt dXm

dt

+ QXm = Vm , t # t,

where

Vm (t) =

I Ulm(t),

t E [0, ti),

V 2m(t), t E ('C, T].

Then, by virtue of (31.45), (31.46), (31.54), and (31.56), in L[0, T ] problem (31.1)(31.3) is equivalent to the operator equation v = PKv,

(31.57)

and the system of determining equations (31.39) reduces to PmVm = Pm9CUm,

(31.58)

where

PKv =

Pix 1v,

t E[0,ti),

P2x2u,

tE('r,T].

Here, solution (31.48) of the boundary-value problem (31.1)-(31 .3) and the solution u o (t) of the operator equation (31.57) are related as follows:

Section 31

Equations with Piecewise-Continuous Right-Hand Side

v?(t) = dxa^) + Qxi(t),

303

tE [ 0,T),

0

U? (t) =

dx2(t)

x?(t) _

§1u°(t) + H1(t),

tE [ 0,'c),

x20(t) = Cg2u0(t) + H2(t),

te ( 'C, T].

+

Q xz(t),

tE ( 'C,

T I, (31.59)

There exists an analogous relationship between the solutions of the operator equation (31.58) and the approximate solution (31.37), (31.39): dx (t ) Ulm (t) =

dt + Qxlm(t),

to [ 0, C),

dx2m(t) V2m(t ) = dt + Qx2m(t), tE (ti, T],

(31.60) xlm(t) = GIUm(t) + H1(t),

tE [ 0,t),

xlm(t) = C72Um(t) + H2(t), tE ( t, T].

Since Ulm (t) and Ulm (t ) are trigonometric polynomials of degree m, the operator equation (31.58) can be rewritten as follows: t E [0, 'C), 1 Plmx l Um , U„1 = PmKUm = (31.61) P2mx2Um , tE(t,T].

Thus, we have constructed the operator equations (31.57) and (31.61), which are equivalent to the original problem (31.1)-(31.3) and the system of determining equations (31.39), respectively. Since equation (31.50) satisfies the required conditions, to complete the proof of Theorem 31.1, i.e., to establish the solvability of (31.39) and validity of (31.52) and (31.53), we can use the method proposed in [Vai2] and developed in [SaR2], where analogous results were obtained in the case of periodic systems of ordinary differential equations without impulse influence. Examples. First, we consider a very simple scalar differential equation

dx _ f (t, x) = x2 - 2 cost • x - 2 sin t, t E [0, tc), dt f2(t,x) = x2-(2+sint)•x+cost, te(it,27t],

(31.62)

304

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

x(0) = x(2n), (31.63) x(n+0)-x(7t-0) = 4. (31.64) We seek an approximate solution of the above problem in the form of a trigonometric polynomial according to (31.37) with m = 2 and co = 1. Then it follows from from (31.36') and (31.37) that apt) = a^2i + a22) - a2^^ - 1q, 2

1) = ai2^ + 1 q, 2 x12(t) = (a02) +a22) -a21) -2)+(a(2) +2) cos t

+ a41) cos 2t + b11) sinlt + b2) sin 2t,

t E [0, it), (31.65)

b12) sin t x22 (t) = ap2) +a (2) cos t + a22)

cos 2t + b221 sin 2t , t e (n, 21c].

Points (31.40) are determined as follows: 3n 5n 7n t1 = 8, t2= 8, t3= 8t4= 8

(31.66) 911 llaz 13n ti - 8 , t2 = 8 , t3 = 8 ,

1570 8

t4

Thus, the determining equations (31.39) take the form -(a(2) + 2) sin t; - 2a21) sin 2t, + bf1) cos t; + 2b21) cos 2t; L`

(a ,2) + a22) - a21) - 2) +(a (2) + 2) cos tj

2

+ a21) cos 2t; + b,() sin t, + by) sin 2t,, - 2 cos t! [(a p2) + a22) - a21i - 2) + (ai2) + 2) cos t1 + a21) cos 2t, + b,(1) sin tj

+b1 sin2t;]-2sint ., i=1,2,3,4, -ai2^ sin t'j - 42) cos ij - 2x22) sin 2ij + 2b22) sin 2ij _ (a02) + a(2) cos ij + bb 2) sin ij +

- (2 + sin ij) (aQ2) +

2)

ai

a2

2) cos 2ij +

b22) sin 2ij )2

cos ij + b1^2) sin ij + a22) cos 2i

+ b22) sin 2ij) + cos ij, i =1, 2, 3, 4.

(31.67)

Section 31

Equations with Piecewise-Continuous Right-Hand Side 305 Table 1 initial value

approximate value

initial value

approximate value

aol)

-

-1.5.10-11

a(2)

1.9

2.0

ail)

-

2.0

a12)

0.0

1.2.10-12

a(1)

-0.1

8 .7.10-12

a22i

0.1

-6.8.10-12

bi1>

-0.1

3.0.10-12

M, 2)

1.1

1.0

b21)

0.1

1.0.10-12

b22)

0.1

-7.5.10-12

The exact solution of (31.62)-(31.64) is as follows:

x =

(x1(t)=2cost ,

tE[0,9),

1 x2(t)=2+sint,

t E(1t,21c).

The unknown coefficients (31.65 ) satisfying (31.67 ) are presented in Table 1. Remark 1. The solution of (31.67) is based on an ABS-method (see [AGS] and [Jen]. Remark 2. By using an ABS-method with other initial values, we obtain another solution of (31.67), namely, aa1) = -0.24670,

a(' = 1.85868 , a21) = 0.03222,

bit) = -0.05326, b21) = -0.23316,

ao2) = 1.50395, a(2) = -0.14132, a22) = 0.28156, bi2) = 0.47541, b22) = 0.24488. As the second example, we consider the second-order differential equation

d2x x2 - x2 - x - 2 sin 2t t E [0, 7t), dt z =

-xsint+x2 - 1-2sint, ta (7t,21t

(31.68)

Collocation Method for Boundary-Value Problems with Impulses Chapter 4

306

Table 2 m=2 irn'r/a/ value

4° «<»>

4'> *f° 4°

approximate value

initial value

' 1.8-10-12 "

"1" 1.

_-2.5 10~12_

"1"

"1.0"

1.

.10.

"1"

-4.2-10" 13 "

1.

.-3.010"12.

.-1.5-10" .

«i2)

"1"

"-1.4-10" 13 "

1.

10

4 2)

"1"

"-2.110 - 1 3 "

1.

.-1.5-10 - 1 2 .

».

-1.0.

"1"

" -510-13 "

1.

.-7.910"14.

' 12

1.

"1"

1.

1.0

"1"

« ( o 2)

1.0 '

"1"

approximate value

(2)

b?

"

1.0

_

1.

-3 10 - 1 2 .

"1"

-2.4 10 -13 "

1.

. 1.4-10-12 .

['}

with the conditions x(0)

=

X(2K),

(31.69) X(0) = x(2n), and, for t = n, x(x + 0)

E

x(x-0)

2

JKT-0)

0

\-E\ i(T + 0)

Rewriting (31.68) as a system offirst-orderdifferential equations, we get *i,i(0 *i(0 =

/G[0,7t),

'1.2(0.

*(0 = '2.1(0

/e(jt,2jt],

x2(t) = ■X2.2W

(31.70)

Section 31

307

Equations with Piecewise -Continuous Right -Hand Side Table 3 m=3

aal) ail)

initial value

approximate value

0 0.011

[ -2 . 1 .10-12

0.01

]

a31)

bil) b21)

b31)

[ 0.98

r 3 . 7.10-12

]

IL-3.2.10-12

0 . 01 -0.01

- 9 . 3 . 10-13

[ -0.01 ,

[ -4 . 8.10-12

1 .01

1 .0

1- 1.01] 0.01

[ -1.0 ] 1 . 6.10-12

[ -0.01 ]

[ _1.1.10-11

0 . 01

[ -1 . 2 2.10-12

[-0.01 ]

a(22)

[ 1.0 1.6.10-12

1 : 0.01

[ -1.8.10-13

0.01 1 1 0.01 0.01

bit)

b22)

-1 . 5.10-13 [ -4.6.10-13 1.0

1 -0.01 0.0

1-2.8-10 -12

1 0.01

[ 2.0.10-12

1.6.10-12

0.01

1.0.10-13

1 : 0.01

[ -4.1.10-13

b32)

x1,2

t E [0, t),

xi1-x12-x1 , 1-2sin2t

m=

x2,2

z2(t)

3.0.10-12

0.01

a(32)

2.2.10-12

xl(t) _

[ -3.1.10-12

0.01

ail)

[1.0]

0.011 1 0. 01

1 0.01

1.0

0.99

1.0

0.991

ao)

4 .2- lo-' 3

1 1.01 a21)

initial approximate value value

t E (7n , 27c].

-x2, 1 sin t - x2 1 -1- 2 sin 2t

Conditions (31.69) and (31.70) take the form

(0)]

x1,1(0) x2,1(20 [x1,2

- [x22 (271 )

x2 2 (7G + 0) - xl, 1(7E - 0)

[ x2,20 +0)- x1,2(71 -0)

2

- [OJ.

308

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

The approximating trigonometric polynomials are as follows: M

x lm (t) = am +

xyn(t) =

(ak) cos kt+bk') sin kt), tE [ 0, 76),

ap2) + 7. (ak2)cOskt+bk2) bksin kt

tE (76, 276).

The unknown coefficients calculated for the cases m = 2 and m = 3 are presented in Tables 2 and 3. For comparison, the exact solution of the problem is as follows: 1(t) _ scot+cost cos t - sin t]'

t e [0, 7t),

x(0 = X2 (t) = rl+sint]

L

t E [76, 276).

Cost

32. Construction of Solutions of Two- Impulse Systems

Example 32.1. On the interval t E [ 0, 1 . 5 ], we consider the elementary semihomogeneous two-point two-impulse boundary - value problem 1 t 1 t dx e dt 1+t2 -1 + [1]'

t # TI = 0.5,

t]x

1

1 1

t * '62 = 1, (32.1)

l x (__o)+ I 1 ' ] X(1+0) = 0, 2

2

(32.2) 1 1 1 1 1 x(1-0) + 10 2 2 0

L

0 1 0 2 1x(0) + 20 81

1 x(1+0) = 0, 10

6 1 x (2 )

0.

(32.3)

Section 32

Construction of Solutions of Two-Pulse Systems

309

Let us construct the Green function for the corresponding homogeneous boundaryvalue problem (32.1)-(32.3). The normalized fundamental matrix of the homogeneous system of differential equations (32.1) has the form 1

*(0 =

f

.-'

1.

(32.4)

»

It follows from (30.11 ) that, in the case of the example considered , we have p = 1 and m = 2. Therefore, according to (30. 12), the domain of definition of the required Green function is decomposed into N1 = 12 subdomains. By using formula (28.9), we obtain

fi(t)( E- K-1 B2fi(2)L2L1 )(D-1(s),

-fi(t) K-1B2fi(2) L2L1

b(t) K

05 t5 $S.1;

(D-1(S),

oStSI5s51;

-1 B2fi(2)L2fi-1(s),

-fi(t)K-1 B2fi

(

2) fi

05t
-1(S),

OWL, (K-1B2fi(2)L2L1-E)
-fi(t) Li K-'B2fi

(

2)

fi-1(s),

)L2)'O-1(s),

1<s:5 2

1
B2fi`2) L2L1)

fi-1(s),

-fi(t)L2 E- L1K-1B20()L2)fi-1(s), fi(t)(E- I..LLK-1

05s5I5t51;

15t5s<_1; 2

L2fi-1(s),

O(t) L1K-1B2fi(2) fi-1(S), fi(t) L1L2(E - K-1

0<s5tS0.5;

fi-1(s), B2fi(2) )

-fi(t)L2LIK-1B2c I ) fi-1(s),

05s52, 15t52; 2 <s<1
where L1 fi-1(2)D11Clfi 2/, L2 = fi-1(1)D'z1C2fi(1), \

(32.5)

310

Collocation Method for Boundary-Value Problems with Impulses Chapter 4

K= Bl+ Bt =

r o n

Di =

" o n l_0.5 l j

r o

i/6]

[20/81

1J

B2 =

(_0.5 l j ,

&2*H)k^

C2 =

r 1

11

5

0.5J

L°-

C, =

,

D2 =

r

0.5"

I

[0.5

0J

ro.i

1 "

L0

0.1

Substituting these values of the matrices B,, C,, and D, (i - 1,2) and (32.4) into (32.5), we determine the Green function in the explicit form: f 1 I" 1 ' I T 225 - 1 3 0 i r 1 - s i 1 — — L r2, 4 8 5 [ - ' 1JL-234 368 J [ s l j l + s i f 1 fir260 130iri

-si

1 0<s < / < - ; 2

1

485L-' 1JL234 117 J L j

l j l +s

2

1 r 1 riro 6501ri -si i —

1 T l f i r 3770/9 — 485L-' l J L - 3 3 9 7 / 9

i

— ^ 2,

485L-' 1JL° 585jLs

0
1 Jl + s -520iri 502 ][s

2

-si

1 — L r2, l j l +s

0
1 r 1 fir2169/2 -2489iri - s i 1 — -J-T,

2425[_-' 1J L

216

1 f 1 fir485

— J 485L-' 1J L 0 G(f,s) = i

r !

,-iro

108

1391 i n

485 L - ' 1J L° -377J[s

i

i j l + i2

1 —L^,

1 -^f<s
l j l +s

-538/5 J|_s

2

-si

1 1 3 — - = , - < f <1 < 5 < - ;

l j l + s2 2

1 T 1 r 1T9720 48601 Tl - s i 1 — — ^2 2425[_-' 1JL7580 3780j|_s l j l + s 1 f 1 ' i r O 48601 Tl - s i 1 — — ^ r2,

485L-' 1JL° 3780jLs

1 T1 —— 485L-' 1 T1 —— 485|_-'

'IT

3658

l j l +s

-48521T1 - s i

2

-£s
2

f i r 13981/45 - 2 1 9 6 / 5 i n

485L-' lJL 4143/45

1 3 l<s£-; 2

1 0<sS-
—-h-, l j l + s2

13911 Tl - s i

—

1 Tl ——

IIs

-si

-377_||_s

1

0<>t<s<

2

1 —^, 1JL9667/3 -4251JL^ l j l + s2 f][ 4143 -48421 Tl - s i 1 — ^2 , 1JL9667/3 -3766j|_s l j l + s

2

1 3 0<s<-, l
2

2

3 l<s
(32.6)

Section 32

Construction of Solutions of Two-Pulse Systems

311

Finally, by using relation (28.10), we determine the solution of the semihomogeneous boundary-value problem (32.1)-(32.3):

3/2 t2 +t xt(t) = JG(t,s)f(s)ds =

0

0 < t < 1 2'

t+1

3/2 t2-0.8t+2.4 x2(t) = J G(t, s) f(s)ds = 2 <- t <- 1, 0 -1.4t-0.8 11

3/2

t2 +6t+9

x3(t) = JG(t,s)f(s) ds = , 1 <- t :5 2 0 6-8t

where G( t, s) is the Green function defined by (32.6), and f(t)= (t, 1).

Example 32.2. On the interval t E L0, 2] , we consider the two-pulse inhomogeneous differential equation (31.1)-(31 .2) with the five-point boundary conditions

0 1 1 0.85833 1 0 x(0) + x(0.2) + x(0.7) 0.5 1 1 -13.01667 0 1 0 1 0 0.16667 + x(1.2) + x(1.5). (32.7) 1 0 0.24691 1

By virtue of (30.4), we have

11.015 8.401666 - 68.33787 # 0.

det -30.216281

-29.251473

Then, according to (30.12) with p = 4 and m = 2, the domain of definition of the Green function is decomposed into Nt = 24 subdomains, in which, according to (30.5), we have

312

Collocation Method for Boundary-Value Problems with Impulses Chapter 4 Tl [-t

f l IT 0.6132 -0.6209' lj{[-0.7053 1.0590 . T 0.5517 -1.17191 L-0.6247 1.6624 J

] Tl -si I 5(0.2S)\ ——r, v '}[s I \l + s2

1 0<:S^t<-, 2

[" 1 11 IT -0.3869 -0.6209' |_-f l\\[-0.7053 0.0590 . T 0.5517 -1.1719"! ^ I Tl -si Li 8(0.2-s)\ — ^, v L-0.6247 1.6624 J '\[s l j l + s 2

1 0S;SiS-, 2

I" 1 / ] fro -0.7918" l-t lJlLO 1.8688 . T 0.3420 -0.42261

) |"1 -si

i

i

5(0 7 j)

*'* l'

"L-0.3293 -0.4707j - - }L, 1 J l T ? ' °"4 r I rirr-0.26H 0.2398i l-t l\\l 0.3125 -0.2946J T-0.3907 0.5756 1 +

G(t,s) = \

} [I -si

i

i

3

[ 0.3694 -0.6356JS(1-2-^} , 1 I T ? ' ° * ' 4 ! * * 4

r 1 ti rro.6992 -1.2598" ~[-t lj|[0.1042 -0.0366 I" 0.6133 -1.83951 1 fl -si i + ~™.-, ~ , ~ . 6(0.2-5» , — - 2, , [0.0957 -0.1363J '\ls l j l +j

i 0<s^-^t^\, 2

r i f i r r i 3.5440"

\_-t l J J L o 0.3098_ r0.2875 0.45731 +

} |~1 -si J S)

[0.OIO9 0.0149r - \[s

i

i

I*'*'*1'

iJlT?'

r 1 /"j rro 3.5440 '

[-» lJJLo -0.6903. f0.2875 0.45731 +

Lo.0709 0.0749j

) Tl -si

i

i

5(a7 j)

2*'*'*U

- JL, 1 J l 7 7 '

[" 1 fl jr-0.3173 1.6450" ~ L-t l J J L 0.2081 1.2794_ r-0.3176 0.61401

5(1 2 i)

l f l -si

1

U-0.0824 0.103lj - - }[, i J l T ^

i

!

3

2*'* **4

Section 32

Construction of Solutions of Two-Pulse Systems

313

T l tl jT-4.6893 1.6450" \_-t l j [[-3.6472 1.2794_ +

r 4.3081 -6.13271 M -si i 8(0.2-5)} — - ^2 , v >s |_ 3.3507 -4.7699 J [s l j l + s

i 0<s<-, 2

3 l
r 1 n j r o 13.9386" [-t lJUo 10.8412 +

[3.1917 3.3701]

} |~1 -si \ 8(0.7-s)M ——=, v L2.4825 2.6212J ' j L ' l j l + s2

i 3 -Ss
G(t,s) = r 1 t i n 10.3641 1.6655

l-t

1J1L 7.2832 2.2954 +

T -3.708 4.64011 } Tl -si 1 8(1.2-s» —-^r, I-2.8840 3.6090J \[s l_]l + .s2

3 l£s
[" 1 tl fj"9.3641 1.6655" [-/ l j [[7.2832 1.2954_ +

r-3.7080 4.64011 } [1 -si 1 8(1.2-s» —-^, L-2.8840 3.6090J )\_s l j l + s 2

3 1 < r <5< -. 2 (32.8)

Therefore, by virtue of (30.9) and (32.8), the solution of the semihomogeneous twopulse five-point boundary-value problem (32.1), (32.2), (32.7) is as follows: t2 +1

3/2

*,(*) = jdt,s)f(s)ds 0

x2(t)

x3(t)

3/2 = $G(t,s)f(s)ds 0 3/2 = $G(t,s)f(s)ds 0

=

, [t + l .

