Topics in nonlinear analysis & applications

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Author: Donald H Hyers; George Isac; Themistocles M Rassias

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Topics in

NONLINEAR ANALYSIS APPLICATIONS

This page is intentionally left blank

Topics in

NONPAR ANAlksiS APPLIOJIONS Donald H Hyers Department of Mathematics University of Southern California, USA

George Isac Department of Mathematics and Computer Science Royal Military College of Canada

Themistocles M Rassias Department of Mathematics National Technical University of Athens, Greece

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 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.

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd. A11 rights reserved. This book, or parts thereof, may not be reproduced in anyform 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-2534-2

This book is printed on acid-free paper.

Printed in Singapore by UtoPrint

CONTENTS Preface CHAPTER 1

IX 1

Cones and Complementarity Problems 1. INTRODUCTION 2. CONVEX CONES • Normal cones • Regular and completely regular cones • Well based cones • Isotone projection cones • Galerkin cones 3. COMPLEMENTARITY PROBLEMS • The explicit complementarity problem • The implicit complementarity problem • The generalized order complementarity problem 4. EXISTENCE THEOREMS • Galerkin cones and the generalized Karamardian condition • Galerkin cones and conically coercive functions • Variational inequalities and explicit complementarity problems • Isotone projection cones and complementarity problems • Comment • Complementarity problems and condition (S) S-variational inequalities and the implicit complementarity problem • Heterotonic operators and the generalized order complementarity problem • Topological degree and complementarity

1 2 4 8 10 16 23 27 30 39 42 47 47 54 59 69 81 82

•

95 104 117

CONTENTS 5. SOME SPECIAL PROBLEMS IN COMPLEMENTARITY THEORY • Boundedness of the solution-set • Solution which is the least element of the feasible set • The cardinality of the solution-set • Nonexistence of solution • Sensitivity analysis • Nonlinear complementarity and quasi-equilibria 6. COMPLEMENTARITY AND FIXED POINTS 7. REFERENCES

132 132 135 139 139 139 139 140 148

CHAPTER 2

167

Metrics on Convex Cones 1. 2. 3. 4. 5. 6.

INTRODUCTION HILBERT'S PROJECTIVE METRIC THOMPSON'S METRIC WORKING WITH TWO CONES MONOTONE SEMIGROUPS AND METRICS ON CONES REFERENCES

CHAPTER 3

167 170 208 220 228 233 241

Zero-Epi Mappings 1. 2. 3. 4. 5. 6.

INTRODUCTION ZERO-EPI MAPPINGS ON BOUNDED SETS ZERO-EPI MAPPINGS ON THE WHOLE SPACE ZERO-EPI MAPPINGS ON CONES ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION ZERO-EPI MAPPINGS AND COMPLEMENTARITY PROBLEMS 7. ZERO-EPI MAPPINGS AND k-SET CONTRACTION 8. REFERENCES

VI

241 243 255 271 284 304 315 318

CONTENTS

CHAPTER

4

325

Variational Principles 1. 2. 3. 4. 5. 6.

INTRODUCTION PRELIMINARIES CRITICAL POINTS FOR DYNAMICAL SYSTEMS VARIANTS OF EKJELAND'S VARIATIONAL PRINCIPLE THE DROP THEOREM STRONG FORMS AND GENERALIZATIONS OF EKELAND'S PRINCIPLE 7. EQUIVALENCIES 8. EKELAND'S VARIATIONAL PRINCIPLE FOR VECTOR VALUED FUNCTIONS 9. APPLICATIONS • Existence of solutions for minimizing problems • Coercivity condition • A global variational principle on cones • Density result • Mountain pass lemma • The Bishop-Phelps theorem • Clarke's fixed point theorem • Borwein's c-principle •
433 444 444 452 454 463 464 477 480 484 488 494 497 519 539

CHAPTER

555

5

325 328 343 355 365 379 426

Maximal Element Principles 1. INTRODUCTION

555 VII

CONTENTS 2. PRELIMINARIES 3. VARIATION ON ZORN'S LEMMA • Applications • Comments 4. NEW MAXIMAL ELEMENT PRINCIPLES • Applications • A fixed point theorem for ordered sets • Maximality and solvability • Variable drops and general solvability • A drop theorem • Lipschitzianess tests • Maximal element principles and general NewtonKantorovich processes • Comments 5. PARETO EFFICIENCY 6. REFERENCES

633 640 641 672

Symbol Index

685

Subject Index

691

VIII

556 562 567 575 576 608 608 609 618 624 626

PREFACE "Pourquoi aspirons-nous a aller de I 'avant, toujours de I 'avant? -Parce que la vie est seulement dans le devoilement." Leon Tolstoi -Journaux et cornets Vol. Ill (1905-1910) Bibliotheque de la Pleiade, pp 232 For a long period of time a great deal of mathematical activity has been devoted to problems and theories associated with linearity. This activity has been supported by the interest to extend several known results of linear algebra from finite-dimensional spaces to infinite-dimensional spaces. By combining linear structure and topological structure, the modern nonlinear analysis arrived at its present form and structure. This, in turn, provides an interplay of functional analysis, differential equations and geometry which are associated with mathematical models that are defined in physics, economics, biology, chemistry, engineering, etc. The reality and complexity of several practical problems, imposed the necessity to study nonlinear problems. It is unanimously accepted that for several nonlinear problems, linearization is not always the right approach. We believe that this fact has influenced an impressive body of research in nonlinear analysis after 1960. The initial development of nonlinear analysis was justified by the study of nonlinear functional equations. In particular, topics like Fixed Point Theory, Topological Degree, Nonlinear Fredholm Alternatives and Surjectivity Theorems, Monotone, Pseudomonotone and Accretive Operators, Bifurcation Theory etc. are now considered as classical subjects in Nonlinear Analysis.

IX

New domains, as for example, Variational and Quasivariational Inequalities, Non Smooth Analysis, Convex Analysis, Variational Principles, Complementarity Theory etc. are now subjects of active research interest. The variety of fields in nonlinear analysis is now very impressive. The present book, Topics in Nonlinear Analysis and Applications is a selection of five chapters: Cones and Complementarity Problems, Metrics on Convex Cones, Zero-Epi Mappings, Variational Principles, and Maximal Element Principles. Every chapter is, in some sense, independent, but there are several common elements. An attempt, however, has been made to present the material in an integrated and a self-contained way. The chapters presented in this book represent new directions in research. In every chapter, we try to discuss results with the proofs that are considered (to us at least) significant and related to some new ideas in nonlinear analysis, as well as to some new mathematical models or practical problems. Because of these aspects, every chapter is not exhaustive; hence the title of the book, "Topics in Nonlinear Analysis and Applications". It is hoped that every chapter can be considered as the subject of a stimulating graduate course or as a source of new research in nonlinear analysis for a self-study or seminar. Some 500 references have been cited here, including preprints. As a rule, we have studied the original sources and in some cases have retrieved some forgotten but useful results. Each chapter begins with an introduction, and ends with a list of references. The book concludes with a symbol index and a subject index. The first chapter deals with the notion of convex cones and with the nonlinear complementarity theory. The most important kinds of convex cones considered in the nonlinear complementarity theory, and in the study of other problems in nonlinear analysis, are discussed. In particular, the concepts of normal cone, regular and completely regular cones, the well based cones, the isotone projection cones, the Galerkin cones etc. are all studied. The complementarity theory is related to optimization and game theory. Generally, it is related to the study of economical equilibrium or to the study of equilibrium as considered in some problems in elasticity or in

X

engineering. The complementarity theory is a cross-point of several theories of nonlinear analysis and of applied mathematics. In this chapter, one can observe a mutual contact of the specialists working in complementarity theory, and its applications. In Chapter 2 we can observe how the ordering defined by a convex cone can be used to define a metric on the cone or on some of its remarkable subsets. In particular, we study the Hilbert's projective metric, the Thompson's metric, a metric defined by two cones, and a metric defined by some monotone semigroups acting on the cone. An essential property of projective metrics on cones is the fact that some linear or nonlinear mappings become contractions with respect to these metrics. Using the projective metrics, one can obtain several interesting existence theorems for nonlinear equations or for positive eigenvalues without compactness. An application to the complementarity theory is also presented. Chapter 3 is devoted to the study of zero-epi mappings. The concept of zero-epi mapping, introduced in 1978 by the Italian school, is similar to the concept of topological degree, but is strictly more refined and simpler. In particular, we present the concept of zero-epi mapping with respect to an open bounded set, to an arbitrary subset and to a convex cone. This concept was applied by the Italian school to the study of several classical nonlinear problems. Furthermore, in this chapter, we present other kinds of applications. For example, the study of Nash equilibrium for a couple of mappings, the study of nonlinear complementarity problems in infinite dimensional spaces, and the study of nonlinear complementarity problems depending of multi-parameters. We conclude this chapter with the concept of zero-epi mapping associated to k-set contractions. The concept of zeroepi mapping is an elegant and efficient mathematical instrument in nonlinear analysis that has not previously been presented in a book form. In Chapter 4, we present several results related to some important modern variational principles. We study the Ekeland's variational principle, its variants and its generalizations. We study the Ekeland's principle as a particular case of a more general principle, which is an existence

XI

theorem of stationary points for a generalized dynamical system. It is well known that the Ekeland's principle is equivalent with other principles, as for example the drop theorem, the Caristi-Kirk fixed point theorem, the flower petal theorem, and others. We prove these equivalencies among these principles. We also present the Ekeland's principle for vector-valued functions and for some general topological vector spaces. A very recent result concerning the density of points satisfying the s-Ekeland's principle, is also presented. For generalizations of Ekeland's principle we present: the generalized strong forms of Ekeland's principle, the Borwein's s-principle, the Borwein-Preiss principle, the Deville-Godefroy-Zizler principle, a constructive variational principle proposed recently by A. D. Ioffe, etc. The second part of the chapter contains applications to the study of several problems considered in nonlinear analysis. For example, the study of coercivity, the study of fixed points, some density results, the study of surjectivity, etc. We also present a global variational principle on cones, the Bishop-Phelps theorem, the Clarke's fixed point theorem, the mountain pass lemma, the Zabreiko-Krasnoselskii theorem etc.The last part of this chapter is devoted to the study of the drop property and its applications to the geometry of Banach spaces. The final chapter provides a study of the main maximal element principles recently discovered in nonlinear analysis. We begin this chapter with a variation of Zorn's lemma and an interesting characterization of the fixed point property by ordering. In the second part, we present several maximal element principles, such as the Bremdsted principle, the BrezisBrowder principle, the Altaian principle, the Turinici principle etc. Moreover, emphasis in this chapter is given to applications. For example, we apply the maximal element principle to study the solvability of abstract equations, to study the drop property, etc. We present some Lipschitzianess tests, a few general drop theorems, some results about variable drops and some general solvability theorems. We show how the maximal element principles can be applied for the study of some general NewtonKantorovich numerical processes. In the last part of this chapter, we study

XII

the Pareto efficiency. The subject discussed in this chapter has not previously been treated in a single unified source. We wish to thank Professors A. D. Joffe, M. A. Noor, J. -P. Penot, who read parts of the manuscript of the book and provided some very useful comments. We are especially grateful to Mrs. Viorica Isac for typing the whole manuscript, with enthusiasm and supporting it with kindness. We also wish to thank F. Sakellaridis and E. Alexiou for their assistance in proofreading the manuscript in English. Last, but not least, it is a pleasure to acknowledge the superb assistance that the staff of World Scientific Publishing Co. has provided in the publication of this book. Donald H. Hyers George Isac Themistocles M. Rassias January 1997

XIII

CHAPTER

1

Cones and Complementarity Problems 1. Introduction The concept of convex cone is now systematically used in Nonlinear Analysis as an important mathematical instrument. Especially, after 1960, convex cones have been used in many mathematical aspects as in the study of positive solutions of some nonlinear equations, in the study of positive eigenvalues of nonlinear operators, in the study of periodical solutions of some differential systems, in the study of bifurcation of solutions of nonlinear equations depending on a parameter etc. Another domain, now under development, strongly based on the concept of convex cone, is the Complementarity Theory, Complementarity Theory represents the study of Complementarity Problems defined in Optimization, in the study of price equilibrium in Economics, in the study of some problems in Elasticity, in Engineering, in Game Theory, in Fluid Mechanics in Stochastic Optimal Control etc. The Complementarity Theory is a cross-point between fundamental mathematics and applied mathematics, between the Linear Analysis and Nonlinear Analysis, as well as numerical methods and the topological methods. There exist now three books on this subject (Cottle , R. W., J. S. Pang and R. E. Stone [1]), (Isac, G. [12]), (Murty, K. G. [1]). Several results and methods that have been developed in Nonlinear Analysis are now currently used in the study of complementarity problems. In this sense, we cite in particular Fixed Point Theory, Topological Degree, Zeroepi Mappings and Regular Mappings etc. It is interesting to remark that the converse is also true, that is, Complementarity Theory can be used to obtain new results in Nonlinear Analysis. In this sense, we cite the fact that the Complementarity Theory can be used to obtain new fixed point theorems. Complementarity Theory is now a large research domain, but we will I

CONES AND COMPLEMENTARITY PROBLEMS present in this chapter only some significant results concerning the relations between the Complementarity Theory and the Nonlinear Analysis. We selected the subjects that best interested us.

2. Convex cones We denote by E(i) [respectively, by [E,\\ |) and (H,<,>) ] a locally convex space [respectively, a Banach space and a Hilbert space]. Every locally convex space is supposed to be Hausdorff. We suppose that the reader knows the main properties and concepts currently used in the theory of topological vector spaces. If E is the topological dual of E we denote by {E,E*\ a duality between E and is*, realized by a bilinear form <,>:E x E —>R (the field of real numbers), satisfying the separation axioms: 1) (x0,y) = 0 for ally =E implies xQ=0, 2) (^^o) = 0 for all x e E impliesyo = 0. Let £ be a vector space over R which is endowed with an order structure defined by a reflexive, transitive and anti-symmetric binary relation " < ". We say that £ is an ordered vector space over R if the following axioms are satisfied: i) x 0| satisfies the following properties: c,) E++E+ c £ + , c2) AE+ Q E+ for all X e R+, C3 )£ + n(-£ + H°}-

If (£(r),<) is a topological vector space endowed with an ordering satisfying axioms (i) and (ii) and the following axiom is also satisfied:

2

CONVEX CONES (iii) E+ is closed in E, we say that (is(r),<J is an ordered topological vector space. Let E be again a general vector space. We say that a non-empty subset K c £ is a pointed convex cone if and only if K satisfies the following properties: k{)E + KQS, k2) XK^K;K for all X e R+t k3) K(X-K) = {0}. If only (&i) and (k (k22)) are satisfied we say that A" is a convex cone. It is verified without difficulty that each pointed convex cone K czE defines, by virtue of "x (x+y)Az<(xAz) + (yAz), 4) xAz=y/\z x l\z=y Az == =0^>(x+y)Az 0^>(x+y)Az ==0,0, (xV z) z) ++ (y (y Aw), Aw), 5) u=x + y = z + vw => => uu ==(xV 6) x + y= xVy +xAy, 3

CONES AND COMPLEMENTARITY PROBLEMS 7) V{xVv,z}=V(jtVy ) vVz) = V{x,j>)z}, 8) xAy = 0V a, /3>0,(ax)A(fiy) = 0. We define, for every x e E, x+ = 0 V x, x = 0 V (-*) and \x\ = x V (-x). We have |x + y\ < \x\ + | v|; \x+ - y+1 < \x - y\ and \x~ - y~ | < \x - y\. The proofs of the formulas presented above and some other details about the vector lattices can be found in (Peressini, A. L. [1]), (Schaefer, H. H. [1]). Let (E,E*\ be a dual system of locally convex spaces and let K c= E be a pointed closed convex cone. The dual of K is denoted by K and it is defined by K' = j v e E* \ (x,y) > 0 for all xeK}. We can show that K' is a convex cone in E*. It is easy to verify that K is pointed, if and only if K - K is dense in E. The polar ofK is denoted by K° and it is defined by if0 = {v e E' I (x, v) < 0 for all x e JTJ. It is true that tf° = -£"*.Therefore by the Bipolar Theorem (Peressini, A. L. [1]) we have that K*' = (Z*)* coincides with the erf E,E' J - closure of the given cone K in E. For the results which we intend to present in this chapter, it is necessary to introduce some important classes of cones. Let £(T) be a topological vector space, K a E a pointed convex cone and Aa E a non-empty subset. The K-saturated hull of A is denoted by [A] and is defined by [A] = (A + K) (\A - K). If A =[A], then A is K-saturated. We can also verify that A
4

CONVEX CONES The following classical theorem characterizes the concept of "normality". Theorem 2.1 I) IfE(x) is a topological vector space and K aE a pointed convex cone, then the following assertions are equivalent: 1) K is normal for z, 2) there is a neighborhood basis "W of zero for r consisting of sets V for which 0<x 0 such that /Jpp(x) ^p(x +y)for all x,yeK, K, 7) there is a family S of seminorms generating the topology r, such that for every peg S there is /jp > 0 such that jup(p(x) + p(y)) < p(x + y) for allx,y e K. III) IfE{x) is a normed vector space \E,\ |), the normality ofK is equivalent to the following assertions: 8) there is an equivalent norm | |, on E, such that 0 < x < y implies

HI^IMI9) there is a constant S>0, such that 0<x
implies S\\x\\ < \\y\\,

10)there is a constant 8 > 0, such that Jmax{||x|,|>|j < ||* + }'|| for all x,y eK. Proof Some of these equivalencies are proved in (Peressini, A. L. [1]), and

5

CONES AND COMPLEMENTARITY PROBLEMS in (Schaefer, H. H. [1]) and the others are easy to show. ■ Normality has some important consequences. Theorem 2.2 Let E(x) be a topological vector space and K a E be a pointed convex cone. IfK is normal for t, then every order bounded subset ofE, is r-bounded. In particular, every order interval in E is x-bounded. Proof A proof is in (Peressini, A. L. [1]). ■ Theorem 2.3 IfE(x) is a locally convex space ordered by a normal pointed -K' ' -i -K convex cone K, then E =K -K. Proof A proof is in (Peressini, A. L. [1]), and in (Schaefer, H. H. [1]). ■ If D c E is a non-empty subset, we say that D is < directed (resp. > - directed) if for every x,y e D, there exists z e D such that x,y < z [resp x,y>z]. We say that D is directed if D is < - directed or > - directed. Theorem 2.4 If D is a directed subset of locally convex space E(f), ordered by a normal cone K cz E, and if the filter ^(D) of sections ofD converges for the weak topology o\E ,E), then &~(D) converges to the same limit for z. The most important results, obtained in the theory of ordered topological vector spaces, are obtained supposing that the cone of positive elements is a normal cone. Examples 1. Let E = (Rn,<,>) K x K=\x

n

be the Euclidean space.

The convex cone

(x )eR for alii =12....n\ is a normal cone. In this i)e. ={ \x = (x.)eK" {x[x i)leR "\x, x,2ZO for alii =12,..,n} topological vector space, every closed pointed convex cone is normal.

6

CONVEX CONES 2. Let X be a compact topological space and E = g (XR) the vector space of continuous functions from X into R. The pointed convex cone K = ( / e (?(X, K) I f(x) > 0 yfrr all x eX\ is a normal cone with respect to the topology defined by the norm | / | = sup||/(x)|| x e x\. 3. Let (Q,//) be an arbitrary measure space and E = LP(Q.,/J) where p > 1. The convex cone K = \feL{Q,ju)\f>0,ju-almost almost everywhere) is a normal cone with respect to the topology defined by the norm

u/h[lj/(or^(o]1/p-

4. Let (H,<, >) be a Hilbert space and J (H) = {IT:// -^ i / I T continuous and linear }. The convex cone K = { T e A (H) \ T is self-adjoint and (T(x),x)> Ofor all xe H}, is a normal cone with respect to the uniform convergence topology on d (H). 5. If E is one of the following topological vector spaces lp, lx,c , Co , each with its usual topology, then in every space, the pointed convex cone formed by the sequences x = {xn}ni_N, such that x„ > 0 for all n e ./V, is a normal cone. Let E = t> be the vector space of functions with compact support in R" and having derivatives of all orders. The convex cone K = if e Z>\f > O], is not normal with respect to the Schwartz topology onZ>. 6. Let Q c R" be a bounded domain. For every k e Nandpe [1, + oof, let PF*(Q) be the vector space 0j(fl) = {/ e Lp(Q)| Daf e Lp(n) for all aeN"

such that \a\ < k\,

where Daf is the derivative of / in the sense of distributions. The space Wp(Ci) is a Banach space with respect to the norm

"/ C ■■

nliip

ii J lip

r

=- l 7y.lz^/l = (r,\ ■pp) ^ iIW JaJft"h.(a)

Vp -UP

i

HP

«—i II

= y \D f\ i—i

Li«is* z * ii

J

J

Hi (0) 7

CONES AND COMPLEMENTARITY PROBLEMS The convex cone K = I / e W* (fi) | / > 0 on Qj is not normal with respect to the topology defined by the norm | | . • Regular and completely regular cones Let E{x) be a locally convex space and JT c E a closed pointed convex cone. Definition 2.2 We say that K is regular [resp. sequentially regular] if every net [resp. every sequence] monotone increasing and order bounded of elements ofK, is convergent. Definition 2.3 We say that K is completely regular [resp. sequentially completely regular] if every net [resp. every sequence] monotone increasing and topologically bounded of elements ofK, is convergent. From definition, we obtain immediately the following result. Theorem 2.5 In a locally convex space, every pointed normal completely regular convex cone is regular. Examples 1. If / is an arbitrary non-empty set, we consider on the vector space R1 the product topology. The convex cone K = R+' is a regular cone. 2. If £ = I00, the convex cone K = /+°° is not regular. Theorem 2.6 Let E{x) be a locally convex space and K c E a pointed convex cone. IfK is normal and weakly complete {resp. weakly sequentially complete), then it is completely regular {resp. sequentially completely regular). Proof Since K is weakly complete, it is weakly closed, and hence it is closed with respect to the topology x. The normality of K implies also, that E* = K* -K*. Let {xt}.eI be a net of elements of K. Suppose that {*,}. is

8

CONVEX CONES monotone increasing and topological bounded. For every/ e it, the net of real numbers {/(■*,)}. ., is a Cauchy net (since it is monotone increasing and bounded). Hence, the net {;c,}. , is a weakly Cauchy net and because K is weakly complete, it is weakly convergent to an element x* e K. From Theorem 2.4, we deduce that {*,}. is x-convergent to the same limit x* and therefore the theorem is proved. ■ Corollary 2.7 In a semireflexive locally convex space, every pointed, closed normal convex cone is sequentially completely regular. Proof Since every semireflexive locally convex space is x(E,E ^quasicomplete, we have that every semireflexive locally convex space is weakly sequentially complete, and the Theorem 2.5 is applicable. ■ Corollary 2.8 In the space Lp(Q,/j), with 1 < p < oo, every pointed closed normal convex cone is completely regular. We recall that a locally convex space E is nuclear if and only if there exist an equicontinuous sequence {/„}„eA, <=£* and a sequence {^„}neN c/j1", where If = ix = (x„) e /, \xn > 0 for all n eNj such that for all continuous semi-normp onEand a l l x e £ , w e have x < K*)
Corollary 2.9 In a nuclear space, every closed pointed normal convex cone is completely regular. Remark 2.1 In (Isac, G. [1]) several results are presented, concerning the relation, between the regularity and normality of a pointed convex cone. An interesting method to define regular cones, is to consider the cone defined by a biorthogonal system. 9

CONES AND COMPLEMENTARITY PROBLEMS

Let E(x) be a locally convex space. A couple of sequences

K{x ,f„} n

neN,

where {*„}neA, <= E and {*„}neA, <= £* is a biorthogonal system if and only if, /„(*„) = 8nm for all «,w e N. If {*„,/„}neA, is a biorthogonal system defined on E, then the set K = \x e E | /„(*) > 0 /or a// n e N) is a convex cone. The cone K is pointed, if the sequence {/„}neN is total in E* , i.e. linear span[{fn}neN)

= E'.

If E(j) is a Frechet space, the biorthogonal system {*„,/„} neAr with ix an Schauder basis and the cone K associated to the system {f» } N tota^llZJ is

{*«>/«} ), eN is normal, then K is regular (Isac, G. [1]). We denote by ME" ,E\ the strong topology on E* defined by a duality (E,E*\.

The definition and the properties of this topology are in (Schaefer,

H. H. [1]). We also cite without proof the following result. Theorem 2.10 [Karlin] Let \E,\ \\j be a Banach space, such that E* is piE',E\-separable. trable.

Then, with respect to the topology piE',E\,

closed normal cone KczE

every

is completely regular and regular.

Proof A proof is in (Karlin, S. [1]). ■ • Well based cones Let E(x) be a locally convex space and {0} * K c E a convex cone. Definition 2.4 We say that a non-empty convex subset B ofK, is a base for K, if each nonzero element x e K has a unique representation of the form x-Xb for A > 0 and b e B.

10

CONVEX CONES Remark 2.2 Since a base B for a convex cone must be a convex set, it is clear that 0 g B. If B is a non-empty convex subset of E and iST c E is a convex cone, we say that K is generated by B, ifK = (J XB. ASO

Proposition 2.11 Let E(z) be a locally convex space, B a E a convex subset and K 1 and c\, Ci e B such that X0cl - (/l0 - l)c 2 . Since we obtain a contradiction of the uniqueness property for base representations, we must have 0 ^ . To show the converse, we suppose that B satisfies the stated conditions. If Xybx = A2b2 for bh bi e B and positive real numbers X\, A^, then we have 0=

1 / t i — A*2

(Xfa - X~2A b22)b2e) Ae JLif X\ {A>\b\ \fX\* *X2. XQ. .

It follows that X\ = Xq. and hence b\ = 62 ■ Consequently, B is a base for K.

m

Proposition 2.12 Let E(x) be a locally convex space and K
rl{i)r\K=B.

Proof Iff is a strictly positive continuous linear functional on E, the linear manifold / - 1 ( i ) contains f-\l)f]K, \K, 0 €f~l{l) and l K=u{x(r {l)C\K)Us:) uJAfr'wn*) A=U{>V vy\j w i-j

\X>0\. \x>6\. >0).