0 < t £ 0.5,

[Y-0.8f + 2.4~ =

,

0.5 < t < 1,

[ -1.4/-0.8 [V + 6/ + 9 =

, [

1 <, t <, 1.5,

6-8/

where the Green function G(t, s) is defined by (32.8). Example 32.3. Consider the two-impulse nonlinear differential equation on the interval [-1, 1]

314

Collocation Method for Boundary-Value Problems with Impulses d2x = (x-t)2 dt2

x(-0.3-0) +

1-x

Chapter 4

t*T1 =-0.3, t*TZ=0.3, (32.9)

x(- 0.3-0) - x(-0.3+0) - x(-0.3+0) = 0,

x(-0.3-0) +

x(- 0.3-0) + 2x(-0.3+0) = 0, (32.10)

x(0.3-0) + x(0.3-0) - x(0.3+0) - x(-0.3+0) = 0, -5.01729x(0.3-0) + x(0.3+0) + x(0.3+0) = 0 with the two-point boundary conditions x(-1) + x(-1) - x(1) = 0, x(-1) + x(1) = 0. (32.11) To solve this problem , we use the algebraic collocation method . According to this method, we seek an approximate solution of the m-pulse ( m = 2) problem (32.9)(32.11) for the differential equation of order n = 2 on the interval [ a, b] _ [-1, 1 ] in the form of an algebraic polynomial of degree N + n: n-1 N 1: Q ,tn+k, xN(t) = xjN(t) = PNgjtq + q=0 k=O

j = 1,2,...,m+1 = 3

tE [ tij-l, tij],

( to=a, tm+1=b).

In particular, for N = 0, we have x10(t) = Pool +P011t+Q01t2, tE [ -1,-0.3],

x20(t) = p002+P012t + Q02t2, x30(t ) = P003+P013t+Q03t2 ,

tE [ -0.3 ,0.3], (32.12) tE [ 0.3, 1],

where Q0, , i = 1, 2, 3, are unknown coefficients . One can verify that the homogeneous equation d2x dt2 0,

t*'rl=-0.3, t*'V2=0.3,

with the homogeneous conditions (32.10) and (32.11 ) has only the trivial solution. If we choose the coefficients poij, i = 0, 1, j = 1, 2, in (32. 12) in the form

Section 32

315

Construction of Solutions of Two-Pulse Systems poll = 0.425 Q01 + 1.158524Q02,

pool = 0.2125 Q01 +0.579262Q02 , P002 = 1 .966808 Q02,

P012 = -0.095114Q02, (32.13)

P003 = 1.02375 Q01 + 1.780119Q02 + 1.91 Q03, P013 = -0.7875 Q01 +0.579262Q02 -2Q03, then the approximate solution (32.12) satisfies the additional conditions (32.10) and (32.11) for any Q01, Q02' and Q03 Substituting (32.13) in (32. 12), we obtain x10(t) = (0.2125 + 0.425 t + t2) Q01 +(0.579262+ 1.158524t)Q02 , x20(t ) = ( 1.966808 -0.095114t+t2)QO2 , (32.14) x30(t) = (1.02375 -0.7875t)Q01 +(1.780119+0.579262t)QO2 + (1.91- 2t+t2)Q03• By using the roots t1 , t2, ... , tN of theChebyshev polynomial TN+1(x) = T1(x) of the (N + 1) th degree and the formula i(j) = tij

(t-a)+tij-1(b-t)

j=1,2,...,m+1=3,

b-a

which transforms these roots into the points to(j) j1(j) ..,tN c=(tij-I,tij), j=1, 2,...,m+1=3, respectively , we specify three collocation points, namely, tol) = -0.65, toe) = 0, io3) = 0.65 (32.15) (one point in each of the intervals (-1, - 0.3), (-0.3, 0.3) and ( 0.3, 1) ). Substituting (32.14) in (32 .9), for values (32.15) of the independent variable , we construct the determining equation , whence we get Q01 = 0.547822,

Q02 = 0.543758,

Q 03 = 0.774272.

Therefore, the required approximate solution (32.14) has the form

316

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

x10(t) = 0.431391+0.862781t+0.54782212, tE [-1,-0.3], x20(t) = 1.069468-0.051719t+0.543758t2, tE [-0.3,0.3], (32.16) x30(t) = 3.007647- 1.664976t+0.774272t2, tE [0.3, 11. The deviation of the approximate solution ( 32.16) from the exact solution x1(t) = e 1-t+t,

x2(t) = e0.000105-t

+t

x3(t) = e1-t +t,

does not exceed 0.3, i.e., exi(t)-xio (t)I <-0.3, i = 1 , 2, 3, tE [ -1, 1]. Assuming that N= 1 in the approximate solution XN(t) and using the roots to and t 1 of the Chebyshev polynomial TN+ 1(x) = T2 (x), one can obtain the following six collocation points: i01) = -0.897488,

i1(3) = - 0.212132, to3) = 0.402513,

i1(') =

il(2) =

-0.402513, 0.212132,

il(3) =

0.897488 (two points in each of the intervals (-1, -0.3), (-0.3, 0.3) and ( 0.3, 1)). In this case, the approximate solution has the form

x11(t) = 0.363748+0.608676t+0.129789t2- 0.1 1882t3, x21(t) = 0.992324+0.000221 t+ 0.503805t2-0.167834t3, (32.17) x31(1) = 2.691205 -1.649879 t + 1.181005t2 - 0.238603 t3, and the corresponding error does not exceed 0.02.

APPEIIDIX THE THEORY OF THE NUMERICAL -ANALYTIC METHOD: ACHIEVEMENTS AND NEW TRENDS OF DEVELOPMENT M. Ronto, A. M. Samoilenko, S. I. Trofimchuk

Al. History of the Method of Periodic Successive Approximations In the early 1960s, the scientists of the Kiev school of nonlinear mechanics headed by Academician Mitropolsky extensively investigated the problem of nonlinear oscillations. In the course of these investigations, one of the authors of the present paper suggested a method for studying T-periodic solutions of nonlinear systems of ordinary differential equations dx d t= f(t, x),

x,fE R",

f(t, x) = f(t+T,x), (1)

x(O) = x(T). In the works of Samoilenko [Saml] and [Sam2], the method indicated above was called the numerical-analytic method for the investigation of periodic solutions. In the subsequent publications of the mathematicians of Kiev and other mathematical schools, it was called the numerical-analytic method of periodic successive approximations or the Samoilenko numerical-analytic method. The scheme of the method is extremely simple and deals with the investigation of the equation t

T

x(t,z) = z + j f (s, x(s, z)) ds - TJ f ( s, x (s, z )) ds. (2) 0 0 Each solution x(t, z) of equation (2) is a T-periodic function [i.e., x(0, z) = x(T, z)], and if the third member on the right-hand side of equation (2) is equal to zero, then x (t, z) is also a solution of (1). The converse statement is also true. Thus, the method con317

318

Appendix

sists of two main parts, namely, the "analytic" part, i.e., the solution of equation (2) by the method of successive approximations, and the "numerical" part, i.e., the solution of the transcendental equation T 4(z) = T j f(s, x(s, z )) ds = 0. 0

(3)

In the operator form, the algorithm of the method can be described as follows: Denote by f the Nemytsky operator generated by the function f (t, x) :

f : C[O,T] --> C[0,T], fx(t) := f( t,x(t)), tE [ 0, T]. We introduce the mappings t J : C[0,TI ---> C[0,T], J x(t) = jx(s)ds, tE [ 0, TI,

0 P: C[O,T]-3 CPT[ R"], Px(t ) = x(t)- T[x (T)-x(0)], tE [ O,T], Q: C[O,TI - C[ O,T], Q := I-P. It is easy to verify that P is a projector that maps the space C[ 0, T] = C(C[O, T], R" ) of all continuous vector functions into the subspace

CPT[R" ] of T-periodic functions. Then, according to the basic idea of the method, the boundary -value problem (1) is equivalent to the system

x = z+P.ifx, Q.Ofx=0. In the last system (the first equation of which is integral, and the second of which is determining), the unknown is the pair (x(.), z), and x(t), x(0) = z, is the required T-periodic solution. ALL Basic Assumptions. Let us formulate the key ideas of the method. It was established in [Sam!], [Sam2], [SaRI], [SaR4], and [SaR6] that the method "works" under the following assumptions: (H1.1) The function f (t, x) = (f1(t, x), ... , f"(t, x)) is defined, continuous, and Tperiodic in t in the domain ( t, x) E R X D, where D is a domain in R.

Appendix

319

(H 1.2) The function I f (t, x) I = (If, (t, x) 1, ... , I f"(t, x ) 1) is bounded in its domain of definition by the vector M E R+ and satisfies the Lipschitz condition in x nxn: with matrix KE R+

If(t,x)I <_ M, If(t,x')I -If(t,x")I <- Kix - x"I. (4) (H1.3) The set Dp = {xE R": S(x, (3) CD } # 0, (5)

which consists of points x E R" lying in D together with their (3 = T Mneighborhoods , is nonempty. (H1.4) The maximal eigenvalue ? max (K) of the matrix K is such that Tkmax (K) < q,

(6)

where q = 3.1 ... (see also Section A1.2. 1). Here, we have used t h e notation I (x t, x2, ... , x") I _ Ox 1 I, I x2 1, ... , I x" I) , and the inequality between vectors is understood as componentwise . The use of the "pseudonorm" I x I instead of the standard norms II II in R" and the application of estimates of the form

Ilf(t,x)II -f(t,x')II <- KIIx - x"II, KE rat, is explained by our intention to obtain the most convenient applicability conditions. In what follows , it is convenient to use the term "periodic boundary-value problem" rather than "periodic solution of a system of differential equations" and consider the segment [0, T] instead of R in the domain of definition. Consider a sequence of T-periodic functions {xm(t, z)} that depend on a parameter z E III;" and are defined by the recursion relation t

T

xm(t,Z ) = Z+ J [.f(S,xm_t(S,Z)) - TJf(ti,x,"_i(ti,z))dc ]ds, (7) 0

0

where in = 1 , 2, ... and x0(t, z) = Z. Obviously, xm(t,z) = xm(t+T, z) and xm(0,z)=z for any mE N.

320

Appendix

The following statement on the convergence of this sequence and on the properties of the corresponding limit function x*(t, z) is true: Theorem 1. Let the T-periodic boundary -value problem (1) be such that conditions (HI. 1)-(H1.4) are satisfied. Then, for every z e D the following assertions are true: (i) the sequence { xm (t, z)} is uniformly convergent:

lim xm (t, Z) = x*(t, z);

(8)

(ii) the function x*(t, z) is the unique T-periodic solution of the equation

x(t) = z+ f if(s,x(s)) - T f f(ti,x(t))d[] ds 0 0

(9)

in the domain D or, which is the same , it is a T -periodic solution of the "perturbed " boundary-value problem dx dt =f(t,x) - 0(z), (10) x(0) = x(T), where the "perturbation term " has the form T

(z) = T if .f(t, x*( t, z)) dt; 0 (iii) the following estimate is true: kx*(t,z)-xm (t,z)I < - a1(t)Q m(E-Q)-IM,

where a I (t) = 2tl l T I <_ 2, Q= T K; \ / q (iv) the function A(z) : DS-4 R" is a Lipschitz function.

tE [ 0, T], (11)

Appendix

321

For a solution x"(t, z) of problem (10) to be a solution of the original periodic boundary-value problem (1), it is necessary and sufficient that z satisfy the determining equation

A(z) = T j f ( t, x*(t, z )) dt = 0 (12) 0 (a system of n equations for n unknowns). The following question arises: Does this rather special technique used for the construction of the determining equation allow one to find all T-periodic solutions of problem (1)? The statement below shows that the determining equation (12), in fact, "detects" all T-periodic solutions of (1) in the domain under consideration. Theorem 2. If conditions (H]. 1)-(H1.4) are satisfied, then the roots z = z * of the determining equation A(z) = 0 determine the initial values of all T-periodic solutions x (t) of problem (1) lying in D and such that z E DP . Of course , in practice , the computation of the function A(z) is not always possible, except for the case of autonomous systems where A(z) = f (z). Therefore , we must restrict ourselves to the construction of the first approximations T

Am(Z) =

Tf f(t,x m(t, Z))dt,

(13)

0

where m is a certain natural number. The statement below shows how one can infer the existence of solutions of the exact determining equation (12) on the basis of the properties of the approximate determining function Am (z) . Theorem 3. Suppose that conditions (H1. 1){H1.4) are satisfied and, in addition, (i)

there exists a convex closed domain D1 C D a such that, for some fixed integer m > - 1, the approximate determining equation

Om(Z) = 0 has a root z = zm of nonzero topological index in D,; (ii) the following inequality holds on the boundary aD1 of the domain D1: inf I Am(z)I > 2Qm+1(E-Q)-1M. ZEat7, 3

322 Appendix Then the T -periodic boundary-value problem (1) has a solution x = x* (t) with the initial value x' (0) = z * E Dl . The numerical-analytic method also allows one to specify domains that are free of Tperiodic solutions. Theorem 4 . Suppose that conditions (H1.1)-(H1.4) are satisfied and a subdomain D2 C D P is such that

^Am(z)I > Zsup K(E+3K(E-Q)- ')]p(z,aD2) +

[

3Q»`- I(E

-Q)-'M (14)

for some m E N and Z E D2. Then the boundary-value problem (1) has no Tperiodic solutions with initial values from D2. Obviously, inequality (14) for all m and z E D2 with ">" replaced by "< is a necessary condition for the subdomain D2 to contain a point that may serve as the initial value for a periodic solution of problem (1). A1.2. Improvement of Applicability Conditions. Later, the numerical-analytic method of periodic successive approximations (for brevity, called below the "method") was developed in various directions. Part of the work was devoted the extension of its original scheme to more general classes of boundary-value problems; a survey of these investigations is given in the forthcoming parts of the present paper. However, it should be noted that some new generalizations of the method substantially differ from its original version and can be regarded as independent techniques inspired by [SaR1], [SaR4], and [SaR6]. We discuss this question in more detail in the final part of this survey. A considerably lesser part of the work was devoted to the extension of the field of applicability of the classical version of the method, i.e., to finding more exact convergence estimates, weakening conditions (H1.1)-(H1.4), etc. The importance of such investigations is also explained by the fact that the corresponding results can also be applied to other types of boundary-value problems. Below, we consider some of these results in more detail. A1.2.1. A Priori Estimates. The improvement of the a priori estimates appearing in the theorems of Sec A1.1 is, first of all, based on the more precise determination of the constant q in inequality (6). (It is clear that the greater this constant, the wider the field of applicability of the method.) This problem was studied in numerous works. In the list below, we outline the evolution of the problem with necessary comments. (1). 1965: Samoilenko ([Sam!], [Sam2]), q = 3.1.

Appendix

323

(2). 1976: Samoilenko and M. Ronto ([SaR4], [SaR6]), q = tt. (3). 1982: Samoilenko and Laptinsky [SaLa], q = 3.416... . (The proof has a gap, which can be eliminated with the use of the Krein-Rutman theorem [KrR].) (4). 1985: Zabreiko and Evkhuta ([EvZl], [EvZ2]); E. Trofimchuk [Tro2], q = 3.416... . (The a priori estimates obtained are rather complicated because they require the computation of A m (I - A)-1, where A is a certain integral operator.)

(5). 1992: Kwapisz [Kwa3], q = 10 = 3.1622. (6). 1996: M. Ronto and Meszi ros [RoM], q = 10• 3 (7). 1996: M. Ronto, A. Ronto , and S. Trofimchuk [RRT], q = 3.416... (A method is suggested that allows one to obtain simple a priori estimates.) (8). 1996: M. Ronto, A. Ronto, and S. Trofimchuk [RRT], q = tin. (This is a hypothesis; see also E. Trofimchuk [Tro2], where, in particular, it is proved that q < 2n.) Note that the results of [Saml], [Sam2], [SaR4], [SaR6], [Kwa3], and [RoM] also contain simple a priori estimates for the method. Problem 1. Find an analog of estimate (6) under the condition

If(t,x)- f(t,x")I s K( t)Ix -x"I,

(15)

where the components K. (t), i,j = 1, 2,... , n, of the matrix K(t) are nonnegative summable functions. The second condition important for us is assumption (H1.3). Clearly, it would be the optimal solution if one succeeds in discarding this condition completely. One can also try to diminish the quantity

M

T 2

M.

An obvious improvement is attained if we replace the first inequality in (4) by

If(t,x)I 5 M(t).

(16)

324

Appendix

Then we can choose 0 as follows:

p =

max 11- t I f M(s) ds + t J M(s) ds . Tt :E[0,T] \ T 0

The estimate from Lemma 3.1 in [SaRI, p. 13], which has the form t

f

f(t)

I +T

T f f(s)ds]dt

2(t- r)(1 - t-ti) max If(t)I V to[ ti,ti+T], T

tc[T,T +T]

(17)

was improved by M. Ronto and Meszaros [RoM] for ti = 0 as follows:

f [f(t) -Tf f(s) ds dt

]

2a1(t) I mo ]f(t)-t mm f(t)I

Here, a 1 ( t) = 2t (1 - t), 1 a 1(t) <- 2 . Thus , (H1.3) can be replaced by the condition T(H1.3'), which is, in fact , (H1.3) with R = T M', where

M' = 1

L

max

2 (t,x)E[O, T]xD

f( t, x) -

min

f(t, x)

(t,x)E[O,T]xD

J

Note that we always have M'<- M, and M'= M if and only if max f (t, x) min f (t, x). tE[0,T]xD tE [0,T]xD

Finally, M. Ronto and S. Trofimchuk [RoT2] showed that (H1.4), together with the sublinearity condition lf(t,x)I <- Klxl+r(t) `d( t, x)E R"+1 r,(t)E L1[0,T] , i= 1,2,...,n, allows one to discard assumption (H1.3).

Appendix

325

Note that similar improvements of the estimates were also made for other boundaryvalue problems. In the subsequent parts of the present paper, we give, in particular, several new estimates for second-order equations (see Section 3.1) and difference equations (Section 3 .7; see also M. Ronto, A. Ronto, and S. Trofimchuk [RRT]). In this direction of development of the method , we should also mention interesting results obtained in [RoT2] . It turns out that, in the course of investigation of the problem of existence of solutions in a "large " domain D (say, in a ball of great radius ), in certain cases , we can restrict ourselves to the verification of conditions (H1.2)-(H1.4) only in a neighborhood of the boundary aD (e.g., a sphere). Inside the domain D, where the required T-periodic solution of equation ( 1) lies, f (t, x) may not be a Lipschitz function. The following statement is true (Theorem 20 in [RoT2]): Theorem 5. Suppose that, for all xi: IIx;II > d and almost all t E [0, T], the function f (t, x) satisfies the Caratht;odory condition in R x R" and the Lipschitz condition

IIf(t,x0-f( t,x2)II <- l(t)IIx,-x211, where l(t) E L 1 ([0, T]), l(t) > 0. Also assume that \t T max 11 - t I J l(s)ds+ - J l(s)ds < 1 1Elo,T3 ` T/0 T t 2 Then (i) the many-valued determining function A(z) : R" --> Comp (

Rn)

is well de-

fined; (ii) there exists r> 0 such that, for valued;

1

I z > r, the function A (z) is single-

(iii) if, in addition,

A(z)

*

0

V

z : 11z II = r,

deg (A(z), aBr( 0), 0) * 0,

then equation (1) has a T-periodic solution x(t): [0, T] -> Br(0), where Br(0) is the ball of radius r centered at the point x = 0.

326

Appendix

A1.2.2. Connection between the Determining Equations and the Symmetry Properties of the Differential Equation. In many cases , by using the symmetry properties of the function f(t, x) in equation ( 1), one can simplify the investigation of the determining equation (12). Following Samoilenko and M. Ronto [SaRI , p. 37], we consider the case where the function f (t, x) is odd in t, i.e.,

f(-t,x) = f( t,x).

(18)

Theorem 6. Suppose that the right-hand side of equation (1) satisfies conditions (Hi.1)-(H1.4) and (18). Then all solutions x =x(t) of system (1) with initial values x(0) E Dp are T-periodic. Proof. This statement is an immediate consequence of the fact that the integral mean value of an odd periodic function over the period is equal to zero and, therefore , A (z) = 0 for all z e D0. This simple theorem can be regarded as the first step in the use of property ( 18). Theorem 6 can be significantly strengthened. Problem 2. Prove the existence of a T -periodic solution of equation ( 1) in the case where condition (18) is satisfied, assuming only the continuity of f (t, x). Hale [Hall] used more general symmetry conditions for the function f (t, x), assuming that there exists a matrix Q with the properties

Q2 = 1, Qf(-t,Qx) = -f( t,x).

(19)

This condition includes , in particular, the cases of odd and even functions f (t, x). Modifying Hale's results (Theorems 6.5 and 6 .6 in [Hall , pp. 60-61 ]) applied to the scheme of the numerical-analytic method, we obtain the following strengthened version of Theorem 6: Theorem 7. Suppose that conditions (Hi.l)-(H1.4) and the symmetry condition (19) are satisfied. Then a solution x(t, z) of the integral equation (2) satisfies the relations

Qx(-t, z ) = x(t,z), f(-t, x(-t, z)) = -Qf(t,x(t,z)) for all z such that Qz = z.

Appendix 327 Proof. By virtue of Theorem 1, there exists a unique T-periodic solution x (t, z) of equation (2) for all t E (- 00, oo ). Let us replace t by - t in (2) and multiply both sides by Q. As a result, we get -t

T

Qx(-t,z ) = Qz + Q J f( s,x(s,z ))ds + TQ f f( s,x(s,z))ds. 0

0

Assume that z satisfies the equality Q z = z. Replacing s by -u and using property (19) and the fact that the integral mean value of a T-periodic function f (s, x(-s, z)) can be computed over any interval of length T, we arrive at the identity t

-T

Qx(-t, z ) = z - f Qf (-u, QQx(-u, z )) du - -t f Qf (- u, QQx (-u, z)) du 0 0 = z + f f(u,Qx(-u, z)) du 0 t

- J f (u,Qx (-u, z)) du

-T T

= z + f f(u,Qx(- u,z))du - T

0

J Qf(u,QQx (-u,z))du.

0

Since a solution of the integral equation (2) is unique , we get Qx (- t, z) = x(t, z). Further, we obtain

f(-t, x(-t, z)) = QQf(- t, QQx (-t, Z)) = QQf (-t,Qx(t,z)) = -Qf(t ,x(t,z)), as required. Note that an analog of this result for the Cesari -Hale method (see Section 2.2) is given in Corollary 6.1 in [Hall , p. 61]. Corollary 1. Suppose that the conditions of Theorem 7 are satisfied and the matrix Q is diagonal, i.e.,

Q = diag ((; t,a29...,an), a.€{ -1,+1}. Denote by k- and k+ the number of "-1" and "+ 1" entries in the matrix Q, respectively, so that k- + k+ = n. Then the vector

328

Appendix a

\> " Y " a 2 - - . ~ y - an\ =

z = [ ~Y~

z{ah,...,ait,),

which depends on k* numbers a (| , a, 2 ,..., a ^ e R [the indices 'i.t2.••-.'*+ specify the position of a "+1" enrry « ^« s«n'e5 o,, a 2 , . . . , on] and satisfies the system of k~ equations in k* variables

'-^^U J y;(Jf ^

z(a .

, a . 2 ,..., s + )))rf, = 0,

i = 1, 2 , . . . , n,

o determines the initial value for a T-periodic solution of system (I). Thus, the symmetry condition (19) allows one to reduce the dimension of the determining system (3) from n to k~, which obviously simplifies the scheme of the method. Let us consider the following example: Example 1. Let us investigate a r-periodic solution of the second-order equation

* " = / ( ' . x, A where f{-t, -x, y)= -f(t, x, xf). We rewrite this equation as the system dx d-t=y< (20) ^=fU,x,y). at It is easy to see that the right-hand side of (20) satisfies condition (19) with the matrix Q = diag (-1,1). Therefore, any vector of the form z = (0,a) is the initial value of a T-periodic solution of system (20) whenever hypotheses of the type (H1.1MH1.4) hold and the number a satisfies the scalar determining equation T

±]y(s,(0,a))ds 0

T

= j.]y(s,z(a))ds 0

= 0.

(21)

329

Appendix

Other interesting examples that demonstrate how property (19) can be used to substantially simplify the investigation of determining equations can be found in [Hall, Section 7]. It is worth noting that these arguments can also be used for the investigation of the characteristic equation of the averaging method (see [Hall, Section 7]).

A2. Relation to Other Investigations It is natural to guess that the numerical-analytic method should have certain common features with some of the other numerous approaches available for the investigation of boundary-value problems for ordinary differential equations (see, e.g., Conti [Con], Mawhin [Maw3], [Maw4], Capietto, Mawhin, and F. Zanolin [CMZ1], Boichuk, Zhuravlev, and Samoilenko [BZS], Kiguradze [Kig3], Farkas [Far2], and Rouche and Mawhin [RouM]). Indeed, the equations of the method and even its basic techniques are often closely related to equations and techniques of other methods proposed long before it (the Lyapunov-Schmidt method [Maw4, p. 27] or the Cesari-Hale method [Hall], [CeH], [Cesl]-[Ces3], which appeared almost simultaneously). However, the different objectives pursued by the authors resulted in essential differences in the results obtained. A2.1. Lyapunov-Schmidt Equation. Let us show how the main equations (2), (3) can be obtained by the direct application of the Lyapunov-Schmidt scheme to the periodic boundary-value problem (1). Following Mawhin [Maw4], we first recall how the Lyapunov-Schmidt equations are derived. It is reasonable to use the Lyapunov-Schmidt method for the investigation of the equation Ax

=

Nx,

(22)

where A , N : X - Z are continuous operators in Banach spaces X and Z, the operator A is linear and irreversible , and N is a nonlinear operator satisfying the Lipschitz condition with sufficiently small Lipschitz constant . ( In the case where A is invertible , by virtue of the Banach fixed -point theorem , equation ( 22) has a unique solution and can be studied by the method of successive approximations.) For the operator A, we require that there exist projectors P : X -*X, P (X) = ker A, and Q : Z -* Z, Q (Z) = Im A. We set X = kerA®X,, Z = ImA®Zt,

(23)

and denote the restriction A : Xt - Im A by A Then, according to the Banach theorem on the inverse mapping, the inverse operator A i t : Im A -* X t exists. Let x = k + v be the decomposition of an XE X corresponding to (23). Then

330

Appendix (22) a Av = N(k+v) p [Av=QN(k+v), (I-Q)N(k+v)=0] [v = Ai1QN(k+v), (I-Q)N(k+v)=0]. (24)

We fix an arbitrary k E ker A. By the assumption of the smallness of the Lipschitz constant for N, the first equation in this system has a solution v = v (k) (this solution can be found by the method of successive approximations). It only remains to find solutions k* of the bifurcation equation (I-Q)N(k+v(k)) = 0 to obtain all solutions k* + v (k") of the equation under consideration . Note that the reasoning presented above can obviously be given the local character by restricting the operators A and N to a certain neighborhood in X. In the special case of problem ( 1), it is natural to set X = {x(t)E C1 [0, T] : x(0) = x(T)j, Z = C[0, T],

Ax = t, (Nx)(t) = f (t,x(t)). d Then, obviously, T kerA = {x0E R"}, ImA = {v(t)eC1OTI: T f v(t)dt = 0

0 and, therefore, we can choose (Px)(t) = x(0),

(Qx)(t) = x(t)_

T

((I-P)x)( t) = x(t)-x(0),

T

T

T

f x(t)dt, ((I-Q)x)(t) = f x(t)dt. 0 0

Rewriting the Lyapunov-Schmidt equations (24) in this notation , we arrive exactly at equations (2), (3). Furthermore, according to the Lyapunov-Schmidt scheme, we should first solve equation (2) with respect to x (t, z) by iterations , fixing z e R"= kerA and assuming that the nonlinear Nemytsky operator f (t, x) satisfies the Lipschitz condition with sufficiently small constant . The next step of the method is to study the bifurcation equation. This completely coincides with the procedure used by the numerical -analytic method.

331

Appendix

A22. Cesari-Hale Method for Weakly Nonlinear Differential Equations. Let us return to problem (1). Assume that conditions (H1.1MH1.3) are satisfied. Instead of inequality (HI.4), we use the following more restrictive condition: (H2.1):

TXmax(K)<2.

Consider the sequence of ^-periodic functions depending on a parameter z e R" and defined by the recursion relation

ir

' r xm(t,z)=z+

j

f(s,xm_x(s,z)--\f(i,xm_x(x,z)dx

m = 1,2,...,

x0(t,z) = z,

ds, (25)

where the vector i,m_i is chosen at each step so that the integral mean value of function (25) satisfies the following relation:

Sx

Jf*z) = j\xm^z)dt

= z.

(26)

0

[In (25), the components of the vector %m_ { determine the lower limits of integration for the corresponding coordinates of f(t, x).] The existence of such mean values \m_x was proved in [Hall, p. 51]. Note that, although \m_x is not uniquely determined at each iteration step, the successive approximations xm(t, z) are well defined independently of the %m_\ chosen. Indeed, if there were two distinct functions xm(t,z, %'m-\) and xm(t, z, £„_)), then their difference would be a constant because dxm{t,zA'm-l) dt

=

<**„«. 2. &-l) dt

Since both functions have the same mean value z, we can conclude that die constant mentioned above is equal to zero. Thus, the iteration process (25) is well defined. The following statement on the convergence of sequence (25) and its limit function is true: Theorem 8. Suppose that the T-periodic boundary-value problem (1) satisfies conditions (HI. 1)-(H1.3) and (HI.4). Then the following assertions are true:

332

Appendix (i) a sequence { xm (t, z )j of the form (25) uniformly converges to a certain function x*(t, z), i. e., limm_xm (t, z) = x*(t, z); (ii) one can find a vector E* such that the function x(t) = x*(t, z) is a unique (in D) T -periodic solution of the integral equation

x(t) = z + f [f(sx(s)_ 5f(tx(t) dt]ds , o i.e., it is a T-periodic solution of the perturbed boundary -value problem dx

dt

= f(t,x) - A(z),

x(O) = x(T) such that Sx (t) = z, where T A(z) = T f f(t, x*(t, z))dt ; 0

(iii) the following error estimate is true:

^x*(t,Z)-xm (t,Z)I <- 2Qm (E-Q)-1M,

tE[0,T],

where Q = T K and q=2. q Corollary 2. Suppose that conditions (HI.1)-(H1.3) and (H2.1) are satisfied. Then the roots of the determining equation A(z) = 0 determine all T-periodic solutions x(t) of system (1) lying in D and such that Sx(t)=zE Dp. Note that, in our presentation, the statements of Theorem 8 and Corollary 2 and their proofs are suitable modifications of Theorems 6.1 and 6. 2 in [Hall , Section 6].

Appendix

333

Proof of Theorem 8. By virtue of Lemma 3.1 in [SaRl, p. 13], for any continuous function f(t), the following inequality holds for all t r= [t, ti + TI: r T +T

f [

f(t)-T j f(s)ds dt

T

al(t-'t) max I f(t)I < T M, tE[t,s+T] 2

Z

where al(t) = 2t(1-T).

Therefore, for all t e [ r, ,r + TI, we have MT Ixk+l(t,Z)-ZI < - 2 Hence, the iteration process (25) is well defined for all z e D p. To estimate the difference 8k(t, Z) = xk+1(t° Z) -xk(t, Z), we first note that, according to (26), we have

SSk(t, z) = 0,

dSk(t, z) = h(t), k = 1, 2, ... , dt where h (t) = f(t, xk(t,z ))-f(t,xk_1(t,z))-Sf( t,xk(t,z))+Sf(t,xk_ 1(t,z))-

Since Sk(t, z) is a T-periodic function and Sh(t) = 0, one can indicate a vector such that t k(t, z) = j h(s) ds .

Consequently,

334

Appendix

I s k(t, Z ) I t

l

g*+T

J f(t, xk(t, Z))- f(t, xk-1(t, z))) - -

1

T 4.

(.f(S, xk(S, z))-.f( s, xk-1(s , z)))ds dt

KT < 2 I Sk-1(t, Z) I

The subsequent steps of the proof of Theorem 8 are similar to the proof of the principal lemma in [SaRl , p. 16]. Summarizing the arguments presented above, we can draw the following conclusions: In spite of the coincidence of the differential equations for iterations within the framework of the numerical-analytic method and the Cesari -Hale method , the integral equations of these methods are different. In the numerical-analytic method, the parameter z determines the initial values of Tperiodic solutions , whereas in the Cesari-Hale method, it determines their integral mean values. Therefore , solving the determining equation in the numerical-analytic method, we immediately determine the required T-periodic solution , whereas in the Cesari-Hale method, solving the determining equation , one obtains only a statement on the existence of a T-periodic solution with mean value z, the initial value of which (or the vector is still to be found. The Cesari-Hale method does not give a constructive procedure for the calculation of the vector in (25). Moreover, since ^m _1 varies from step to step, the a priori estimates of the rate of convergence given in [Hall] are weaker than those of [SaRi]. Conceivably , this is the reason why the investigation in [Hall] is focused on systems of differential equations with a small parameter ICI << 1. Note that, in the latter case, the numerical-analytic method and the Cesari-Hale method, in fact, coincide with the averaging method. Remark 1 . The convergence estimates in Theorem 8 (q = 2) are more precise than those in Theorem 6.1 in [Hall, Section 6], where q = 0.5. A2.3. Cesari Scheme for Essentially Nonlinear Systems of Equations. There arises the following question: Is it possible to modify the method so that restrictions on the right-hand side of system ( 1) of the type (H1.3) and (H1.4) can be omitted ? It turns out that this can be done in several different ways. One such modification proposed by Kwapisz [Kwa4] will be discussed in the subsequent parts of this survey . Here, we dwell upon the Cesari method [Cesl] that solves an analogous problem for the Cesari-Hale method described above . Since, as shown above , the numerical -analytic method and the Cesari-Hale method are quite similar , by analogy we can improve the method of periodic successive approximations and its modifications.

Appendix

335

The idea of the Cesari method is well described in [Hall], [RouM], and, especially clearly, by Knobloch in [Kno]. Following these works, we briefly describe its application to the solution of the periodic boundary-value problem (1). In the Cesari method , one fixes not the mean value z but 2m + 1 n-dimensional vector parameters

(a0, a1, b1, ... , am, bm)= a E R(2m+1)n that determine the initial function

x0(t,a ) = ao+ m (akcos2^t+bksin2 t . T k=1 \

J

The integral equation of the Cesari method has the form x(t) = x(t, a) t

T

= x0(t,a )+ f f(s,x(s,a))ds-Tf f (s, x(s, a )) ds + F(f)(t). (27) 0 0 According to the idea of the Cesari-Hale method, the operator F should be chosen so that the function

J

J As, x(s, a)) ds - T f (s, x(s, a)) ds + F(f)(t) 0 0 is T-periodic and does not contain the first 2m + 1 harmonics. In particular, if m = 0, we arrive at the integral equation of the method of periodic successive approximations. In different notation, equation (27) can be rewritten as t

x(t) = x0+

J

T

f(s,x(s))ds- T f(s,x(s))ds+PmLfx, 0 0

J

or, which is the same, x = xO+Lfx-PmLfx = x0+[1-PmLfx, (28) where

336

Appendix t

[Lfx](t)

T

= f f(s,x(s))ds- f f( s,x(s))ds 0

T 0

and

[Pmx](t) =

I

C

k=0

Akcos

T

2

kt+Bk sin 4t

is the truncated Fourier series for the function x(t). Assume that x = x( t) is a solution of equation (1). Then x(t) is obviously a solution of problem (1) if

1T - f f(s,x(s,a))ds = 0, T 0

(29) dx^ta) =

C

d PmLfx ('), dt

m? 1.

Elementary calculations show that the determining equations (29) can be rewritten as follows: A(a, t) = dx dt, a ) - Pm.f (t, a ) = 0.

(30)

Equating the coefficients of the corresponding harmonics , we obtain from (30) a system of (2m + 1)n scalar bifurcation equations with the same number of unknowns. In [Kno], conditions were obtained that guarantee that the iteration process xm+1(t, a ) = x0+ [I-Pm ]Lfxm,

m= 0, 1, ... ,

is well defined and convergent in CPT. Assume that the following conditions are satisfied: (H2.2) The function f (t, x) is defined and continuous in the domain

W = {(t,x):05t5T, IxI„= max Ixi15A l i=1,...,n (H2.3)

.f ( t,4(t))E

L1[O,T] Vo (t) e CPT, RR (t)ISA;

(31)

Appendix

337

(H2.4) there exist ko > 0, k 1 > 0, and l > 0 such that

If(t,x)I, <- kplxl_+k1,

If(t,x)-f(t,y)I„ s 1 I x -yI_

for all tE [0,T], (t,x), (t ,y)EW; (H2.5) the number m is so large that x(m)•1 < 1, x(m)(koA+ki) < A (A
R(2m+1)n

is such that I xo(t, a )I 5 A-x(m)(k0A+k1). Then the iteration process (31) converges to the unique solution t; = l;(t, a) of equation (28), and l;(t, a) is a periodic solution of the differential equation (1) if and only if A (t, a) = 0 in (30). Conversely, if the function t:, = 4(t) E CPT satisfies the differential equation (1) and is such that I ^(t)I < A,

then

4 _ 4(t, a) and A(a, t)= 0.

Pm b( t) = x0(t, a),

338

Appendix

Remark 2. In [Kno], in fact, a more general case was considered where ko = ko(t), k 1 = k 1(t), and l = l (t) in condition (H2.4). It is hard to argue with the remark of Hale [Hall, p. 131] that the investigation of the determining equation (30) is extremely difficult. One of the possible ways here is to approximate the vector a = a0 by the vector determined by the m th-order Galerkin approximation of a solution of equation (1). This approach was used by Cesari in [Ces 1 ] for studying the equation x'+x3 = sint.

Also note that, in the original paper [Cesl], Cesari used the L2-norm, which distinguishes this work from [Kno] cited above. Many interesting studies were inspired by [Cesl], and it would be useful to analyze their results from the viewpoint of the numerical-analytic method. In this connection, we mention the monograph [Maw4] and the vast bibliography therein, and also the works [Hall] and [Ces2]. A2.4. Mawhin Theorem. It is rather beneficial to compare the integral operators of the Cesari and Cesari-Hale methods with the integral operator Mj introduced by Mawhin in [RouM, pp. 101-107]. In particular, for the differential equation (1) without the separated linear part, the operator Mj is defined as follows: T T Mix(-) = T f x('t)dt+TJJf(t,x(t))dt

0

0 T + f [.ft , x('0) - T f f((Y, x( a)) do dti

0

0

T

t

T

0

0

0

T f f [ it. x(c)) -T f f(a, x(a))da ] dt] dt. It is proved in [RouM, p. 102] that all T-periodic solutions of equation (1) coincide with the fixed points of Mj if det J * 0. The following interesting result was obtained in [RouM] with the use of the operator Mj: Theorem 10. Let fI E R" be a bounded domain. Suppose that the differential equations of the family

Appendix dx = Ef(t,x), dt

0<E< _ 1,

339 XE R °,

tE[0,T], (32)

have no T-periodic solutions with initial data x ( 0) E aQ and, moreover, the set II(E) = {xE(t)} of all T-periodic solutions of equation (32) with initial values in Q is uniformly bounded, i.e., I XE(t)I < M

VE, V xE( t)E TI ( E): xE(0)E S2.

We set T

AC=0(z ) = T- ' f f(s, z)ds * 0 0 for all z E an . In this case, if deg (AE_0(z ), ac , 0 ) * 0, then the periodic boundary-value problem (1) has at least one solution. We borrowed the formulation of this statement from [RoT2], where it was proved with the use of the numerical-analytic method. A more general version of Theorem 10 for systems of k th-order differential systems can be found in [RouM, p. 1861 (Theorem 3.17). Note that a statement close to Theorem 10 was first proved by Mawhin in 1969 for locally Lipschitz equations with the use of the Cesari method (see [RouM] and [Maw!]). This theorem can also be regarded as a consequence of the theory of coincidence degree (see Rouche and Mawhin [RouM, pp. 225], [Maw4] and Gaines and Mawhin [GaM2]).

A3. Application of Modifications of the Numerical -Analytic Method to the Investigation of Various Boundary-Value Problems As indicated above, the numerical-analytic method was first introduced as a tool for the investigation of T-periodic solutions of systems of first-order differential equations. Later, numerous papers appeared where the scheme of the method was extended to new classes of problems. In this section, we analyze results obtained in this direction. Numerical-analytic algorithms for boundary-value problems that are more general than periodic problems are considered in the subsequent parts of this survey. A3.1. Second-Order Equations. In applications, it is often necessary to study a Tperiodic boundary-value problem for the second-order scalar differential equation

340

Appendix

d22 dt

f

+0) 2 x =

dxl

xe R.