11

CONES AND COMPLEMENTARITY PROBLEMS Applying Proposition 2.11, we obtain that f~l(l)C\K~)K is a base for K . Conversely, if B is a base for K, by Zorn 's Lemma, we obtain the existence of a linear manifold 40 which is maximal (with respect to inclusion) in the class of all linear manifolds in E containing B but not 0. It is easy to verify thatu?o is a hyperplane. Hence, J0 = / - 1 ( l ) for some linear functional / on E. Since fijb) = 1 for each b in the base B, it follows that / is a strictly positive linear functional and B = / _ 1 (l) D K. ■ We cite without proof the following classical result: Theorem 2.13 [Krein-Rutman] If (E,\ |) is a separable Banach space, then every closed pointed convex cone K
12

CONVEX CONES Theorem 2.14 [Klee] Let E(T) be a locally convex space and K a E a pointed closed convex cone. The cone K is locally compact if and only if it has a compact base. Proof A proof of this theorem is in (Isac, G. [1]) and in (Klee, V. [1]). ■ We denote by (■^(r)>{pa}ae^)

a

locally convex space with the topology r

defined by a sufficient family of seminorms, (Marinescu, Gh. [1]). We recall that the family of seminorms {pa} aeA is sufficient if for each a,/3 G A, there is a y e A such that pa{x),pp{x) < pr(x),

for all x e E, and,

for every x =£ 0, there exists a e A such thatpa(x) *■ 0. Theorem 2.15 [Isac] Let l£(r),{/? a }

I be a locally convex space and

Kd E a closed convex cone. The cone K is well based if and only if there exists a continuous linear functional f such that, for every a e A, there exists a real number ya>0 with the property: x r yYaP aPa{x)
(2.1)

Proof If K is well based, then there exists a bounded convex set B0
K=

\JAB0. A>0

A classical separation theorem implies the existence of a continuous functional / such that: i) f{a)> 1 for all a BT0 , ii)X0)
= com>({0} U5 0 ) since, for every b e B, we have b = (1-A)0 + Xa\,

13

CONES AND COMPLEMENTARITY PROBLEMS where ax e B0 and 0 < A < 1 . (If A > 1 we have/6) ==Af{a Ma{) >1 which is x) > impossible). Because M is a bounded set, we deduce that B is bounded. Let a e A be an arbitrary element . Since B is bounded, there exists /? a > 0 such that pa(z) < /3a for all z e 5. Let x e K\{0} be such that pa{x) # 0.

if 1 X x) *A PAX))

In this case, there exists /^ > 0 such that — —p— e 5 .

' 1t ( x Yi

I fia T T =T-P* D..(X) I ,

E> Pa 1-

x Av/ \A x\Pa\ V)

put ya =—,

*x

W h l c h im

We have

f X ^ I >= A> P I I e S / —TT x = ^ - ^ " Pa .Pa\x V \Pa{ )J

If We

Pa

we obtain the inequality (2.1). We remark that (2.1) is

Ha

satisfied also if pa(x) = 0 since / is positive on K. We prove now the converse. For this, we suppose that there exists/ s E satisfying (2.1). Since, the family [pa] . of seminorms is sufficient, we deduce that/is strictly positive on K. Because / is continuous and K is closed, we have 0 $.B where B = ix e K\ f(x) = l j . Using again, the fact that the family ip„\

, of seminorms is sufficient, we have that B is bounded and the

y+a ) a^A

theorem is proved. ■ For normed vector spaces Theorem 2.15 has the following form: If K is a closed convex cone of a normed vector space (El |), K is well based if and only if there exists a continuous linear functional f such that \\x\\
14

CONVEX CONES Proof Let B be a bounded base for K such that 0 g 5 . There exists a closed circled and convex neighborhood V of zero such that VV\B = (/>. Let/?^ be the Minkovski's functional of V. We have V = Ix eE\ pv(x) < \\ which impliespv (b) > 1 for all b e B. Let {x,}i(E/ and {>',}.e/ be two nets in E such that 0 < JC, < yt for all i e / and limjy, = 0. Since for all i e /, xb yt e K there exist at, $ e R+ and bt, i',e B such that xt = afti and yt = pt b't. Because 0 < a,6, < pp\ for all i e I and B is a base for K we have that 0 < at < pi for all i e I. Using the Minkovski's functional pv and the facts that lim_y(- = 0 and py^b',) > 1, we can show that lim/?,. = 0, which implies ie/

that l i m a , = 0 .

16/

Since B is bounded, for every circled neighborhood

iel

V e 1/ (0), (where f(0) is the filter of neighborhood of zero in E) we have that there exists a > 0 with the property that X B cz V for all -a < X < a. Because there exists z'o e / such that, for all i > i0, we have 0 < a, < a, we deduce that for every circled neighborhood F e ^ ( 0 ) we have or, bt e Vfor all i > z'o, which implies that lim:c( = 0 and the theorem is proved. ■ iel

We note that it is easy to show that Theorem 2.16 is a consequence of Theorem 2.15 (which was proved several years after Theorem 2.16). Corollary 2.17 In a Hausdorff locally convex space, every locally compact cone is normal. ■ Corollary 2.18 In Rn every pointed closed convex cone is normal. ■ The reader can find several examples of well based cones in (Isac, G. [1]) as well as the proof of the following Theorem: Theorem 2.19 Let E(x) be a locally convex space and K c E a pointed convex cone. IfK is well based and complete, then it is completely regular.

15

CONES AND COMPLEMENTARITY PROBLEMS Proof Since K is complete it is closed and, being well based, it has a closed bounded base B. Let {*,},e/ be a monotone increasing and topologically bounded net of elements of A". Let U be a circled neighborhood of zero and e > 0 such that e B c U. The set |x, | i > /„ j , where r0 is a fixed arbitrary element of/, is bounded. Consider the elements yi = xt -x^,(i

> ; 0 ). For

every i > I'O, there exist bt e 5 and /i, > 0 such that >>, = A,- bt. Since | v, 11 > i0) is a bounded set there exists P > 0 such that |v,|/>/0]c/fl

(J>£5

which implies that |A,|/>z' 0 | is a bounded set.

^-(KASl

We put n = sup! A,,|/>i 0 }, and we consider an index I'I such that X, >/j-e.

For every i > iu we have yi -y, =/?,c,, where c, 6 B and

B{ = Aj - Xt < e. The last relation is a consequence of the fact that B is a base for K. Hence, xt -x(i eU,for all i > iu which implies that {*,}iel is a Cauchy net in K, and, because K is complete, {x,}.

is convergent and in

conclusion, K is completely regular. ■ • Isotone projection cones Let (//,<, >, K) be an ordered Hilbert space. We say that (H,<, >, K) is a Hilbert lattice if and only if the following properties are satisfied: i) H is a vector lattice, ii) J|jc|I = ||x| for allx e H{i.e. the norm \\ \\is absolute), Hi) 0 <x defined on H is a K- local inner product if (x,y) = 0 whenever x, y e K and x A v = 0. The concept of A"- local inner product is inspired from potential theory. Since H is a Hilbert space, we have K* = {yeH\ (x,y) > 0, for all x e f ) c H

16

CONVEX CONES and we can show thattJsrriJ K fl K" j r * **{0}. The proof of this fact is based on some variational arguments (Isac, G. [10]. From this result, we deduce that the following properties for a convex cone K in a Hilbert space H are very important: I) KdK KczKUi.(i.e. Kis sub-adjoint), II) K = K* (i.e. K is self-adjoint), III) K^>K (i.e. K is super-adjoint). Theorem 2.20 Let (//,<, >, K) be an ordered Hilbert space. If H is a vector lattice, the following statements are equivalent: 1) H is a Hilbert lattice, 2) for all x,y e Kwe have <x,y> > 0 (i.e.KczK*), K c K) and <, > is K-local inner product. Proof (1) =>(2). Indeed, for all x,y e K we have \x - y\ = (x - v) + (x - v) = 0 v (x - y) + 0 v (v - x) < x + y, which implies (since H is a Hilbert lattice) that ||x - v|| < \\x + y\\ and by the formula .•a ||x + v| - | x - v | | =4{x,y), we obtain (x,y) > 0 for all x,y e K. For all x,y e H, we have x A y = 0 if and only if jx\ -\y\\- \x\ + \y\. If x,y e K are such that x A y = 0, then |JC - y\ = x + y. Since H is a Hilbert lattice, we deduce that \\x + y\\ = \\x - y\\, which implies ii?

(x,y} = -^\x + yf-\\x-yf\ = 0. (2)=>(1).

Indeed, if 0 < x < y then ||vf -\xf in

iii2

\\1

I +

=(y-x,y

+ x)> 0 [by

\

assumption (2)] and |belli -|bc| =4{x+,x )-0= 0 (since the inner product assumption (2)] and ||x|| -|pc| =4{x ,x~)-0 (since the inner product <, > is JT-local). ■ Theorem 2.21 a) If(H<, >, K) is a Hilbert lattice, then K = K'.

17

CONES AND COMPLEMENTARITY PROBLEMS b) If{H,<,>, K) is an ordered Hilbert space which is a vector lattice and <, > is a K-local inner product, then H is a Hilbert lattice, if and only if, K = K'. Proof a) If H is a Hilbert lattice, then by Theorem 2.20 we have that K a Jf*. The inclusion K* c K is also true. Indeed, x eZ* and x g K imply x > 0 and (x,x~\ = (x+ - x~ ,x~\ = (x+,x~)-(x~

-II2

,x~) = -lx~j <0 < 0 , which

is impossible. Hence, we must have K* czK. Thus, we have K = E*. b) If H is a Hilbert lattice by (a) we obtain K = K". Conversely, if K = Jf* by Theorem 2.20 we have that His a Hilbert lattice. ■ Let (H<, >) be an arbitrary Hilbert space and K c H a closed convex cone. It is known that the projection operator onto K (in the sense of best appro ximation), denoted by PK, is well defined (Zarantonello, E. H. [1]) and, for every x e H, Pg{x) is the unique element of K satisfying

Wx-PJx minllx — vll \x-PK(x)\ =■■ rmdx-y\\. We note the following classical result.

y yen

(2.2)

Theorem 2.22 For every element x e H, Pg(x) is characterized by the following properties: 1) (PK(x)-x,y)>0 >0 for forallyeK, ally 2) (PK{x)-x,PK{x))

= 0.

Proof A proof of this classical result can be found in (Zarantonello, E. H. [1]) or in (Kinderlehrer, D. and G. Stampacchia [1]). ■ Theorem 2.23 [Moreau] IfKczH H is a closed pointed convex cone and x,y,z eH, then the following statements are equivalent: J)z = x+y, xeK, yeK° and <x, y> = 0,

18

CONVEX CONES

2) x = Pfc) andy = i^(z). Proof (1) =>(2). Suppose that x, y, z e H satisfy assumption (1). Since {x-z,u) = (-y,u)>0 for all u e A" and x-z,x {x-z,x) =-iv.x -(y,x)= = 0, applying Theorem 2.22 we obtain that x = Pg(z). Because (Jf°)° = K, by a similar argument, we obtain that y = P - (z). (2) => (1). Indeed, if z e H is an arbitrary element, we set x = P^z) and y = z-x. We have (z-x,u-x) = (z-x,u)-(z-x,x) - x,u) - (z - 3x,x) =(2 =z-x,u) (z-x,u)<0 for < every u e K (using Theorem 2.22). From this inequality, taking u = Ax (with X > 0) we obtain that (/I - l)(.y,x) ^ 0. Since A - 1 can be positive or negative we obtain (y, x) = 0, and, because we showed that (y, w - x) < 0 for all M e A, we deduce that (y, u) < 0 for all u e K. Hence, y € A0. Applying Theorem 2.22 the theorem is complete. ■ Corollary 2.24 PK is a positively homogeneous operator, that is, PK(ax) = aPK(x) for all a eR+ and all x e H. Proof

Indeed, from Moreau's Theorem, we have x = PK(x)

where (PK(X),

P^x))

+PAx),

= 0.

If a e R, then we have ax = aPK(x)+aPKo(x),

where aPKaP (xK(x)eK, =*,

ceP^x) e K° and (aPK(x),aPKo(x)j = 0. Since the decomposition given by Moreau 's Theorem is unique, we deduce in particular that aP ccPKK{x) {x)==PPKK(ox). {ax) ■■ ■ Let (H,<,>) be an arbitrary Hilbert space and KCZHSL fa closed convex cone. The order monotonicity of the projection operator onto a closed convex cone has been studied in (Isac, G. [10]), (Isac, G. and A. B. Nemeth [1-5]). (Isac, G and L. E. Persson [1]) and (Bernau, S. J [1]).

19

CONES AND COMPLEMENTARITY PROBLEMS Definition 2.6 We say that K is an isotone projection cone if and only if, y-x X &K implies PK(y) - PK(x) eK for every x, y e H. The main properties of isotone projection cones are given in the following: Theorem 2.25 If (H,<, >) i) every isotone projection ii) every isotone projection Hi) every isotone projection

is a Hilbert space then: cone KczH is sub-adjoint (i. e. K a K). cone KaH is normal, cone KaH is regular and completely regular.

Proof i) We suppose that K is isotone projection. From Theorem 2.2 S we have ^ ' ( 0 ) = -K*{= K°). If x e K, then -x < 0 and since PK is monotone increasing with respect to the order defined by K it follows that 0 < PEKTX) < Pi(0) = 0, that is, -x e ^ ' ( 0 ) = -K',

i.e. x e K*

ii) Suppose that K is isotone projection. If 0 < x < y we have that 0 < (y - x,y) and 0 < (y -x,x) (since y-xeKcK).KcK). From that we have I*! < (x,y) < \\y\\ , that is

||JC|| <

||y|, which implies that K is normal.

iii) If K is isotone projection, then it is normal and by Corollary 2.7 and Theorem 2.5 we have that K is completely regular and regular. ■ Definition 2.7 A convex subcone F^Kis is called a face ofK ifx e K, y sF andy -x e K imply x e F If A c H is a non-empty subset, we denote by spA the linear space generated in / / b y A, i. e. the linear span(A). Theorem 2.26 For every face F of an isotone projection cone K a H, the space spF is projected by PK onto F and F is an isotone projection cone in spF.

20

CONVEX CONES Proof A proof is given in (Isac, G. and A. B. Nemeth [2],[3]). ■ Definition 2.8 The exposed face of a closed convex set A c H perpendicular to a vector u e His by definition the set F

A{U) = y

e A

(y>u) = sup(x,u)>.

Theorem 2.27 Let K a H be an isotone projection cone. If K0 is an exposed face of K, then PK projects spK0 onto KQ and K0 is an isotone projection cone in this subspace. Proof A proof can be found in (Isac, G. [10]). ■ Examples of isotone projection cones are given in the papers (Isac, G. and A B. Nemeth [2],[5]). Theorem 2.28 1) If [H,<,>,K) is a Hilbert lattice, then K is isotone projection and moreover, P^x) = x+. 2) A Hilbert space (//,<,>) ordered by a closed self-adjoint cone K is a Hilbert lattice if and only ifK is isotone projection. Proof A proof is given in (Isac, G. and A. B. Nemeth [3]). ■ An important and difficult problem in the study of isotone projection cones was to find a characterization of isotone projection cones in a general Hilbert space. For the Euclidean space (R",<,>\, this problem is now completely solved by the following result. Definition 2.9 We say that a convex cone K c R" is correct if and only if, for every face FcK,we have PspF[K) c F. Theorem 2.29 [Isac-Nemeth] Let KczR" be a generating convex cone. 21

CONES AND COMPLEMENTARITY PROBLEMS The following assertions are equivalent: i) K is isotone projection, ii) K is latticial and correct, iii)K is polyhedral and correct, iv) there exists a set of vectors |w, | i si) with the property that (",,«,) ^ 0 for all i *j , ij e I and K = Uut | / e / ] j , (where ( )° denotes the polar cone), v) K is latticial and PK(x) < x+ for every x e R. Proof The evidence of this result can be found in (Isac, G. and A. B. Nemeth [5]). It is a long proof with several technical details. ■ Thus, it would be useful to present a simple approach of this result. We now consider that (H,<,>) is a general infinite dimensional Hilbert space. In this case, the following definition is fundamental. Definition 2.10 We say that a cone K^His weakly correct if and only if for every face FczK, we have Pyj(K) c K. Theorem 2.30 (Isac, G. and A. B. Nemeth [2]) Every generating isotone projection cone in an arbitrary Hilbert space is latticial and weakly correct. ■ The open problem posed in (Isac, G. and A. B. Nemeth [2]) takes the following form. Open problem It is true that, in an arbitrary Hilbert space, every latticial and weakly correct cone is isotone projection ? This problem has been recently studied by S. J. Bemau [1]. He considered the following stronger form of Definition 2.10.

22

CONVEX CONES Definition 2.11 We say that the cone KaH is strongly correct if and only if for each face FcK,we have P^yi^) C F ■ The strong correctness is exactly the correctness which we used in the Euclidean space [Rn,<,>\, since in R" every subspace is closed Using several technical results, S. J. Bernau [1] proved the following result. Theorem 2.31 In an arbitrary Hilbert space (H, <, >) a generating cone is isotone projection if and only if it is latticial and strongly correct. ■ It is interesting to know if a weakly correct cone is strongly correct. • Galerkin cones Let (E,\\ ||) be a Banach space and KczEa closed con vex cone. Definition 2.12 The convex cone K c E is a Galerkin cone if and only if there exists a countable family of convex subcones {JSTn}ngAf of K such that: i) K„ is locally compact for every n e N, ii) ifn < m then Kn c Km

iii)K=XJKn. nzN

A Galerkin cone will be denoted by K(Kn)n<_N. Examples 1. We say that a convex cone K a E has a Schauder basis if there exists a sequence {*„}JneAfN a^KK such that every JC eJfhas a decomposition of the 00 < x>

form x- E
* is a Schauder basis a > 0 for each n eW. Iff*.} It an>0 for each n eJV. n If {x„}nsN is a Schauder basis

r

•i

for every n e N, for JT, then Kn = cone\xx,....,xn) - * ; for K, then Kn = cone{xl,....,x„) = <£lA.ixi\A.i >Q> for every n e N,

A*.l^°

23

CONES AND COMPLEMENTARITY PROBLEMS defines a Galerkin approximation of K. In this case K(Kn)neN is a n<=N Galerkin cone. 2. If the cone K admits a Galerkin approximation by the finite element method, then K is a Galerkin cone. We recall that a point JCO is an extreme point for a convex set A c E if x0 e A and if A \ {x0} is convex, or, equivalently, x e A,y e A andfa:+ (l - t)y = x0 for some t e ]0,1 [ imply x = y = x0. We denote by ext(B) the set of extreme points of B a E. A Banach space E is said to have the Krein-Milman property {KMP, in short) if each closed convex bounded subset of E is the norm-closed convex hull of its extreme points. The class of Banach spaces with the KMP is large enough since it contains all dual spaces that are subspaces of weakly compactly generated Banach spaces, as in particular every dual separable Banach space, and any reflexive Banach space. For more details, the reader is referred to the book (Diestel, J. [1]). If (E,\ \,K) is an ordered Banach space, we say that E is a Banach lattice if and only if \E,\ \,K)

is a vector lattice and

jjcj < |y| => \x\ < \y\ for all x, y e E . Theorem 2.32 Let [E,\ |,JSTJ be an infinite-dimensional separable Banach lattice with the KMP. If K has a closed bounded base, then it has a Schauder basis. Proof Let B be a bounded convex base for K. We recall that

K= [x&E\ x>0}. Since £ is a Banach lattice, K is closed. Since 0 £ B, pick a > 0, such that a < \\x\\ 0 (2.3) (2-3)

Let x be an extreme point of B. For each y e K such that 0 < y < x, we claim

24

CONVEX CONES that y = Ax for some k e ] 0,1]. Indeed, x = y + z with z e K, and we may find A, > 0, A2 > 0, bx,b2 in 5 such thaty = A\b\, z = A^bi. We have _Al^ — iUI + .— ^A2 — 6L2 . x = (A,+A (/Ij + A22)) — *1 /li + A>2

Aj + A2

By virtue of the uniqueness of the decomposition of x, we necessarily get that x is a convex combination of b\ and b2. Hence, since xe ext{B), x = bx=b2 and the claim is proved. In particular, for x, y e ext{E) with x A y 5* 0, the relations 0 < x A j / < x and 0 < x A y < y and the preceding observation yield x A y = /be = juy for some A e ] 0 , l ] j y u e ] 0 , 1]. Hence, x = — y and because x, y e B, we must have x = y. Thus, the extreme points of B are pairwise disjoint and according to the relations x + y = x V y+jc A y and 2(x A y) = x + y - \x -y\ (which are valid in any vector lattice) for x,y eext(B), we deduce 0<x<x+y = \x-y\. (2.4) This forces the set of extreme points of B to be discrete since, by virtue of (2.3), (2.4) yields: a < \x\ < \x - y\\ for all x,y e ext(B). (2.5) Since E is separable and normed, it satisfies the second axiom of countability as does its subset ext(B). Being discrete, ext{B) is then necessarily at most countable. Thus, we may assume that there exists {b„}n^N such that ext{E) - {bn}ni_N. For each x e B, since 0 < x A bn < b„ and b„eext(B), there exists x(n) such that x A b„ = x(n)b„. Since x (n)b„. < bm we get JC(H)||6„|<|&„|, which implies that necessarily x(n) e [0,1]. For each n e N, we have n

n

00 << £ xinb, jc(f)2>,. = (x(l)b, V ... V x(n)b„) + (x(l)b, (x(l)b] A....A x(n)b x{n)bnn) 1=1

= x{\)b,\J....\Jx{n)b x(\)b,V....Vx{n)bn, 25

CONES AND COMPLEMENTARITY PROBLEMS n

since the elements bt are pairwise disjoint Hence, we get 0 < ^ *(*)&,■ < x, 1=1 +00

a relation which implies that ^]x(z')6, = Jc exists and 0<x<x. X .

By

i=i

assumption, B = convext(B), since E has the KMP. Hence, x= lim xk i-»+oo

Z^(*

/L(£)&,, where /(£) is a finite subset with xk ecortv({6,.}.6jv), i. e. xk = with xk econvttbj}.^), i. e. xk = ^ 2 , ( ^ ) 6 , , where I(k) is a finite subset of N, A,(k) e [0,1] and TA^,.(A) = 1. We observe that of N, Uk) e [0,1] and , e£/ ( i )A,(Jfc) = 1. We observe that ' ■ « / ( * )

x A Xk =x A

y ;*i(*h^

Z^AA,(A:)Z>,.< £xA6,= Xx(06,<*.

,-£*,(*)&,£ 6 /(i) £xAA,(*)M ie/(*J

,6/ x/\xk=xA £W x A 6 , = , e J ^x(0fc,SJc. (.*; i e / ( * ) i e / ( * ) i 6 / ( ) i ) (x Axk)< x By continuity of the operator (w,v) -> w A v, we *derive x i=e / (lim By continuity of the operator (w,v) -> w A v, we derive x =jt-»0O lim (x f\xk)< x

jt-»oo

+00

+00 di)b and the proof is complete. and < t, and and therefore therefore x=x x=x = = ^! * (x(/)6,, the proof is complete. ■

i1==i1

Let (E,j I) be a general Banach space and C c £ a closed convex set. We say that a continuous operator (not necessarily linear) P : E —>E is a projection onto C if P(E) = C and /'(x) = x for every x e C. Given a Galerkin cone K(K„)n<=N in E we can show that for every n e N, there exists a projection P„ onto JT„ such that, for every x e Jf, we have lim Pn(x) = x. n—>oo

To show this, we use the following classical result. Theorem 2.33 [Krasnoselskii-Zabreiko] Let D be a closed convex subset of a Banach space \E,\ |). For every positive a, there exists a projection Pa onto D which satisfies \x-Pa(x)\<{\ notes the distance ofx to D.

+ a)p(x,D)

where p{x,D) de-

Proof A proof of this theorem is in the book (Krasnoselskii, M. A. and P. P. Zabreiko [1]). ■

26

COMPLEMENTARITY PROBLEMS Proposition 2.34 If (E,\ |) is a Banach space and K(Kn)n€N a Galerkin cone in E, then, for every n e N, there exists a projection P„ onto Kn such that, for every x e K, lim P„(x) = x. n-»oo

Proof For every K„(neN), we apply Theorem 2.33 considering a = 1. We obtain for every n e N, a continuous projection P„ onto K„ such that pc-/^(x)j <2d(x,Kn) for every x e K. Since A" is a Galerkin cone, for each x e K and each e > 0, there exists n e N with \\x - x„\\ < s where xn eKn. Hence, for every m > «, we get || x -Pm {x)\\ < 2d(x,Km) < 2d(x,K„) < 2\\x -xn\\ < 2s. Thus, lim PJx) = x for every x e K. ■ n->oo

3. Complementarity problems In this section, we will introduce the most important kinds of complementarity problems. Complementarity Theory is considered by several mathematicians as a large independent part of Mathematical Programming Theory, however the reality is different. Complementarity Theory is a collection of deep, interesting and difficult mathematical problems. This new domain in mathematics is a nice research subject, because it has many applications and interesting connections with important chapters of Nonlinear Analysis. So far at least, three books have been published in this subject (Cottle, R. W., I. S. Pang and R. E. Stone [1]), (Isac, G. [12]) (Murty, K. G. [1]). The books (Cottle, R. W., J. S. Pang and R. E. Stone [1]), (Murty, K. G. [1]) are concerned with the Linear Complementarity problems, while the book (Cottle, R. W., J. S. Pang and R. E. Stone [1]) presents the Nonlinear Complementarity Problems, generally in infinite dimensional spaces. In 1961, in the paper (Dorn, W. S. [32]) is obtained the following result. If A is a («,«)-positive definite (not necessarily symmetric) matrix, then the minimum value of the quadratic programming problem 27

CONES AND COMPLEMENTARITY PROBLEMS

•

minimize (x, Ax + q), *~ , , whereV = \xeR"\ Ax+ where V=\xe.R"\ Ax + qq eR"\andq eR"\ and q eR" eR"

(3-1)

is zero. The Complementarity Theory was born by this result. Considering problem (3.1), we define the following problem known as the Linear Complementarity Problem. If A is a («,«)-matrix and q e R" is arbitrary, [find find x. 6R"+ LCP:\ LCP: J I and (x,, (x., Axtt

such that Ax.+q eR" Ax,+qeR" H + +q) = 0.