`'x'dt '

(33)

Equation ( 33) is equivalent to the system dx dt = wy,

(34) dy 1 dt _(Ox + w At' x, y, w).

The direct application of the numerical -analytic method to system (34) shows that the frequency w cannot be very large. For example, even if f (t, x, y) = 0, the following inequality must hold: wT

<

1,

(35)

q where the value of q is given in Section A 1.2.1. This difficulty can be eliminated by the following change of variables proposed by van der Pol: x = z1 sin c t + z2 cos t ,

y = z cos

wt

- z2 s in

wt,

where Jw - w l is sufficiently small. The system of equations obtained can now be investigated with the use of the numerical-analytic method because, in this case , a condition of the type (35) (with f( t, x, y) = 0) takes the form ^w-wIT < 1. q However, there is no need for such a change of variables when Co is sufficiently small or co = 0. For example, a system of this sort arises in the course of investigation of torsional harmonic oscillations of a shaft consisting of two bodies with large moments of inertia joined together by a nonlinear elastic brace to one of which a sinusoidal torque is applied [BoM], [SaRI]. In particular, in the case where P t, x, y) satisfies the Lipschitz condition (H3.2) presented below, by direct calculation, we obtain the following conditions for the convergence of an iteration process of the form (7) for system (34):

341

Appendix

K2 <2

(36) ,r2 v

K2T < 1. 4q 4q According to (12), the determining equation is a nonlinear system of two equations in two variables (xo, Yo) = (z 1, z2). To get rid of the inconvenience caused by the two-dimensional nature of the problem, Samoilenko [Sam3] developed a modification of the numerical-analytic method for the second-order equation (33) with w = 0: d'x d x dt2 = f( t, x, dt ),

x E R.

(37)

As a result, the problem reduces to the investigation of scalar equations (iteration and determining), which is very important . However, as will be shown in what follows [see condition (H3.4)], in this case, the convergence conditions for the method significantly worsen as compared with (36). Assume that equation (37) is defined in the domain (t,x,y)e R1xD = R1x11x12,

teR1,

xe [a,b] = I1, ye [c,d] = I2,

and the function f (t, x, y) is continuous in (t, x, y), periodic in t with period T, and satisfies the following conditions: (H3.1)

1f(t,x,Y)I

< M, M>0;

(H3.2) If(t,xi,Y1)- f(t,x 2,Y2) I < K 1Ix1 -x21 - K2IY1-Y2I; K1,K2>0; 5 5 T2M (H3.3) b-a > 2 , c < -6 TM < 6TM <- d;

(H3.4) 4 K1 + 5TK2 < 1. Let an operator L : C[0, T] -4 CPT= CPT (R) that maps continuous functions f (t) into continuous T-periodic ones be defined by the formula

(Lf)(t) = f [f(ti)-Sf]dt, 0

342

Appendix

where T

Sf = T f f(t) dt 0 is the integral mean. Obviously, the functions xm+1(t, z) given by the recursion relation

, z) Xm + 1(t, z) = z + (L2(f(t, Xm ( t, z), dX (

)J

(38)

)(t) I

m = 0, 1, 2, ... , z(t, z) = z,

where (L2 f)(t) = (L( Lf))(t), are twice continuously differentiable and T-periodic, i.e., xm +1(t, Z) E C2PT.

Below, we give sufficient conditions for the convergence of the sequence

xm+ 1(t, z)

in the space C 1 P. Theorem 11. Suppose that conditions (H3.1)-(H3.4) are satisfied for equation (37). Then the sequence of functions xm+,(t, z ) defined by (38) converges in C1 PT to a function x*(t, z) that satisfies the equation d2x

d x I - 0 (z), - fI dt2 t' x, dtJ)

(39)

where

A(z) = S f

(t,

t, z)1 x*(t, z), dx*( dt J

Furthermore,

I x*(t, Z) - Xm(t, Z) I dx*(t, z) dxm(t, z) I dt dt where

Qm(E - Q)-1zj0,

(40)

343

Appendix

T 2 Kt 2 K2 Q 5 K 5 3 l 3 K2

I x, (t, z) - x0( t, z) 10 Zt 0

-

d xl (t, z) _ dx0 (t, z) dt

dt

< TM 2

0

T 2

5 3

and I x(t, z) - y(t, z) l0 stands for sup, I x(t, z) - y (t, z) I . Moreover, on the set 2 2 rr MI, Il = a+T 4 ,b - T 4

a function A (z) of the form (40) and its "approximation" Am(Z) = S f I t, Xm(t, Z), dxm(t, z)) \ dt \

(41)

are defined and continuous, and

I A(t, z) - A m (t, z) I << Kl I X* (t , Z) - X m( t , Z) 10 + K2

dx*(t, z) _ dxm(t, z) dt dt

K1 < ,Qm(E-Q)- 'zo = dm, K2

where (,) denotes the standard scalar product in the space R2. Corollary 3. Roots z = z * of the scalar equation A (z) = 0 determine the initial values

X(0) = z *,

t d (O ) _ -S J f l s, X* ( s, z*), dx*(s, Z*) )ds

0 \ of a T-periodic solution x = x(t) of equation (37).

344 Appendix Corollary 4. Assume that, under the conditions of Theorem 11, for some m >- 0, the function Am(z) defined by (41) satisfies the inequalities

min Am(Z) <- -dm, ZEII

min Am(Z) > dm.

ZE/,

Then equation (37) has a T -periodic solution x = x(t) such that

I x(t), dx(t) e D dt )

for t r= R t and its initial data satisfy the conditions Idx(0)I 5TM X(O)C= 11, 1 dt 6 Consider some special cases of equation (37). Assume that the right -hand side of equation (37) is a polynomial in x and y, i.e., (42)

f(t'x,y) = Y,py(t)x'y', 1, i

where the functions p, (t) are summable and bounded on [0, T]. For such f(t, x, y), the following statement is true: Corollary S. Let f (t, x, y) be given by a finite sum of the form ( 42) and assume that conditions (H3.1)- (H3.4) are satisfied. Then the initial data

+T4

2

X(O) = zE I1 =

a

2

, b-T4 ,

J

r 56M 56M1 dd0) = Sf t f(s,x*(s,z),dxdt'z)1 dsE c+-,d-- = IZ 0 of the T -periodic solutions of equation (37) are either all isolated or completely fill the interval Ii I. It is clear that, in the first case, the initial data of T-periodic solutions are isolated points of the rectangle D 1 = II x IZ, whereas, in the second case, they fill a certain curve lying in D 1. The last statement is in good agreement with the known property of periodic solutions established for analytical autonomous systems on a plane (see [BaL]).

Appendix

345

Corollary 6. Suppose that conditions (H3.1)-(H3.4) are satisfied for equation (37) and the function f (t, x, y) is odd in t and x for all (t, x, y) E R t x D, i.e.,

f(-t, -x, y) = -f(t, x, y). (43) Then the differential equation (37) has a unique T -periodic odd solution x = x(t). This solution has the form x(t) = x *(t, z) =

lim Xm(t, Z),

where Xm(t, z) is defined by (38) and its initial data are the following: X(O) = 0,

ddto)

_ -SJ fl s, x*(s,0),dxds' 0) Ids.

Note that, generally speaking, the periodic solution whose existence is guaranteed by Corollary 6 is nontrivial. Corollary 6 complements some results of Pliss [Plil] and Heinbockel and Struble [HeS] concerning periodic solutions of systems with certain symmetry. Problem 3. Weaken conditions (H3.1)-(H3.4) so that the oddness condition (43) still guarantees the existence of a T-periodic solution of equation (37). It is easy to verify that the operator (L2 f)(t) can be rewritten in the form

(L2f)(t) = J f(i)(t - ti)dti - T J f ( ti)(t 2T - ti dti. (44) 0 0 / By using representation (44), Chornyi [Choi] , [Cho2] weakened conditions (H3.3) and (H3.4) as follows: 2 (H3.3*) b-a ? TTg , c <- -MT < 4T <_ d;

(H3.4*) T

K, + TK2 < 1. 16 4

346

Appendix

On the basis of the last relations , the corresponding corrections can be made to the formulation of Theorem 11 and its corollaries. Note that condition (H3.4*) is more restrictive than relations (36), which are sufficient for the application of the numerical-analytic method to a system of two differential equations of the first order equivalent to (37). A comparison of Corollary 5 with Example 1 shows that a minor reduction of the Lipschitz constant from (36) to (H3.4) enabled us to omit the determining equation (21). Example 2. The analysis of a periodic solution of the equation 2 dx + w2x = sinXt dt2

(45)

shows that the indicated reduction of the Lipschitz constant is important from the viewpoint of the realization of the method and corroborates that estimate (H3.4) is rather sharp. Indeed, equation (45) has a unique T-periodic solution with period T = for all

wT

<

2n.

(46)

For wT = 2 it, all solutions of equation (45) are unbounded. On the other hand, conditions (36) and (H3.4*) for this system take the following form: wT<2q=6.42..., (4 7) wT<4. [One can easily verify that, in this case, condition (H3.3*) also leads to condition (47).] Thus, the determining equation (21) appearing in Example 1 cannot be omitted. Remark 3. The equation

d2x = 0 dt2 shows that Corollary 6 only guarantees the existence and uniqueness of an odd T-periodic solution , i.e., some other solutions may also exist. Let us show that either of conditions (H3.3) and (H3.4) guarantees the existence of at least one odd T-periodic solution. In the proof of the corresponding statements, the

Appendix 347 Schauder fixed-point theorem and the Leray-Schauder continuation method, respectively, are used. Let x denote the least upper bound of the functional I

J(f) =

max tE[O,1]

1

J (t- u)f(u)du+tJ f(u)udu O

(48)

0

over the set of all odd continuous 1-periodic functions f (u) such that I f (u) I <- 1. Note that the modulus sign in (48) can be omitted because, if necessary, one can always replace f(u) by -f(u). Since the function

$(t) = (L2f)(t) is 1-periodic , it attains its maximum value at a certain point t = t * of the interval L 2' 2 1. Then t. I d4(t*) = 5 f(u)du + J f( u)udu = 0 dt f

0

0

and, hence, t*

t"

1

t*

f

fi(t) = t* J f(u) du - 5 uf(u)du + t* f(u)udu = - 5 uf(u)du. 0 0 0 0 Therefore,

K <_ I $(t*) I <-

1/2 1 J udu = 0

By using the estimate obtained in [Cho2], one can show that

K <,

1 16

Since the number K is important for the investigation of specific equations, below, we determine its exact value.

348

Appendix

In what follows , we denote by COPT(R) (CEPT(R)) the Banach space of all odd (even) T-periodic functions continuous on R with the supremum norm. Furthermore, we set t T (A f)(t) = J (t - u)f(u)du + t j uf(u)du,

0 (Bf)(t) =

(49)

0

f(u)du + j u f(u)du. 10 0

(50)

T

Obviously,

dt(Af)(t) = (Bf)(t), and the operator A coincides with the restriction of the operator L 2 to the subspace COPT(R) C CPT(R), where CPT (R) is the Banach space of all T-periodic continuous functions. Lemma 1. For all f(t) E COPT ( R), the following estimates are true:

I(Af )(t)I

2

<- 21f(t)I0, (51)

I (Bf)(t)I < 4 If(t)I0, z where the constants T and T are unimprovable. Furthermore, 32 4

Af(-t) = -(Af )(t), . (Af )(t) = f(t), (Af)(0) = (Af)(2) = (Af )(T) = 0,

Bf(-t) = Bf(t). Proof. Taking the oddness of f(t) into account, by direct substitution of t = 0 and t = T into (49), we obtain (A f)(0) = (Af)(T) = 0.

Appendix

349

Furthermore, T Af(-t) = J (-t- u)f(u)du - T Juf(u)du.

0

0

Performing the change of the variable u = - v in the first integral, we get T Af(-t) = J (-t+v)f(v)dv - T f uf(u)du = -(Af)(t).

0

0

Further, 2

d- (A .f)(t) = f(t) and

(Af)(2) = J2( 2-u)f(u)du + 2 Juf(u)du. 0 0

(52)

Since T

T/2

$ u f(u)du = 0

0

T

$ u f(u)du + $ u f(u)du, T/2

by setting u = v + T in the second integral , we obtain T

T/2

0

!

5uf(u)du =

f uf(u)du +

0

-T/2

0

(v + T) f(v + T) dv

T/2

Juf(u)du + 0

0

-T/2

T/2

0

j v f(v + T)dv + T f f(v+T)dv -T/2

0

= 2 $ uf(u )du + T $ f(v+T)dv. 0

Therefore,

-T/2

(53)

350

Appendix T/2

(A f)(2) = 2 /

J

f( u)du -

0

T/2

T/2

5

5

0

uf(u)du + 0

0

uf(u)du + 2 f f(v +T)dv

-T/2

7, T/2

2

J f(v+T)dv = 0.

- T/2

Let us estimate the norm of

(Af)(t) = 4(t). Assume that I 1 (t) I attains its maximum at the point t = t'. By virtue of the odd symme try fi(t), we canof assume that t T E 0, 2 ]. Consider the following two cases:

t*E

10, T ] 4

and

t E

_T _T 4' 2

In the first case , taking into account that (B f)(t*) = 0 and, hence, t'

T f(u) du = - f uf(u)du,

Jo

To

we get t`

0(t

^ T

J (t* - u) f(u) du + 0

du = -J u f(u) du. T J u f(u)0 0

Therefore, T/4

2

(t')I < Julf(u)Iodu <- If(u)Io32 0 In the second case, we have

Appendix

t

where t

E

*= (t*

T)

_2

+

351

T = t

2

T +2

f

T

0] and 4 (t `)= ^(t + . [_4 , 2)

Let us calculate the value

''(t) _ t + T). We have t+T/2/

'Y(t) =

T

f (t + T - u ) f(u)du + t+T l2 f uf(u)du 0

\

0

T T Tfuf(u) du+2fuf(u)du

0

0 T/2 T, t+Tl2 T

+ f 0

(t

+ 2 - u) f (u)du +

f

T/

2

(t +-u)f(u)du.

Changing the variables u = v + 2 in the last integral , we obtain

T T/2 \ `Y (t) = 2 f uf(u)du + f (T - u I f(u) du

0

0

/

T/2

T

T

+ t f f(u)du + T f uf(u)du + f (t-v)f(v+T)dv. 0 0 0 According to (52 ), the expression in square brackets is equal to zero and, therefore, T/2

`Y (t) =t

f0

f(u)du+

Since dT (t) = 0, we have

T

T

t

f uf(u)du + f (t-v )f(v+T)dv.

0

0

Appendix

352 T/2

T

j f(u)du + I f uf(u)du + J f(u + )dv = 0. 0 0 0 Therefore,

4(t) = `P(i ) _ - j v f(u +T2 )du 0 and

I4,(t')f <

MZ

2

If(t)10.

2 Let us show that the constant 32 in estimate (51) is unimprovable. We introduce

the function (.. TI

(54)

f(t) _ - M, te[-T,0),

and extend it onto R in the T-periodic and odd manner. Then T T/2 T 2 J u f(u)du = f Mudu - f Mudu = M[ 8

( T2T2J] 2

- 2)0 0 T/20

M4

Therefore, for t E [o,?. ] , we have

$(t) = (A f)( t) = f M(t - u) - T MT2 0

0

_ - M vdv - tMT = Mt2 - MTt = Mt(2t-T). 4 4 4 4 r

The function 4 (t) takes its maximum value in modulus at the point t = T. This yields

Appendix

353

|(A/)fI]| = 4*4H = 32^ . H \4)\ 2 J)

Furthermore, it follows from (50) that T (Bf)(-t) = J f(u)du + T Juf(u)du.

0

0

Performing the change of the variable u = -v in the first integral, we obtain t IT (Bf)(-t) = J f(v) dv + T f uf( u)du = (Bf)(t).

0

0

Note that the extremum value of the function

(Bf)(t)

is attained in the interval

L 0, T ]. In view of (53), relation (50) yields t

0

(Bf)(t) = f f(u)du + 0

T/2

f f(u)du + f uf(u)du

-T/2

0

T/2 T/ 2

= T f uf(u)du 0

Therefore, for all t e

I (Bf)(t)I s

-J

t

T/2

f(u) du = - f u f(u)du

t

0

+ f ( -_ i)juu.

t

L 0, 2 l , we have

2 t2 T/2( 2u )du TM2 +M J (1-T t

Mt2

T

+M

L

2

42-

-t -Tl

t2JJ 2T t2 + M4 - Mt

_ MT (2t _ 1 2 MT MT MT MT r _T l 8 + 8 = 4 b'fe 10'21 2 \T 2l + 8 Furthermore , by using again a function 1(t) of the form (54) as an example, one can show that the constant T in (51) is unimprovable because

354

Appendix

(Bf)(0) 4T This completes the proof of Lemma 1. Corollary 7. On the set of all odd continuous 1-periodic functions f(u) such that If (u) 15 1, the least upper bound of functional (48) is equal to

x = 32 .

Theorem 12. Suppose that, in the domain IxISa,

IYI

Sb, tER,

the function f (t, x, y ) is continuous, periodic in t with period T, and bounded in modulus by the number M. Also assume that the following inequalities are true: T2M TM a> < 32 , 4 -b. Then, under the symmetry condition (43), the differential equation (37) has at least ( one odd T -periodic solution x(t). Moreover , x(O) = x f x(T) = 0.

Proof. By virtue of Lemma 1, any solution x(t)E COPT of the integral equation

x = Afl t, x

(55)

l ,Li

is, in fact, the required one. Obviously, equation (55) is equivalent to the system of two integral equations x = A f (t, x, y),

(56)

y = Bf(t,x,y)• In the direct product W = COPT(R)x CEPT(R), we consider the closed convex set Z,,1 = {(x,Y)E W: Ix(t)I S a,

Iy(t)I 5 R V tE

R}.

Appendix

According to Lemma 1 , for the compact operator

355 A B , we have

(A B : W-4 W,

and if sup IAf(t,x, y)I (X, Y) E Za.9

(57)

sup

I Bf(t, x, y)I :5 I ,

(X, Y) E Za.6

then the Schauder fixed-point theorem [Tre, p. 411] implies the existence of at least one solution of system (56). Since,forany (x(t),y(t ))E Za,O,

If(t,x(t),y(t)) I <_ M, we conclude that, according to Lemma 1 , for condition (56) to be satisfied , it suffices that the following relations hold: T2 M MT 32 4 The theorem is proved. Consider several examples. Example 3. Note that, for a system of the form 2

dt2

= f x,

d^

+g(t),

(58)

where the function f I x, dt I is odd in x and g ( t) is odd, the conditions of Theorem 12 can be simplified. 2 Thus, the inequality a >_ 3 M can be replaced by

Appendix

356

T2 max I f(x, y) I

a >_ IxlSa, lyl5l3 + max I (Ag)(t)j, 32 te[o,T] and the inequality TM <_ b can be replaced by

T max I f (x, y) I R> Ix15a, Iy15P + max I(Bg)(t)I• 4 te[0,T] Corollary 8. The differential equation

d2x + /(*) = *('), dt2

(59)

where the functions f (x) and g (t) are odd and g (t) e COPT, has at least one odd T -periodic solution x(t) such that T) x(0) = x( 2 = x(T) = 0,

provided that there exists a > 0 such that

T2 Max I f(x)I a >_ IxISa + max I(Ag)(t)j. 32 te[0,T]

(60)

Example 4. Let us apply inequality (60) to the investigation of the existence of a periodic solution of equation (45). We establish that a T-periodic solution exists if one can find a such that 2

w

2

2

2

2

Twa+ T
i.e., T2w2 < 32 and Tw < 4,/ - 5.656. The last estimate is in good agreement with the unimprovable inequality (46). Example 5. Let us apply (60) to the analysis of a 2n-periodic solution of the equation 2 d+x2= T £sint. I

(61)

Appendix

357

The inequality T2a3 32 + e <- a is solvable if the function µ (a) = T2a3 32 - a + e has a positive root, for which it is sufficient that µ((Xmin) < 0.

Then 2 2

µ (amin ) =

3T amin 32 - 1 0,

whence 32 44-2 amin =

3T2 = 3

Therefore, = T2 32 4V-2 - 4-F2 + e = - 8^ + e < - 0. µ(a1°) 32 3T2 TNF3 T73 3T[3 The last inequality holds if

e<

8 f 3T f 3nj3-

0.3465319.