(3.2) (3.2)

We consider R" with the Euclidean structure. In 1963, G. B. Dantzig and R W. Cottle generalized Dom's result to the case when the principal minors of the matrix A are positive (Dantzig, G. B. and R. W. Cottle [1]). The result announced in 1963 by Dantzig and Cottle was generalized in 1964 and 1966 by R.W. Cottle to a certain class of nonlinear functions (Cottle, R. W. [1], [2]). Also, in 1965, C. E. Lemke proposed the LCP as a method for solving matrix game( Lemke, C. E. [1]). One of the first important papers on complementarity problems is Ingleton's paper (Ingleton, A. W. [1]). In this paper, the importance of complementarity problems in engineering applications is shown. After 1970, the Complementarity Theory has known a strong and ascending development based on several important results obtained by R. W. Cottle, B. C. Eaves, Karamardian, O. L. Mangasarian, R. Saigal, F. J. Gould, C. B. Garcia, G. More, M. Kojima, N. Megiddo, I. Kaneko, M. Kostreva, J. S. Pang, M. S. Gowda, G. Isac etc. The name "Complementarity" has as its origin the following fact. A solution JC* of problem LCP is said to be nondegenerate if at most n components of 2«-components vector (x', Ax* + q) equal zero. Otherwise, it is a degenerate solution.

We denote by N„ ={1,2,...,«}.

If x ' ^ l x * } ' '

is

nondegenerate solution of problem LCP and y = Ax + q, then the sets

28

COMPLEMENTARITY PROBLEMS A = \i\xi > 0> and B = \i\y* > OJ, (where y* = ly*\ _ ) are complementarity subsets of N„ (that is, A-Nn\B). ,5 The problem LCP can be generalized for nonlinear mappings. If f:R"^>R" Rn is an arbitrary mapping, the Nonlinear Complementarity Problem associated to / and to the convex cone R" is: {find find xJC*t e Rl R" such that JK f(x,) f(x,)J e R" NCP:V NCP-.V 7 and(xt,f{xt))-0. 0 The study of economic equilibria (Ann, B. H. [1]), (Manne, A. S. [1]), (Nagurney, A. [1]), the study of some problems in elasticity (Isac, G. and M. Thera [1]), (Isac, G. and M. Thera [2]), (Goeleven, D., V. H. Nguyen and M. Thera [1]) or in lubrication (Kostreva, M. [1]), (Oh, K. P. [1], [2]) have been important motivations for the development of the study of nonlinear complementarity problems. An interesting collection of nonlinear complementarity problems are also presented in (Dirkse, S. P. and M. C. Ferris [1]). The Complementarity Theory is now a wide domain of Mathematics. If we consider, in particular, only the Linear Complementarity Theory, we remark that it is a rich mathematical theory formed by a variety of algorithms and a variety of applications in applied sciences and in technology. On this subject alone, there are probably more than one thousand papers published (see for instance the references published in (Cottle, R. W., J. S. Pang and R. E. Stone[l]), (Isac, G. [12]), (Murty, K. G. [1]). Nevertheless, Nonlinear Complementarity Theory is now in developing. Our principal aim in this chapter is to present Nonlinear Complementarity Theory as a new and fascinating field for applications of Nonlinear Analysis. For this aim, we selected only some representative results. We will show that Complementarity Theory can be considered as an interesting source of new problems for Nonlinear Analysis, and, also, Complementarity Theory can be used to obtain new results and new concepts in Nonlinear Analysis. Finally, since the Complementarity Theory has many applications in Economics, Applied Sciences, Engineering, Optimization, 29

CONES AND COMPLEMENTARITY PROBLEMS Game Theory, etc, this domain can be considered as a new opened door for new applications of Nonlinear Analysis. The relations between Complementarity Theory and Nonlinear Analysis are deep. Not only Nonlinear Analysis is used to study the Nonlinear Complementarity Problems, but also to study and solve the Linear Complementarity Problem. We consider now the most important classes of Complementarity Problems. • The explicit complementarity problem Let \E,E)

be a dual system

of locally convex spaces and K cz E a closed convex cone. Given a mapp i n g / : K-^E , the explicit complementarity problem associated to / andK is: r find xxn0 eK find e K such such that that ECP(f,K):\ r and {x (x00,f(x == 0. 0. 0. f(x(x0)0) eeK' K and ,f(x00)) )) = Models 1) Let c = (c,) e K", b = (6,) e Rm be arbitrary vectors and A = {ay) an (m, «)-matrix of real numbers. Consider the primal linear program minimize (c, A x) xe^ I F 5 ' xe9 (PLP): (PLpy.l • ! . ?9,f = \x sR" eR" |IxxeR? eR? and and Ax-b Ax-b € Je eR?) /H

(

and its dual (DLP): (DLpy.l

maximize (y,b) y^7i y ^{ ?7zZ = jx [x eR sRm | x eR? eR™ and A'y-c

eeR^V /^j.

A fundamental result in Linear Programming Theory is the following: Theorem 3.1 If there exist x0 e 7\ and y0 e ? 2 such that (c,x0) = (y0,b), then x0 is a solution of the problem PLP andy0 is a solution of the problem DLP. By this result, we can associate to the couple of problems (PLP, DLP) a By this result, we can associate to the couple of problems (PLP, DLP) a

30

COMPLEMENTARITY PROBLEMS linear complementarity problem. Indeed, considering the slack variables u eR" and v eRm such that Ax - v = b and A'y + u = c and denoting -A'' xx~\ u [u] [c1 0 -A' z= ; wM = ; q = z= ; w = V ; q = -b and M - A 0 }, 'we obtain the following y. -6 A 5

_>>J

|_vj

L J

\_ °

linear complementarity problem

,n+m find zeR"+n such that

+m +m LCP[A,q;Rn+m+m eRl+m ,w=Mz Mz + q qeR" eR^+m LCP(A,q;RT ):):- zzeR" ,w=Mz

and and (z, {z, w) w) = = 0. 0. The main contribution of Complementarity Theory to Linear Programming is the fact that the complementarity problem transforms an optimization problem in an equation. 2) Consider the convex program minimize minimize f{x) fix) ? = {xeR:\gi(x)<0;i 0;i = l,2,...,m}.

(3.3)

In this programming problem, we suppose all the functions convex and continuously differentiable. The Lagrangian function L(x,u) for (3.3) is given by m m

:) +++ YY M L{x,u) f{x) \uJiSi{x). L{x,u) ==f{x) u: ?ig(x). i{x).

2^ i=i i=i

Hence, u = («,) e Rm and the Karush-Kuhn-Tucker necessary conditions for optimality can be written as

31

CONES AND COMPLEMENTARITY PROBLEMS dlix,

u)

^ ^ -

dx dxj3 9L(x,u)

l,2,...,n ==hhj(x,u)>0;j j{x,u)>0;j 7 == l,2,...,n

= l,2,-,m . - ^du- W h* (x,u)>0;i . ) *««-!*".» n+i

t

(3.4)

x > 0, u > 0 x>0,u>0 n

m

U, ==Ii

i=\

U ^Xjh,(x,u) a«J X "MAA++,/(*' £ X ; 7 J , ( * > " ) = 0 and ( * > " ) = 0. 0

x

If we set z == '

u

and h(z) [^(z),...,A„(z),An+1 (z),...,^n+m h(z) = [fy(z),...,/z„(z),/z n+1(z),...,A +m (z)]', then the si

system (3.4) can be stated as the following complementarity problem: n+m find zeR zeRn+m such that that n+m such +m m +m n +m ECP(h,R"+m ECP(h,R" ):-2 ): • zeRl z e ^+m+,h(z)^R , /J(Z) + e J^

and{z,h(z)) and(z,h(z)) = Q. 0. 3) In the books (Isac, G. [12]), (Cottle, R. W., J. S. Pang and R. E. Stone [1]), some other mathematical programming problems which can be transformed in complementarity problems are also presented. We cite for example the Quadratic Programming Problem, some Saddle Point Problems (some max-min problems) etc. (see (Isac, G. [12])). 4) Consider a two-persons game (a bimatrix game), that consists of two players. The first player solves the program maximize maximize ((xx, Axx2 )

G (G.

i &-\

x *A

f

H

(*W) -

n

w—\

1

j=y

and the second player solves the program

32

1

i x isx2 given ■: £z*j ^ =. <*cx'x iSR: e R" *]Xj:== -1 1 where >, where is given,

COMPLEMENTARITY PROBLEMS (x1,, AA22xx2 ) maximize \x

te):

(G2)

V

m

=H

= •{ x 2 zR™ e Iff VJ ]__2 x 2 = 1 ,>,where where given, '.xxx1lisisisgiven, 7^ 1 = 7=1

where AhA2 are two suitable matrices. This game will be denoted by 2 ( < J , , G 2 ) . A pair of vectors (x^x ) is a Nash equilibrium point for (Gl5G2) if and only if: nO xl solves (G/) given x 2 and n2) xl solves (G2) given x,1. The problem which we have in this case is to find a Nash equilibrium point for (GX,G2)- We can prove (see (Isac, G. [12])) that the bimatrix game (Gl5G2) is equivalent to the following complementarity problem +m 'find find X X<ER" eR"+m suchthat that + such

Y=MX + qeRn++m and (X, (X,Y)Y) = 0 where whei -e \x ~ v1 LCP[M,q,Rn++m): m)-\x X = \X,<\7 ;? = [~*V=K ,Y LCP(M,q,RT -ee Lv2J L* J x

L* J

M=

0

_-w

\r \

\_y

-A o

We denoted by e a unit vector of appropriate dimension. A generalized model of the model LCP( M,q,Rl+m\ presented above can be used to solve some polymatrix games (Isac, G. [12]). 5) An important field of applications of the Explicit Complementarity Problem is in Economics. The Explicit Complementarity Problem is much used in the study of equilibrium of diverse economical phenomena. Several

33

CONES AND COMPLEMENTARITY PROBLEMS models in this sense are presented in the books (Isac, G. [12]), (Cottle, R. W., J. S. Pang and R. E. Stone [1]). We selected in this sense the following example. Consider an economy or a sector with production and, to simplify this we will restrict ourselves to an economy with competitive behavior thought but with no price distortions. Suppose that our economy has m commodities and n activities with constant returns to scale production. Our model is based on the assumption that production is characterized by a linear technology matrix with fixed input-output coefficients. For i =\,2,...,m and j = l,2,...,n, we denote: p = (pi) the vector of prices, b = (b,) the vector of endowments, d{p) = (di (p)) the market demand functions, which are assumed to be point-to-point and continuously differentiable but not necessarily linear, v = (y7) the vector of activity levels, c = (cj) the vector of unit costs of operating the activities, A = (<3y) the technology matrix of input-output coefficients consistent with unit production, where ay > 0 (resp. atj < 0) denotes an output (resp. input). It is known that there exist several ways to characterize an equilibrium, but we use here the Scarfs definition (Scarf, H. E. and T. Hansen [1]). A price vector p, and a vector of activity levels y, constitute a competitive equilibrium if: 1) no activity earns a positive profit, c- A'p, > 0, 2) no commodity is in excess demand, b + Ay, - d(pt) > 0, 3) no prices or activity levels are negative,/?. > 0, y* > 0, 4) an activity earning a deficit is not used and an operated activity has no loss, [c- A'p*} -yt = 0 , 5) commodity in excess supply has zero price and a positive price implies market clearance, p[{b + Ay, -d[pt)) = 0. We note that the vector c of operating costs represents factors of production that are exogenous to the economy or sector under certain considerations. If (1) - (5) describe a general equilibrium problem of a closed economy,

34

COMPLEMENTARITY PROBLEMS then the cost vector c = 0, because all prices will be determined simultaneously and no single price will be exogeneously given. In this case, the demands d,{p) for i=l, 2, ....,m are functions of all prices in the economy, i. e. both product and factor prices. Furthermore, these demand functions will usually be specified in a manner consistent with individual household utility maximization, that is dt(p) = T)x* where x\ is h

the Kth household's utility maximizing demand of the commodity i. Households excess demand are given by dip) -b. We remark that if the demands satisfy each individual household's budget and there is nonsatiation, then p'd(p) = p'b, and the demand function d(p) are homogeneous of degree 0 in all prices. We observe that, when c = 0, conditions (1) - (5) determine only relative prices, that is, if the vector pt represents equilibrium prices, so does Xp* for any scalar X > 0. Hence, we are free to normalize the prices. Now, we observe that we can associate to this model the following complementarity problem: find

z&Rn++m such that

\F(z)eR:+mand(z,F(z)) *); = 0 U"l

ECP(F,R^m):\ iwh \>here z =

"Ud

F(z)

\c-A'p

and D

b + Ay-d(p)

Remark 3.1 The problem ECPi F,R?m i is a nonlinear complementarity problem since d(p) is generally a nonlinear mapping. 6) Variational inequalities form another source of nonlinear explicit complementarity problems. The study of variational inequalities is an important domain in Nonlinear Analysis with many and important

35

CONES AND COMPLEMENTARITY PROBLEMS applications. Given a locally convex space E(x), a mapping f:E—>E E* and a closed convex subset D of E, the variational inequality associated to / and D is the following problem:

(

Vl{f,D)

find x, x, £e D D such such that that ( x - xx.,. ,f(x / ( xt)). ) ) > 00 for all x xeD. eft (x

When the set D is a closed convex cone, we have the following result. Theorem 3.2 Let E(T) be a locally convex space, K c £ a closed convex cone and f:E—>E a mapping. An element x, e K is a solution of the problem VI(f,K), if and only if, x, is a solution of the problem ECP(fK).

r

Proof Suppose that x. e K\s a solution of the problem VI(f,K). We have (x - x.,f(x.))

> 0 for ally e K

(3.5)

and if y e K is an arbitrary element and x= y + x,, then, from (3.5), we obtain (y,f(xt))> 0 for ai\yeK, (3.6) which implies t h a t / * , ) e K\ If we consider* = 2xt in (3.5), we deduce (*.,/(*.)) >0 and if we put in (3.5) x = 0, we have (*♦,/(*.)) <0 and hence (x,,/(x»)) = 0 which means that x, is a solution of the problem ECP(fK). Conversely, if x, is a solution of the problem ECP(f,K), we have (*,,/(*.)) = 0 and (x,/(*.)) > 0 for every x e K. We obtain immediately ( x - x , , / ( x » ) ) > 0 for all x e K, that is x, is a solution of the problem VI{f,K). ■ From Theorem 3.2, it follows that the problem ECP(fK) is essentially, a 36

COMPLEMENTARITY PROBLEMS variational inequality but with respect to a convex cone. Because the most important results obtained about variational inequalities are obtained assuming that the set D is compact or bounded, not all results about variational inequalities are applicable to explicit complementarity problems. Hence, for complementarity problems a new theory must be developed. A more concrete example is the following. We consider a differential equation of the form

M-

' ^ - ■== Ax(t); Ax{t);t>0 t>0 W ■ dt dt x(0) = x0,x0 &K

(3.7)

where K cR" is a closed convex cone and A : R" -> R" is a mapping. In some practical problems, as for example the study of economical systems, we are interested to find x(t) such that x(f) e K and we are interested to study the problem find x such that find such that

• x(t) x{t) e K for all t > 0, x(0] x(0) = xn0 and

(3.8) (3.8)

ldx(t] x(t))\>0, > 0,for for allt>0 allt>0 and and all all yeK. yeK. / — ^ -- Ax(t), y - x(t)

A*

By (3.8), we obtain an explicit complementarity problem. 7) An interesting application, of Nonlinear Complementarity Problem in infinite dimensional vector spaces, is the study of the post-critical equi librium state of a thin elastic plate. Consider the mathematical model constructed by means of the classical nonlinear description proposed by Von Karman for plates undergoing large deflections relative to their thickness. Suppose that Q is a thin elastic plate (the thickness is supposed to be constant) resting without friction on a flat rigid support. The material is supposed to be homogeneous and isotropic. From the mathematical point of view, Q is identified to a bounded open connected subset of R2. The •37 37

CONES AND COMPLEMENTARITY PROBLEMS plate Q is assumed to be clamped on yi c y and simply supported on Y2 = Y \ Yi, where y is the boundary of Q which is supposed to be sufficiently regular. Suppose that a lateral variable load XLa,{a = 1,2, as it is indicated in (Cimetiere, A. [1]) where X is positive and increasing representing the magnitude of lateral loading, is applied to the boundary of Q. Because Q is a thin elastic plate, we observe that if X exceeds a critical value, termed the critical load (specific for each plate Q), then the plate deflects out of its plane, and we say that it buckles. Consider the Sobolev space, A*,,

A,.

2 2 22 (Q),v;,y i,2 equipped with the H2(Q) = \\uueL e L(Q)\ (Q) | -— >; _ ^ _ -€L Vij = 1,21 ec L (C1), dx{ dxtdx} norm |*|La/oy anc* l e t E De m e closed subspace of H2(Q) defined by

dz

=o, J

2 2 0,a.e, B-\, E = \zeH
dn

exterior to Q and —(•) denotes the exterior normal derivative. The middle dn of the plate is referred to an Euclidean coordinate system (0; x\ ,x2, z), where the z-axis represents the vertical displacement of the plate. Consider the closed convex cone K = iz e E | z > 0, a. e. on Q j . The cone K represents the vertical admissible displacements. For a fixed A the post-critical equilibrium of the plate Q is governed by the following complementarity problem find z eK e.K such that
(3.9) (3.9)

(z,z-AL(z) + C(z)) = 0, where X > 0 is the intensity of the lateral load (XLa), L : E -> E is a selfadjoint linear compact operator and C : E -> E is a nonlinear continuous compact operator connected with the expansive properties of the plate. The operator C is positively homogeneous of order p = 3, it is the Frechet

38

COMPLEMENTARITY PROBLEMS derivative of the mapping z -> — (z,C{zfj and it satisfies also the following properties: i) (z, C{z)) > 0 for each z e E, ii) z e E and (z, C(z)} = 0 imply z = 0. The operator T(z) = z - X L{z) + C(z) is the Von Karman operator, which is very important in elasticity. For further details see (Cimitiere, A. [1]), (Isac, G. and M. Thera [2]), (Goeleven, D., V. H. Nguyen and M. Thera [1]). 8) The Explicit Complementarity Problem is the mathematical model used in the study of some contact problems in Mechanics, of some free boundary problems, of some problems in lubrication, of equilibrium of traffic flows and to maximize the oil production. (Isac, G. [12]). Also, the Explicit Complementarity Problem has interesting applications in Structural Engineering, in Fluid Mechanics, in Electrical Circuit Simulation, etc. (Isac, G. [12]), (Cottle, R. W., J. S. Pang and R. E. Stone [1]). 9) Finally, we remark that any nonlinear perturbation of a linear complementarity problem is a nonlinear complementarity problem. • The implicit complementarity problem Let (E,E*) be a dual system of locally convex spaces, K a E a closed convex cone and D c E a nonempty subset. Given the mappings /:£>—»£* and g:D^> E, the implicit complementarity problem associated to fg and K is find x0 e€ D such that ICP(f,g,Kh ICP(f,g,K):< Ag(x g(x00)eK,f{x ) eJT,/(xo) 0)(=K e f and ,(g(*o),/(*o)) {{g[x = 0. O. o),J{x0)) = This problem was considered for the first time by A. Bensoussan and J-L. Lions [1], A. Bensoussan, M. Goursat and J.-L. Lions [1] and studied by I.

39

CONES AND COMPLEMENTARITY PROBLEMS Capuzzo Dolcetta and U. Mosco [1], by U. Mosco [1] and other authors, namely J. S. Pang [1], [2], D. Chan and J. S. Pang [1], M. A. Noor [2], I. Capuzzo Dolcetta, M. Lorenzani and F. Spizzichino [1], G. Isac [4], [9], [12], G. Isac and D. Goeleven [2], etc. Models The implicit complementarity problem was introduced by A. Bensoussan and J-L. Lions as the mathematical model of some stochastic impulse control problems (Bensoussan, A. and J-L. Lions [1]). Generally, the implicit complementarity problem is related to the theory of QuasiVariational Inequalities (Capuzzo-Dolcetta, I. and U. Mosco [1]) which are currently used in the study of some stochastic optimal control problems or in the study of various free boundary problems associated to some differential operators. 1) Quasi-variational inequalities and the implicit complementarity problem In the study of some practical problems, we have the following situation. We have two spaces H and E such that E(f / ,v-u) for alive E,v< M(u), where/is an arbitrary element in H. Considering the duality (E,E*),

if

a(y) satisfies some continuity properties, we remark that the identity a(u,v) = (A(U),V) defines A as an operator acting continuously from E into E*. About the equivalence between the problem QVI(f,M) and the following implicit complementarity problem 40

COMPLEMENTARITY PROBLEMS find find UGE u e £ such that eK,f ICP(f,M,A):• :\ M(u)-u M(u)-usK,fICP(f,M,A):-

- A(u) eK' eK* and

(M{u)-u,f-A{u)) = (M{u)-u,f-A{u)) = 0, 0, one can consult the papers (Capuzzo-Dolcetta, I and U. Mosco [1]) and (Mosco,U. [1]). 2) The optimal stopping problem By a finite Markov chain we mean a system consisting of: i) a finite set of states S= {l,2,...,n}, ii) and (w, rt)-matrixPo = ipij), ij = l,2,..,n. The matrix P0 is called the transition matrix and satisfies the following conditions: iu)py>0, 1
iv

n

)Z#/ = L1-z'-w-

If /? a n d / a r e two functions on {l,2,..,n} interpreted, respectively, as a payoff and a fee, we set

v/ = ^-{|/^ ) + J P X4 )}

(3-10)

for every i e {1,2,..,n}, where r is a Markov time (Scarf, H. E. and T. Hansen [1]), y(k) is the state of the chain at the time k and E is the expectation. We define also the value v* by v* = supj v] | r e M> for every i e {1,2,...,«}, where the supremum is taken over the class M of all Markov times for which the right-hand side of (3.10) makes sense. A Markov time z is said to be optimal ( in the sense that it corresponds to a Bayes stopping strategy) if v* = vj , for all i e {1,2,...,«}. The search for v* and T, if they exist, is known as the optimal stopping problem. This problem was considered by E. B. Dynkin [1]. In (Schroder, J. [1], Chapter II,

41

CONES AND COMPLEMENTARITY PROBLEMS Theorem 16) it is proved that v is the minimal solution of the following system of inequalities in R" (the functions/? and/are considered as vectors in J O [p
"/^{L/,M+/VM}

(3-12)

for every i e {1,2,..«} and all re M. Considering that R" is ordered by R"+, we have that v* satisfies necessarily the Implicit complementarity problem u-p&Rl,K> • u-f-P0u Pnu eR" and (u-p,u-f- / • P0u) = 0. • The generalized order complementarity problem Let (E,\ \,Kf be an ordered Banach space which is a vector lattice with respect to the ordering defined by K. Suppose given m nonlinear or linear mappings

f„f2,..,fm:E ^ E.

The Explicit Generalized Order Complementarity Problem associated to the family of mappings {/.}" and to the cone K is find x0 e K such that EGOCP[{fi}lvK)-,fi(x0)eK,i eK.i = \Z,..,m and A(x0,fl(x0),...,fm(x0)) *o - 0. Also, given n mappings fx,f2,..,fn:E->E - > E and a non-empty subset D a E,

42

COMPLEMENTARITY PROBLEMS the Implicit Generalized Order Complementarity Problem associated to the family of mappings {/;}* and to the set D is i,

IGOCPi

^

\

[find xn eD such that

\", ,D '/<=!' \V.irt )

\A(fl(x0),f2(x0)...,fn{x0))>/„(( -c0o)) ) = 0.