Thus, for the indicated values of e , equation (61) always has at least one odd 2 n-periodic solution x(t) such that x(0) = x(tt) = x(2n) = 0. It should be noted that equation (61) with e = I was investigated earlier by Cesari in [Cesl], where a rather complicated proof of the existence of an odd 2n-periodic solution is given. Problem 4. Verify whether the operator L 2 preserves the property of symmetry of the third kind, which means that the odd function f (t) is such that

f(t+T) = -f(t). Prove that equation (61) has a 2n periodic solution with symmetry of the third kind. Let us show that an estimate of the type (H3.4) guarantees the existence of a T-peri-

358 Appendix odic solution of equation (37). In fact, we assume that the function f (t, x, y ) satisfies a condition weaker than the Lipschitz condition, namely, we assume that f (t, x, y) is a function of sublinear growth, i.e.,

If(t,x,y) I <- plxl+glyl+r (t) (62) for all t, x, y e R. To establish the existence of a T-periodic solution of equation (37), we apply the Leray-Schauder continuation method with respect to a parameter £ E [0, 1 ] to the parametrized system x = eL2f ( t, x, y),

(63) y = £Bf(t,x,y)• For £ = 0, this system has the trivial solution x = 0, y = 0. Let us obtain a priori estimates uniform in £ e [ 0, 1 ] for the solution (x(t, a), y( t, a)) of system (63). We have

Ix( t, e)I < £ IL21 (pIx(t,e)I + g Iy(t,e)I)+IAIr(t),

Iy(t,e)I <- IBI (plx(t,e ) I+gly(t,£)I)+IBIr(t), where t

t to t2 z(u)du + IL2Iz(t) = f -u+ 2 T 2T 0

T 2 -u I z(u)du, T ft 1 t-

2 T/2/ 2u I B I z(t) = T j uz(u)du+ f l- T +l^ z(u)du. 0

1

\

Performing elementary but cumbersome calculations (we do not present them here), we obtain

L21 1(t) 16 4( I t 2 I Consequently,

2,

to [ 0, T].

359

Appendix 2

I I L2 I z(t) I <_ 161 z(t) Io. Similarly, for tE

1

0, -T ], it can be shown that

2 IB11(t) =

2 2t +T_ t,

0

tE ['2]'

and, therefore,

IIBIz(t)I <- 4T Iz ( t)Io, tE

10,

T

Note that, by virtue of the symmetry properties of z (t ) and y ( t), it suffices to establish estimates only on the interval t E L0, T ] . Finally, we get

I x( t , e) 10 I Y(t,e)Io

T2 16p T 4p

T2 I x(t, e)10 16q 1 (r T ( IY(t,e)J0) + \r2 4q

where r1, r2 > 0. Standard reasoning shows that , in the case where the greatest eigenvalue of the matrix in the last inequality satisfies the relation 2 max =

(64)

6p+4q < 1,

the quantities I x (t, a )1 o and I x (t, a )1 o are bounded by a number that depends only on T, p, q, and r(t) and does not depend on e. If relation (64) is satisfied , then the Leray-Schauder continuation theorem guarantees the existence of at least one T-periodic solution of equation (37). Problem 5. Examine the case where p = p(t) and q = q(t) in relation (62) with the use of the a priori integral estimates obtained in the course of investigation of system (63). Note that, for the equation

360

Appendix d2x dt2 =

.f(t, x),

(65)

one can obtain stronger results . In this case, system (63) is equivalent to the equation T

X(t) + J K(t, u) f [u, x (u)]du = 0, 0 where

K(t, u) =

[ u(i--L I, 0< _u5t<_T, T) tIl-U ,

)

05t5U:5T,

because T J K(t, u)x(u)du.

Ax(t)

0

Obviously, K(t, u) = K( u, t) >- 0. Let us determine the eigenvalues of the operator -A. Let -Ax - —x.

Then d2x = Xx, dt2

x(0) = x(T) = xl 2

J = 0.

One can easily verify that X > 0 , and the eigenfunctions have the form

x,, (t)

=

sin

t.

(66)

361

Appendix

Taking the boundary conditions into account, for the eigenvalues of the operator considered, we get

e

n2, n e N .

{

Therefore, all eigenvalues of the operator -A are positive and _ T2 max 4n2 2 By virtue of the Hammerstein existence theorem (see [Tri , p. 266]), for p < T2 or, 2 which is the same, 4T2 < 1, the nonlinear integral equation (66) has at least one con-

tinuous solution. If, in addition,

If(t,x)I <_ PIxI, then, according to the uniqueness theorem IV in [Tri, p. 27], the solution indicated above is unique. In other words, the following statement is true: Theorem 13. Suppose that f (-t, -x) = f (t, x) and the following conditions are satisfied:

If(t,x)I <- P IxI + r,

(67)

pT2 4Tt2

< 1.

Then the differential equation (65) has at least one T-periodic solution . If r = 0, then this solution is unique (x(t) 0). Remark 4. The analysis of the differential equation (45) shows that the estimates in 22

Theorem 13 are optimal. Indeed, we have 4ti < 1, whence T< 2n

.

Remark 5. Condition (67) can be somewhat weakened (see the table in [Tri, p. 269]).

362

Appendix

Problem 6. Obtain sharper conditions for the convergence of the iteration process (38) by the calculation of the spectral radius of the linear operator associated with this scheme (see Theorem 1 in [Trot], Lemma 17 in [RoT2], and Lemma 3.2 in [RRT]). Note that the basic scheme of the numerical- analytic method was extended to the case of periodic solutions of systems of second -order differential equations in [Str] and [SaR2]. Various modifications of the numerical -analytic method were also applied to the Tperiodic boundary-value problem for equation (33) in the case co * 0. Thus, in [MKG], such modifications of the method were used for the proof of existence theorems for Tperiodic solutions of equation ( 33) both in resonance and nonresonance cases. In the nonresonance case, in addition to conditions of the type (H3.1)-(H3.4) sufficient for the convergence of the iteration process

qT xm+

1(t) =

Zw

j Q(ti)f(ti, xm(ti), d 0

xo(t) = v ( t),

Iv(t)I :5

m(t

)sin w(t -ti) dti,

21 MTq,

m=0, 1,...,

dv(t) I < MTq, dt 2

1, 0!5'r:5 t;

Q(r) =

-1, t «<_ q T,

rather restrictive arithmetic conditions were imposed. Namely, it was assumed that wTq = (2p - 1)n and (2p - 1 , wq) = 1 for T-systems of the first class (according the terminology of [MKG]), and wTq = r7t, r = 2k, q = 2s - 1, and (coq, r) = 1 for 2Tsystems of the first class. Here , p, q, r, and k are natural numbers, and the expression (m, n) = 1 means that the natural numbers m and n are coprime. (Note that these conditions are meaningful only for natural numbers w!) In the resonance case ( T-s y stems of the sec on d c lass [MKG ] ), one has T = where no is a fixed natural number. In this case, under the conditions

a> wMT, b>-MT,

T(1+K2) < 1,

it was proved that the following iteration process is convergent:

Ztsr^o

w

Appendix

363

Xm+1(t, c, 0

T

=

x0(t, c,

0+

\

f Q1(ti)f(ti, xm('t, c, $) , dXm 0

T,

c, $) )sin w(t- 't) dt,

m = 0,1,2,...,

1-T, 0<-T:5 t;

Q1(t) =

t

t
where

x0 (t, c, $) = ccos ( cot +0) = y(t),

Iy(t)I <- a-1, w

dy(t) dt

<- b-MT.

The last two relations are equivalent to the inequality

c S min

aw-MT b- M w w

.

In the nonresonance case , the sequence of iterations converges to the required solution (i.e., it is not necessary to solve any determining equations ). In the resonance case, 2ttn0 the problem of existence of -periodic solutions is uniquely related to the problem of w solvability of the equation T

A(c, ^) = T f f I T, x*(ti, c, $), dx*( c, ^)Jsinw(t_t)dt sin = 0, where c*(t, c, 4>) = lim -+-x. 0' c' 0). In [Choi] and [Cho2 ], a theorem on the existence and uniqueness was proved for a Tperiodic boundary-value problem in the case where x E (R, I I . II) in the nonresonance equation (33) under the following conditions:

yT = (K1 + (oK2) < 1, y = 1 2wsinw 2

364

Appendix

and

Jjf(t,xi , xl)- f(t, x2, x2)jj

K1lixt-X21(+ K2IIxi-xx11•

In [PeR], a new scheme of the numerical-analytic method was suggested for secondorder differential systems of the type (33) with Co * 0. This scheme is applicable regardless of whether the entries of the Lipschitz matrices K1 and K2 are large or not. According to this scheme, T-periodic successive approximations are constructed in the form

xm+1(t, 4) = A(t)4 + w I f sin w(t-s)f(S, Xm(S,),m')dS 4 dxdss o T

w'1(t) f sinw (T-s)f(s, xm(s,j),dxm(S, ds 0 T ( dxd( s, )1 -'P2(t) f cos w(T-s)f s,xn(s, 0, J ds, m=0,1,2,..., s

(68)

0

where 'P ( t) = ('P1 (r),'P2 ( t)) is a differentiable n x 2n matrix formed of n-dimensional blocks satisfying the relations 'I'1(T) = In+'P1(0), 'P2(T) = 'P2(0), 'I' (T) = 'P (0), 'I2(T) = In+'P2'(0), and A (t) I„ cos (0t, 1 1„ sin wt I -'P(t) ['((o, T) - I2i1' w /

(D (co, T) =

In cos wT In sin wT -wIn sin WT

In cos wT

where In is the n -dimensional identity matrix. It was shown that, under a global Lipschitz condition of the type (H3.2), there always exists a matrix 'P(t) ("sufficiently small" in the mean) such that sequence (68) converges to a differentiable T-periodic function x* (t, 4). The limit function is a solution of equation ( 33) (x a R) if and only if the parameter 4 e R 2 n satisfies the determining equation

365

Appendix T

0

[I2. - b (w, T)]4 =

ds'

^') ds lJ

w., sin w(T -s)f (s, x*(s, 4), dx

J

j cosw(T-s)fI s, x*(s, ),_(ds )ds

0 Note that various types of second -order differential equations were also studied in the works of Shlapak [Shill, [Sh12], Samoilenko and Perestyuk [SaP2], Sobkovich [Sobl], [Sob2], Shovkoplyas [Shol], [Sho2], M. Ronto and 0. Martynyuk [RoMa], 0. Martynyuk [Marl], [Mar2], 0. Martynyuk and S. Martynyuk [MaM], Korol [Korl ], [Kor2], and M. Ronto and S. Trofimchuk [RoT2].

A3.2. Applications of the Method to the Investigation of Differential-Functional Equations A3.2.1. Delay Differential Equations. Mitropolsky, Samoilenko , and Martynyuk [MSM], Martynyuk and Samoilenko [MaS], and Martynyuk [Mard] first applied the numerical-analytic method to the investigation of T-periodic solutions of delay differential equations of the form

x'(t) = f(t, x(t),x(t-A )),

x,fE R". (69)

Here, A is a positive constant that characterizes the delay in the system, and the vector function f (t, x, y ) is defined in the domain (t,x,y)E (-oo,00)x Dx D (70)

( D is a closed bounded domain in ]R") and periodic in t with period T. The principal results are proved under the following conditions, which are standard for the method: (H3.5) For all {x, y } C D and tE R , we have I f (t, x, y) I <- M, where ME R+; (H3.6) for all {xj, y;}j=1 2 C D and tE R, we have

If(t,x1,Y1)-f(t,x2,Y2)I <- K1Ix1-x2I+K21Y1-Y21, where { K 1, K2 } C L ( R") are nonnegative matrices;

366

Appendix

(H3.7) D p * O, where D S : = { X E R': S (x, (3) E D } and P:= T M ;

(H3.8) the largest eigenvalue ? max (Q) of the matrix C T 3Y A)21 Q = (KI+KZ) 3+2TA T

is less than the unity. Since we consider T-periodic solutions of system (69), we can assume without loss of generality that 0:5 A!5 T. Theorem 14 ([MSM], Section 2, Theorem 1.1). Suppose that conditions (H3.5)(H3.8) are satisfied. Then, for every z e D p, the sequence of T periodic functions

Xm(t, Z)

r

1T

= Z + j f (t, xm- 1(t,Z), xm -1(t-A, Z)) - J f( s, xm -1( s , z), xm - 1 (s - A, z)) ds d t T 0

0

converges to the limit function x =x*(t, z) uniformly in t E and z E D and, furthermore,

kx*(t,Z)-xm(t,Z)I <_ 2Qm(E-Q)-1M.

Indicating a condition for the solvability of the determining equation

A(t,z) : =

T f f(t, x *(t, z ), x*(t - A, z )) dz = 0, 0

one can establish the following statement: Theorem 15 ([MSM], Section 3, Theorem 1.2). Suppose that conditions (H3.5)(H3.8) and the following two conditions are satisfied: (H3.9) for some m e N, the approximate determining function

Appendix

367

T

Om(Z) :=

T Jf

(t,xm (t,z),xm(t-

A,z))dz

(71)

0

has an isolated singular point z = z,,,o of nonzero Leray-Schauder index; (H3.10) there exists a closed convex domain D1 C D p in which z = z,,,o is a unique solution of the approximate determining equation Am(z) = 0, and the inequality inf IA. (Z)I > TQm(E-Q )_ 1(K1+K2)M 3

Z E S,

holds on the boundary S 1 of the domain D1. Then system (69) has a T-periodic solution x = x * (t) with initial value x * (0) _ xO* E D 1.

Note that the justification of the averaging method for T-periodic delay differential equations is a consequence of the application of the method to systems in the standard form X' (t) = ef (t,x,x(t -A)),

(72)

where E is a small parameter. Corollary 9 ([MSM], Section 6, Theorem 1 .3). Suppose that the function f (t, x, y) on the right-hand side of system (72) is defined in domain (70) and satisfies conditions (H3.5) and (H3.6). Also assume that the averaged system

d4(t) = EX0 ( (t),^(t-A)), dt where T

X0(x,y) :=

Jf(t,x,y)dt,

T0 has an isolated equilibrium position 4 = 40 : X0(40, 40) = 0 and the index of the point 40 is different from zero. Then, for sufficiently small E <- E0 , system (72) has a Tperiodic solution.

368

Appendix Let us indicate several interesting applications of Theorems 14 and 15.

Corollary 10 ([MSM], Section 3, Theorem 1.4). Suppose that the function f (t, x, y) in the scalar equation (69) satisfies conditions (H3.5) and (H3.6) in the domain -oo- 1, function (71) satisfies the inequalities m min Am(Z) 5 -T Q (E-Q)-'(K1+K2)M, a+2M5z5b--M 3 1-Q

min Am (Z) ? T Qm (E-Q)-'( K1+K2)M. 3 1-Q a+TM5z5b-TM 2

2

Then the scalar equation (69) has a T -periodic solution x = x*(t) with an initial value satisfying the relation

a + T M <- x*(0) <- b-TM. 2 2 Corollary 11 ([MSM], Section 3, Theorem 1.5). Suppose that the right-hand side of the system x'(t) = f(t,x(t),x(t-T )),

x,fE R", (73)

is defined in domain (70) and satisfies conditions (H3.5) and (H3.6) and the following symmetry condition: (H3.11) For all x, y, the equality f (t, x, y) = -f(- t, x, y) holds. Then a T -periodic solution x = x*(t, z) of system (73) passes through every point z r= D o at time t= 0. Example 6. In [MSM, Chapter 1, Section 6], in the course of investigation of a 2nperiodic solution of the equation x(t) = EIx2(t)-Sint-x(t-A)cost+0.001 , 05t:5 2tt, (74)

Appendix

369

with e 2 0 by the numerical -analytic method, certain subsets 521 and S2 2 in the plane of parameters (E, A) were introduced. These subsets are such that there are no 27t-periodic solutions equation (74) for (E, A) E 521, and there is at least one 2n-periodic solution of equation (74) for ( e, A) E 52 2 . Moreover, it was established that 27t-periodic solutions do not exist for small a satisfying , say, the inequality (4+sinA)2E2 < 4(e2+2)(0.002+3E2-ecosA), but appear as e reaches a certain critical value E = E (A). In [MSM], the method was also used for finding T-periodic solutions of the secondorder scalar equation d2 x = f(t,x(t),x(t-A),x(t),x(t-A)), dt2 systems of neutral type x'(t) = f(t, x(t),x(t-A ), x'(t-A)), and countable systems of delay differential equations of the form (69) with x, fE ! . If the function f in equation (69) can be represented in the form f (t, x, y) _ g (t, x, y, x, y), where g satisfies certain additional monotonicity conditions, g(t,x,Y,u,v) <- g(t, x, y, u, U), for all x<- x , y!5 y, u ? W, and v ? u, then one can combine the numerical-analytic method with the method of two-sided approximations ( see [Kurp], [KuT], and [MSM, Chapter 1 , Section 12]). In [Yan], the numerical- analytic method was applied to equation (69) in the case of a right- hand side of the special form f(t, x(t), x ( t -,r) - x(t)). Problem 7. Construct a scheme of the numerical-analytic method for nonperiodic boundary-value problems for the delay differential equation (69) that takes the specific character of delay into account. A3.2.2. Functional-Differential Equations . Note that, by the change of variables

x(t, z) = z + f y(s, z) ds - T J y(s, z) ds, 0 0 the integral equation (2) can be reduced to the form

(75)

370

Appendix

C

At, z) = f r, z + $ y(s, z)ds y(s, z)dsl. (76) T 0J l o

j

The change of variables (75) transforms T-periodic functions into T -periodic ones. Therefore, having found a solution of equation (76), we obtain a solution of equation (2) by the application of transformation (75). Thus, the investigation of a T-periodic solution of the boundary -value problem (1) can be reduced either to the integral equation (2) or to the integro-differential equation (76). Furthermore , the operator on the right-hand side of (76) also possesses the required property of transforming T-periodic functions into T-periodic ones. For simplicity , we assume that x E R" and f (t, x) is continuous on the set

(t, x) E [0, T] x R.