Models The Generalized Order Complementarity Problems are important since they are the mathematical models for several practical problems. We present a few examples. 1) The Generalized Complementarity Problem (Cottle-Dantzig) In 1970, R. W. Cottle and G. B. Dantzig [1] defined the following problem. n

where m =-y.> ^ jm J;

Suppose given a function f:R"^Rm Rm

m > n;

M

f = [fj\ \"

/i=I

and/ 7 :/?" —>Rmj. The Generalized Complementarity Problem

is to find ze R" such that for j =l,2,..n the following relations are satisfied : 0

Zj>0

■f/(z)>0,i

(3.13)

= l,2,..,mj

ny/(z)=<>mj mj

'J

i=\

This problem was considered in [24] with fJ =qJ + Mjz; j =1,2,..,«, where M J is a (mj, «)-matrix, qJ eRmj

n

and ^jmmJi=m.■ m. If we denote M

m, =max[/«y] and we define the mappings Tx,T1,..,Tnu:Rn -> R" as the columns of the following matrix

43

CONES AND COMPLEMENTARITY PROBLEMS

//to /ato-zAto- « Z.'to /22W-« «

- « - «

//•(*) Z/to z/"to //w

z*to /iW - «

where a is an arbitrary real number such that a > 0, then the problem (3.13) is exactly the problem EGOCP{Tx, Tt,...,Tm, R"). The problem (3.13) has been studied in (Cottle, R. W. and G. B. Dantzig [1]), (Ebiefung, A. and M. Kostreva [1]) and (Szanc, B. P. [1]). 2) The Mixed Lubrication Consider the case of mixed lubrication in the context of a journal bearing with elastic support. In this case, we have two operators T}(X) = H0 + a + L(X) and

0H\ +-VI T2(X) =-R{X)
3

'xrf Kdty

*n

where R(X) = V-e H (x,y,t)VX. The variables have the following signification : X is the contact pressure, L is a linear integral operator, a is a constant (the whole-body approach, depending on problem), Ho represents surface separation when the load is zero, H is the film thickness, U is the entertainment velocity, G is a piezo-viscous coefficient, x is the spatial variable in the direction of rotation, v is the transverse spatial variable and V is a positive constant depending on the problem. The experiments show that the physical phenomenon is described by three regions: 1) X > 0, TX{X) = 0, T2(X) > 0; (solid-to-solid contact) 2) X = 0, 3J(X) > 0, T2(X) > 0; (cavity point) 3) X > 0, TX(X) > 0, T2(X) = 0; (lubrication point)

44

COMPLEMENTARITY PROBLEMS Considering E = Hl(Q) (definedover L2(Q)) and K= [u eH'eHl(Q)\u>0, a.e. o n Q j , the problem to compute the pressure is to solve the problem EGOCP{T\, T2, K). This problem was considered in (Kostreva, M [1]), (Oh, K. P. [1], [2]) and (Isac, G. and M Kostreva [1]). 3) The Discrete Dynamic Complementarity Problem. This problem was defined by Harrison and Reiman in 1981 [see the references of (Mandelbaum, A. [1])], and, it is important since this model is a unifying framework for fluid and diffusion approximation of stochastic flow network. Consider E = Rm endowed with the Euclidean structure and the ordering defined by K= R+ . Suppose given x = |x(0),x(l),...,x(«),...] a sequence in Rm with x(0) > 0 and & an (m, /M)-matrix. The Discrete Dynamic Complementarity Problem is, find the sequence y- {y(0),y(l),..,y(n,..)\ •)} such that i) z(n) = x(n) + TRy{n) > 0, for all n = 0,1,2,.. ii) ^(0) = 0, Ay(rt) = y(n) - y(n -1) > 0 and

(3.14)

Hi) (z(n), Ay(ra)) = 0, for all n = 0,1,2,... (we consider by conventiony{-\) = 0). From (ii) we have that each coordinate y}, = |v y (w)|« = 0,1,2,..j of y is nondecreasing and from Hi) we have that yy (« - 1) strictly increases to^,{M) only when zfri) - 0 for j = 1,2, ,m. Consider the vector space S = (x|x:N-> Rm) ordered by K = [x GS\ x(n) > 0, for all n eiVJ where A^={0,1,2,....,n,....}. 5" is a Frechet space with respect to the topology defined by the family of seminorms {pr}ri.N,

where pr(x)=

sup ||^(")|. 0<,n<,r

Also, S is a vector lattice and K is a normal and regular cone in S. Now, we define the following operators: Ti(y)(n) = x(n) + &y(n) ; for aline N T2 (v) = {0,^(1) ~y(0), ...,y(n) -y(n - 1),...} 45

CONES AND COMPLEMENTARITY PROBLEMS The Discrete Dynamic Complementarity Problem is exactly the problem IGOCP{T\,T2,D), where D = [y eS|.y(0) = o}. This problem was considered in (Mandelbaum, A. [1]), (Isac, G. and D. Goeleven [2]), (Isac, G. [11]) and (Isac, G. and M. Kostreva [1]). 4) The Global Reproduction of an Economical System with Several Technologies Consider an economy consisting of n production sectors. Every sector is constrained to use the production of the others. The demand of the sector ]{j = 1,2,3,...,«), for its technology k (k = 1,2,3,..,m) for the sector i is given by the function / / ( * , ) , where Xj is the level of the gross activity performed in the sector j . We suppose the following hypothesis: 1) all the functions /§*(■*,) are continuous, 2)//(0) = 0 f o r a l U M 3) 0 < uj < vj implies /<*(«,) < f.f(v.),

for every k =\,2,..,m.

The balances between total activities and final demands for the technology k are given by -

+ 'I. f»('('My, .*'P'tt'') » i

Xj

n n

=

/=i

0.15) (3.15)

l,2,..,n. i = 1,2,.., n. We define / / ( * , ) = [//(*, )f=] for every j = 1,2,....,» and k

n

F {x) =x-Y,fjk(xj) 7=1

X.) X J

for every k= l,2,..,m; where x =

{xv...,xtt)'.

We say that a final demand v° = (v,0,..., v°) is attainable if the set n Sy0={xeRn+\ * I F\x)-yo>0,...,Fm(x)-y°>0\ -J> - v °°> : >) is non-empty. For this >uj '+ I model, the problem is to show that given v° > 0 the problem %,.. IGOCPi T\,T 2,---,T +\ has a solution x° > 0 which is the least element of Tm,R K)

46

EXISTENCE THEOREMS it is Sy0, where 7j(x) = Fi(x)-y°,...,T Fmm(> (x)-y°. X -y * - y°, m(x) [* = F production x° is realizing v° with a minimal social cost.

0

In this case, the

The Complementarity Theory studies several kinds of problems associated to a particular complementarity problem. The most important problems are the following: 1. the study of solvability, 2. the study of existence of a solution which is the least element of the feasible set. For example for the problem ECP(f,K) the feasible set is ?={xeK\f{x)eK*}, K 3. 4. 5. 6.

the study of the solution-set from the topological point of view, the study of the cardinality of the solution-set, important to study when the solution-set is a bounded set, the study of a complementarity problem must be finished by a sensitivity analysis.

4. Existence theorems In this section we present several existence theorems based on some special ideas. Every idea can be considered as the starting-point of new results in Complementarity Theory. • Galerkin cones and the generalized Karamardian condition Let (E,F) be a duality between two locally convex spaces, K F a mapping. We denote by if* the dual of K with respect to the duality < - , - > . Theorem 4.1 [Karamardian] The problem ECP(f,K) has a solution if: 1) the function (x,y)->(xj(y)) is continuous onKxK, 2) there exists a compact convex set DczK such that for each x e K\D there exists z e Z> such that (x - z,f(x)) > 0.

47

CONES AND COMPLEMENTARITY PROBLEMS Proof A proof is in (Karamardian, S. [1]). ■ Karamardian 's Theorem was generalized by M. S. Bazaraa, J. J. Goode and M. Z. Nashed, and by G. Luna (see the references of (Isac, G. [12]). Another generalization is in (Isac. G. [5]). An analysis of Karamardian's theorem proves that assumption (2), in the case of infinite-dimensional spaces is a strong one, because it implies a compact localization of a solution of the problem ECP(fK). Using the concept of Galerkin cone we will introduce a generalization of Karamardian's condition, which is more flexible. Let (//,<,>) be a Hilbert space and K{K^)n&N a Galerkin cone in H. Definition 4.1 We say that a mapping f: K —» H satisfies the condition (GK) (the generalized Karamardian's condition) with respect to the Galerkin cone K(Kn)n&N, if there exists a countable family {D„\ [A, neJVN of subsets ofK such that: 1) for every n e. N, Dn is a convex compact subset ofKn 2) for every x e K„\D„ there exists y e D„ such that (x - y,f(x)} > 0. The next result assumes that condition (GK) is satisfied with an equibounded family {Dn} neNN ■ This assumption is naturally satisfied in several situations. Examples : 1) We assume t h a t / : K -> H has the following form (used in the study of some problems in elasticity theory), f(x) = x - AL(x) + T(x), for every i • , , 1 (x,L(x)) xeK,K where Lr is an operator such that — = supV-^>0 is well P x*K ||x| defined (for example if L is a linear self-adjoint compact operator, as in the study of elastic plates), X > p and T is a JT-coercive operator, that is,

48

EXISTENCE THEOREMS lim

(x,T(x)) X

M_u* xsK

\ " = +oo.

wr

In this case, condition (GK) is satisfied with an equibounded family {Dn}neN with respect to every finite rank Galerkin approximation (JSTn)neAr of K, (i.e., for every n <= N dim(jr„ - A"„) 0 such that for all x e K satisfying r0 < \\x\\ (x,7{x)) X we have =-^- > 1 > 0, which implies

IN 2

P

1+M^)>A>(fi^W)

H2

P~

\\4

for each x e K satisfying r0 < \\x\\. Hence, (x,f(xf) > 0 for each x G K satisfying r0 <\\x\\. If (ifn)neAf is an arbitrary Galerkin approximation of K of finite rank and for every n e N, we take D„ =B (0,r0) f\ Kn, where B(0,r0) ~ [x sH\\xl
then we have that condition (GK) is satisfied

(takings = 0) with the equibounded family {Dn} n e V. 2) We assume again that/ has the form, f(x) = x- AL[x) + T[x), but where L is a bounded linear operator such that ||L|| < —, with X > 0 and T is a strongly iT-copositive operator, that is, there exists m > 0 such that for all x <=K, (x,T(x) - r(0)) > m\\xf. We suppose that 7(0) * 0. In this case we have A(X,L(X))<

<xA|X||X||

^(JC,X)

for all x

E

Jf, which implies,

(x,x- XL(x) +T(X) - r(0)) > m\\xf, that is +m\\x\\ +ffj|bc|| . ,i(u;y-i-m|p ^ , X - ^ ( X J + r(*)) 7\XJ; :>(^,r((

I-

.

49

CONES AND COMPLEMENTARITY PROBLEMS Il7f0)|| If we set r0 =■ " " and if (K„)neN is an arbitrary Galerkin approximatioi 4 m

of K of finite rank, then since for all x e K \ B(0,r0) we have (x,x-AL(x) + T(x))>(x,T(Q)) + \\x\\\\T(0)\\>0, we deduce that conditioi (GK) is satisfied with respect to (Kn)neN when {Dn}neN 'neA

neN

is equibounded

and Dn = B(0,r0) V\Kn for every n e.N. Definition 4.2 We say that f ' : K -+ H is a (ws)-field if fix) =x - J\x)foi every x e K, where T: K —> H satisfies the following properties: 1) T is norm continuous (not necessarily linear), 2) for each weakly convergent sequence {*„}neA, ci K the sequence (r(x n )}>neN has a strongly convergent subsequence. Remark 4.1 The concept of (ws)-field can be considered in an arbitrar] Banach space (E,\\ p. When the space E is reflexive, in particular when 1 is a Hilbert space, the condition that / is a (v»>.y)-field implies that T is i completely continuous operator. Let cp : R+ -> R+ be a mapping such that

0 for every t > y, when y eR+. The following concept was defined and studied in (Weber, V. H [1])Definition 4.3 We say that an operator (generally nonlinear) T: K-> Hi. ; implies \\T\x)\\ < c). If the mapping f :K -> H satisfies the following assumptions: 1) is a (ws)-field (where fix) = x - T (x)),

50

EXISTENCE THEOREMS 2) condition (GK) is satisfied with an equibounded family {£)„ } or condition (GK) is satisfied and T is (p-asymptotically bounded with lim (pit) * +oo, supposing that this limit exists, <-»+o°

then the problem ECPffiK) has a solution. Proof Since K(Kn)neN

is a Galerkin cone we can suppose that for every

neN there exists a projection Pn onto K„ such that for every x e K, \imPn(x) = x. Using assumption (2) and applying Theorem 4.1 with n-»oo

respect to every cone Kn we deduce that for every n e N the following complementarity problem has a solution

{{

ECP{f,Kn):

find xxnnn eeKKnnn such find suchthat that x„« ~~~ T(x„) eK eK" K"n nand and (x„,x„ (x„,x„ T(x T(x = = 0. n)) (T(x„) n) { n> n ~ -{ n)) =n)) °- 0. T X

e

a

n

d

X

X

T X

Let {*„}n£N be the sequence where xn (for every n e N) is a solution of problem ECP(f,Kn).

We will prove that {x„}

N

is bounded. First, we

observe that, for every n e N,x„e D„. Indeed, if x„ e K„\ Dn, then y„ eDn exists such that, (x n - yn, f(xn)) > 0, which implies, (*«->-»/W)>o.

(*„./(*»))-(^/(*»))>o (*.,/W)-(J^/(*,)}>O *»./(*»))- {ym/M >o

/(*X

and hence, (^,/(*-)>
N,

is bounded. Also, if condition (GK) is satisfied

with Tbeing (p-asymptotically bounded and r->+oo lim q^t) * +<x>, then {x„}nsN is also bounded, since if we suppose that {x„}neN is not bounded there exists a subsequence \x„\

l "lc)ksN k<=N

of KLeiV {*„}neAr such that lim |x^ || = +oo. Then, 4->+a>l

51

CONES AND COMPLEMENTARITY PROBLEMS

H*JIMklD-

for every k e N such that r<||jc„J we have, ll(j!r % |
implies,

Ik If=(*%. I%* I = (*%.^x^*)))>)>***I%*)IHkl I%*)l)l•l •IkIk*I I*^(Ikl)lkl' *^(lk MlkIlk IlkI II.*(4.1)^ Ikf=(**A 00 = +°o. that is, \\x pr„n I < (bc_ cMx I)) and hence lim ©ipc. -*» n tii MI my IKIh-" ' neiV is bounded last relation is impossible. In conclusion, we have that {*„}neAf

Kl

and then it admits a weakly convergent subsequence j x„t | keN whose limit xt belongs to K, since K is weakly closed. The sequence \xn\

IteJV

is

strongly convergent. Indeed, since/is a (ws)-field we find a subsequence {yj}j.jeNN

of

W*keN 6*

s u c h t h a t A:=

HZ^j) j-t+oo

c»M n») ^ well defined ({?{v,)}.^ '\y, jeN

is strongly convergent). Using the weak lower semicontinuity of the norm

2 Nir= M*)> M^)}

>i\yj || || and the relation ||yy||2 = \Vj,liyj\\,

wr

for every j e Nv/e get L2

II II 2 / \ ||^|| 2 << l iyjm \\x,f+oo" " ->+a>'VJW

(4.2) (4.2)

For every j e N we set, xXjJ == Pj(x.) PJ(x.) ■ Then, by the convexity of Kj, we have 111 1 _- -f f(\\-- 1) ll }/j 6eKj. Zj 1
j7 "

V

J; ;

Since, yj is a solution of the problem

ECP[f,KjJ (and this is true for every/ e N), we have ((zj z , --y^ltvyj~ .y,,yyy, --T\yjj)> r■nyj ( j ; ) ) >- i 0 from which we deduce [*j-yj,yj-nyjp ffor o r eevery/ v (xj,yj) (xj,7{yj)) every./ (xy, vy)1 >- (xj,l{yj)) (*v »7t>'j))for ery/ eee 7N.

M'JU and x , we obtain

Because {?[)>/)}

JeN

t

52

and {xj jeN

(4.3)

are norm convergent respectively to ^

EXISTENCE THEOREMS |JC,| >(x >(x \\xftf>(x.,A). t,A). t,A).

(4.4) (4.4)

|2

From (4.2) and (4.4) we have |x*| = (x*,A), that is IlyJ}

feN

is convergent

to ||JC*||. But, since the norm of a Hilbert space has the Kadec-Klee property, we obtain that \y\ is strongly convergent to x». Finally, because jzN

x* e K, we show thatX**) X*.) eK ee K A"*and and (*.,/(x.)> ^xt,f(xt)) ==00..0. Indeed, let z e K be x../"(x. an arbitrary element and denote Pfz) by z, for everyj e JV. We observe that Izj - y •} is strongly convergent to z - *♦, and we have ieN

fa - yj y,,f ,f(yj))\Vi > every; A , for every/ €T.JV. (zji■■~ypf(yjf) *>0,^ 0for every j e€ N.

(4.5) (4.5)

Taking the limit asj tends to +oo in (4.5) we obtain, (z-x (z-x*,f(x,)) > -xt,f(x,))>0, >(0, for t,t[x, each zZ e€ K, = K, which is equivalent to f(x*) (x.,f(x.)) = 00 (see / ( * . ) ee K* * and /x«,/(x»)) fix, =C Theorem 3.2). ■ Remark 4.2 It is evident that, if in Theorem 4.2 assumption (2) is replaced by the following assumption: 2b) condition (GK) is satisfied with a family [Dn }nsN such that (3rt > 0XV« Opn e N)(Vx eD e£>„)(||x| > r,) (3n r,) n)(jx\\ > then the problem ECP(f,K) has a solution JC, and x»5* 0. Theorem 4.2 is ap plicable to the model (3.9) which is used in the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions (Isac, G. and M. Thera [2]). Other results, similar to Theorem 4.2 are also proved in the papers (Isac, G. and M. Thera [2]) and (Goeleven, D., V. H. Nguyen and M. Thera [1]). An interesting generalization of Theorem 4.2 for arbitrary closed pointed cones (not necessarily Galerkin) is in (Zhou, Y. and Y. Huang [1]). We

53

CONES AND COMPLEMENTARITY PROBLEMS remark also a generalization of Theorem 4.2 for reflexive Banach space; proved in (Thera, M. [1]), with a proof which, modulo some details basec on duality mapping, is similar to the proof of Theorem 4.2. • Galerkin cones and conically coercive functions The concept of conically compact coercive function which we introduce now, can be used to study the explicit complementarity problem with respect to a Galerkin cone. The conically compact coercivity is based on the concept of conical bornology (Isac, G. [8], [17]). A bornology on a non-empty set X is a collection g" of subsets of X such that i) S covers X, i.e.,X= i.e.,X ( J 5 , i) SCOVQXSX, Beff

n

ii) if 5, ,B2 ,..,B„ e S(n e AO then B,. eff, ii) if5, ,B2,..,B„ e S(n e AOthen (J#, e^» =i iii) if B e g"and C c B then C € g". iii) if B e g"and C c B then C e f. Let E(T) be a locally convex space. For every x e Ewe denote: <2b\x) [BaCE|EIBBisisclosed closedbounded, bounded,convex convexand andx x&B> £B) & b(x)==\B (?c(x) ==[B #.(*) [5 cc £E || 5B isis convex convexcompact compact and and xx g€ B\. B\. £(0)] we set K0 (B) = \J AB. KB. If x * 0 and For every 5B € &(0) [resp. 5B e &(0)] Aao (x) [resp. [resp.5B ee &(*)] &(*)]we wedenote denote 5B e€ (? gbb(x) KK,xX(B) (B) B}. A> }. (5) == {Ab {^A ++ (1 - A)x | A A> >0,0,bb e€e55} Definition 4.4 We say that a subset A
54

EXISTENCE THEOREMS Every conically bounded set is conically compact. The converse is not true. In (Isac, G. [8]) is proven that in a locally convex space E a set A c E is (x)-conically bounded (resp., (x)-conically compact), if and only if it is (0)conically bounded (resp. (O)-conically compact). In consequence, we will say shortly "A is conically bounded" whenever A is (x)-conically bounded for some x e E, and similarly for conically compact sets. The family of conically bounded sets (or conically compact sets) forms a bornology on E. This interesting bornology was studied in (Isac, G. [8]), (Bahya, A. O. [1]) and applied in Functional Analysis (Bahya, A. O. [1]) and in Optimization (Isac, G. [17]). Let (*-\ (JS,|| I)IDbe a Banach space , KczE K c E a closed convex cone and/: f.K->R K -> R HI a function. The level set of order X (X (X eR) eR) of the function/is by definition LW(f){xeK\f{x)<x). (f) =: [xxeK € K\ f(x) ix(f) fix <X\. <X We recall that the function/is coercive over K (in the classical sense), if and only if, the level sets Lrff) of/for every X are /,:= inf f{x) f[x) < +oo. bounded sets. We suppose that -oo < /*:= JCSA XEK

Definition 4.6 We say that f is conically compact coercive, if and only if, for each X > f. the level set Lx(f) is a conically compact set. X>f. The next result was proved by G. Isac and M. Thera in [2] and it is a conically compact coercivity test.

NII)

Theorem 4.3 Let K be a convex cone in a Banach space \E,\ E. ||J and {fi\^Li>\Sj\ ._ two families of mappings from K into R. Suppose that the WZA*j}% i=V

>i

following assumptions are satisfied: 1) f is lower-semicontinuous and positively homogeneous of degree pt >0 for x}, for every i e{\,2,...,m e{\,2,-,m\), {\,2,...,m\}, 2) f{x) > Ofor each x e K\ {0} and each i e {\,2,...,m\}, 3) for each] e {l,2,...,m2}, there exists r, rj > > 0 such that

55

CONES AND COMPLEMENTARITY PROBLEMS *^ /M * < 00,, {L2,..,™,}} and lim \^J^-< rrjj < vaax[p max{/?,.t |11i ee {L2,..,/«,}} so, 0 0 Mh-* xeK xeK

lljcfl HP

II II

«2

_ isa « ^ then, for any at > 0 and $ > 0 the function hix) h[x) == Z «,.^(x) -YjPjgj Z A/#7 i=i

;=i 7=1 7=1

.=1

conically compact coercive function. Proof A proof is in (Isac, G. and M. Thera [2]). ■

The next result extends the classical Weierstrass variational principle to conically compact coercive functions. This result was proved in (Isac, G. and M. Thera [2]). Theorem 4.4 Let (E,\\ ||J be a Banach space and K a E a closed convex

w;, '"2

cone. Let {/a;, {/•}"!, and IgA _ be lower-semicontinuous mappings from K into R that satisfy assumptions (1) (2) and (3) of Theorem 4.3. Iff: K -» i? tf->J? is lower-semicontinuous such that f(x) > Z ^.«
;=i 7=1

x e K, where a, > 0 and p) > 0,for each i andj, then f 'achieves its minimum on every locally compact convex pointed subcone K0 c K. Moreover, the set of minimal points off is compact. Proof Let h be the function from K into R defined by A(*) =

I '"I

i=\

/=!

-z W*) tn->

hfi\X)

7=1

i=i

By Theorem 4.3 we have that h is conically compact coercive and since f(x) > h(x) for each x e K, we have t h a t / i s conically compact coercive too. Hence, for each locally compact convex subcone K0 of A" we have that x mf{h(x)\xeKQ} is finite, which implies that inf{f(x)\xeK {/(*)|xe* F "0) o0} is also

M

56

EXISTENCE THEOREMS

[-

M

finite. Moreover, for each £>0 £ > 0 the set L f(x) + e6 \ LE£ = \x eiKKQ0 fix) < inf fix) eA0 0 )J [ ' ' ixi=K is non-empty and compact, since it is a closed and bounded subset of KQ , which is locally compact. Therefore f ] Le is non-empty and compact and e>0 £>0

each element x0 of this intersection satisfies f{x = mmf(x). f(x). 0) /(*b)= xeK

■

0

We apply now the concept of conical compact coercivity to the study of the solvability of the explicit complementarity problem. Theorem 4.5 Let H(<,>) be a Hilbert space, K(K )ni_N a Galerkin cone V n »/n<=N inH,f:K^H H-> R two Junctions. If If, the the following following assumptions assumptions K^>H and O : H^R are satisfied: are satisfied: 1) fis a (ws)-field {wherefix) T(x)), (w/zereX^) == *x -- T(x)), 2) fis the gradient of®, 3) , +co, (x,f(x)) x Jx|-»+oo xsD xeD

"

then the problem ECP(fK) admits a solution x*e K, which is a cluster point of a sequence {x„} .N, where for each nn eN, solution of.ofECP(fK ECP(f,Knn). eN, x„ isIs aasolution * . .nineN' Proof By virtue of assumption (3), we have that O achieves its minimum on every cone K„. That is, for every nn eN x„ e K„ such that eN there exists x„ O(x„) = ®(x,) - min min0>(x). (;c). xeK„ xeKn

I I \ In particular, since (/(*„)>*-*«) In particular, since (fix\x„ ),x-x,x-x, n)x

every xx (f(xAx-x n)>0 0 (f(x „),x-x„)> i for for every n),x-x n)>

'

X X n) X ®(xxn+t(x-x ))-®(x n+t ®(XX-x n n( ~ n))~®{ +n -w\xnn) we get we get /lim— -*u+ t

/-->o+

t

which implies implies (by ee JST„ K„ which (by Theorem Theorem 3.2) 3.2) that that

57

CONES AND COMPLEMENTARITY PROBLEMS (/(*„ ),*,,) = 0. 0. This means that x„ solves ECP(f,Kn) an< /f{x„) ( * . ) eK* « < n and (/(*„),*„} hence we have .2 \\xn(={T{xn),xn). (4.6; (4.6; kf=(%,)>*»)■ By assumption (4), the sequence {*„}neN neJV is bounded and therefore admits i weakly convergent subsequence |JC„ | izN

whose limit x* belongs to K

Since/is (w.s)-field, we may find a subsequence \yt)te/V ( „ of \xn }

ietf

sucl

that A = lim T(xn)exists, and, by using the weak lower-semicontinuity o: l'-f-W0 I'-f-WO

the norm, (4.6) yields 2

W 2t< \\x f+00 l->+oo"

(4.7 (4.?:

Set xt = PK (x.). Then, since Kt is convex, we have that r =

> <

<-D»

+ # (?)*< ( ' - ?)»

belongs to Kt. Hence, since v, solves ECP(f,Ki), we have (/(y, ),z, - yS > 0 ;/(y,.),z,.-y,.)>0 from which we derive

>*,^ n ^ p

((y,,*,) y,*,) > ( % ) , * , . ) .

M^U

(4.8; (4.s;

ftL*

Since [T{yi)}ieN and {*,},l eA , norm converge respectively to A and*., w< etf ieAf obtain (from (4.8)), II 22, =(x,,x.)>(/i,x,). ||x.| \\x,f=(x„x.)>(A,x ( * , , * . ) > (i4,JC») t).

{ML,

(4.9; (4.9

N.

Finally, from (4.7) and (4.9), we have that {||y,|}ieA J-JV tends to |x.|, whil<

ji s w e a k l y convergent to x.; hence, since the norm of a Hilbert spac< {yi)ieN 'ieJV UL*

58

EXISTENCE THEOREMS has the Kadec-Klee property, we deduce that WheN {>;},ieN„ tends to x* with respect to the norm and by continuity off, we get lim/(>>,.) * ) «=//(*»). ( * . • Let ;->oo ;'->oo

'fr) = /(*) •

z e K be an arbitrary element. Then z = limf,., where f.Si^PM* = PK (z) and since i-»oo l'->00

'

v,- solves ECP(f,K,) we have ft-*./(*)) (£,. ->',,/(>',■))*o > 0 for eachieN. /' sN. Thus, taking the limit as i tends to +00 in the last inequality, we get (z-x.,f(x (z- x,,f(x,)) t))>0 > 0 x*,J[x,)>U for each z e K which implies (by Theorem 3.2) that x solves ECP(f,K). ■ Other results similar to Theorem 4.5 are proved in (Isac, G. and M. Thera [2]), (Isac, G. and M. Thera [1]). • Variational inequalities and explicit complementarity problems Considering Theorem 3.2, we remark that another way to obtain new existence theorems for explicit complementarity problems, is to find new existence theorems for variational inequalities, with respect to unbounded closed convex sets. Several authors have recently worked in this sense, and some interesting general results were obtained by J. S. Guo and J. C. Yao in [1]. We present now some of their results. Let (E,l I) be a Banach space K c E a closed non-empty set a n d / : K -»£* a mapping. We recall that the variational inequality associated with/and K is Ind xxtt eK find e K such such that that

(

Vl{f,K):

x-xtt,/(*.)) ,/(**)) ^ 00 for for each each x xeK eK )) > \ (x-x

We say that / is continuous on finite-dimensional subspaces, iff is conti nuous onK f)F for every finite-dimensional subspace F ofE. We consider on F the restriction of the topology of E. Theorem 4.6 [Ky Fan] Let X be a compact convex set in a topological vector space. Let A be a closed subset of X x X with the following properties: 1) (x, x) e A for every x e X, 59

CONES AND COMPLEMENTARITY PROBLEMS 2) for anyfixedy e X, the set Ix eeX\{x,y) X\ (x,y)<£A <£ A\ is convex (or empty). {xeX\(x,y) ZA} Then there exists a point y0 e Xsuch that XxlyAcA. X x jy 0 } c A. Proof The Proposition is Lemma 4 and is proved in (Fan, K. [1]). ■ The following result is a variant of the classical Hartman-Stampacchia's Theorem.

>.l It

Theorem 4.7 Let (E,\ |) be a Banach space KcEa

non-empty weakly

compact convex set and f : K -» E a mapping. If for each sequence [xn\ a-KKweakly convergent to an elementx we have neN

lim inf(y (y~x - xnn,f(x„))<(y-x,f{x)},yeK, , f(xn)) < (y - x,f(x)), y e K, then VI(fK) has a solution. «-+<»

Proof If v e K is an arbitrary element, we denote D(y) = lxeK\(y-x,f(x))>0} {xeK\(y-x,f(x))>0}. Since, for every v e K, we have y e D(y), we conclude that D(y) is non empty for every y e K . Let {xn}neN be an arbitrary sequence in D(y). neN Since K is weakly compact, by Eberlein's theorem, {x„}nsN neN subsequence J x% J

keN

has a

weakly convergent to an element x e K. We have

,fx))>0 (y --xnk x„ (y-x ni,f(x k ,f(x„ t)) > 0 for each k e N. nk))>OforeachkeN.

(4.10)

From the assumption and (4.10) we deduce that (y-x,f{x))>0 (y-x,f(x))>0. y-x,f(x)} Therefore, x e L\y) and hence L\y) is weakly countable compact. By Eberlein 's Theorem, L\y) is weakly compact and hence weakly closed for every y e K. We put S = {(x,y) e K x K\\(y-x,f(x))>0\ (y --x,f(x)>0 x,f(x)) > o}. Since, for each

60

EXISTENCE THEOREMS v e K, we have D{y) L\y) = = ixeK\(x,y)eS\ |x e{xeK\{x,y)eS} K\ (JC, V) GSJ we obtain that for eachy e tf, the \xeK\(x,y) set \x eK\(x,y) G^J iyyeK\(x,y)eS) <=K\(x,y) gs} is tzK\lx,y) e=S sS is weakly closed. The set {yeK\(x,y)£S} empty or convex for each x e K. Applying Theorem 4.6 we obtain an element xt e K such that {x*} {x.}xKczS, x K c S , i.e., (y-x„f(x (y-x )\> 0 for all -Xt,f(x \xKczS t,t[Xttt))>0 2 U 7 E JC and the proof is complete. ■ Theorem 4.8 Let K be a non-empty closed convex set in a Banach space [E\ I) andf: K K^E* -»£* an arbitrary mapping. Then x, e K is a solution of the problem VI(f,K), if and only if, there exists a set D a E such that int(D) is non-empty and the set K f]D satisfies the following properties.: 1) K f]D is bounded closed and convex, 2) xt eKC\'mt(D), x eEKHD K C\D. f)D. (x-x.,f(x (x-x„f(x )) allxsK X . , t/))>0 I X . I ) ^>0for t Proof Suppose x* e K is a solution of VUf,K). Let p > 0 be a positive real number such that |x,|Q t,f(x t))>0 = Ax Ax +(l-X)x, +(l- A)xt eKf)D. &K\\D. By assumption (3) we have (z-x -x(z-x J(x, >U t

which implies A-(x-x A(x-x-x„t !x-xt,f(x*)}>0 t,f(x t,f(x*)}>Q t))>0 t,f(xt))>0 ^ u for all (x. >U and hence (x-x x e K, that is, xt is a solution of VI(fK). ■ Theorem 4.9 Let (E,\\ : * :.. l II)||j be a reflexive Banach space, K c:E a non-empty

closed convex set andf: K—> K-+E E i a mapping. Suppose that there exists a set D a E such that int(Z)) is non-empty and the following assumptions are satisfied: 1) Kf]D is non-empty bounded, closed and convex,

61

CONES AND COMPLEMENTARITY PROBLEMS

kL*<=*n£

2) for each sequence {x„}neN a KC\D weakly convergent to an elementx, neN (y-x„,f{x„))<(y-x,f(x)),yeK, we have liminf(j> - x:n,f(x . , / n*))- ) <{ys y - xx,f(x)), , / x , yv ee K, i, n-*0, (x-u,f{x))>0, C >U,

such that

then VI(f,K) has a solution. Proof Let L\K) f]D. Since Theorem 4.7 is applicable for/and L\K), EKK)==KKC\D. there exists JC. G D(K) such that (x-x,,f(x.)) (x-x.,f(x.)) > >0for 0 for allxVtorallx

(4.11)

If x. e KC\int(D) we have that x» is a solution of VI(fK) by Theorem 4.8. Suppose now that x. eKf)dD Then, by hypothesis there exists eKHdD ( X ,-- Kw,/(x„)) , / ( X . ) ) > 00 . From (4.11) and the last ine u eKf)int(D) such that (x, -u,j u , > ^u quality we have (x [xt♦t -- ww,/(x,)) u,f(x.)) K be an arbitrary element. ,/lx.l) = = U0 . Let xx ee K There exists X e]0,l[ such that Ax + X)u e L\K) (since + (l - (l-A)ueD(K) UGK flint(D)). We have 0 < (A(x-u) + u-x.,f(x.))x,)) = Xix 0< A.(x-u,f(x.)) A(x- u,f(xt)) = A(x-xt,f(xt)), = MX-X..

+(w-x.,/(x.)) +(«-x.,f(x.))

nx.

which implies that x. is a solution of VI(f,K). ■ Theorem 4.8 and 4.9 have interesting applications relating to the solvability of the explicit complementarity problem.

N I)

Theorem 4.10 Let (E,\ fl) be a Banach space, KczE KczE a closed convex cone and ff:K^E : K -> E an arbitrary mapping Then x. e KK is a solution oj X. G —* ECP(f,K) if and only if there exists a set D c E such that int(£>) is nonDczE empty and the set Kf]D satisfies the following properties: rni 1) KCiD K H D is bounded closed and convex, 62

EXISTENCE THEOREMS eKClint(D), 2) xx,t eJSTI eKf\int(D), 3) (x-xt,f(xt))>0forallxe ))>Oforallxe Kf)D. tor all x € Kf]D Proof The theorem is an immediate consequence of Theorem 4.8. ■

NII)

Theorem 4.11 Let \E,\ <, J) be a reflexive Banach space, K E a completely continuous operator. Suppose f-.K^-E that there exists asetDczE such that int(D) is non-empty and the following assumptions are satisfied: 1) K C\D is non-empty bounded closed and convex, 2) for each x e K fl dD there exists ueKf] u e K f] mt(D) int(Z)) such that (x-u,f(x)) >0, (x-u,f{x)) ./ * then the problem ECP(f,K) has a solution. Proof Since the completely continuity of / implies assumption (2) of Theorem 4.9, the theorem follows from Theorem 4.9. ■ Corollary 4.12 Let (E,\ |] be a reflexive Banach space, K a E a closed convex cone and f:C:K-+E K —> E a completely continuous operator. Suppose that there exists r > 0 such that for each x e K with \x\ = rr, there exists be = ueK

such that \\u\\ x-u,f(x))>0. > 0. Then the problem ECP(f,K) \\u\\ < < rr and {x-u,f{x)^ has a solution. ■ For the next corollary we need to introduce some definitions. Let \E,\ I) be a Banach space, KczE a closed convex cone and f:K^E' f:K—>E* a mapping. We say that/is : i) weakly coercive if lim \x,f(xj) +<x>; x,f(x)) = +co; ||x|->+oo xsX

(*./W == +°o +oo;; ii) coercive if lim -—^-r.—W-MMl xeK M xeK

63

CONES AND COMPLEMENTARITY PROBLEMS Hi) a-copositive if there exists an increasing function a : [0,+<»[ —»[0,+oo[ with a (0) = 0 and lim air) = +co such that (x,f(x) - /(0)) >M4#I) ||x||a(||x|) (*,/(*)-/(<>)) r—>-KO r-»+co

for all x e K. If cif) = kr for some k > 0, then/is said to be strongly copositive. Corollary 4.13 Let \E,\ ||l be a reflexive Banach space, K a E a closed convex cone andf: K—>Ea completely continuous operator. If one of the following conditions is satisfied: 1) lim lim inf/x,fix)) infix,f(x))>0, (*,/(*)) >0, w->-**> xeK

2) fis weakly coercive, 3) fis coercive, 4) fis a -copositive, 5) fis strongly copositive, then the problem ECP(fK) has a solution. Proof If, condition (1) is satisfied, then there exists an r > 0 such that > (x,f(x)\ > 0 for all x e K with \\x\\ \\x\\ >irr . In this case the corollary follows x 2 T\x))*v ;*./(*)> *o from Corollary 4.12. Since, about conditions (1) - (5) we have the implications (5) => (4) =>(3) =>(2) =>(1), the corollary follows. ■ Other interesting results about the solvability of the problem ECP(f,K) can be obtained supposing that the mapping/ is a pseudo-monotone operator. In this sense R. W. Cottle and I. C. Yao [1] worked recently. We present some of their results. Let (H, <, > ) be a Hilbert space, K c Ha non-empty subset a n df-.K^H. / : K^>> K-> H. We say that f is pseudo-monotone, if and only if, for any x and y in K, we have (xx-y,f(y))>0 - y, > 0 implies (x --y>j\*) y,f(x)) *> 0. This concept was defined (x-y,f(x)}>0 v. f(y)) nv 2 and studied by S. Karamardian in [2] (see also (Karamardian, S. and S. Schaible [1]).

64

EXISTENCE THEOREMS We say that / is continuous on finite-dimensional subspaces if it is continuous on KC\F for every finite-dimensional subspace F of H with A"f]F s* ■ It is well known that a monotone mapping (in the sense of Browder and Minty) is pseudo-monotone, but not conversely. Pseudo-monotone maps are also studied in (Karamardian, S. and S. Schaible [1]). Proposition 4.14 Let (H, <, > ) be a Hilbert space, KczH a closed convex subset. Iff: K —» H is a pseudo-monotone mapping continuous on finitedimensional subspaces, then x* e K is a solution of the variational inequa lity (u --x,f{x))>0,forallu&K x,f(x)) > 0,for allueK (u-x,f(x))>0,foraUueK (4.11) (4.H) if and only if (u-x [u-xtt,f(u))>0,forallueK , f{u)) >0,forallueK

(4.12)

Proof Suppose that*, satisfies (4.12). Consider an arbitrary element u e K A, << 11put putx^ xi ==AXu (1-X\x* . We have x% e K and from and for every 0 < X w ++ (1-A)x* ))>0. > 0. Computing the limit as X \->0 (4.12) we deduce A(u-x„f(x A(u-xt,f(xxx)) —» 0 and using the continuity of / on finite-dimensional subspaces we obtain (u-x Therefore x* is a solution of (4.11). Conversely, ' « - Xt,f{x . , / (t^>0. J C . ) ) >> 0 . supposing that x* e K is a solution of (4.11) and using the fact that / i s ,f(u)}>0K « for all u e K, that is, x« is a pseudomonotone we have (u-xt,f(ufj>0 solution of (4.12). ■ Theorem 4.15 [Cottle-Yao ] Let (H, <, > ) be a Hilbert space andK a H a closed bounded and convex subset. Iff :K -> H is a pseudo-monotone mapping continuous on finite-dimensional subspaces, then there exists xt e K such that {x-x,, [x-x.,f(xf(x 0 for all x e K. t))>0 t)) > >U Proof Let "P be the family of all finite-dimensional subspaces F of H with Kf]F * 0. For every F e 7 let PF be the orthogonal projection of Honto F. KC\F*. Let ffFF == PPFF°f°f be the composition of Pp and / By Hartman-Stampac65

CONES AND COMPLEMENTARITY PROBLEMS chia's theorem [Hartman, G. J. and G. Stampacchia [1], [Lemma 3.1]] or [(Isac, G. [12]), Theorem 3.1], there exists for every F e ? a n element xF eKf]F such that )}>0, ueKC\F. (u-xFFx,f(x ueKftF. F))>0, F F,f(x F))>0, u, for all ueKftF.

(4.13)

We denote by KF = |x G \ F a G e "p\ for every F e ?. For F e ?, we denote by KF the weak closure of KF . The family JJT/| F e ? } has the finite intersection property. Indeed, for F, G e "P, let M e "? be such that FU G a M. Then <0 * KM a KF f] KG. Since, H is reflexive and K is closed and bounded, we have that K is weakly compact. Because KF c K for all F e ?, it follows that f]KF * tf>. Let x. e H ^ F - Consider an FFe? e?

Fe?

arbitrary element u e K, and F e ? such that u s F. Since K° if£F is weakly compact and x. x. e e f]K which K nneN„ CczJTp n *F,; , there exists a sequence {*„} iMnetf F Fe? Fe?

converges weakly to x. (We used also Eberlein's theorem ). By Proposition 4.14 and formula (4.13), we have (u\u-xnx,f(u))> Q, 0, for forallaWnzN. n e N. n,f(u))>

(4.14) ;4.i4)

The function («-*,/(«)) (u - *x,f(u)} , / (u is weakly continuous in x . Computing the limit in (4.14) as n -> +oo we have, (u-x.,f(u)) (u - x,,f(u)} > 0, 0 for all u e If and applying again Proposition 4.14 we have («-x„/(x,))>0, (u-x,,f(x,Jt))>0, for X, ) ^ t all u e K and the theorem is proved. ■ The importance of the next result is the fact that it gives some necessary and sufficient conditions for the existence of solutions to the variational inequality problem for unbounded sets, when the mapping / is pseudomonotone. 66

EXISTENCE THEOREMS Theorem 4.16 [Cottle-Yao ] Let (H, <, > ) be a Hilbert space and K a closed convex subset in H and f:f: K -> —> HH a pseudo-monotone mapping which is continuous on finite-dimensional subspaces. Then the following statements are equivalent: 1) there exists x* e K such that (x - x*,f(x.)j x, ,/(*.)) > 0, for all x e K,

(4.15)

2) there exist u e K and a constant r> r > \\u\\ \\u\\ such that(x-u,f(x))>0 (x - u,f{x)} > 0, for all x € Kwith \\x\\ IWI ==r,r, 3) there exists r > 0 such that the set Ix ejK"||jc| 0,

\\x\\ = r, there exists u e Kwith \\u\\ < r

4) there exists a closed, convex set E in H such that int(is) i±ty, Kf)E is nonempty and bounded and, for each x e K f] d(E), there exists usKf] int(£) such that (x - u,f(x)) > 0, 5) there exists a non-empty closed, bounded convex subset B of K with \X&K(B) * <> | such that the following condition is satisfied: for each x e dK{B), there exists u e int^i?) such that (x - u,f{xf> > 0. Proof (1) implies (2). Indeed, let*. eJTbea solution of (4.15). Choosing r > 0 such that |x*| < r and considering u = x*, we remark that (2) follows from the pseudo-monotonicity of the mapping / (2) implies (3). This implication follows immediately. &H\\x\
67

CONES AND COMPLEMENTARITY PROBLEMS dK(S) = B\mt dJS) B\mtKKBciB\(Kf]'mtE)cz(E\intE)r\K BciB\(Kf]'mtE)cz(E\intE)r\K = KC\^{E)KC\^{E) We con YmtE)cz(E\mtE )HK == ueKC\int{E)czmt clude that, by (4) there exists u eKC\'mt(E) a mtKK(B) (B) such that (x - u,f(x)) > 0, that is (4) implies (5). (x-u,f(x))>0, (5) implies (1). By Theorem 4.15 there exists xt e B such that ( x - xx.. ,,/(x.)) (x / ( x . ) ) > 00,, ffor o r aall l l xx ee£B. .

(4.16)

If x e K is arbitrary, we have two possibilities: i) xt eint^(5). In this case, there exists A e]0,l[ such that A)x, € B. Then by (4.16), we have l/ (l (xx- -xx. », /,(/ x( x. ), ) )>>00. . Thus, Ax + {\(\-A)x, 0. ( x - xx..,,//((xx..))) >>0. ii) x. eded K(B). K(B). By the condition in (5), there exists u eint^(5) such that Therefore by (4.16), we have (x. (x,-w,/(x.)) -w,/(x,)) .-u,f x. = = (0. Consider A e ]0,1 [ such that Ax + (1 - A)u e B. By (4.16), we have 0 < (A(x u - x.,/(x,)) x.,/(x.)) x. ,/(x,)) -= A(xA(x - u,f\x u,/(x»)) u,/(x»)) -= /l(x A(x-x»,/(x.)) - x,,/(x.)). (x - u) + uA(x-x»,/(x. t)j Hence, (^xx --x..f xx,„ ,/(x,)) / ( xX. . ) ) >>>00 for all x e A" and the theorem is proved. ■ ((xxt-u,f(x.))>0. .-w,/(x.))>0.

As an immediate consequence we have the following result for explicit complementarity problems. Theorem 4.17 Let (H, < >) be a Hilbert space, K c H a closed convex cone and f: K —> H a pseudo-monotone mapping which is continuous on finite-dimensional subspaces. Then the following statements are equivalent: 1) the problem ECP(f,K) has a solution, 2) there exist u e K and a constant r > \\u\\ (x - u,f(x)\ > 0 for \\u\\ such that {x-u,f(x))>0 all x € Kwith M=r, ||x|| = r, 3) there exists r > 0 such that, for each x e K with \\x\\ Ml ==r,r, there exists \u\\ << ? r and(x-u,f(x))>0, u eK with \\u\\ ( x - w,/(x)\ > 0,

68

EXISTENCE THEOREMS 4) there exists a closed convex setEcH such that int(E) * (j), Kf]E is nonempty and bounded and for each xxeKfl^E), eKf]o[E), there exists u e Kfixnt(E) such that ((x* - «u,f(x)) ,/(*)) > >00, ueKr\'mt(E) 5) there exists a non-empty closed, bounded, convex set B c:K with intjy(5) BdK non-empty and satisfying the following condition: for each x e dg(B), dniB} there exists u e intx(2?) such that (x - u,f(x)) > 0. ■ 'x-u,f(x))>0. The next result, which is a particular case of Theorem 3.1 proved in (Guo, J. Sh. and J. Ch. Yao [1]) can be considered as a variant of Karamardian's Theorem [Theorem 4.1] for pseudo-monotone mappings which are continuous on finite-dimensional subspaces. Theorem 4.18 Let \E,\ ||j be a reflexive Banach space andK a E a closed convex cone. Let Letf: f :K — ->» EE be a pseudo-monotone operator which is continuous on finite-dimensional subspaces. If there exists a bounded subset D ofK such that for each x e K\D there is u e D with (x-u,f(x))>0, (x-u,f(x))> )> 0, then, the problem ECP(fK) has a solution. ■ Some other existence results for variational inequalities with respect to unbounded sets can be used to obtain new existence theorems for explicit complementarity problems. In this sense we cite the papers (Guo, J. Sh. and J. Ch. Yao [1]), (Yao, J. C. [1]) and (Hadjisawas, N., D. Kravvaritis, G. Pantelidis and I. Polyrakis [1]). Results on complementarity problems, based on variational inequalities are also proved in the papers (Noor, M. A. [1] [7]), (Noor, M. A. and Th. M. Rassias [1]) and (Noor, M. A., K. I. Noor and Th. M. Rassias [1]). • Isotone projection cones and complementarity problems The con cept of isotone projection cone was introduced in 1985 by G. Isac, with the aim to use the geometry of cones in Hilbert spaces, in the study of comple mentarity problems. This interesting class of cones was studied by G. Isac and A. B. Nemeth [1] - [5] and recently by S. J. Bernau [1]. Using isotone

69

CONES AND COMPLEMENTARITY PROBLEMS projection cones we will study the complementarity problem by some special iterative methods. Let (H,<,>) (#,<>>) be a Hilbert space, K a H a closed convex cone and f, g : K —» H two arbitrary mappings. We consider in this section the following complementarity problems: ECP(f,K)

((

find ee K K such such that that find xx0n G )sK'and(x f(x0o)) f(x00)eK'and(x )eK'and(x0> ,f{x )) 00,f(x 0)) ( f(x

W)

and

=0

find Ind x0 eeKK such that r(x0Q)eK,f(x )eK,f(xQ0))eK'and ICP{f,g,K): ICP(f,g,K):< g(x eK'and eK •
If x is an arbitrary element in / / w e denote by Pg(x) the projection ofx onto K, i. e., J|JC - /i(x)| /ff(x)|| = min|x mm|x - y\\.

•.|*-M*)l

y<=K"

Proposition 4.19 The element x.eK x* e K is a solution ofECP(f,K) if and only ifx, is a fixed point for the operator F(x) =Pn[x-fix)] in K. r{x)=P K[x-fix)]inK. Proof Indeed, if x. e K is a solution of ECP(fK), then we can show that xt satisfies properties (1) and (2) of Theorem 2.22 which implies that x xx.=Pn[x.-fix,)]=F(x,). = F(x.). Conversely, if x, eG K and x. * ~AX*)]> x, == Pg[ t =PK[X. -fix,)] Pdx.^{x,)], then since Pg[x.-fix.)] Pn[x.-fix.)_ satisfies properties (1), (2) of Theorem 2.22, we deduce that x, is a solution of ECP(fK). ■ Proposition 4.20 The problem ECP(f,K) has a solution, if and only if, the mapping *>(*) <$(x)===PPK(x)-f{P«(x)), - f(PKK(x)), defined for every x e H, has a fixed PK(X-AP (X) K(x)

70

EXISTENCE THEOREMS point in H. Moreover, if x0 is a fixed point of O, then x* Xt =Pg(xo) =PK(XO) is a solution ofECP(f,K). Proof Suppose that JC0 is a fixed point for the mapping <S>, that is x = P X P x PK(XQ) we have Xt K and o = K{ xt €e K *b WO)o )~-f\/ (K{Wo)) b ) )■ If we denote xx*t =— Pg(xo) x0 = x* xt- - f(xt), = j{Xt f(xlx»), or x, x, -x - 0x0 = f{xfix,) By Theorem 2.22 we obtain t), t). or (/(*♦), v) > 0, for every y e K, that is /(*«) f(xt)eK\e K*. (f{xt),y)>0, Using again Theorem 2.22 we deduce (f(Xt),Xt) (/(jt '[Xt),Xt) (. and hence x. is a t ),jt,) = 0 solution of ECP(f,K). Conversely, suppose that x* Xt e€ K K is a solution of ECP(f,K). We denote JC =x» - / ( * » ) and from Theorem 2.23 we deduce x00=Xt-f(xt) = Xt~f(Xt that Pg(x = xtx,t, and thus PK(XO) 0) — P ®(xo) K{X0)-f(PAX0))=X*-f{x*) = 0, ®(*b) = == PK(xo)-f(PK{xo))=X*-f{x*) ^r(«6)-/(^(«b))=**-/(x.) = X*b.

that is, x0 is a fixed point of O. ■ We present now a. projection iterative methods for a sum of two operators. If A is a non-empty subset of H, we denote by a(A) the measure of noncompactness oiA (Sadovski, B. N. [1]), i.e. a(A) = inf{r > 0 | A can be covered by a finite family of subsets of H whose diameters < r}. Let D be a subset of H and f:D-t H a continuous mapping. fFe say /^af / is condensing (see (Sadovski, B. N. [1])) if, for every non-compact bounded setAczD we have a[f[A^ < a(A). A)
(*.\ ID

Theorem 4.21 [Browder] Let (E,\ ||j be a uniformly convex Banach space and let D a E be a closed bounded convex subset. IflfT:D-+D T:D -> D is a nonexpansive mapping, then Thas a fixed point. Proof The original proof is in (Browder, F. E. [1]). ■

71

CONES AND COMPLEMENTARITY PROBLEMS Theorem 4.22 [Sadovski] ^t Let (E,\\ (E,\ |) be aBanach space andlet Dei Ebe E, I) mapping, a closed bounded convex subset. IfT.D^D lfT.D^Disa D is a condensing condensing mapping, then Thas a fixed point. Proof The original proof is in [127]. ■ Theorem 4.23 Let (H,<,>)be a Hilbert space ordered by an isotone (//,<,>) projection cone K ez H. Let f : K —> H be an arbitrary mapping. We suppose that f has a decomposition of the form f(x) = f(x) + f2(x) + d, where f is monotone decreasing, f2 is monotone increasing and d e H. Let O : H—> Hbe the operator defined by {x)-f{P -fK2{x)) {PK{x))-d.
r) = W-/> W - A W -«

Given xWo < Vo, yo, consider the sequences i{x„} {^n}nzN X eH with x*o 0,yQ eH 0 ^ ni_N, UMneAf neAr n)neN' ieN defined by X *„ = PAxA- PK{x PK{x (xHn))-ffAPAvA)-d. {PKK{y (yHn))-d, +i = n)-f{P fAPAxJ]n)-f x(PKK{x 2(P n+\ P p P p Xx x n))-d PK{y )-fl( K{K( yn+\ == yj PK{y V.K{yn))-fA -K{yn))-fA U\fAx. n/V n))-f2(PK( n)-A(PK(y
If the following assumptions are satisfied: i) x00<x <xi andyr is is non-expansive, (a) non-expansive, (b) <X> is is condensing, (b) condensing, is continuous continuous and and dim dim /7<+oo, H/ <+oo, (c) <& is then there exists x such that: e N, 1) xn<x< y„,for every n neN, 2) xtt =PK{X) =Pg(x) is a solution ofECP(f,K), P x V \\x'*~~Pg * nJ||p^nhn ~~xxn»II--for every K(nfi\^\\y (* hnn\\,jor forevery everyn eneN neN i\ -x ,