Then a solution of equation (76) can be found by iterations, provided that the Lipschitz condition (15) is satisfied and the spectral radius r(B2) of the operator B2=KA:CPT-+ CPT, where (Kx)(t) := K( t)x(t); t T (Ax)(t) = + T j x(s)ds , (1_ ): x(s)ds 0 0

t E [ 0, T] ,

is less than the unity : r(B2) < 1 . Note that, for the convergence of the iteration process (7), it is sufficient that the condition r(B1) < 1, where BI=AK, be satisfied. [In the particular case K(t)= K, one has r(B1) = r(B2).] Generally speaking, the operators A and K, do not commute and , therefore , sufficient conditions for the convergence of the iteration processes for the solution of equations (2) and (76) may be different. An equation of the form (76) was chosen by Augustynowicz and Kwapisz [AuK] and Kwapisz [Kwa3 ]-[KwaS] as the most convenient for the investigation of boundary-value problems for a functional -differential equation of the neutral type:

Appendix

371

x`(t) = f(t, x(a(t)), x`(13 (t))) (77) Ax(O)+Bx(O) = d, where a, P : [0, Tj - [0, T ] , JA, B, B -' j CL (R7), and d E I[8". Here, the function f E C([0, T] x R" x R") satisfies the condition

If(t,x1,z1)-f(t,x2,Z2) I :5 Klxt-x21 +Llzi-z21 with some constant matrices { K, L } C L (R"). In this case, problem (77) was reduced to the problem of fording a solution y = y * (t, z) of the equation

y(t, z) a(0

-T (t) l y(s, z)ds = f t, z + a(t)q(t) + T 0

T

aTt)

j y(s, z)ds, y(R(t), z) (78)

a(t)

where

(p(z) :=

TB

-1[d-(A+B)z],

by using the method of successive approximations with the subsequent calculation of the parameter z e R" from the determining equation

(p(z) - T j y*(s, z) ds = 0. (79) 0 In [AuK] and [Kwa5], certain conditions that guarantee the convergence of successive approximations in the solution of equation (78) were indicated. Namely, the following statement is true: Theorem 16 ([AuK], [Kwa5]). Suppose that the spectral radius of the operator Kw = 2 K sup (T - a(t)) a(t) + L T te[0,T]

(80)

is less than the unity. Then equation ( 78) has a unique solution y = y * (t, z) for any z e W', which can be found by the method of successive approximations independently

372

Appendix

of the choice of the initial approximation. Furthermore, if z = z* is a root of the determining equation (79), then the function t x(t, z) = z* + f y(s, z*)ds

0 is a solution of the boundary-value problem (77). Note that the solvability of the determining equation was not investigated in [AuK] and [Kwa5] . Instead, a "nonresonance" case was considered [i.e., the case where det (A + B) # 0], for which a modification of equation (78) was constructed that does not require the solution of the determining equation. It should be noted that this idea was also suggested by Trofimchuk and Kovalenko [TrK] and Ronto [ Ront] independently of [AuK]. The following statement is true: Theorem 17 [AuK]. Suppose that det(A + B) * 0 and the spectral radius r (KW) of the operator K, defined by (80) is less than the unity. If, in addition, the greatest positive eigenvalue of the matrix TI (A+B)-1BI (I-KW)-1K sup I E - a(t)B-'(A+B) tE[O,T] T

is also less than the unity, then there exists a unique solution of the boundary-value problem (77). Note that the methods discussed above can be used for establishing the existence and uniqueness of a solution of the boundary -value problem for functional-differential equations of a more general form (see [AuK]), namely, x(t) = f(t,x(at)),...,x(a,,)),x(Rt)),...,x(Rq))), Ax(0) + Bx(T) = d. In this case, operator (80) takes the form

KW = 2 i I Ki

sup (T -ai(t))ai (t)] +

Li,

Ti=1 L tE[O,T] j=1

where {K1}? 1 and {Li}?=1 are the matrices from the coordinatewise Lipschitz condition

373

Appendix f (t, x1, ... , XP, xp+1, ... , xp+q) - f (t, x1"'..., X", XP+1, ... , X p+q )

P

q

KilxPi-xp+i Ki I xi-xil + i=1 i=1

A3.2.3. Equations with "Maxima. " For h > 0, the equations with "maxima"

x'(t) = f

Ct,

x,

max x(s)^, tE R,

xE D C R", (81)

S E [t-h, t]

where, by definition,

max x(s) = I max x1(s),..., max x„(s) SE[t -h,t] SE[t- h,t]

SE[t-h , t]

J

are an interesting particular case of differential -functional equations with delay. The interest in such equations has arisen comparatively recently. [As far as we know, at most 40 works were devoted to the subject , the most recent of which appeared in 1995.] As Myshkis indicated in his survey lecture [Mys2 ], "A version of differential equations of complex structure with delay is presented by equations whose right-hand max x(t + 0) ... The specific features of these sides contain expressions of the form -hs0<-o equations are still inadequately studied. " In [BaS2], equation (81) was first studied with the use of the numerical -analytic method. In that work, T-periodic solutions of the T-periodic system (81) were sought according to the iteration algorithm Xm + 1( t, z) t = Z + J f I s, xm(s, z),

max Xm (ti, z) ds

0 l TE[s-h, s]

T - t f f (S, Xm(s, z),

max xm (ti, z) ds, (82)

T T E [S-h, S]

111

where m = 0, 1, 2, ... and xo(t, z) = z. It was assumed that the following conditions are satisfied:

(H3.12)

T zE Dp , where f3= 2M and M =

tEoz (

maE oxolf(t,x,y)I;

374 Appendix (H3.13)

r((T + 4h )L) < 1, where L = L I = L2 in the Lipschitz condition I.f(t,x1,Y1)-.f(t,x2,Y2)I <- LiIxt-x21+L2IYi-Y21, which is assumed to be satisfied for all (t, x, y) E R x D x D ;

(H3.14) h > 0. It turns out that, under assumptions (H3.12)-(H3.14), one can apply all techniques of the numerical-analytic method to the investigation of T-periodic solutions of (81). In particular, the iteration process converges , and one can investigate the corresponding determining equation. Note that the operator ( Bx)(t) = max x(t + 0), where I is a compact subinterval in 061 R, maps periodic functions into periodic functions with the same period . Moreover, if x E CPT(R ), then Bx E CPT( IR). Indeed, the continuity of the function max x(t + 0) eEi in the variable t is a consequence of the continuity of a superposition of continuous functions and the following property of maximum:

max x(9) - max y(9) I <_ max I x(9) - y(0) I. eE/

oEI

OEI

Thus, the right-hand side of equation (81) satisfies an inequality of the form

I Bx-Byb <- Kjx-yj0, where K = E. Note that the last inequality is used in the case of differential -operator equations of the general form (see Section 3.3), a particular case of which is differential equations with "maxima." Then, according to the convergence condition (93) of the numerical-analytic method for differential-operator equations , we can replace assumption (H3.13) by the condition

rl 3Lr+2LZ I < 1.

If, in addition , L I = L 2 (which was assumed in [BaS2] ), then, obviously, we obtain the condition

56

r(L) < 1.

375

Appendix

This condition is weaker than (H3 . 13) because it does not take the sign and value of h into account. Simultaneously with [BaS2 ], Sarafova and Bainov [SaB2], [SaB3] applied the numerical-analytic method to the scalar differential equation Y (t)+y(t)+ q max T

E [t-h, t)

(83)

y(ti) = fo(t),

where foe CPT(R ), q > 0, and h is a positive constant. In the investigation of equation (83), the numerical- analytic method was also used to obtain a possibility to consider arbitrary values of q . The corresponding generalization was obtained by Perestyuk [Perl] . By using the operator of generalized averaging

T(r [M^,f](t) = f(t) J e ^(T-T)f(ti)dT, =-(1 +q), 0 Sarafova and Bainov [SaB2 ], [SaB3] considered the sequence of periodic functions

xm+1(t,z)

= z - q f e- (t +q)(t-r) Lmax [yo(s) +xm(S,z)- (yo( ti) + xm(t,z))] [T-h,,r]

0

- MC max (yo(s)+xm (S, z))-(y0 ('r)+xm (i , I sE[T-h,T]

where y0 = y0(t) is the unique solution of equation (83) for h = 0. Denote a := 2hgr(1+q)a2)+2],

R := suplyo(t)I,

d(X) := *p(l + X ) a ^ | j , where

2 (ea.(T-t) -1) (ext - i) It)-

X(eI,T -1)

The following theorem was proved in [SaB2] and [SaB3]:

z)),} dT,

376

Appendix Theorem 18 ([SaB2], [SaB3]). Suppose that the following conditions are satisfied: (i) Equation (83) is given on the interval [a, b];

(ii) the interval [ a, b] is such that bb

2-

as > dX;

(iii) for some integer m > 1, the function

Am(Z)

= (I +q) z + qM4

max (YO(s) +Xm(s,Z))- (YO(t)+Xm(t,Z))]

SE [t -h, t]

satisfies the inequalities Z) < - R (2hq)(Y m+1 min Om ( a+d)L Sz<- b-d)L 2 1-q

min Om ( Z) > a+dA 5z5b - d; 2

R (2hq)am+l 1-q

(iv) the number h is so small that a < 1. Then equation (83) has a T-periodic solution x = x(t) such that x(0) E [a + d;,, b-d,] and x(t) =

lim Xm (t, X(0)). m-a-

The conditions of this theorem seem to be rather restrictive. In connection with this, we formulate the following problem: Problem 8. Prove that the condition a > 1 guarantees the existence of at least one T-periodic solution of equation (83). Later, differential equations with "maxima" were studied with the use of the numerical-analytic method by Shpakovich and Muntyan [ShM], Muntyan [Mun], and Yuldashev [Yul]. Thus, in [ShM], the system of integral -differential equations

x (t) = f 1s, x,

t max x(s), f cp(t, s, x(s))ds se [t-h, t]

t-h

Appendix 377 where the functions f (t, x, y, z) and (p (t, s, y) are T-periodic in the variable t, was considered. The result obtained in [ShM] coincides with the results established in the present paper for differential-operator equations (see Section 3.3) if we set (Ax)(t) = f ( t, x, (Bx)(t)) ,

(Bx)(t) = [(B lx)(t), (B2x)(t)] =

t max x(s), j cp(t, s, x(s)) ds s e [t-h, t] t-h

and use an inequality of the form

r( L^+TL2K) < 1 2 3 in estimates. In [Mun], the second-order system

x'(t) = A.x+ p t, x(t), y(t), max x(s), f cp(t, s, x(s), y(s))ds s e [t-h

, t] t-h t

y'(t) = z t, x(t), y(t), max x(s)9 j cp(t, s, x(s), y(s))ds s e [t-h, t]

t-h

was studied with the use of a generalized version of the method suggested in [Perl]. In [Yul], the numerical- analytic method was used for the investigation of the problem of existence and construction of periodic solutions for systems of first -order integro-differential equations with "maxima." A3.3. Differential- Operator Equations. It turns out that the investigations performed in Sections A3.2.1 and A3. 2.3 can be considered from a common point of view. We have seen that the numerical - analytic method allows one to study periodic solutions of the corresponding equations in a quite simple manner , and the intrinsic complexity of the equations considered is often ignored. Thus , when studying equations with "maxima," we did not even examine the statement of the initial -value problem and its solvability. In fact, the most part of the investigation reduced to the investigation of the integral equation of the method and the resulting determining equation. In [Zav], Zavalykut noticed this feature of the numerical -analytic method for the first time and introduced so-called differential-operator equations. Below, we present the principal results of [Zav].

378

Appendix Let A be a continuous operator defined on the space CPT(R, D) of continuous T-

periodic functions taking values in a closed domain D : A : CPT(R, D) -4 CPT(R, D), i.e., (Ax)(t) = (Ax)(t + T), -cc < t < 00. We consider the general problem of finding T-periodic solutions of the equation x'(t) = (Ax )(t),

to

R.

(84)

Let us consider several important special cases. A3.3.1. Functional-Differential Equations of Retarded Type. Consider the following equation: (85)

x'(t) = .f (t, xt),

where (Ax)(t) = f ( t, xr) , xr( 0) = x(t - 0 ), 0 E [-ti, 0 ], and the function f(t, y): R x C[- ti, 0] -•*V is T-periodic in t and continuous in the collection of its variables. Since x t+ r = x r for x (•) E CPT( R, D), we have (Ax)(t) = f(t,xr) = f( t+T,xt+r) = ( Ax)(t+T). Delay equations and equations with "maxima" are particular cases of (85) because one can set

g (t, x(t ), x(t -0) = A ( t, xt), hl t, x(t), max x(ti) 111 TE[t-s,t] J)

f2 (t, xr),

where .f1 (t, xt) = g(t, xt( 0), xt(-'c)),

f2 (t, xt) = hI t, x(0),

max xt(u) I . uE[-T,0] ))J

A3.3.2. Equations of Advanced Type . Note that equation (85) can be an equation of advanced type for < 0. Although equations of this type are less investigated than those of retarded type (see, e.g., Gyori and Ladas [GyL]), they do not differ from the viewpoint of the application of the numerical- analytic method of successive periodic approximations.

Appendix

379

We also note here the following particular case of the operator A : (Ax)(t) = f ( t, x,

x(tl ), x(t2), ... , x(tp)),

where t1 , t2, ... , tP is an arbitrary finite sequence of points of the interval [0, T]. A3.3.3. Integro -Differential Equations. In (84), the operator A may have the following form:

(Ax)(t) = f ^t' x(t),

j e ',o^i_

T^cp(ti, x( ti))dti I, tE I[8,

where ?0 > 0 and the function (p(s, x) is continuous and T-periodic in s. [Here and above, we have assumed that the continuous functions g(-), h (), and f () are T-periodic in the first variable.] Remark 6. In [Zav], an equation of the following form was investigated: x '(t) = f (t, x, Bx1 (86) where the operator B possesses the same properties as the operator A in (84 ), and the continuous function f is assumed to be T-periodic in the first variable . It is obvious that, by setting (Ax)(t) = f ( t, x(t), (Bx )( t)), we can make the considerations more transparent. When studying equation (84), we assume that jAx-Ay6 5 Lix-ylo, where L = { L ;j >- 0 : i, j = 1 , 2, ... , n }. Then the T-periodic boundary-value problem for equation (84) is equivalent to the integral equation x(t, z) = z + (Nx)(t, z). (87) Here, the operator x H z + Nx defined by the formula

(Nx)(t) = J (Ax)(s)ds--t f (Ax)(s)ds 0 0 maps the space CPT(R, D) into itself whenever z belongs to the set Dp, where

380

Appendix (3 = T M, M =

sup

2 x E CPT(1R, D)

(88)

I Ax 10.

Moreover, since

INx-NyIO <- TLIx-yI0, the operator N is contracting in the complete metric space CPT(1R, D) if the inequality

r(L) < |

(89)

is true. Consequently, equation (87) has a unique solution x = x*(t, z) and, by virtue of the standard argument and properties (see, e.g., [KVZRS]), this solution is continuous and possesses the Lipschitz property with respect to the parameter z. Now analyzing the determining equation T A(z) = T J (Ax*)(t, z)dt = 0 0

by the methods described above , we determine the value z = z* [such that A(z*) = 0)] for which the function x = x' (t, z *) is a solution of the differential-operator equation (84). As indicated above, in applications , one often deals with equations of the form (86), where the function f (t, x, y) : R x D x D 1 -> R" is T -periodic in t, possesses the Lipschitz property in the variables xe D, ye D 1 C Y,

If(t,x1,y1)-f(t,x2,y2) I <- L1Ix1-x2I+L2Iy1-y2IY, and

IBx-Byb s KIx-yIO. (90) Here, (Y, Y+, I • I Y) is a Banach space with a given cone

Y+ C Y of nonnegative

elements and the abstract operation of taking modulus I I Y: Y -4 Y + (for details, see [RRT], where some results of [SaR2] are generalized ), and L1 E L (R", R") , L 2 e L ( Y, ][8" ), and Ke L ( R", Y) are linear continuous operators that are positive in the sense of Krasnoselsky with respect to the conic sets Y+ and 1R+. In this case, condition (89) takes the form [Zav]

Appendix rl 2[L1+L2K ]I

381 <

1

(91)

because

If (t, x(t), (Bx)(t)) -f(t, y(t), (By)(t))

s L 1I x(t)-y(t)I +L2I (Bx)(t)-(By)(t)I. Therefore,

If(t,x(t), ( Bx)(t)) -f( t,y(t), (By)(t)) I0 <- L 1Ix(t)-x(t)Io+L2I (Bx)(t)-(By )(t)l o <- [L1+L2K]jx-yi0. Estimate ( 91) can be improved with the use of one of the ideas from the thesis of Muntyan (see [ShM] and [Mun] ). Indeed, let the iteration process be defined by the relation xm+1(t, z ) = z + (Nxm)( t z), m = 0, 1 , 2, ...,

(92)

xo(t,z) = z. Then

I x1(t,z)-xo(t,z)I <_ al(t)M. Assume that

Ixm(t,z )- xm-1(t,z)I < al(t)c,

cE R.

Then I xm+l(t,Z )- xm(t,Z)I

/ \T T 5 11-T I j I (Axm)(s)-( 'm-1)(s)Ids + T J I(Axm)(s)-(Axm-1)(s)Ids \ 0 t < (1 T/f [

0

L1k xm

-xm- l

I +L2KIxm - xm-l lojds

382

Appendix

Trr I+L2KIxm-xm _tIo + t [LqI xm-xm -l

}ds

t

<- (i_f) t

0

Llcat(s)+L2Kc]ds + t L

t

[

tcal(s)+L2Kc]ds

L1ca2(t)+TL2Kca1(t) <_ [?Li+?L2K] ai(t)c = Qal(t)c.

By induction, we can conclude that Ixm+1(t,Z)-xm(t,Z)I <- a(t)Q'M. Thus, the iteration sequence (92) converges whenever

r(^L]+jL2K^

< 1.

(93)

Note that condition (93) is more precise than (91). Let us now prove several auxiliary statements necessary for what follows. In many cases, these statements are of independent interest.

Lemma 2. Let L = (kiLj ), . =1, where {Lj } m are m x m matrices with nonj =1 negative elements, be an m2 x m2 matrix, and let {ki}'j C R+ be arbitrary nonnegative constants. Then the spectral radius of the matrix L is determined by the relation m

r(L) = r

E=1 kiLj .

(j

Proof. I. First, we consider the case where the components of the matrices Li and all numbers ki are strictly positive. In this case, by virtue of the Perron theorem (see [Gan, Chapter XIII, Section 2]), the matrix

G = kiLj j=1

has an eigenvector b all components of which are positive and

Appendix

383

Gb = r(G)b.

Let us form the vector a = col (k t b, k 2 b,... , k m b) with positive components. In view of the last equality, it is easy to verify that

La = r(G)a. Further, since a matrix with positive components cannot have two linearly independent nonnegative eigenvectors (see Remark 3 in [Gan, Chapter XVI, Section 2]), by using the Perron theorem again, we conclude that r (L) = r(G).

II. Now consider the general case where some components of the matrix Li and the numbers ki may be equal to zero. Let A (e) e }^ i =1 be the matrix all components of which are equal to e > 0. Then the matrices Li (e) = Li + A (e) and numbers ki (e) = ki + e satisfy the conditions of the first part of the proof. Therefore,

m

r(L(e)) = r Yki( e )Li( e) . i=t As is known, the spectral radius of a finite-dimensional matrix continuously depends on the parameter. Therefore, passing to the limit as e - 0 in the last equality, we complete the proof of the lemma. The last statement can be generalized as follows:

Lemma 3. Let P = (KiLL)m. and Q = (LLKi)m. , where Li and K; are m x m matrices with nonnegative components, be m2 x m2 matrices. Then the spectral radii of the matrices P and Q are determined by the relations

m

r(P) = r

E

m

E

LJKi . KJLJ r(Q) = r i=t i=t

Proof. It suffices to repeat the reasoning used in the proof of Lemma 2 with the vectors a and b defined as follows:

Appendix

384 (i) For the matrix P, m

b > K,L, = µb,

a = (bL 1, bL2,... , bLm).

i=1

Then aP=µa. (ii) For the matrix Q,

KJI b = vb,

a = col (K1b, K2b,... ,Kmb).

t=t Then Qa = va. The following statement is very helpful in the proof of the convergence of various sequences of functions: Lemma 4. Suppose that there exist sequences of vectors {am}m 2:1 C 1[8+ and {bm }m,1 C 1R

and nonnegative continuous functions (p 1 and (p 2 such that, for a

certain sequence of functions {xm: M E NJ C C([0, T], Rn), the following inequality holds for all m, beginning with a certain number mo >- 1: Ixm+l(t)-xm ( t)I < am(p1 ( t)+bmcp2(t), 0<-t<-T.

If, in addition, for some 2n x 2n matrix Q with nonnegative elements, the inequality (am+i) 5 (am) bm+1

bm

is true and the spectral radius of Q is less than the unity, then there exists a uniform limit of the sequence {xm}n E!C x*(t) =

lim xm(t). m-+-

Proof. Let cp(t)=(cp1(t),(p2(t)) and cm=col(am,bm). Since em+j= Qjcm, by virtue of the given estimate of I xm+ 1 ( t) - xm (t ) I we have

Appendix

385

//

Xm+1(t)-X-(t)I < IXm+At )- Xm+k-1(t)I+...+IXm+1 ( t)-Xr(t)I

< (p(t)(Qk-1

+Qk+2+...+E)Cm

< (p(t)(E-Q)- 1Cm = (p(t)( E-Q)-1Qm-mOcmO

for all t. Since r (Q) < 1, the last relation implies that the sequence { xm : m >-1 } is fundamental, which completes the proof. Remark 7. Estimate (93) is a simple consequence of Lemmas 2 and 4. Indeed, the conditions of Lemma 4 are satisfied if we set cp1(t)= a2(t) and cp2(t)= al(t) and define the matrix Q according to the formula 3T Ll T2 L2K Q = 10 6 L1 3L2K Since 2 T Q < 3 Li 6 L2K Q *

L1 2 L2K by virtue of Lemma 2 we obtain

r(Q) <- r(3 L1 +T L2K)• Furthermore, since

Q *

3TLI 10

6L2K

L1 9TI2K , in addition to (93), we obtain the following useful sufficient estimate:

r(10L1+ 9 L2K)<1; this estimate complements inequality (93).