72

EXISTENCE THEOREMS 04) if lim Ik, -JC - xfl fl= 0 tfzera then lim lim PiV(x„) =x*. x*. K(xn) = v n-»oo n-»oo" n-»°o"

«—>oo n->oo n_>oo

'

Proof By Proposition 4.20 we obtain the existence of the element x if we show that <1> has a fixed point which satisfies (1). First, we prove the following relation: (4.17) (4.17)

or *n ** *■+! *„+i ** J'-n V*n ^* >»» y„,ffor allnn eeJV. N. *„ aU

Indeed, by assumption (i) and since K is isotone projection we have for «=0 < *}1 <"n+2 ^>"n+l n ++1, ^nl+ )i ))) -- ^d 2 == ^ rPK{Xn l ()* -„ /+ i.+)l)-fl{PK{Xn*l))~fz{PK(y„+l))-d W *<„ +2 n++2 ^ ^ ( (^M.i)-/ l ) - / 1 (^(y» . ^ ^ . +)i))-/2W*. ) (pA - ^ x^n ^ i))-d , ) ) - ^ s^(y ^PK{y n+i)-fi{PAyn+i))-f 2 + + i))-rf = = ^^nny++ 22i-n+

Hence, we have

x0 <x, <....< xn <....< yn <..
We show now that f for o r e v eerr M[xxn>y ®{[ n>yn] yy "" ee ^ ^ n]0)0) E K ^ „ ] 0 ' '

Indeed, let x e [^„,7 n ] 0 be an arbitrary element. increasing and/i is decreasing we have

Since f2 and PK are

73

CONES AND COMPLEMENTARITY PROBLEMS *„++,, = = 'K\» iPK{Xn)-fl(PK{*n))-A(P*{y«))-d *„ W -/,(',iW)-/aM^ PMM-fi[rMM)-MW.)Y ^ fPA* W - /-MW « W * ) ) --M*/ i ( ' iKW «' \x ) --a <

Pr

v. PAy»)-fi(PAyn))-f2(PK(x«))-d -

=yy„ = n++ui> C that is, <&([* ^{[x„,y ]0n)^[x ynn+1+i],v0^[B+x4c[x K+i.JWiJoRJWu. Bn,y n ] 0c_)c[x B ,v fl ] 0 , for every n e N. Since Kis n'-^n n+v n^yn]o regular (being isotone projection) there exist w =n—><*) lim V x„, Vv ==n~>oo lim v„ and we have M < v. Moreover, we can show that

O([ ,v]o0 )c[ ,v]o0 < KMM ) =MM

4 18 <(4.18) - )

Indeed, for every n e N and x e [u, v] we have, Indeed, for every n e N and x e [w, v] we have, PAx. - TAFAx.)]-

*~.=^-/fey-Ay-''

<
< ^ v
and taking limits we deduce that u <
Since K is normal and

eH\v

The set [w,v]0 = ix ["> Jo u< x < v\ is convex. Since K is normal and closed, we have that [«,v]0 is closed and bounded. From assumption (ii) and using Browder's Theorem or Sadovski's Theorem or the classical Brouwer's Theorem, we obtain that O has a fixed point in fn,v]0and conclusion (1) is proved. Conclusion (2) is a consequence of Proposition 4.20. To prove conclusion (3) we use the fact that PK is isotone and we />,(*„)£*. :X, !*/»,(?„) i have [using also (1) and (2)] P K(xn) < x, < PK(y„) which implies 0< x, PKn)

'*-WPW.-W

(4.19)

Since K is normal with the property that the normality constant is P = 1 we 74

EXISTENCE THEOREMS deduce from (4.19) that \\x,-P (x ]\<\\y -x„\\, for K(xn)\<\\P K(y n)-P l h - PKMI *\\p K{y n)-K nPE(xmi n ^ I k - * J every n E N. Now, from assumption (3) and since PK is continuous, we have (4) and the theorem is proved. ■ One can use a similar method for the Implicit Complementarity Problem. Theorem 4.24 Let (//,<,>) be a Hilbert space ordered by an isotone projection cone K a H. Suppose, given two continuous monotone decreasing mappings f,h : K —> H. Given x0, yo e K with XQ < yo , consider thesequences {xn}neN >{yn} neN defined by: neN neN' *»+l = = PPK[ K[XX*n ~ KXxn)- /(*»)] h{yn)> *»+i f{xn)\ + hH? \)>nh n\ r ^ - -^ % v j ) - .-/ /{ynW )\ +] /z^„ ■yn+ yn+i i ==^ Ps[y» +%) for every n - 0,1,2 If the following assumptions are satisfied : i) xa<xxandy\ <>-0 ii) if dimH = +oo, the mapping 0{x) <3>(x) ==h(x) h(x)++ PJx-hlx)-f PK\xPK-[x-h{x)-f{x)] h[x) - /(*)] hlx) X) is nonexpansive or condensing, then the problem ICP(f,I-h,K) has a solution x* E K such that xn<x,oo n-»oo

n->oo n->oo

Proof The proof is similar to that of Theorem 4.23, however, we are using instead the mapping O(JC) 0>(JC) -= *(x) h(x) -PKh[x) - f(x)], f(x)] defined for every Hx) ++ PPKK[x[x-h(x)\x-Hx)-flx) xeK.K. xeK. n ny« for Since x0,y0 E E K, £ then the fixed point JC. of O satisfying xnn<x <xt
Definition 4.7 Given aeR

such that 0 < a < 1 and Tu T2 : H-> H two 75

CONES AND COMPLEMENTARITY PROBLEMS mappings, we say: 1) T\ is (a) -concave if for every x e H and every X such that 0 < X < 1 we have A rTaT x(x) < T^Xx), x(x) 2) T2 is (-a) -convex if for every x e H and every X such that 0 < X < 1 we aa have T2(Xx) < \XX 7 T2(x) Remark 4.3 (1) Notions similar to (a)-concave and (-ar)-convex operators were studied by M. A. Krasnoselskii [1] and A. J. B. Potter [1]. (2) We can show that T\ is («) (a) -concave (or T2 is {-a) (-a) -convex) if and only aa a if T 7j(//jc) p~ T2(x) for every x e Handp> 1. l(px)

) be a Hilbert space and andKc KcH an isotone projection cone. Suppose that the mapping ®(x) =
every x e H,

associated with the complementarity problem ECP(fK) has a decomposi tion of the form ^(x) = Tl(x) + T2(x), where T\ is increasing and (a)concave, and T2 is decreasing and (-a)-convex {T\ and T2 are not necessarily continuous). Given u0 e K and po > 1 such that l u T Tu u + + a a Mo~ ■ o *> ^ \{\{o)o) T2T(u
MneN

de nedb

fi

rx0 -= pp00 uu00 u

yyo = y"o"o Mo o 0 =

•Xn = TT1{x + T22(y (ynn__i)i) 1{x n_ n_])])+T = TTll{y (ynn__ii)+T )+T22(x (xnn_,) yyn n = for all n eJV. <EN. Then the following statements are true: Then the following statements are true: 76

76

EXISTENCE THEOREMS 1. the sequences {xnH}}n&N ,{ynn}neN neN neN are convergent, neN nzN'lSnlneN 2. lim xn = lim yn, n—>oo n—><x>

n—>co n—>co

3. the element xt = lim xn = lim yn is a solution i ofECPi ofECP(f,K), n—>oo

lx„ 4. \x | j nc „--x*j< - xx*| * J<< /jU / 0Q 1

1

n—>oo*

• ||UQ|| for every every nn eeJ VN. Ug ^br .

//oa".

Proof We will prove this theorem in several steps. e NM have [x neNwe [x„,y ]00 cn-i,y [*„_„ (a) For every every>„_,]„ n,ynn\^[x n-\ ;/,,_,]„ Indeed, xxo^yo' 0 <>"o and, from assumptions, we have *i = Ti(xo)

+ T

2{yo) = ^(-"o^o) + T2(ju0u0),

Tl(MoU0)>jUoaTl(uQ),

ftll

T2{HQU0)>

a Mo ^ T2{u0),

which imply x

x \ * *Mo"[T > >"o / / o Ca[[K K _ 1'"o] " o ] = /V Mo\M o = = *o» o> ^ " [ ^ ( l"{u o )0 )+ - 1 - rT2("o)] 2(U0)] 2:

yi Tx{yQ)) ++ 7 T 7i = = 71(^0 22^(x0Q)•).

Also from assumptions,

?i(«)^K("o). ^(/"o^oj^K^^o)T2(/J.Q\)<^T2(U0).

Thus we have tt = ^ o ) + T2(x0) < K[3i(«o) + r 2 («o)] 5 K [ / 4 ~ % ] = /"o"o = ^o ■ Vx = TM + T2{x0) < Mol^uo) + T2(u0)] < K [ / 4 ~ % ] = Mo"o = ^o •

Now we prove that xx < y>\. We know that Now we prove that xx
\ = Zi(* 0 )+ ^(j'o) = ^ ( / ^ " o ) + ^ ( / W o ) *i = Ti{xo)+T2{y0) = T$Mo$ + TI{MOUO)

77

CONES AND COMPLEMENTARITY PROBLEMS y\ = Ti(yo) + T2(xo)=Ti(Mo"o) But since JU^UQ*
an

+T

2(Moiuo)-

^ because Tx is increasing and T2 decreasing we

lw s T (juM u ), T XU0\)o) ^ 7iO"o x Qo)> 0 ^(y"o T

2{MOUO)^T2[MOUO),

whence xxl *
(4.20)

From (4.20) and by induction since Tx is increasing and T2 decreasing we obtain *„_,x < x„_ <xx„ < nyoo

Indeed, we have

n—>a>

xnn < xtt < y, < ynn, for all n EN

(4.22)

which implies *„+1 = Tx{xn) + T2(yR) < Tx(x.)+T2(yt)< Tx(y„) (y„)++ TT22(x (xnn))== yyn+x n+x and computing the limit we get xn<xt<

Tx(x (x,) +T
Now we prove that for every n e N, we have 78

(4.23) (4.23)

EXISTENCE THEOREMS Mo2a"y„^x„.

(4.24)

We prove this relation by induction. Indeed, for n = 0, we have 2 u x Mo2yyo =Mo MO2(Mo (MOUO) o) = -"
If (4.24) is true for n, we have

Mo2a"+l■y-yn+l = ^ +i(y, [nhT 7 i 2((^Xn))]+ r 2 ( , „ ) ]==Mo ^2a""++i1 T7l;(y( ny)n+) + ^Mo^T " + 1 r22(x„) (,„) A** n+l=Mry <7j(/.o 2 ^ B ) + ^

+ 1

- T ^ y ^ T ^

+ ^ . T M

^T1(x„)+T (xn)+T22(y„) (y„) = x„ xn+V +V Hence (4.24) is true for every n e N. From (4.24) and (4.23) we get 2 0, 0 < y. -t
(4.25) (4-25)

and passing to the limit we obtain that><» that^* = x*. So, we have that x* is a fixed point for the mapping O and from Proposition 4.20 we obtain that x* is a solution of ECP(f,K) because x* e K. To show conclusion 4) we observe xn <
11 In — p|«00||,, and

2a"

/i2

)

the theorem is proved. ■ The importance of Mann's iterations in the fixed point theory is well known (Mann, W. R. [1]), (Ishikawa, S. [1]), (Desbines, J. [1]). This iterative method for non-expansive mappings has the property that it is convergent to a fixed point when the classical successive approximation method is not convergent. The following result will be cited.

79

CONES AND COMPLEMENTARITY PROBLEMS Theorem 4.26 [Ishikawa] Let D be a closed subset of a Banach space \E,\ I) and let Tbe a non-expansive mapping from D into a compact subset ofE. If there exist X\ and {f„}™=1 such that 1) 00
2) fX 2) ix

°

n=\

3) xn eG D, for jor all n <=N, where wherex^\= tnl{x„), x^-i= (l-t„)x„+ t„T(x„), then {xn}neN converges to a fixed point x, e D ofT. Let K a H be a closed convex cone and h : K —» H a mapping. We recall that h is said to be compact if every bounded subset of K is applied by A in a relatively compact set. We have also the following iterative methods to solve the problem ECP(fK). Theorem 4.27 Let (H,<,>) be a Hilbert space, K a H an isotone projection cone and h : K —> » H a mapping. If the following assumptions are satisfied: 1) h is non-expansive and compact, andh(x) h(x)<
n G N where 0 < t„ < a < I, ^tn

= co and xx is an arbitrary element such

n=\

that 0 < x\ < Pg(b), is convergent to a solution ofECP(I-h, K). Proof From Proposition 4.19 we deduce that the problem ECP(I-h, K) has a solution if the mapping T(x) = Ps(h(x)) has a fixed point in K. If we consider the set D - [x G JST| 0 < x < PK{b)\, the properties of PK and the assumption that A" is an isotone projection cone, then we observe that all

80

EXISTENCE THEOREMS assumptions of Theorem 4.26 are satisfied for T and D. Hence {xn}neN

is

convergent to a solution of the problem ECP(I-h, K). ■ Remarks 4.4 1) Other authors have used iterative-projection methods to solve the problem ECP(f,K) but imposing in some cases strong conditions to have that some mappings are contractions (Noor, M. A. [1], [2],[4], [5]). 2) Other results based on isotone projection cones are in (Isac, G. and A. B. Nemeth [4]). • Comment We defined in Proposition 4.20 the operator 0>{x) ==PPKK(x) Q>{x) (x) -f[P -f[PKK(x)) (x)) for for every everyxx ee H, H, and we studied the problem ECP(f,K) using the fixed points of O, i.e. points XQ such that = O(x 0 ). xx00 = ®(xo)-

(4.26)

If for an arbitrary x we put equation (4.26) in the form x - <£>(x) = 0, we have x-PKK{x) x-P {x) ++f(P f(PK{x)) {x)) == 0. 0. (4.27) (4.27) Denoting T(x) = f(PK(x)) + (x - PK(x)), we obtain for equation (4.27) the form rorm

T(x) = 0.

The operator T is exactly the operator named by S. M. Robinson the "normal operator". Robinson defined this operator in R" in 1992 in the paper (Robinson, S. M. [1]), while we used the same operator but associated to a fixed point problem in 1988 in (Isac, G. [6]). Moreover, we used this operator in an arbitrary Hilbert space. The same operator was used in (Shi, P. [1]) in the study of solvability of the variational inequality

81

CONES AND COMPLEMENTARITY PROBLEMS

„>;

(A(u),vv -u)>(f,v-u), -») for all v e D (A(u),

(4.28)

where A : H -+H (H is a Hilbert space), / e H and D c H is closed and convex (Shi, P. [1]). In (Shi, P. [1]) it is proved that the solvability of variational inequality (4.28) is equivalent to the solvability of the following equation A(PD{x)) + x-PD{x) = f. (4.29) This equation is named in (Shi, P. [1]) the Wiener-Hopf equation. A similar operator as the normal operator it seems, was used by M. Kojima in [1] and by M. Kojima and R. Hirabayashi in [1] in the study of some problems in nonlinear programming. ■ • Complementarity problems and condition (S) + Let ( £ , | |) be a Banach space and K a E a pointed closed convex cone. Given a mapping / : K -> -» E*, we consider the problem: ECP{f,K\

((

[find find x, eJT e K such that

' [f(x.) f(x.)eK'and(x.,f(x.)) e K' and (*.,/(*.)) = = 0.

In this section, we consider the case when E is a reflexive Banach space and f(x) = = 7j(x) - 7^(x), T2(x), with T\ and T2 satisfying special conditions. Such a case seems to be frequently used in many practical problems. We denote by (w) - lim the limit with respect to the weak topology and by (w*) - lim the limit with respect to the weak star topology. Definition 4.8 Let D be a subset ofE. We say that a mapping T : D —>£ satisfies condition (S) [S)++, if for any sequence {x {x„} n } BgAf ngAf c D with (w)- lim xn=xt, (w*)- lim T(xn)= u e £ * and limsup(xn,7T(jc„))<(*,,«) n—ken

82

EXISTENCE THEOREMS we have that {xn)n&N has a subsequence norm convergent to x,. The property (S)+ was defined by G. Isac and studied by G. Isac and M. S. Gowda in [1]. Recently, this condition was extended to multivalued mappings by P. Cubiotti and Y. C Yao [1]. The class of operators satisfying (S)+ is sufficiently large. We recall that a weight is a strictly increasing continuous function 0 : R+—> R+ such that O(O) 0(0) = 0 and lim O(r) = +oo. Given Given J:E-> J:E->

/->+oo

aa weight weight O, O, aa duality duality mapping mapping on on E E associated associated to to O O is is aa mapping mapping E E J(x) = - lx* sE* (x,x*\ = = |x|||x*| \\x\\ix*land be* tI = 0(|x||)> 22 '' such a«
t

where | ||, is the norm on E . The duality mapping is studied in the book (Cioranescu, I. [1]). Examples 1) If £ is a Hilbert space and
p c£c H„ 2) E = £,(n,/j), L |"JL/.= [y\Pdx 2))ifIf If B E= = LPp(C1,M), {a>n), \u\ \^u\ dx Lf=\ j[

\

I l/J-2

J[u) = ./(wj = \U\ |W|

l/p li1,p

pp __ 11 . !1< < ?P << +oo, +00, (,-) 0(r) then ,\
• U. U.

Vp

p r ~\ T , \\u\\ = £ |Z}«||J

3) If the domain Q c R" is bounded, E = and md = W^ WQ' (n), {CI) , \\u\\ \\u\\ == ££ |A"lz/ |A"lz/ andmd 3) If the domain Q c R" is bounded, £E = Li=l Li=l L;=i 2 d ( du P- du \ 1
A duality mapping is a monotone operator and it is strictly monotone if E is strictly convex. If (E,j [E\ ||) is a reflexive Banach space with IE*,|| \E*\ ||J strictly convex, then a duality mapping associated to a weight function O is a demi-continuous point-to-point mapping (see Cioranescu,I. [1]). If (E,j I) is a Banach space, then a duality mapping on E is a point-to-point [E\

83

CONES AND COMPLEMENTARITY PROBLEMS mapping and norm continuous, if and only if, the norm of E is Frechet differentiable. Definition 4.9 A mapping T:E^>E E' is said to satisfy condition {S)+ if for any any sequence sequence \x \xnn ]] NN^E ^E which which converges converges weakly weakly to to x, x, in in E E and andfor for which which x limsup(x„ - x.,T(xn)) < 0, we have the norm convergence °f{ „}neN to x,. n-*oo

The condition (S)+ is currently used in nonlinear analysis and it was introduced by F. E. Browder [2] - [5]. The importance of condition (5)+ is the fact that one can verify this property, under suitable concrete hypotheses, for some maps of a Sobolev space W"'P(Q) into its conjugate P W^m'p\Q) (where P' p'=^—), of the form = p-V 7>' T{u)= £ ( - il ) H DaTa(x,u,...,Dmu)

"")

a<m

(Browder, F. E. [5]).

Proposition 4.28 If a mapping T ; E -> E' satisfies condition (5)+, then it satisfies (S) . Proof Let {*„} {x„}nn6jv ^N be a weakly convergent sequence to x, in E such that [T(x„)} {T(x„)}nnt^_NN is weakly-star convergent to u e E* and l i m c i i n / -v T% -v i \ -rf* ■/•mi - . \ Yimsup(x <(x,,u). \imsup(x ))<(x.,u). nn,T(xnn)) n—>co

We have {xr-x.,T{x -x.,T{xnn)) )) which implies

=

(x {x ,T{x ,T{x ))-(x„T{x ))-(x„T{x )) M**)) nn

nn

n

lim sup(x„ - x,, T(xn)) = lim sup(x„, T(xn)) - lim (x,, T(xn)) n-»oo

<(xt,u)-(x.,u) ,u)-(x,,u)

84

n-»oo

U)

= 0.

EXISTENCE THEOREMS Since (S)+ is satisfied for Twe have that {x„}n<EN is norm convergent to x*.

■

We say that T: E —> E is strongly p-monotone if there exists a continuous strictly increasing function p: R+ -> R+ such that (x - y,f(x) - /(v)) f(y)) > p(\x pQx - y§), yfj, for all x,y e E. p(0) = 0 and (x Proposition 4.29 Any strongly p-monotone mapping T : E —> ■E'E satisfies condition (S)+. Proof Let {x„}neN be a sequence weakly convergent to x. in E and such that limsup(x„ -x*,T(xn))

< 0. Since we have

fl-»oo

4k _ X* II) - (Xn ~ X'. r (*« ) - ^ ( ^ )) = (Xn ~ X*. r (*« )) - (*« - X* > ?t*« )) _ r X _TX X TX X X 4k * * II) (*» " **' ( « ) i * )) = (*« * > ( n )) ( n ~ *»?t*« )) we obtain fl-»oo

we obtain

0 < liminf/o{|xn - x t | ) < limsup/?(||xn - x,|) 0 < liminf /J(|JC„ - x t |) < limsup/?(||xn - x»|) < lim sup(x„ - x,, r(x„)) - lim(x„ - x., T{x,))

is convergent to zero. ■

Corollary 4.30 Any strongly p-monotone mapping satisfies condition (S)+. Proposition 4.31 Let (E,\\ |) be a Banach space which is Kadeg and such that E is strictly convex. If J is a duality mapping on E associated to a weight 0, then J satisfies (S)+. 85

CONES AND COMPLEMENTARITY PROBLEMS Proof Since. E' E* is strictly convex, we have that J is a point-to-point c mapping. Consider a sequence {x„}neA ^ such that (w)- lim;cnn =x,, = xt, nf.rN <=.£ (VIM - lim J(*„) (w*\ J(x„) ==uu and and limsup(;c„, \imswp(xnJ(JC„)) ,j(xn)} <<(x*,u). (x*,u). From Fromthe thedefinition definitionofof Jwe have

(x = X (x„,j(xn))-(x.,j(xn)) (*„n-x.,J(x - * „ j(*„)-j(*.)) = (*«»' n)-j(x.)) *J *»,

-(,„,j(xO) + (x.,J(,.)) = [(k«)-^.«)]-[k|-«^«] [(kl)-^.«)]-[k|-«^«]

n\\-(xn,j(x.))] + p(xk nl 4 ■||^I-(x.,7(x ),)>] «-<*„, 44M -NI-MW)] +[|j(,.)L .\\x + [|j(*.)|.

that is, (*„ - *x.,j(x . , J fn)c ) - J(*.)> 0(|x.|)][h| - N ||x.|] J(x.)) > [0(K|) - 0(|M)][|k|| | ] > 0, which implies

(4.30)

0(||,.||)][||,„||-|k.||] 0(||,.||)][||,„||-|k.||] (K||)-0(|,.||)][K«-|h|] P [a>(K||)-0(|,.||)][|K||-|h|] n->oo

< lim sup(x„, J(xn)) - lim (x,, j(xn)) - lim (x„ - x,, J(x,)) <(xt,u)-(xt,u)-0, = o, that is we have

JS[*fcD-«0*D]W-NJS[*fcD-*0*D]W-N-= 00

We show that (4.31) implies that {||*J}n6A , is convergent to neN

<4 - 31 > |JC,|.

To show

this, we prove that every subsequence of {||x„||]f„e* Uhas a subsequence convergent to ||x,| [106]. Indeed, let {IkJ)<)keN W LineN' iV

The s e

1

uence

x

{ nt}k.keN N

convergent to x*. Hence, ||x„41||

86

keN

is

be a subsequence of

bounded since {x„k}^ keN

isi s weakly

has a convergent subsequence. We

EXISTENCE THEOREMS denote this last subsequence by {|xx|}

. The sequence {||xxx||} |}

must be

||JC*|. convergent to |x*|. Indeed, if we suppose the contrary we have lirn||xf[|x|== pcj pcj ++ e, e, with with cc *^ O The mapping mapping <

limO(Jjc1.|) = 0(|jc.| limO(||x 0(|x.| + c) and O(|;c,|| (|h|)][|K.|-|x.|] = «c^0, ikajoflx,.!)-^!)] mi-MH«c*o. which is a contradiction of (4.31). Hence, {x„}n€N is weakly convergent to x*, {|x„||} „ is convergent to |JC,| and since E is Kadec, we obtain that [xn}}neW J satisfies [S)+. ■ neW is convergent to x* and we have that Jsatisfies Let K c= E be a closed pointed convex cone. Definition 4.10

We say that a mapping T2:K^>E* satisfies Altman's

condition with respect to TX:K -> E* for r > 0, if for every x e K with |;c| = r we have (x,T2(x)) < (je,7j(;c)y. |x| Remark 4.5 If E is a Hilbert space and Tl(x) = x for every x e E, then we obtain from Definition 4.10 the classical Altman's condition for T2 . This condition is an essential assumption in several known fixed point theorems (Altaian, M. [1], (Shinbrot, M. [1]). In the next result, we use Altman's condition to obtain an existence theorem for the problem ECP(f,K) with respect to a locally compact convex cone. The following generalization of the classical Hartman-Stampacchia theorem will be used in the next result.

87

CONES AND COMPLEMENTARITY PROBLEMS Theorem 4.32 [Hartman-Stampacchia] Let yE,\\ \E,\ |) ||j be a Banach space, E its topological dual and let C be a compact convex set in E. If E' f\C^>E*E' is a continuous mapping, then there exists x* e C such that (x --x,,f(x.)) x,,f(x.)) > 0, for every x e C. Proof This variant of the classical Hartman-Stampacchia theorem is a particular case of Theorem 3.1 presented in (Isac, G. [12]) and proved in (Holmes, R. B. [1]) and (Kinderlerer, D. and G. Stampacchia [1]). ■ Theorem 4.33 Let (E,\\ [E,\\ ||j |j be a Banach space, K azEE a locally compact convex cone and TX,T2:K—» :K —•E' > E'i two mappings. If the following assumptions are satisfied: 1) T\ and T2 are continuous, 2) T2 satisfies Altman's Altman 's condition condition with with respect respect to to T\for T\for some some rr > > 0, 0, then the problem ECP{T ECP(T\T , K) h has a solution x. with |x.|| < r. { 2 M||x»|| < r. Proof Since K is locally compact, the set Kr = jx ix e K\ \X\ |X| < 0, for all x sKr (4.32) (4.32) We show now that x. is a solution of the problem ECP(T\ - T2, K). Indeed, about x. we have two possible cases: i) ||x.|| 0, for all x e K, which is equivalent to the fact that ECP(T{X- - TT22, ,K). x. solves the problem ECP{T K). ii) |x,|| = r. In this case from (4.32) we have ( 0 - x»,7](x.)- rT22(x,)) (x,)\ > > 0, 0, that is , (x.,7](x.)- r2(x»)) < 0 or (x.,7;(x,)) (*,2j(x.)) < (x. - T2(xt)), and using

88

EXISTENCE THEOREMS assumption (2) we obtain (x*,Tx(xt)-T22(x(jc.)) = 00 (x.,7;(x,)-r = t))

(4.33) (4.33)

The proof will be finished if we show that Tx(xt)-T2{xt)

e K*. K. Indeed, from (4.32) and (4.33) we have (X,7J(JC,)- T2(xt)) > 0 for all Kr hence all

x ezK,i K, that is, we have 7j(x*) - T2 (x*) e^T'a e K and the theorem is proved. ■ 7j(x t )-^(x.) Corollary 4.34 Let \E,\ |) \\j be a Banach space, K a E a locally compact convex cone. If T:K —> E is a continuous mapping andfor some r> Owe have (x,T(x)j > 0 for all x e K with \x\ = r. then the problem ECP(T,K) has a solution x* with p»| < r. ■ The first main result of this section is the following theorem. Theorem 4.35 [Isac-Gowda ] Let (E,\\ \E,\ I) |) be a reflexive Banach space and let K{K } be a Galerkin cone in E. Suppose given two continuous V nn/neiV mappings TX,T2:K->E* :K —> » E . If the following assumptions are satisfied: 1) T\ is bounding (i. e. maps bounded subsets into bounded subsets) and satisfies condition (S) with respect to K, 2) T2 is a (ws)-compact operator, 3) T2 satisfies Altman's Altman 's condition with respect to T\ for some r > 0 with respect to K, \\x4 such that |x*| < r. Proof Let xn be a solution of the complementarity problem ECP{TX - T2, Kn) obtained by Theorem 4.33, with \\x„\\ < r. We have ft x-- Tr22)(x \xJxn)) (T )(x„) < and and (x(xa,{T 0. n) eEKl 22 a,(Txx-- TT n)) ==0.

(4.34)

89

CONES AND COMPLEMENTARITY PROBLEMS Since {xn}neN and (7l(x„)l J7j(jc„)|l n e # are bounded, there exists a subsequence ne/i I xs(n) I

n*N

($ '■ N—> N, strictly monotone) such that KT

^-J-^-** t t n ) - ^ *

md T ^

u i( «n))-^» W'))~^~*

(4(4.35) 35)

-

T x

for some x. eG K A and u e E*. £*. By the (ws)-compactness of T2 there exists a further subsequence of j x^ x^B\ J

nzN

I ^2\xMn))}

n<=N

denoted again by I x^/B\ >

, for which

'*ss n o r m convergent to some v e E*

(4.36)

Then x, eeJf l f and (7J - T2 \x^ Moreover,we wecan canshow showthat that )(^^(n))] —' '' > aw-- v.v. Moreover, K'. In fact, let x e JA^ f ^B ) and m > n. Then /x,(7J »u-- v e A*. /x,(?; - T2ix^m))\ > 0 holds since (7J - ^^)(^( ^ mj ))e e ^ m) ) <= c *"J(„) A*(n) by definition. Letting m m -HO, we obtain (x, uw-- v)v)>>00for forany anyxx e£ A^,N A ^ N.. Since Since (J(JA^/„\ ^ „ \ isisdense denseininA,A, this this proves uw -- v £ A .. From these and the equations (4.34), (4.35) and (4.36) we obtain X (*(K«),7i(x«K«))) = ((*«K-)'( «K-)'(7J " r 4 ^X ^" ) ) ) + ( ^x ^W) ''rr22((x^^w) ) ) ^ <**>v)^ (*•>")

(since x. G A" A and u-v w - vee A"). A"*). This Thisyields yields jx^x \x^n\ }Jisisnorm normconvergent convergenttoto x,x, since Tx} satisfies (S)+. Therefore, (?; - T2^x^j

is norm convergent to (T (j[{ -- T2)(xt) = u-v u- v eeK', l',

hence the second equality in (4.34) gives (x,,(7; -T2)(x,)) = 0. Thus, we have shown thatx. is a solution of the problem ECP(T\-T2JT)

90

■

EXISTENCE THEOREMS Condition (S)+ can be generalized for multi-valued functions. Let (E,\ |) ('• be a Banach space and DczE D a E ia closed convex subset. Let <, > be a duality betweennEE andIE'. E*. E,E' Definition 4.11 77K; The multi-valued function T.D^E' -f>E' rL (Z. \i. e., e., T:D->2 ) is

said to satisfy condition (S)+ if for any sequence {(*„,/„)} of Tsuch that, {xn}nsN

is weakly convergent to xx*,{f n}„eNN *nf»s nB convergent to some /f*. e E' E tand limswp(xn,fn)<(x,,ft), .,/..

in the graph s

' weakly-star we have that

n->oo

{x„}neN has a subsequence norm convergent to x*. Definition 4.11 has been proposed by P. Cubiotti and J. C. Yao in [1]. Consider now the Generalized Variational Inequality associated with T and D (briefly, GVI(T,D): ( xt,f)eDxE* , , / , ) e Z ) x E\ £ ' such that find (x GVI(T,D): GVKX.D):- f,f, eleT(x,)and eT(x,)and (y-xt,f ,f)>0,forallysD. for all yeD. t)>0,forallyeD. IfK is a pointed, closed convex cone in E, we can show, by a similar proof as in the case of single-valued functions, that GVI(T,K) is equivalent, in the sense of the next lemma, to the Explicit Multivalued Complementarity Problem (denoted briefly by EMCP(T,K): find (xt,ft)eKxE iKxE*E'» such that EMCP{T,K)\- f, eT(xt)ClK' )C\K' EMCP{T,K):\f

and

{x.,f.) = 0. (*.,/.) 0.

* N II) *

Lemma 4.36 Let (E,\ |) be a Banach space andKaE KdE a pointed, closed

91

CONES AND COMPLEMENTARITY PROBLEMS convex cone and T :K \K —*> EE' a multi-valued function Then a point (xt,yt) ) eKxE* EMCP(T,K), if and only if it solves GVI{T,K). eKx E' solves s Proof A proof is in the paper (Cubiorti, P. and J. C. Yao in [1]) ■ We use the following general result. Theorem 4.37 [Cubiotti-Yao] Let (E,\ \E,\ ||j be a reflexive Banach space and DczE a non-empty convex and weakly compact subset. If T is an upper semicontinuous multi-valuedfunction from D into E such that : 1) for each x e D, T(x) is a non-empty, closed convex subset ofE , 2) T satisfies condition (S)+, 3) T(D) is bounded then the problem GVI{T,D) has a solution. Proof A proof is in (Cubiorti, P. and J. C. Yao in [1]). ■ Applying Theorem 4.37, we obtain two interesting existence results for the problem EMCP(T,K) in a reflexive Banach space. We recall that a multi valued function T :K —> E , (where K is a convex cone) is bounding if for every bounded subset DczK,we D cz K, we have have that that T(D) T(D)isis bounded. bounded. Theorem 4.38 Let (E,\\ ||j be a reflexive Banach space andK a E a closed convex cone. Let T be an upper semicontinuous multi-valued function from K into E such that : 1) for each x e K, T(x) is a non-empty closed convex subset ofE*, 2) Tsatisfies condition (S) , 3) T is bounding, 4) there exists a non-empty bounded closed convex set D ccz K K such such that thatfor for each x eK \ D and eachfe T(x) T\x) one has sup(x - y,f) > 0. y<±D yzD

Then the problem EMCP{T,K) has a solution xt <=DxE*. <E\

92

EXISTENCE THEOREMS a Proof Let {D {Ai} n} neA , be a;sequence of bounded closed, convex subsets of E

such that DcD cncD D mm for every n,m e N with n< m and [JD„= K. D c r,DD.,D n For each n & N, N,byby Lemma 4.36 and Theorem 4.38 there exist xn e Dn and f/„n eT(x eT(xnn)) such such that that (x„ (xn --yy,,ffnn)) < < 0,./or 0, for a//y ally eDnn. By assumption (4), the sequence {xn}n€N lies in D. Since D is bounded and E is reflexive, there exists a subsequence of {*„} {x„}neN weakly neA,,, denoted again by {xn}nsN, convergent to a point x* eD. Since 1\D) is bounded, we may suppose that the sequence {/„}neJV is weakly convergent to a point /». We have (x (JCnB - xt,fn) < 0, for all n e N, and therefore lim sup(x„, /„) = lim sup(x„ -x„f - * , ,n/)„ ) + + lim (x, n—>oo

n—>oo

«—>oo

J ni

,fn)<(x,,f).

By assumption (2), we have that the sequence {xn}ntneN _N has a subsequence, denoted again by {x„}„ {*„}„eAf eA, > which is norm convergent to x*. By (Berge, C. [1], Theorem 6, pg. 112) the graph of Tis Tis closed, hence /* / . e T(xt). Let > y> ee KK be bean anarbitrary arbitrary element. element. There Thereexists existsnn00 ee NN such suchthat that>>> ee D^. D^. We have, (xn-y,fn)<0, n0, and hence 0 > limsup(x„ -y,f -y,f„) n) n->oo

= limsup(x„ - x, ,/„) + lim(x, -y,fn) lim(x. -y,f„) n-»oo

= (x. - %/♦)

«-><»

(since limsup(x„ -x»,/„) -xt,f„) = 0). Because (x* -y,f*)

< 0 for all v e -ST, by

Lemma 4.36 we have that (x»,/,) (x.,/,) is a solution of the problem EMCP{T,K). U

be a reflexive Banach space andKc andKcE E a closed Theorem 4.39 Z,ef (l?,|| |) fee convex cone. Let T be an upper semicontinuous multi-valued function from K into E such thai that : 'E\ , 1) for each each xx ee K, K, T (x) is a non-empty, closed convex subset ofE 93

CONES AND COMPLEMENTARITY PROBLEMS 2) Tsatisfies condition (S)+, 3) T is bounding, 4) there exists a weakly compact convex subset D of K such that mtK(D) * and for each x edK(D), f sT\x) there exists y e int^(D) intK(D) such that {x-y,f)>0, (x-y,f)>0, :E'. then the problem EMCP(T,K) has a solutionn x.x, eD> sDxE'. then the problem EMCP(T,K) has a solution x. sDxE'. Proof By Theorem 4.37, there exist xt e D and / . e l\xt) such that Proof By Theorem 4.37, there exist x* e D and / . e l\xt) such that (x, -u,f)<0, for all u & D. (x, - w, / . ) < 0, for all u & D. We have two cases: We have two cases: i) x, e intK[D). In this case, for each z e K i) x, e intK[D). In this case, for each z e K real number X such that x, + A(z-x.) real number X such that x, + A(z-x.)

(xt-z,f)<0. (x -z,f)<0. ii) x, t edK[D).

\ D, we \ D, we eD. eD.

By assumption (4), there exists y ii) x, edK[D). By assumption (4), there exists y (x, - v,/,) > 0. Therefore, by (4.37) we obtain (x, (x, - v,/,) > 0. Therefore, by (4.37) we obtain (x,

(4.37) (4.37) can can By By

choose choose (4.37) (4.37)

a positive a positive we have we have

e intx(£)) such that e mtg(D) such that -y,f.) = 0. -y,f.) = 0.

Let z eK\D (\D be an arbitrary element . We choose A. e]0,l[ such that Az + (l - X)y e D. Using again (4.37) we deduce

0>(xt-[Az + (l-A)y\,ft) = A(xt-z,ft)

= A,(x.-z,f.) A(x.-z,f.) +(i-*X (l-X)(x.-y,f,)

and the theorem is proved. ■ Theorems 4.38 and 4.39 were proved for the first time in (Cubiotti, P. and J.C. Yao [1]). We finish this section with the remark that other existence results for the problem ECP(f,K), based on condition (S)1 are proved in the

94

EXISTENCE THEOREMS paper (Isac, G. and M. S. Gowda [1]). It seems that condition (S)+ is a deep property and a good substitute of compactness. Probably, other new existence results for the problem ECP(f,K) can be obtained using this condition. • S-variational inequalities and the implicit complementarity problem Let yE,E*\ be a dual system of locally convex spaces and K a E a closed convex cone. Given a subset D a E and mappings S:D->K and T\D^> E*, the Implicit Complementarity Problem associated to T, S and K is: find x0 e D such that ICP(T,S,K):< ICP(T,S,K):\ «)- T(x0) eK* and

(S{x ),T{x 0. (S{x000),T(x (S(x ),T{x ))0)) == 0. 00)) If the problem ICP(T,S,K) is defined, the S-Variational Inequality associated to this problem is:

((

S-VI(T,S,K):K):

[find find x0n e D such that T(x0n) eG K* and (x-S(x (x - S(x e K. Q),T(x Q))>0,forallxsK. 0), T(x Q)) > 0, for all x x&K.

Proposition 4.40 The problem S-VI(T,S,K) is equivalent to the problem ICP(T,S,K). Proof Indeed, if x0 is a solution of the problem S-VI(T,S,K), then S(x0) e K and we have (4.38) (x - S(JC S(x0),0), T(x0)) > 0, 0, for for allxeK. allxeK.K. (4.38) (4.38) Let u e K be an arbitrary element. If we put x = u + S(x0) in (4.38), we obtain (u,T(x0)} > 0 for every u e K, that is we have T\x0) e iC. If we put

95

CONES AND COMPLEMENTARITY PROBLEMS x = 0 again in (4.38), we have (S(x0),T(x0)) < 0, and, since (S(x0),T(x0)) > 0, we deduce (s(x0),T(x0)) = 0. Conversely, let x0 be a solution of the problem ICP(T,S,K). We have S(x0) G e K, 1\X T\XQ) e Jf* and (S(x ^5(X00),T(x ),7(JC = 0, 0 , which implies 0 )) = Q) G Q)) (x - S(x0), T(xQ)) > 0, for all x G e K. ■ To study the solvability of the problem ICP(T,S,K) we need to consider also the following variational inequality:

({

find xxQ00 G eD GD Dsuch such suchthat that that S-VI(T,S,D):D) \(x(x- S{x S(x00),T(x00)) > 0, for all x eD. c X has a Theorem 4.41 [Browder] A mapping T0:D—>2D, where D cz fixed point if the following conditions are satisfied: 1) X is a locally convex space and D is non-empty, compact and convex, 2) the set T0(x) is non-empty and convex for all x G e D, 3) the preimage T^\y) I^\y) = [x e £>| v e T0(x)) is relatively open with respect to to D, D, for for ally ally ee D. D. Proof A proof is in (Zeidler, E. [1], Proposition 9.9, pg. 453). ■ Theorem 4.42 Let D cz E be a non-empty compact convex set, T-.D^E' T:D->E' DczEbe and S :D —>K two mappings. If If, for every x e Dwe have (S(x), T(x)) < (x, 7\x)), T(x)), then the problem S-VI{T,S,D) S-VI(T,S,D) has a solution. Proof If the problem S-VI(T,S,D) does not have a solution, then {Vx G D)(3u eG D)(u (Vx D)(u -- S(x), S(x), T(x)) T(x)) < < 0. Let T0:D -> D be the multivalued function defined by

96

(4.39)

EXISTENCE THEOREMS T0(x) = \u e D\ (U - S(x), T(x)) T[x)) < o\, o}, for every x e D. We remark that T0(x) is non-empty and convex for every x e D. Since Tand S are continuous, the mapping v->(x-5(v),7'(v)) is continuous, too, and we have that ^(y) 1^\y) = jx [x GD\ y eJ 0 (x)j = {x eD\(y-S(x),T(x)) E, T : K —> E continuous mappings. If the following assumptions are satisfied: 1) 1) there there is is aa number number r> r> 00 such such that that Sy Sy Kf Kf cc JJ ff ,, and 2) there is an element u00 sK such that S\UQ) e K, p(« 00 ) < rr an (x -- S(u S(u00), where (x ), T[x)^ T[x)^ > > 0, 0, for for all all xx ee K K satisfying satisfying rr < < \x\ \x\ < < max{r,r max{r,r00}, }, where rr00 is is aa number number such such that that sup|p(«)| supj|p(«)| uu ee Kf Kf \\ < < rr00,,

3) (s(x),T(x))<(x,T(x)),forallxe for all x e Kf, then the problem ICP(T, S, K) has a solution x* e Kf Kr such that |S(x,)|<max{r,r |5(x,)|<max{r,r00}. }. Proof Since K is locally compact we have that Kf is a compact convex set. 97

CONES AND COMPLEMENTARITY PROBLEMS Applying Theorem 4.42 with D= Kf,we obtain an element xt e Kf such that (x (4.40) (JC -- S(x S(x,), T(xt)) >> 0, 0, for for all all xxee K?. K*. (4.40) t), T(x.)) We have that S(x>) e K and two cases are possible. |S(jc.)(j << rr.• ^x If x e Kis K is an arbitrary element, then there is a sufficiently i) Wx*)! small X € ]0,1 [ such that w + (l(l -- A)S(x,) X)S(x,) ee Kf. Kf. If If in in (4.40) (4.40) we we put put w== Ax + x = w we have X(x A(JC - S(x.), r(jc,)) T(x,)) > 0, that is , (x - S(x.), T(x,)) > 0 for all aU x e€ K and by Proposition 4.40 we have that x> is a solution of the problem ICP(T, S, K). ii) ||iS(jc.)|>r. In this case, we have r <\S{x^< max{r,r0} and, by assumption (2) we obtain (S(x.)-S(u00),T(x.))>0 (S(x.)-S{u ),T(x,))>0 > 0

(4.41)

and since for every x e Kf we have (x-S(xt),T(xt))>0. {x-S(x,),T(x.))>0.

(4.42)

We deduce (using (4.41) and (4.42)), (x-S{u0),T(x.))=(x-S{x.) )j(x.))=(x-S{x.) ?(X,)H++S{x.)-S(u S{xt)-S(u0()),T(x.)) ),T(x.)) that is we have,

==

{x-S(x {x-S(xt),T{x t),T{xt)) t))

+ S(xt(S(x.)-S(u ),T(x,))>0, >("o) 0),T(x.))>0, W>2 (x - S(u0),T(x,)) K?. ,7t*.)) > 0, for all x e K?.

(4.43) (4.43)

If x e K is an arbitrary element, then there is a sufficiently small A. e ]0,1[ such that v = Ax + (l - X)S(un) e K?. Now , if we put x = v in (4.43) we

98

EXISTENCE THEOREMS obtain (x - S(u0),T(xt)) > 0, 0, for all x e K.

(4.44) (4.44)

Since H^wo)!! < r we can put x = S(u0) in (4.40) and we deduce (s(u0Q)-S(x )-S(x,),T(x.))>0 t),T(xt))>0 >>0

(4.45)

From (4.44) and (4.45) we obtain (JC S(x,), T(x,)) T(x,)) >> 0, 0, for for all all xx ee K. K. (x -- S(x,),

(4.46)

Since S(x-) e K, from (4.46) and Proposition 4.40 we obtain that x* is a solution of the problem ICP(T, S, K) and the proof is finished. ■ We extend now Theorem 4.43 to Galerkin cones. Definition 4.12 Let K(Kn) be a Galerkin cone in E. We say that S :K -► ^ EE is subordinate to the approximation (K„)n£N of the cone K if there exists n0 e JV such thatfor every n>n0we have S(Kn) c Kn. Some examples of subordinate mappings are given in (Isac, G. and D. Goeleven [1]) Definition 4.13 We say that S : K -> E is r-subordinate to the approximaDefinition tion (K„)neN if there exist r > 0 and «0 sN such that for every n >n0we have s(K^.)czKn,

where K% = {x sK„ s K„ |\\x\\ | \\x\\<
K then it is IfS:K^>Eis ofK, JT->£is subordinate to the approximation1 (tf„) neJV °f (*.U > r-subordinate for any r > 0.

99

CONES AND COMPLEMENTARITY PROBLEMS Remark 4.6 If S : K -» -> E is continuous and r-subordinate to the approxi mation {Kn)neN,

then s(Kf)^K. SIK^CK.

Indeed, if x * e Kf, then we have two

cases: a) ||*|| 11*11 <oo

|*„ - JC| \x\ x\\| < < rr--1*| Since, for every

for for every every nn > > nnuu which which n > max\n ,n }, we have 0 x maxfo,^),

implies implies ||x*nn||<<||xxnn--x*| |++|x| |x|
continuity that S(x) e K. b) ||*|| and rr << ||*„||, ||*„||, then then |)jic|| = r. If for every n e N, x„ e K K„, x„n = xx and m lim x n—>oo

( r \ r _ r _. for all considering the sequence yn = T.—r. -en xn, where 0 < en £ E a«<5? and TT::KK— -+ »E E be be strongly strongly continuous continuous mappings. If the following assumptions are satisfied: I) 1) S S is is r-subordinate r-subordinate to to the the approximation approximation K^K^ K^K^ 2) there exist such that |p(«o)| < r, 5(t^) e K and 2) there exist m m e e N N and and u u00 e e K Km m such that \S(u^ < r, S(UQ) e Kmm and (x (x -- Sfa),7(*)) Sfa),7(*)) > > 00 for for all allxx €e K„ K„ satisfying satisfying rr < < \\x\\ \\x\\ < < max{r,r„}, max{r,r„},where where rrnn is sup|||S(tt)|| \u eKf\ eKf\ max.[n 0,m], n>max[n 0,m], 3) (Six), 7f *)), for all xsKf, (S(x), T(x)) < (*, (x, T(x)), then then the the problem problem ICP(T, ICP(T, S, S, K) K) has has a a solution solution x* x* such such that that ||*.||< ||*.||< r. r.

100

EXISTENCE THEOREMS Proof We remark that for every n>max{« 0 ,m} all the assumptions of Theorem 4.43 are satisfied for every problem ICP(T, S, Kn) and hence we have a solution xn for each of these problems. Since, for every x*n (with «>max{« 0 ,/w|) we have be!\
''

II " I I

(

"insN

sequence. Because E is reflexive, \xn\ has a weakly convergent sequence. Because E is reflexive, \x*n\ has a weakly convergent subsequence j x* \ . We denote again this subsequence by \x*n \ and subsequence | x* \ . We denote again this subsequence by \x*n \ and we put xt = (w)- limx*n. We have that xt e K and IbtJ < r , since Kf is we put xt = (w)- n->oo limx*n. We have that xt e K and IbtJ < r , since K? is closed. Hence S(xn->oo t) e K. Let x e K be an arbitrary element. For every closed. Hence S(x e Jf. Let x e A" be an arbitrary element. For every n > max{«0,/n}, wet)have n > max{«0,/n}, we have (Pn{x)-S(xn),Tix:))>0, >0, (4.47) (Pn{x)-S(x:),T{x:))>0, (4.47) where {P„}nc.N is a sequence of projection (P„ is a projection onto Kn). Since 5 and T are strongly continuous, computing the limit in (4.47), we obtain (x - S(x,), T(x,)) > 0, for all x <E z K.K. (4.48) The proof is finished since from (4.48) by Proposition 4.40 we have that x* xt is a solution of the problem ICP(T, S, K). ■ We consider now the particular case when S(K) c K. Theorem 4.45 Let \E,\ ||j ||) be a Banach space, K a E a pointed locally compact convex cone and S :K^> K; T : K -> E continuous mappings. If the following assumptions are satisfied: 1) {S{x), T(x)) < (x, T(x)), for all x e K, 2) there exists r > 0 such that for every x e K with r < \\x\\ there exists an 0,0, elementvxx €G Ksuch that ||vj| \\vx\\< x,T[x))> || x*xt ||\\<
CONES AND COMPLEMENTARITY PROBLEMS Proof We denote by Dn = ix e K\ \X\ < n\. Since K is locally compact we have that for every n e N, D„ is a convex compact set. We apply Theorem 4.42 with D ~Dn and we obtain a solution xn for the problem S-VI(T,S,Dn). Thus we have for every n eN there is xn e Dn * such •< suchthat that

(4.49) (4.49)

(s(xn)-v,f[x;))
neJV

isisI bounded. InIndeed, supposing the contrary we have

that, for every k > 0, there exists n e N such that k < pc* . If k > r, then there is a natural number n such that r < & < * * < « .

For this x*n, by

assumption (2) there is an element v . e K such that v . < r and x

II xn II

n

(S(x'nn)-)-v (s(x Vx.,r{x n))>0. x:,r{x:))>o.

(4.50)

But, since v^. < r < |JC*| < n, from (4.49) we have (s(x*\ - v^., jix*)\ < 0, which is a contradiction of (4.50). Hence, lx'\

:}."IneN

\

K is locally compact the sequence |x*J \

subsequence \x* \ keN

"IneN

is bounded and because

has a norm convergent

x'„ .\ now show that x. is a solution Let x, x. == lim x' n .We k^yoo "'

of the problem ICP{T, S, K). Indeed, if v € K is an arbitrary element, then there ism e JV such that for every n > m we have, v e Dn and for every nk>m,v £ £>„t and ^(x* t ) - v,l(x^)^ < 0. Using the continuity of 5and Twe obtain (Sfc,)- v.T^x,)) < 0 for all v e K, that is x. is a solution of the problem S-VI(T,S,K), which by Proposition 4.40, is equivalent to the

102

EXISTENCE THEOREMS problem ICP(T, S, K). Obviously, by assumption (2) we must have |JC»| < r || and the theorem is proved. ■ Corollary 4.46 Let (E,\ |) be a Banach space, K a E a pointed locally compact cone and S : K -> K ; T : K -» E continuous mappings. If the following assumptions are satisfied: 1) (S{x), T{xj) < (x, T(x)}, for all x e K, 2) there is a number r > 0 such that for every x e K with r < ||JC|[ we have

(S(x),T{x))>0, then the problem ICP(T, S, K) has a solution x* such that \\ x* || < r. Proof We apply Theorem 4.45 with vx = 0 for every x e K satisfying \x\>r. U Corollary 4.47 Let (E,\\ ||j be a Banach space, KaE

a pointed locally

compact cone and S : K -» K, T : K —> E continuos mappings. If the following assumptions are satisfied: 1) (S(x), T(x)) < (x, T(x)), for all x e K, 2) there exist a number ro > 0 and u0 e K such that for every x e K with r0 < \\x\\ we have (S(x) - u0,T(x)/ > 0, then, the problem ICP{T,S,K) has a solution x* such that ||JC*|| < 1 + max{r0,|w0|}. Proof If we denote r = max(r0,|w0|| + 1 , we have r > r0 and r > ||u0||. Now, we can apply Theorem 4.45 since assumption 2) of this theorem is satisfied with Vf = UQ for every x e K with ||x|| > r. ■ Remark 4.7 Condition 2) of Corollary 4.47 is satisfied if T is semicoercive with respect to S in the following sense:

103

CONES AND COMPLEMENTARITY PROBLEMS / (S(x)-u0,T(x)) ) N W (3«o eK) ti. i lim urn -t-ou . . (3u0<=K) „i n~ W / == +oo v

/lim W ' ) - . ^ ' » - J .