Appendix

386

Let us formulate the final result, e.g., for equation (84). Theorem 19. Suppose that condition (89) is satisfied and D 9 * 0, where the vector R is defined by (88). Then, for every z e Dp, the sequence xm (t, z) defined by relation (92) uniformly converges to a T periodic function x*(t, z) =

lim xm(t, z) m-^ _

for t E R, and

Ix*(t,z)-xm(t,z)I <- Qm(E-Q)-'Mal(t). Furthermore, the function T O(z) = T J (Ax*) (t, z) dt 0

is continuous on the set D a, and its zeros [i.e., the roots z * of the determining equation A (z*) = 0] determine a T periodic solution x = x*(t, z*) of equation (84). In this case, the determining equation A (z) = 0 can be investigated by the methods presented in Section A1.2. Let us also indicate several other works concerning the investigation of differentialoperator equations with the use of the numerical -analytic method. In [Zav] , a theorem concerning averaging of T-periodic systems of the standard form x'(t) = e f (t, x, Bx) which is similar to Theorem 7.3 in [SaR2] , is proved. Zavalykut and Nurzhanov [ZaN], under the influence of the works of Samoilenko and Ronto [SaR3] and [SaR4], considered the problem of finding T-periodic solutions of the equation x'(t) = cf(x,Bx),

(94)

called autonomous in the sense that its right-hand side does not explicitly depend on time. However, since the "nonautonomous" equation (86), as has been shown, can be rewritten in the "autonomous" form (84), there arises the question about the correctness of the extension of the notion of autonomous systems from the field of ordinary differential equations to differential-operator equations. In [ZaN], it is assumed that the operator B "preserves the autonomy" (?!) of equation (94). Apparently, the statement that the operator B preserves the autonomy of equation (94) should be read as (Bx)(t + t) = (Bx)(t) whenever x(t + µ) = x(t), where the

387

Appendix

period µ is arbitrary and not fixed. [Recall that the value of the period µ = T was fixed earlier.] Problem 9. Is it possible to strengthen inequality (93) to the condition

r( T Lt +T L2K) < 1,

where q = 3 .41... ? There are many works devoted to the investigation of various integro-differential equations with the use of the numerical-analytic method. The pioneer publications (see [Vakl]) appeared soon after the method had been suggested. Later, the majority of authors that carried out research in this field considered this problem in various aspects. The analysis of the publications mentioned above reveals two main trends in this field. The first trend relates to the investigation of equations of the form x'(t) = f(t, x(t), (Ax)( t)),

to R, (95)

where the expression for the operator A : CPZ(R) - CP7(R) contains an integral operator of the Fredholm type. In these works , the scheme of the numerical -analytic method presented in Section A3.3 for the differential -operator equations (84) is repeated for specific operators A. For this reason , we omit a detailed analysis of these works. Thus, in [Vak2] , the operator h(t)

(Ax)(t) =

J0 cp(t, s, x(s))ds,

where h(t) and (p (t, s, x) are functions T-periodic in t and s, is considered. In [Gos], the operator b (Ax)(t) = J cp(t, s, x(s))ds, (96) 0

where 0<- a<_ b:5 T and cp(t, s, x(s)) is a function T-periodic in t, is considered. In the case of equation (95) with operator (96), according to the results obtained in Section A3.3, the numerical-analytic method is applicable if the conditions D d # 0 and

r(

3T

+ 2K(b-a) T) < 1 (97)

388

Appendix

are satisfied. Here, it is assumed that a condition of the type (90) IAx-Ayb < Kl x-yI0 and the condition

I
(Nx)(t) =

T

f [s, x(s), (Ax)(s)]ds - T f f [s, x(s), (Ax)(s)]ds 0 0

f

and

(Nx)(t)-(Ny )(t)I <- VIx(t)-y(t)I• Here, the linear operator of comparison V is defined as follows: b (Vu)(t) = L 1(Au)(t)+ L2Kai( t) f u(s)ds.

Thus, condition (97) (see Theorem 6 .2 in [KVZRS]) can be replaced by the assumption that r(V)

<

1.

Since

(Vat)(t) = [

5

LI + L2K a1 (s)ds at (t) a

for all t E [ a, b ], condition (98) is definitely satisfied if

(98)

Appendix

389

rl 3 Ll + L2K f al(s)ds I < 1.

(99)

l

a

J

It is obvious that inequality (99), which coincides with the corresponding estimate from [Gos], is more precise than (97). Further, in [Nur], it was assumed that t+a

(Ax)(t) = f cp(t, 0, x(0))dO, 0

where (p (t, 0, x) is a function T-periodic in t and 0. The straightforward application of the results of Section A3.3 gives the following sufficient conditions for the applicability of the numerical -analytic method: Ds*f, r(3IiT+1L2KTIaI) < 1.

(100)

These conditions are completely analogous to conditions (97). Note that, in [Nur], the following estimate was obtained: 2 l / a \ T )2] < 1, 3r(Lt+LZK[aI 1+ 2T I+ T(1-T)+3T (l _i

which is more rough than inequality ( 100). Furthermore, it was assumed in [Nur] that a e [ 0, T] , whereas in ( 100) the value of a is arbitrary. Also note that, later, Butris [But2] studied a simpler case where

(Ax)(t) = j g(s, x(s))ds. t-T

Other publications related to this trend were mentioned in Section A3.2.3. Some of them will be indicated in what follows (in the section devoted to the investigation of "impulse" systems). The second group of investigations relates to the investigation of equation (95) under the assumption that the operator A has the form r (Ax)(t) = f cp(t, s, x(s )) ds.

(101)

0

The theory of the numerical-analytic method presented in Section A3.3 can be applied to equation (95) with operator (101) if the function cp(t, s, x ) is T-periodic in the variables t ands and, in addition, for all m = 0, 1, 2, ..., we have

390

Appendix T f cp(t, s, x," (s, z))ds = 0,

(102)

0

where { X. (s, z) }m z 0 are successive periodic approximations defined by a relation of the type (92). If we omit condition (102) and construct a scheme of the method in the general case, we inevitably face the necessity of finding a solution of a system of 2n nonlinear transcendental equations with n unknowns (the vector z e R"). It is clear that such a system may have no solutions in the general case. In this connection, we pose the following problem: Problem 10. Find conditions [e.g., certain symmetry properties] for the functions f (t, x, y) and cp ( t, s, x) under which relation (102) is always satisfied. We note that, unfortunately, this problem was not solved in the original work of Vakhabov [Vakl] and in the subsequent works of Sarafova and Bainov [SaB1], [SaB4], and [BaSI]. In connection with the application of the numerical-analytic method to other integrodifferential equations, we mention Alymbaev's paper [Alyl], republished later by Nurzhanov and Alymbaev [NuA], where so-called autonomous systems of integro-differential equations t+i x (t) = f x, f cp(t - s, x(s)) ds t

J

were considered. We also mention [Aly3] and [Wuj], where the equation

x'(t) = f t, x(t), f cp(t, s, x(s)) ds

was considered, and [Aly2], where periodic solutions of autonomous systems of integrodifferential equations with infinite delay were analyzed. A system of Volterra integro-differential equations in the implicit form

X 'M = f I t, x(t), X'(t), f cp(t, s, x(s), x'(s))ds l 0 was studied in the thesis of Turbaev [Tur]. Equations of the form

391

Appendix h(t)

2

d 2 + 0) 2 x = f t, x(t), x (t), f cp(t, s, x(s), x'(s))ds dt

0

were studied by Mitropolsky, Khoma, and Gromyak [MKG] and Chornyi [Cho2]. In [Cho2], a T-periodic solution of the integro-differential equation 2

dt2 =

kt+h(t)

f

t, x(t),

X (t),

J

qp(t, s, x(s ), z (s ))ds

kt

where h(.) is a T-periodic function and k is an integer, was analyzed. A3.4. Abstract Scheme of the Method. In Section Al, we suggested one possible "operator" formulation of the method. Moreover, the interpretation of the so-called Lyapunov -Schmidt equation given in Section A2.1 enabled us to discover a certain relationship between the numerical-analytic method and the Lyapunov-Schmidt equation. (To avoid misunderstanding, we deliberately speak of the Lyapunov-Schmidt equation and not of the "Lyapunov-Schmidt method" because, as far as we know, the latter notion is not used in the literature.) Numerous developments and applications of the numerical-analytic method to various classes of boundary-value problems make us think that it would be useful to present the method in the following abstract form: Let C(D) be the space of continuous vector functions y: D -4 R", (x^, ... , xm) col (y t (xt, ... , xm), ... , yn(X i, ... , xm)), Dc Rm. Denote by K(D) c C(D) the space of functions from C(D) that satisfy the additional condition U(y) = col (UI (y), ... , U„(y)) = 0.

(103)

Taking into account that the main purpose of the method considered is the investigation of solutions of boundary-value problems, we call ( 103) an abstract boundary condition. It is obvious that, for example , in the case of periodic boundary conditions , K(D) is the space of periodic functions. Let A : C(D) -* RA be an operator with domain D (A) = C (D) and range R A = A C(D) c C( D), which gives the equation y = Ay. (104) The problem is to find a solution of equation (104) that satisfies the abstract boundary condition (103), i.e., a solution y e K(D). If A were an operator of the type A : K(D) -4 K(D), then it would be possible to investigate the solvability of equation (104) in the space K(D) by using the general the-

392

Appendix

orems on the fixed points of operators . However, this is impossible in the general case where RA contains points from C(D)\K(D). In such a situation, along with equation (104), one should properly introduce an auxiliary equation z

=

Az

(105)

so that DA = RA = K(D).

It is natural that there should be a certain relationship between the operators A and A, namely, the following conditions should be satisfied: (i) every solution of the abstract boundary-value problem (104), (103) is also a solution of equation (105); (ii) every solution z e K(D) of equation (105) that satisfies a certain auxiliary condition

Bz = 0 (106) given on solutions of equation (105) and satisfied by all solutions of problem (104), (103) is, at the same time, a solution of the original abstract problem (104), (103). Since A : K(D) - K( D), the solutions of equation ( 105) under appropriate conditions can be constructed with the use of various iteration processes , in particular, by the method successive approximations Zk+1 = A Zk , k = 0, 1, 2,... , in which there is a certain freedom in the choice of the zero approximation zo. In the numerical-analytic method, this freedom is used to satisfy condition (106). In the method considered, the analytic apparatus of successive approximations of the Picard type is used for finding the limit function z * = lim k,„ A zk, and the auxiliary (algebraic) equation (106) is solved by numerical methods. Note that, in a special case of problem (104), (103), it is not difficult to construct both an extension of the operator A, i.e., the operator A, and the operator B (see, e.g., [SaR6, Chapter 1]). Let us now present results concerning the application of the method to autonomous systems of differential equations and impulsive equations.

393

Appendix

A3.5. Autonomous Systems of Differential Equations. Consider the case where the right-hand side of the system 4 = f (t, x), x,f E ]R'2,

is independent of t. It then follows from the main equations of the method (see relations (1) and (2)) t

T

x(t,z) = z + J f(s,x(s, z))ds - T J f(s,x(s, z))ds,

(107)

T

A(z) =

J As, x(s, z)) ds = 0,

T0

(108)

that, in the autonomous case, we have

x(t,z) = z, A(z) = f(z) = 0. Therefore, the numerical -analytic method determined by the relation

xm(t,

z) = z +

J L f(s, x -t(s, z)) - T J f(ti, xm-t(ti, z)) dti ] ds m

(see formula (7)) "detects" only stationary solutions of the autonomous system dx = _

dt

f(X),

X E D C )(gn,

f E art,

(109)

x(0) = x(T). In contrast with nonautonomous systems, where the period of the required solution is known, in the autonomous case, in addition to the initial conditions for the periodic solution, one should also determine its period T. Furthermore, the T-periodic solutions x = x(t) of system (109) are not isolated in the extended phase space. The latter makes the theorem on indexes completely inapplicable. This suggests the idea of selecting a certain solution x = x0(t) among all hypothetical nontrivial T-periodic solutions x = x(t + (p) according to a certain rule and constructing successive approximations on the basis of this solution. In this case, it is also natural to change the system of parameters determining the T-periodic solution and consider the vector (C) I C2, ... , C, I , µ) instead of the vector of initial values x O = (x10, x20, ... , xi0). Here, (c1, c2, ... , cn-t) are the local coordinates on the hypersurface r

394

Appendix

of codimension 1 that is "transversal" to the required solution [an analog of the Poincar6 section], and µ = T I (2n) determines the period T of a periodic solution. Note that, by virtue of the transversality of r, the initial parameters c 1, c 2, ... , c,, t, t are, generally speaking, isolated in the new coordinates . However, since even the existence of a periodic solution is a priori unknown, the choice of t is neither unique nor evident. To solve this problem, the following approach was suggested in [SaR4]. It is obvious that every periodic solution intersects each of the sets rt = {xE R : fAx )=0, i=1,2,..., n},

where f = col (fl,f2,...>f,). If at least one of the sets I'j is a hypersurface [as a rule, this is true in applications], we can set IF = I'j = {xE Rn: x=x0 ( c), c=(c1,c2, ..., cir1)E Dc}. The next step is to change the variables

x=c(0,y ), 0 = t = t T

(110)

which enables one to study problem ( 109) near the hypothetical solution x = x0 (t ). The function b(0, y) is 2Tt -periodic in the first variable and is determined in every special case by the information about system ( 109) obtained from its first approximation. Moreover, the function cD(0,y): [0, 2n]xD1 -*D satisfies all conditions usually imposed on the change of variables, namely, it is continuously differentiable with respect to the variables 0 and y and satisfies the condition

det [ a(D

L

(0, y)

ay

1J

J

^ 0.

After the coordinate transformation, at time 0 = 0, the hypersurface t turns into the hypersurface f = {yE R": y=y0(c)}, where x0 ( c) = 4(0, y0(c)).

Appendix

395

As a result of the change of variables ( 110), equation ( 109) takes the form

!LY = F(O, y, µ), dt where

F(e,

y, µ) _ [ aca(O, y) 1 E tf(c(e, - y) 1. y))

Dy

Assume that the scheme of the method enables one to find the 2n-periodic solution y = y*(A, c* , µ*) = lim m-4„ ym (9, c*, t*) of equation ( 111), where e Ym(e,c,4) = y0(c)+ J rr F(O,ym-1(0,c,µ),µ)

0 2n - 21n

j F(s,ym-i(e,c,N),!.t)ds,d9, 0

m = 1, 2, ... , yo(0, c, µ) = yo(c), and (c*, µ*) is the root of the determining equation 2n

A(c, µ) _ 2f F(9, y*(0, c, µ), µ)d9 = 0. 0 Then the function x = x*(t) _ OD _L -L \µ*,y*\µ*,c*,µ*JJ is a nontrivial periodic solution of equation (109) with period T * = 2nµ*. Furthermore, x*(0) = xo ( c*) and x*(t ) coincides with the hypothetical solution x =x0(t). It is obvious that x = x*(t + (p ) determines a one-parameter family of solutions of equation (109). With the use of the numerical -analytic method , autonomous systems were first studied by Le Lyong Tai [Lell], [Lel2] and Samoilenko and Le Lyong Tai [SaLl] with the use of the change of variables x = 4) (0, y ) = e Aey + f o ( 0) under the assumption that the value of one of the coordinates of the required solution is a priori known.

Appendix

396

Problem 11. Indicate classes of autonomous systems that, after a transformation of the form (110), admit investigation by the numerical-analytic method. A3.6. Autonomous Systems with Periodic External Influence. First, we establish one general statement concerning the solvability of a nonautonomous periodic boundaryvalue problem of the form dx = f(t, x),

x(0) = x(T), x, f e R. (112)

Definition 1 (Carath6odory conditions). We say that the function f (t, x) defined on the set [0, T] x f2, S2 a lR', satisfies the Caratheodory conditions if (HKI) for every x e S2, the function f(•, x) is measurable; (HK2) for almost all t e [0, T], the function f(t, •) is continuous; (HK3) for all x, y E 92 and almost all t E [ 0, T], the coordinatewise estimates

If(t,x)I <- m(t), If(t,x)-f(t,y)I <- L(t)Ix-yl are true and the components of the vector m(t) and the matrix L (t) are summable on the interval [0, T]. We also assume that the domain f2 satisfies the following condition (HK4) f2 p : = 1z e 92 : B (z, P) c: S2 } # 0, where B (z, R) denotes the convex set of x e lR' such that, coordinatewise, we have

Ix-zl <- R, where

R = max (Km)(t), tE[O, T]

K is the linear integral operator defined on the (classes of) summable vector functions by the formula /

\t

T

(Kx)(t) = 11- t I J x(s)ds + t f x(s)ds, \ T 0 Tt

(113)

Appendix 397 and IxI = I(XI,X2 ,...,4I = Col (Ix1I,Ix2I,...,IxnI),

max I x(t) I = col ( max I xl (t) I, ... , max I xn (t) I ). t

t

t

We introduce a linear transformation - of the space of summable functions into itself as follows: t

T

z(t) = J [x(s)_5 x(u)du ds.

0

0

We also introduce preordering D of the cone of n-dimensional vectors x = col (x1.... , x„) with nonnegative components as follows: xFy t* ( 3iE{1,2,..., n}: x;>y1). Theorem 20. Suppose that the function f (t, x) in equation (112) satisfies the Carathiodory conditions (HKJ )- (HK3), and the domain f2 satisfies condition (HK4). Assume that the spectral radius satisfies the inequality r (S) < 1, where S is the composition of the linear integral operator (113) and the multiplication by a variable matrix L( t), i.e., (114)

Sx = K(Lx).

Assume, in addition, that, for some m , the corresponding operator function (see (13)) T

Am(z) = T

J f(s,xm(s,Z))ds

(115)

0 satisfies the following relation on the boundary a S21 of a certain subdomain 01 c 0: I Om(Z) I > Om(z),

z€

as21 ,

(116)

where T Om(z)

=

T

J L(t) (1-S)-1Sml f(t, z)I dt. 0

398

Appendix

If, moreover, the Leray-Schauder degree deg(Am, a521, 0) * 0, then there exists at least one solution x = x *(t) of the periodic boundary-value problem (112) such that its initial value x *(0) E 521. Proof. As shown in Lemma 17 and Corollary 16 in [RoT2], it follows from the conditions of the theorem that the parametrized integral equation (107) corresponding to the boundary-value problem (112) (see (2)) has a unique solution for every z E 0 p, and a single-valued continuous determining function A(z) of the form (108) is given on Up. In view of our assumptions, the linear deformation of a finite-dimensional vector field Am(Z) of the form (115) into A(z), A(k,Z) = Am(Z)+ ?[A(Z)- Am(Z)], a,E[0,1], is nondegenerate on a 01. Indeed, otherwise, there exist zo E a S21 and ?O E [0,1] such that Am(ZO) = - A,0 [A(ZO)-A.(ZO)]Since , according to Lemma 17 in [RoT2],

IA(Z)-Dm(Z)I < 0m(Z),

we have IAm(z0)I <- IA(zO)-Am(zO)I <- Om(ZO), which contradicts inequality (116). This completes the proof of Theorem 20. On the basis of Theorem 20, one can obtain the following result concerning the existence of periodic solutions of perturbed autonomous systems of the form

d= .f(x)+g(t), t E [0,T], x,.f, g E Rn,

(117)

where the function g (t) satisfies the condition g (0) = g (T ). Theorem 21 ([RoT2, Theorem 24]). Suppose that a function f : Q -4 R n is continuous, satisfies the Lipschitz condition (HK3) with matrix L, and can be continuously extended from 52 to Rn with the same Lipschitz matrix. (We denote the corresponding extension by the symbol f .) Also assume that T),.. (L) < 40 = 3.4161306... ( 118) and the following relation holds for every z E a52 uniformly in h E [0, T]:

Appendix

399

T

I.f(z)I D 00

T

LJ(I-KL)- 'I 8h(t) I dt,

(119)

0

where, by definition, we set gh(t) := g(t+h), where the right-hand side contains the periodic extension of the function g () to the entire real axis denoted by the same symbol. Also assume that, in the domain Q, the function f (•) has a unique zero z0 of nonzero topological index, i.e., ind(f,z0)#0. Then equation (117) has at least one T-periodic solution. Proof. First, we note that, without loss of generality , one can assume that T Jg(t)dt = 0. 0

Otherwise, the system should be rewritten in the form T

T

dx _: .f(x)+ T J g(s)ds + g(t)-T J g(s)ds 0 0 We introduce the one-parameter family of differential equations dx = f (x) + gh(x), h E [0, T]. dt

(120)

Let us now clarify the form of the zero determining functions A0(z,h) constructed by the zero approximation x0(t, z) = z. It is easy to see that, under our conditions, we have T

T

D0(z,h) = T J [ (z)+gh(t)I d t = 0

and

.f (z) + T J g(t) dt = f (Z)

0

V Z E R"

400

Appendix A(z,h) = f( z) `d h e [0, T]

a

for all z e s2. Let us apply Theorem 20 to the T-periodic boundary-value problem for equation (120) in the case m = 0. Assumption (119) implies that relation (116) with m = 0 holds on a 0, where K and S are operators of the form (113), (114), because

f (z)+g = f (z) + 8= 8

V z a a Q.

The right-hand side of (116) T

TL f (I-S)- 1StmIjh(t)Idt, 0

is well defined by virtue of assumption (118), which, in this case, is equivalent to the inequality

r(S) = T xa,(L) < 1 q0 for operator (114). Since A0(z,h) > O0(z) V Z E a S2 and deg (A0(z,h), a 12, 0) = ind (f, z0) * 0, by virtue of Theorem 20 we can conclude that there exists a T-periodic solution P(t, h) of equation (120) such that P (O, h) E Q. Let us show that P(t, 0) : [0, T] --> S2. Indeed, if this is not true, then there is h `e [0, T] such that the inclusion P(0, h) E a S2 holds. However, since P(t, h) is a Tperiodic solution of equation (120) for h = h`, this implies that A(P(0, h'), h") = 0, v which contradicts our assumption. Further, since P(t, 0): [0, T] - 92 and f (z) = f(z) `d z e 12, we conclude that P(t, 0) is a periodic solution of equation (117). The theorem is proved. Example 7. Consider the problem of 2n-periodic solutions of the equation z d x+asinx = e ( t), IT

tE [0, 1],

xE S1 = R /21cZ, (121)

where a>0 and e(t)E L 1 [ 0, 1 ]. Equation ( 121) is equivalent to the system

401

Appendix dx dt =y' dx = -asinx+e(t). dt

It is easy to verify that this system satisfies the Lipschitz condition (HK3) and the sublinear-growth condition I f (t, x) I <- L (t) I x I + g (t ) with the matrix

0 1

L(t) = L =

IO

0,

Therefore, the spectral radius of operator (114) is given by

r(S) = 1 Xmax(L) =

q0

qo

and r(S)<1 whenever a < qo = 11 .669948....

(122)

Under these conditions , the determining function A (x, y) is uniquely defined for all x, y e R 2. By corresponding calculations , we obtain

A0(x, y) =

Y (-asinx+e '

where T e : = T J e(t)dt.

0 Consequently, A Yo = o = e a = b OAsinx of o^x yo )0 If I J I < a, then the approximate determining equation A0(x, y) = 0 has two different roots in S t x R . Without loss of generality , we can assume that e ? 0 . Then the two different roots are the following: x2 = 7c - aresin e . x i = arcsin e , a a

402

Appendix

Note that the validity of the inequality I e I < a is also necessary for the existence of a root of equation ( 121). One can verify that this is true by integrating ( 121) over [0, 1]. In what follows, we consider the neighborhood of the point ( x t, 0). (The second point, ( x2, 0), can be investigated by analogy.) First, we choose O 0. For this purpose, we estimate the integral

L

j(I- KL) -t \1 0l e 1Jdt, 0

where the operator K is defined according to (113), namely, r

t

0

t

(Kx)(t) = (1-t) j x(s)ds + t jx(s)ds. If 91

(I-KL)-t

(W0) = \W2 1, then

J Nt = Kcp2, 92 = aK(pt+leI. The direct calculation yields (P2 = a K2 92 + I e 1. Hence, Since max [K 2 . 1](t) = 6

(P2< aI(p210 K2.1+p I.

,

te[[0,1]

we have

k 2 I0

< 1010 6

6

whenever a < 6. Note that we have restricted ourselves to the simplest estimates . If necessary, the condition a < 6 can be weakened to (122) by more accurate estimation. Combining the last two formulas, we get

Appendix

403

W2 <- K2.1Ie'o66oc +IeI. Since t

f0 K2.1 dt =

10'

we get t

0.6a

t

f (p2(s)ds <- IeI06-a + f Ie(s)Ids. 0 0 Further, we have 1 1 f cpt(s)ds = f Kcp2ds 0

0

1 <- f ([3 . i]i0 66a+KIel ds = f Klejds + Selo

)

0

6.2a

0

because

f 0 K3 1 30 As a result, we obtain t e I ds+ 66 f I" I e"lo a

GO o

ft aK Ields+Ielo 0

0.2a2

6-a

It is convenient to choose the domain 0 in the form of the rectangle bounded by the lines x = ±1t/2 and y = ±c (c>0). Obviously, we have A0(x0,y0) ? 00, provided that a-e" > a f Klelds+ lelo0.2a2 (,-n 0

(123)

c> f ale(s 0

)Ids+Ieio6.6a'

404

Appendix

Moreover, deg (D0 (x,y),

an, 0) = ind (A0(x,y), (x1,0)) 0 1 = sign det

-acosx1 0

)

# 0.

Consequently, all conditions of Theorem 21 are satisfied. Therefore, by virtue of this theorem, equation (121) has at least one periodic solution with the initial value (t = 0) lying in the indicated rectangle. Let us consider a particular case of equation (121). We set e(t)= n cos nt. Then e(t) = sin nt , Ii 1o = 1, and e = 0. One can verify that inequality (123) holds whenever

a < 4.81269..., c = 0.6a + 2 6-a it Therefore, with the periodic perturbation e (t) = it cos nt, there exists at least one periodic solution x(t, a) of equation (121) for all a < 4.81269.... This solution is such that

a Iat

< 6.6a + 2, Ix(0,(x)I < 2•

Remark 8. Let us compare the estimates obtained above with those obtained by Granas, Guenther, and Lee [GGL, Theorem 10.3]. In [GGL], an equation of the form

z

d2

+ g (x) d + a sinx = e(t)

was considered. Condition (B) in [GGL, Theorem 10.3] has the form 1 a+ J Ie(t)Idt <

0

2

This condition is not satisfied for equation (121) with e (t) = it cos nt. Therefore, Theorem 21 complements, to a certain extent, the results of [GGL]. A3.7. Impulsive Equations. The numerical-analytic method was first applied to impulsive systems in 1967 by Samoilenko [Sam4]. More precisely, a system of the following form was considered:

405

Appendix ^

+

co^ = e/(x,|)

A^

=^

dt

,*«,.

= J "(§}

-^ x=x0+

dt

x=x0-

dx £0, dt Ar=Jt0+

*

dx

0,

dt

_

\x=x0-

Within the framework of the modem classification of impulsive systems (see Samoilenko and Perestyuk [SaP3]), this system is called a discontinuous dynamical system. In [Sam4], with the use of the algorithm of the numerical-analytic method, the applicability of the averaging method to this system was justified and approximating solutions were constructed. Analyzing further applications of the method to impulsive systems, we outline the integrity and completeness of the investigation carried out in [Sam4], The subsequent applications of the method were more formal in spirit and established conditions of its applicability to systems with impulses at fixed moments of time [or, in fewer works and not always successfully, at nonfixed moments of time]. The paper [PerS] of Perestyuk and Shovkoplyas (see also Chapter 4.3 in [SaP3]) should be regarded as the first and most important among these works. A somewhat improved version of [PerS] appeared as Section 16 of the monograph [BaSi] by Bainov and Simeonov, where the application of the method to periodic impulsive systems of the form — =f(t,x), dt

t*tj,

Ax\t = H,(x),

0Zt
1=1,2,.,P,

jceQcR",

(124)

0 < / , < . . . < / „ < T,

(125)

x(0) = x(T),

(126)

was considered. In this case, as is customary, it is assumed that, in die compact domain Q, the righthand sides of (124) and (125) are continuous in the collection of variables and satisfy the Lipschitz condition with respect to x, namely,

II/(/,*)-/«,?) II * * 0 l * - y l . !#,•(*)-fliOOl 1 Kifx-yl

i = 1.2

p.

x,yea,

and, moreover, there exist constants A/( and a function m(t) e Lx[0,T] ||//,(x)|| S Mt

VxeQ,

i = 1,2

p,

such that

sup|/(»,*)| < m(t).

406

Appendix

A solution of problem (124)-(126) was found as the limit of the uniformly convergent sequence of functions

Xm(t, Z) = Z + . I f (S, Xm(S,Z))- f H(-) }ds + I Hi (xm(ti.Z)), (127) 0 ` O
m=0,1,2,...,

x0(t, Z) ° Z,

where T

T H(") =

T

p

ll

s> Xm(S, Z ))ds + Hi( xm(ti. Z)) . J f( 0

i=1

J

The following theorem was proved: Theorem 22. Suppose that m (t) =- M = Mi and the following conditions are satisfied: (Il) there exists a nonempty closed set 0 p that belongs to Q together with its R= T MI 2+ p M-neighborhood; (12)

the following inequality is true:

K3T+pK1+pTK0K1<

(128)

If system (124)-(126) has a T -periodic solution x = cp(t, z) passing through a point z E SZ p at time t = 0, then the solution is the limit of the uniformly convergent sequence of periodic functions cp(t, z) =

lim Xm (t,z) = x*(t,z)

defined by relation (127). Remark 9. In fact, in [SaP3] , more restrictive conditions were considered , namely, the number 0' = TM! 2 + 2p M was used instead of 0 and, instead of inequality (128), it was assumed that

T3° + 2pK1 + pT KO < 1.

407

Appendix

In [BaSi], more accurate computations were performed, which enabled one to weaken the requirements imposed on the impulsive system. More precisely, the following, even less restrictive, conditions were suggested in [BaSi]:

p" = TM +MQ, -1+ Tao +QK1 < KOK,(TQ -SI <1,

where Q = sup{I1 - t i[O,t]

)

S = sup

C

+ t i[t,T]: tE[O,t]},

t I a(Tk) + t I T )o
a(Tk):

t o [0, t] J.

T t
Theorem 22 made it possible to reduce the problem of existence of T-periodic solutions to the problem of the roots of the determining function

A(z)

=T

L

J f (s, x*(s, z)) ds + HI (x*(ti, z)) ]. p

i=1

The methods for the investigation of the determining equation A (z) = 0 presented in [SaP3] do not significantly differ from those in [Sam4] and [SaRI] and, therefore, we do not dwell on them. As applications of Theorem 22, the following scalar impulsive systems (x e R) in the standard, in the sense of Bogolyubov, form were also considered in [SaP3]: dx = Ef (t,x), t#t;, 0<-t<-T, dt

(129)

Axlt = EHH(x), i=1,2,...,p, 0
(130)

Theorem 23 . Suppose that the right-hand sides of the T-periodic impulsive system (129), (130) satisfy the same requirements as in system (124), (125). If the averaged system p

dy = Efo(y) £ J f(s, y) ds + Hi(y) dt T o i=1

408

Appendix

has an isolated equilibrium state y = y 0, f o (yo) = 0, and the index of the mapping f0(y) at the point y0 differs from zero, then, for sufficiently small values of the parameter E, the system of equations (129), (130) has a T-periodic solution x = (p(t, E) such that lim"o (p(t,E) =yo. One can easily notice a certain "asymmetry" in the iterative relation (127) caused by a certain lack of regard for the "correction" of the impulsive part. Indeed, only the integrand is corrected in (127). One should expect that, suggesting a new modification of the method according to its general idea [see Section A3.4] such that the above-mentioned "asymmetry" disappears, we can improve the convergence of the method in certain cases. The results of Trofimchuk [Trol] show the validity of such assumptions. For the determination of solutions of problem (124)-(126), the following recurrence sequence of functions xm(t,z) satisfying the boundary condition (126) was constructed in [TroI]: T

f(s,xm(s, z) -K0 fH("')

xm+l(t,z) = z + 0

]

I

ds +

]

[Hi(Xm(tiZ))_ Kii;;U;) ,

0
(131) m=0,1,.. ,

x0(t, z) = z,

where the average f H(m) is determined with regard for the weight coefficients as T

P

7H(m) T J f(s, xm(s, z))ds + Hi(xm (ti, z)) p

i=1

J

for ti = K0T + K j+... + K. (We assume that the constants Ki are not simultaneously equal to zero because, otherwise, the problem becomes trivial.) As is seen, the right-hand side of (131) already has the indicated symmetry property. Furthermore, the practical realization of the iteration process (131), in fact, is not more complicated than that of scheme (127) because the main auxiliary calculations (the computation of the averages F H(-) and f Hm) are the same in both cases. We introduce a piecewise-continuous function M(t): I -4 R + as follows:

[l(KV _0t+

M(t) =

ti

K; 1

T

1

J

Jm(u)du + M1 0 r
+

]

LrKut+jt;
t

t

I.

Appendix

409

Let (3 *= sup,Et M(t). We have *

T

R <- m(u)du + Mi 0 because (a - x) y + x (b - y) <- a b for arbitrary x E [0, a] and y E [ 0, b ] . Assume that the set 520. is nonempty . The functions xm(t,z) belong to 0 for z E QR* because

(132)

1 xm ( t,Z) - zil <_ M(t) <- 0 T.

Prior to the formulation of the result on the convergence of scheme (131), we introduce the following notation: (Kot +

ai(t)

'
I

-1 +i Kj)-i+3Ki Y = tijP max{ 0, [2(Koti =1

l

i=1

J

]

K? } :5 i K^. i=1

Theorem 24. Suppose that the set QV is nonempty and the inequality q = ti / 3 + y < 1 is satisfied. Then the integral equation corresponding to formula (131) has a unique solution x*(t, z) in the domain considered, and it can be found by the method of successive approximations (131). Furthermore, the following estimates are true: x*(t,z)-xm(t,z)11 <_ Q*[gai(t)+y]qm-z(l-q)-1, m?2,

Il x *(t,z)-zII <- M(t) <-

R

llx*(t,z)-x1(t,z)H :5 min 120% 0*(qa*,(t)+y)(I -q)-' The function x*(t, z*) is a solution of the boundary-value problem (124)-(126) whenever the continuous function

IT p Y(z) = io I $ f( s, x*(s, z))ds + YHi (x*(ti, z)) Lo vanishes at the point z = z *.

1=1

Appendix

410

It is interesting to note that, later, considerations similar to those presented above were independently developed by Samoilenko and Teplinsky [SaT2] . In [SaT2], the periodic problem of control for impulsive systems was studied , which required the examination of an integral equation of the form t

x(t,z) = z + f [ f(s, x (s, z) - ul ] ds + [ Hi(x(s,z) -u2 ]. 0

O
(133)

In this connection, versions of "impulse control" with

u2 = 1 I f(s, x(s, z))ds + PLO

U = 0,

H,(x(t,, z)) ,

"differential control" with T

p

H, (x(t,, z)) ,

u, = - J f (s, x( s, z))ds + 0

u2 = 0,

1=1

and "mixed control" with T

p

u1 = T [ J f(s,x(s,z))ds+P H,(x(t,,z)) , 0

i=1

rT

U2

=

p

(s, x(s,z))ds + Hi (x(ti,z)) iP J f 0 i=1

L

were considered. It is clear that the first two versions can be used for the determination of a T-periodic solution of an impulsive system . This was carried out in [SaT2 ], again under the condition of control over the differential part. It should be noted that less accurate estimates of convergence were obtained there . For instance, instead of (128), it was assumed that KOT + 2pK1 < 1 , 2

KO = K1.

Note that the symmetry properties of impulsive systems were used only in [SaT2]. The following statement is true: Theorem 25. Let the T -periodic system of equations

Appendix dX = e f (t, x), t# ti, dt

411 t o R, X E W

Arlt = eH(t) I t_, i E Z,

be such that H(-t) H(t), f(t, x) f(-t, x), and, on the interval (-T, T ), the 22 moments of impulse influence are located symmetrically with respect to zero. Then every point z E S2 p is the initial value of a T -periodic solution of this system. Note that the application of the method to the investigation of impulsive systems with nonfixed moments of pulse influence is justified in [SaP3]. This monograph indicates the possibility of such an application; however, the important technical Lemma 21.3 in [SaP3], as well as the corresponding statements in [HrB2], contains certain inaccuracies. We do not discuss these results here. Problem 12. Correct the inaccuracy in Lemma 21.3 of [SaP3] for systems with impulse influence at fixed moments of time of the form dx = f(t, x), t # tit(x), dt Arlt=ti (x) = Ii(x). Among the other works on this subject , one should also mention the papers of Akmetov [Akm] and Akmetov and Perestyuk [AkP]. In [AkP], the method was applied to the investigation of a periodic impulsive system of the form dx = Ax+f( t,x,y), t#tt, dt

dt = g(t ,x,Y), t#tt, Arl t=t = Bx + It(l)(x,Y),

Aylt_t

= It(2)(x,Y),

where x E R n9 y E R m, A and B are real n x n matrices, and det (E + B) # 0. The results obtained in [AkP] complement and generalize the investigations of [PerS]. We also mention the works of Hristova and Bainov [HrB2]-[HrB4]. In [HrB2], problem (124), (125) is studied with the use of the numerical-analytic method with the twopoint boundary conditions

412 Appendix Ax(0) + Cx(T) = d,

A, C E Rnxn, det (C) * 0.

In these works, systems with fixed and nonfixed moments of impulse influence are considered. We have already noted the inaccuracy of the lemma from [HrB2] in the case of nonfixed moments of pulse influence. In [HrB4] , an impulsive delay system of the form

dx = f(t, x(t), x(t - h)), t * ti, dt

oxl r _-r,. =

Ii (x(ti)),

where x E R n and the sequence of points [ t; a R, ti+t > t; 1, i E Z, is fixed, was considered. In [HrB4], certain conditions were indicated under which a method for finding periodic solutions based on the ideas of the numerical-analytic method and the Galerkin method can be applied to this problem . Simultaneously with [HrB2]- [HrB4], Sarafova and Bainov [SaB4] applied the method to the investigation of periodic solutions of the impulsive system of integro-differential equations

= f(t, x, f cp(t,s,x(s))ds ), t * ti (x), dt 0

Axlt_t(x) = I.(x),

i = 0,±1,±2,....

Existence theorems were proved for both cases t = ti and t = ti(x). We also mention the papers of Gulka [Gul ] and Butris [Butl], where the application of the ideas of the method to integro -differential equations with impulse influence is considered. Thus, the first part of [Butl ] is devoted to the determination of periodic solutions of nonlinear (globally Lipschitzian in the domain of definition ) systems of differential operator equations with impulse influence of the form

dx = f(t, x,Ax), t * ti, dt Axl,=r = Ii(x,Ax). The indicated work mostly follows the ideas of [PerS ]. It should be noted that rather restrictive conditions are imposed on the operator A, e.g., (Ax)(t) = (Ax)(t +i),

jj(Ax)(t)- (Ay)(t)jj < gjjx(t)-y(t)II.

Appendix

413

This disadvantage was partially eliminated in the second part of that work [Butl] where, in a similar way, the system dx b(t) dt = f(t,x, f g(s,x(s))ds), t#t;, a(t) b(q) Axl t=t = 1i(x, f g(s,x(s))ds) a(ti)

with periodic functions a(t) and b(t) was investigated. In [Gul], a combination of the ideas of the method with the results obtained by Tsidylo and Gulka in [TsG] enabled one to prove, under certain conditions, the existence of a unique T-periodic solution of the system

dt

Ij(x(sj-0)) , tit;,

= f t,x, f cp(t,s, x(s))ds+ 0 O<sj
Axl t-t = I;(x),

i,j=0,1,....

Finally, we note that the results of [Sam4] were generalized by Samoilenko and Perestyuk in [SaPI]. Problem 13. Justify the application of the method to problem (124)-(126) in the case where the constants KO and K; in the corresponding Lipschitz conditions are replaced by matrices.

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