°

\H->*» \H-*»

IH| |H|

JJ

Theorem 4.48 Le? (i?,|| |) be a reflexive Banach space and K(Kn)n&N a Galerkin cone in E. Let S : K -> K and T: K -> E be strongly continuous mappings. If the following assumptions are satisfied: 1) S is subordinate to the approximation (Kn))n<=N ofK, neN ,ofK, 2) (S(x), T(x)) < (x, 7{x)), for all x e K, 3) there is a number r > 0 such that for every n>n0 and every x e K„ with r < ||;c|| there is an element vx ei5Tn such that \\vx\\ < r and (S(x)-vx,T{x))>0, )>0, then the problem ICP(T, S, K) has a solution x» such that \\x, || < r. Proof Since, for every n > «o we have S(Kn) c Kn and all the assumptions of Theorem 4.45 are satisfied, we have that, for every n > n0, the problem ICP(T, S, K„) has a solution x*n. Because of the fact for every n > no, we have pc*
has a weakly

convergent subsequence denoted again by I x*J\nneN e=N iV" We put x, =(w)- n—»oo limx*. To finish the proof, we remark that as in the proof of Theorem 4.44 we conclude that x, is a solution of the problem ICP(T S, K). Obviously we have ||x.|| < r. ■ Several existence results about the Implicit Complementarity Problem, also based on S-variational inequalities are proved in the paper (Isac, G. [9]). In the results proved in (Isac, G. [9]) we suppose that T: K -» E' has the form T =T\ -T2 where T\ and T2 satisfy some special conditions. ■ • Heterotonic operators and the generalized order complementarity problem. The Generalized Order Complementarity Problem was intro104

EXISTENCE THEOREMS duced in [(Isac, G. [4]) and (Isac, G. and M. Kostreva in [1])] and studied in (Isac, G. [4]), [11]), (Isac, G. and M. Kostreva in [1]) and (Isac, G. and D. Goeleven [2]). If we analyze the methods used in the study of this problem, we conclude that two methods seem to be natural. One is, to use the concept of ^-isotone operator (Isac, G. and M. Kostreva [1]), (Isac, G. and D. Goeleven [1]) and (Isac, G. and D. Goeleven [2]) and another is to use the concept of heterotonic operator (Opoitsev, V. I. [1]). We now present some results based on the concept of heterotonic operator. Let £ be a Banach space or a locally convex space. Given a closed pointed convex cone K a E, we denote by "< " the ordering defined by K. Suppose that, with respect to the ordering "< ", the space £ is a vector lattice. We denote the sup(:c, v) by x V v and mf(x,y) by x A y. Given m operators Tx,T2,..Tm:E -> E and a non-empty subset D c f i , the Implicit Generalized Order Complementarity Problem associated to the family of operators {7j}™ J and the set D is:

tt

/ (find xt G D such that x [find IGOCP\{T,y?,K,D\\ , Xxv IOOCP\{T.}",K,D\\ ,# ,, ,v ,, ,, N = 0. Tm,{x.)) X hh> V' ' ){A(T {x ),T {x,),...,T (x,)) l t 2 m \' J[A(Tl{xt),T2{xt),...,Tm(xt)) = = 0. 0. In many practical problems, we have D = K. In this case, we denote our problem by IGOCPUT^ ,K\.

(w:>4

Given the problem IGOCPUT^^K.D),, * , £ > ) , we define the operators: H[x) - V H[x) v^x-T ( X - 7l(x),x-T J ( X ) , X2-(x),...,x-T 2 ^ ( J C ) , . .m.(x)y, , X - 7 ^ ( X ) ) ; for for all all x e E, E, G(x) = A(X + TX(X),X + TT22(X),...,X (x),...,x + +T Tmm(x)); (x)); for for all all JJCC Ge E. E. Proposition 4.49 IGOCPUTA^KfDh

The element xt e D is a solution of the problem if and only if, x, is a fixed point ofH or, if and only

if xt is a fixed point ofG.

105

CONES AND COMPLEMENTARITY PROBLEMS Proof The proof is in terms of an elementary calculus based on the proper ties of the latticial operators "A" and "v". ■ From Proposition 4.49, we conclude that it is important to study the operators H and G and to establish some useful properties with respect to the ordering. In this sense we consider the following general case. Let F\, F2l...,Fmbem operators from E into E. We denote F {x),...,F allxx e E, F.{x) (x)) for all A{x) = A(Fl]{x),F22{x),... >Fm m{x)) Fv(x) {x) = v(Fx(x),F2{x),...,Fm{x))

for allxe£.

Definition 4.14 We say that T : E —» E is a heterotonic operator on a set DcE, if an only if, there exists an operator T:E x E —> E such that: i) f(x,x) = T(x) ,for allx e D, ii) t(x,y) is monotone increasing on D with respect to xfor anyy, Hi) f[x,y) is monotone decreasing on D with respect to yfor any x. The concept of heterotonic operator was introduced and studied by V. I. Opoitsev [1]. When we say that T is heterotonic we suppose that f is selected. Remarks 4.8 1) A monotone increasing (resp. decreasing) operator Tis heterotonic. Indeed, if T is monotone increasing (resp. decreasing) we take f{x,y) = T\x) + x-y (resp. f{x,y)=T(y) +x-y ). 2) If T is heterotonic the choice of T is not unique. 3) The sum and the composition of two heterotonic operators is a heterotonic operator. Proposition 4.50 If Ft = /?,. + St;i = 1,2,..,m where i?, is increasing andSt is decreasing, then FA andFy are heterotonic operators. Proof If we take 106 106

EXISTENCE THEOREMS t{x,y) = A(Rl{x) + S,{y),...,Rm{x) + Sm{y)); and Fv(x,y) = V(_RJ(JC) + S1(y),...,Rn(x) + Sn(y)), for all x,y e E, we remark that Definition 4.14 is satisfied for FA, respectively for F v . ■ A more general result, is the following. Proposition 4.51 IfFifor every i = 1,2,..m is heterotonic, then FA and F v are heterotonic, also. Proof In this case we take K{*>y) = ^(Fi{x,y),...,Fm(x,y)); and Fv(x,y) = vJF^x,y),...,F m (x,yj}, for all x,y e E. ■ Suppose now that E is a Banach space. Let || || be the norm defined on E. Definition 4.15 Let a - o~(t) be an increasing real-valued continuous at 0 function from R+ into R+ such that a(0) = 0. Let DaE be a non-empty set. We say that T: D—> E is a-Holder continuous if \T(X) - T(y)j < cr(||x - yfj, for all x,ye.D. We say that Tis of Holder type if there is an increasing continuous function a: R+ -> R+ withCT(0)= 0 and such that T is a-H6lder continuous. Suppose that (E,\ ||,jn is an ordered Banach space such that with respect to the ordering defined by K, the space E is a vector lattice.

'(*.! I.')

Definition 4.16 -/JM£,| | , ^ j is an ordered Banach space which is a vector lattice we say that the norm \\\\is a Riesz norm if: i) I |JC| I = ||jc||,_/br allx e E (the norm is absolute),

107

CONES AND COMPLEMENTARITY PROBLEMS ii) 0 < x < y implies \x\ < \\y\\, for all x, y e E (the norm is increasing). Remark 4.9 It is easy to show that a norm || || on E is Riesz if and only if |JC| < \y\ implies jjcf < \y\, for all x, y e E. Proposition 4.52 If (E,\\ \\,K) is an ordered Banach space which is a vector lattice and the norm \\ \\ is Riesz, then for every set ofm operators of Holder type Fi,F2,...,Fm:E -> E we have that FA andFv are operators of Holder type too. Proof It is sufficient to prove the theorem for m = 2. Suppose that F\ (resp. F2) is cr, (resp. ar2)-H6lder continuous. We have F,(*) VF2(x) = F,(x) Fl(x) + [F2(x) - F,(x)] F,(*)]+ and Fi{y) VF2(y) = F,(y) Ffo) Ft(y) + [F2{y) (y) - F,(v)] F(y)] + which implies Hx)vF2{x)-F,{y)vF2{y)\ v^W| = Fl(x) +

•fcM-[F (x)-F (x)] -F (y)-[F (y)-F,(y)] 2

l

+

l

2

+

++ + = [m -m] +fcM--FH F W] -Fl(y)} (y)Y - m]+teM ~-[F te2W(y)- *i

<2|F1(^)-FI(y)| + JF |F 22W-F W-F22(4 Since the norm of £ is Riesz, we deduce, |F,(x) v F 2(x) (s) - F,( v) v F2(y)\\ (v)| < 2|F,(x) - FlF,(v)| (y)\\ + |F22(*) (x) - FF22(y)\\ (y)| *2a4x-yf) <2cx (||,-y|)^(|x-y|), 1 (|x-y|) + ffjflx aCT2 2(lx-y\^o{$x-y\\), where o(t) = 2cr,(f) + cr2(t), for every t e [0,+oo[. The function a(t) is where cr(r) = 2cr,(r) + cr2(r), for every f € [0,+oo[. The function a(t) is 108 108

EXISTENCE THEOREMS continuous, increasing and cr(0) = 0. We have the same conclusion for FA, since FA(x) = Fx(x) A F2(X) = -U-F^x)) v (-F 2 (x))] for all x e E. ■ By a similar calculus, as in the proof of Theorem 4.52, we can also obtain the following result. Theorem 4.53 If(E,\\ \\,K) is a Banach space which is a vector lattice, the norm \\ \\ is Riesz and for every i=\,2..,m, we have F\ = Rt + Si, where i?, and St are of Holder type, then the operators Fv(xty) = v(R1{x) + Sl{y),...>Rm{x) + Sm(y)) and Fv{x,y) = A(R1{X) + S,( v),...,* m (*) + Sm{y)) are of Holder type, if we consider on Ex E the norm ||(x,.y)|| = ||x|| + ||y||/br all (x,y) e ExE. ■ Definition 4.17 Given a heterotonic operator T: E —> Ewe say that (x*,y*) is a coupledfixed point ofT if f(x* ,y,) = x, and f(yt, x*) = yt. This concept was introduced by V. Lakshmikantham and studied by D. Guo and V. Lakshmikantham [1] and also by Y. Z. Chen [1]. Every fixed point is a coupled fixed point. The set of coupled fixed points localizes the set of fixed points. Definition 4.18 We say that a coupled fixed points (x;y*) of a heterotonic operator T is minimal and maximal on D if for every coupled fixed point {x,y) e D ofTwe have x,<x vo]n 0} is strongly invariant for the heterotonic o]o =={x e E\u{QxeE<\ux°-x<-vv<>}

109

CONES AND COMPLEMENTARITY PROBLEMS operator T: E -» E if u0 < T(u0,vQ) and T(v0,u0) < v0. The next result gives a localization and the existence of a solution of the problem IGOiIGOCPfa}",K,DJ. Theorem 4.54 Let

LE,|| |,JfJ is an ordered uniformly convex Banach

space, which is a vector lattice and the norm \\ \\ is a Riesz norm. Let T\, T%...,Tm be heterotonic operators from E into E. The operator H or G associated with the problem IGOCPUT^",K,D\•a is heterotonic and we denote it by T. If the following assumptions are satisfied: 1) T is continuous, 2) there exist x0 ,y0 e D such that [x0,y0]Q is strongly invariant for T and

[x0,y0]0^D, 3) T is nonexpansive or condensing or continuous and dim is < +a>, then there exists a coupled fixed points (x,,>>t) of T minimal and maximal in [x0,y0]0 and a solution ut of the problem IGOCP[[TtYl,K,D\

such that

x* < u* < y,. Proof

Using the points x0, y0, we define the sequences {x„}neN and b ;

UL* y

x lU.+1 n+i = = T\x T(xni yn),), for all n,y

neN,

K ++ i = T[yn^n), \ far for all neN. We have x0 < xl = T(x0,y0); v, = f(y0,x0) < y0 and using the properties of f we have x, = f(x0,y0) < y, = f(y0,x0),

which implies x0 < x, < y, < yr

By induction, we show that xn <xn+l < vn+I ^ ] , for

110

EXISTENCE THEOREMS every n e N . Since the norm is Riesz and E reflexive, K is normal and regular. Hence, because {*„}neJV in increasing, {y„}niEN decreasing and they are order bounded, there exist x* = lim xn and y* = lim v„. We have n—>oo

n—>oo

x* 0]0. Since we can show that r([x»,^»] 0 Jc[x»,y t ] 0 , using Browder's or Sadovski's or Schauder's fixed point theorem (Zeidler, E. [1]) we obtain that T has a fixed point u* e [xt,yt]Q which is a solution of the problem IGOCPUT^^K^)

■*,D) ,D

and

the theorem is proved. ■ Definition 4.20 We say that a heterotonic operator T : E —> E is a-{concave, convex) operator (0 < a < 1) if, for every 0 < X < 1 and x,y e D, we have a 1) X". XAaf{x,y)
This class of operator was studied by V. I. Opoitsev [1]. Remark 4.10 We can show that condition (1) (resp.(2)) of Definition 4.20 is equivalent to the following 1') (resp.2'): 1') f[/4x,y) < MaT(x,y); for all fi>\ and x,y e£>, 2') f(x,fiy)

> fTaf{x,y);

for all // > 1 andx,y x,yeD.

When (E(r),K) (E(T), k) is an ordered locally convex space with a topology x de-

111

CONES AND COMPLEMENTARITY PROBLEMS fined by a family of seminorms {pa}a^, and K is normal, we can suppose «£/#' that every seminorm/?ahas the following property: 0<xpa(x)<pa(y), 00, for every x, y e K. We suppose that the family of seminorms {Pa} {pa}aSjt *s sufficient (Marinescu, aejt\A Gh. [1]). Theorem 4.55 Let

w.[p

\E(T),

a

}

J be a locally convex space ordered by a

regular normal, pointed closed convex cone K. Suppose, given m heterotonoic operators Tx,T2,...,Tm:E -> E and consider the problem IGOCPi{T^^Kj. :,K).

Denote by T the heterotonic operator H or G associa-

ted ted with with this this problem. problem. Suppose Suppose that that T T is is a-(concave, a-(concave, convex) convex) and and T t is is continuous. If there exist Mo > 1 and UQ > 0 such that computing x0 = fax U0 , y0 = fa H0, Xj = T(xQ,y0) and v, = T(y0,x0) we have that y\^y x0<xl •then there exists x* such that T(x,) = r(jc,,x,) = x, (that

is xt is a solution of the problem

IGOCPUT;}™}

; ,. * K)).

Moreover, x» = lim xn = lim yn where «-><»

n->oo

\x T(x n == x,yn_x) for [** T{nx_n-\^n-\) M allall neN, neN, [yn = T{yn-\>xn-\) far for all and, for any a e/t, we have, Pa{x*-x )<M0 Pa{x*-xnn)<Mo

ff

\

1 v V

neN,

1 \ ^ r Pa{«o) for all n eN

Mo

)

Proof Consider the sequences {*„}n(EA, and i-^n/neAf {yn}n€N defined above. We •neN have xn-\ -x <x ^xnn
EXISTENCE THEOREMS Indeed, for n = 1, (4.51) is true by assumption . Suppose (4.51) true for n and we prove that it is true for n + 1. We have x

n = % , - i . J V i ) *< T(x f(xnn,y„_ ,yn_x)< ) < f(x f{xnn,ynn) = xn+x,

x

*» n+l =T(x T{xnn,y,ynn)) <xn) ^ T{yn_x,xn_x) = yn. We prove now that Mo2a"y„<x "y„ ^ nx„for forall allKnGe #N..

(4.52) (4.52)

= M = 2 2 Indeed, we have JUQ //^(/^"o) • Suppose that (4.52) is Mo2""yy00 ==Mo {Mouo) =/"o' /VMoo =*o*o• true for n, and we show that it is true for n + 1. We have ] +1 2a 2a 2 2a ^o2a2a"+"V„ i i==Mo T(y ,x„) fytfy^x,) Mo y+n+ Mo2a2"* ^f{y„,x ) <^Mo < l(//o f(//o V„^„) <* f(x % T{x < < f j ^ . /i\x l\x z oniHnfy^ ,^y^ ^^) \ V„>*„) n,,x n)„n)) < n*,x V n n»-n-

^Mo ^ 0

—2an+1 'V

\

—2ff"+1

x T[ n>yn) = n*»»^n) = Mo ^0

x

X

n+l^ n+l*« + l^*n+I-

Hence, (4.52) is true. Since we have x0 <xx <..<xn <..00

yt = limyn and xn <xt
for every n e N, which implies for every n e N, which implies an]

2

2a

l 0
Since JST is normal we obtain Since K is normal we obtain 2a paa(y, -xt) f o r {y* -*•)* - Mo2")p "" ajPaM;

a11

hence "n e eN N^a™ d1 all aa e e^ Aand hence

xx*t=y < xt-x x„ xnn-x„, < y„ - xn, for all n e sN N we get n
113

CONES AND COMPLEMENTARITY PROBLEMS pa(xt-xn)
(4.53)

is important. We give now some examples when (4.53) is satisfied. a) Suppose Uo> 1 and Uo > 0. Compute x0 = jUo\;

y0 = MQUo> xi = T(xo>yo) and Ji = fy>o>*o)- If

0 < a < - and /u2"~xu0 < T(u0,u0) . In this case K is normal and if T is heterotonic a-(concave, 1 convex), with 0 < a < - Iand f:Int(K) x Int(K) ->• Int{K), then for every 2

z0 elnt(K) there is 0 < t0 < 1 such that tlQn~az0 < f(z0,z0)

we put in this case w0 = z0 and nQ = f<j""2, then we have M20a-]uQ
V

Also, in this case, we can show that Thas exactly one fixed point in Int(K). Consider now another interesting case. Suppose that T[x,y) = A[F,(X) + F2{ v),G,(x) + G2( v)].

114

EXISTENCE THEOREMS If for some rur2 > 0 and for every 0 < X < 1, we have

3(*) + ^(j>)**fcW + ^W] and

[G,(x) <M-k) + Giljy) ^^{xj :*: ++ G^y}], then f(^iy]>^%y), f\ foe,—y\ >Z"f(x,y), where r0 = max^,,^).

This result is

important since Theorem 4.55 is valid if we replace assumption that " T is a-(concave, convex) "by "there exists 0 < a < 1 such that T\tx,rly} > taf(x,y),forfor all 0 < t < 1 andx, y G J5T. " We study now the approximation of solutions of the problem IGOCP[\TiY',K,D\

'*)

using

cr-H6lder continuity. We denote again by T the operator H or G associated to the problem IGOCP({Tt}", K, D\ . Theorem 4.56 Let D cE be a non-empty closed convex set and T: D —>D a mapping such that I — T is a-Holder continuous. Let {*„}neN , bebe the Mann-Toeplitz iterations associated to T, that is

{

x0 G eDand D and for every n eN,

^n+i=^-an)x„+a„T(xn), «\

where {an}neN=N satisfies the following properties: 1) cto = l

2) 0 < a„ < I, for all n eN and

3

oo 00

=+00 Y.i ) n=0 Z»«„=+oo. -

If there exist «o G Nand r0 > 0 such that r0 < a„for every n>no and {xn\ n€iV is convergent to x,, then x, is a solution of the problem 115

CONES AND COMPLEMENTARITY PROBLEMS

M,

is conver convergent x,, tnen thenx*x. 'sisa asolution solutionofofthe theproblem problem {xn} IGOCP^Y^JCD). D) "n\ Proof We will show that x* is a fixed point of T. Since D is closed we have that x. e D. For every n > n0, we have \\T(x,)-T(xnl

= \\T(xt)- x. + x. - x„

+xn-T(xn}\

. - \\ -))l f | x\\x.-x +

n

< | | x . - x j + <x(|x,-xj).

Because, for every n > n0, we have — < — we deduce #„

\\x.-T(x,)\\ = l{xt-xn)

+

^n

(xn-T{xn))OU (T(x + \T(x n n)-T(xt))l

< ||x. -xn\\ + \\xn - T(xn}\ + \\T(xn)- T(x.)\ 5

II* - * J +

K -*«+l| + |k' - X n I +
= 2|*. -x„\ + o-(|x. - x j ) +—||x„ - xn+l\\ < 2||x. - x j + o(|x. - x j ) + -||x„ - x n+1 |. r o Now, computing the limit we obtain that T\xt) = x, and the theorem is proved. ■ Remark 4.12

Theorem 4.56 has an interesting consequence for the

problem IGOCp({Tt}",z) , when K is regular, since, in this case we have

116

EXISTENCE THEOREMS Since a„ > 0, when T(xn) -xn<0, for all n e N, we obtain that ix„) „ is a decreasing sequence in K and hence it is convergent. The condition 1\xn) - x„ < 0 when T= His equivalent to 7X*n) > 0 for all i = \,2,...,m that is xn is feasible. We recall that the feasible set of the problem IGOCPUT,}",K,D\K,D\ is the set F = [x BD\ T,(X) eK,for

alii = L2,...,m}.

From Theorem 4.56, we obtain the following result. Corollary 4.57 If all the assumptions of Theorem 4.56 are satisfied, ifK is regular and for every n e N, x„ is feasible, then {xnfneN is convergent to a n} solution of the problem IGOCPl IGOCPUT^^K). We finish this section with the remark that other results about the problem IGOCP({i;}™,K,D) are proved in the papers (Isac, G. and D. Goeleven [1], [2]) and (Isac, G. [4]). • Topological degree and complementarity The topological degree considered as one of the most important mathematical instrument in nonlinear analysis can be used in Complementarity Theory. With the topological degree we can study both the Linear Complementarity Problem and the Nonlinear Complementarity Problem. In relation to the Linear Complementarity Problem, the topological degree has been effectively used to study the existence of solutions and the cardinality of the solution-set (Kojima, M. and R. Saigal [1]), (Howe, R. and R. Stone [1]), (Howe, R. [1]) and (Garcia, C. D. [1]) and also to study the stability of solutions (Ha, C. D. [1]). The Nonlinear Complementarity Problem was studied by the topological degree in papers (Goeleven, D., V. H. Nguyen and M. Thera [1]), (Pang, J. S. and J. C. Yao [1]) and (Isac, G. , V. Bulavski and V. Kalashnikov [1]). For the discussion in this section we make use of some basic results from degree theory. Those readers who are not familiar with this theory can consult the books (Lloyd, N. G. [1]) and (Rothe, E. H. [1]).

117

CONES AND COMPLEMENTARITY PROBLEMS We now recall some properties of topological degree. Let D be a bounded open subset of R" andy a point of R" . The closure of D is written D and its boundary 3D . We denote by &{p) the linear space of continuous functions from D into R" . If F E ) and v e R" is such that y €F(cD), we denote by deg(.E, D,y) the topological degree associated with F, D and y. (In this situation, we shall say that deg(£,D,y) is defined). Recall that if X and Y are topological spaces, two continuous mappings / and g from X into Y, are said to be homotopic (f~g) if there is a continuous mapping H:[0,\] xX^>Y, such that H(0,x) = f(x) and H{\,x) = g{x) for all * e X We now give the most basic properties of degree. 1) If deg(is,D,y) *0, then the equation F(x) = y has a solution in D. 2) Suppose that F,G e
0 < t < 1, then

and F = G on cD, then deg(F, D, y) = deg(G, D, v)

(provided thaty gF(<2))). 5) Ify £F(dB), F e^{p) andD is the disjoint union of open sets Z), (i = I, m

2, ...,m), then deg(F,D,y) = £deg(F,D,.,>>). i=i

6) Ify * F{cD), F eS(D), K c f l is closed andy £ F\K), then deg(F, D, y) = deg(F,D\K,y). A proof of these properties can be found in (Lloyd, N. G. [1]). The following result is useful for our discussion. For the proof see (Krasnoselskii, M. A. and P. P. Zabreiko [1]) or (Rothe. E. H. [1]).

118

EXISTENCE THEOREMS Theorem 4.58 [Poincare-Bohl] Let D c R" be an open bounded subset and F,G&e{p)5) Consider the homotopy H(x,t) = tG(x) + (l-t)F(x), 0
We say that LCP(A,q,R") is stable

at xt (or that x, is a stable solution of LCPiA,q,R"\)

if x* is an isolated

119

CONES AND COMPLEMENTARITY PROBLEMS solution ofLCPiA,q,R") 5>0

and corresponding to every e > 0, there exists a

such that {x. + eB)C\SOL{A',q') * <j>, for all A'eW{nn)(R) andand

q' t=Rn satisfying \\A'-A\\ + \\q - q\ < 8. We denoted by B the unit ball in R" and by SOL(A',q') the solution-set of LCP(A',q',R"). The stability simply means that when (A',q') is close to (A,q), LCPiA',q',R")

will have solutions near x..

Theorem 4.60 Let A e7K, AR)%n be a P0- matrix and x, be a solution of LCP\A,q,R").

Then x, is stable if and only if it is isolated.

This theorem was proved for the first time by M. S. Gowda and his proof is based on topological degree (Gowda, M. S. [1]). Theorem 4.61 Let A e 7K, „\(R) be a P0-matrix and q be any vector in S". Then the cardinality of the set of solutions of LCP(A,q,R") is either zero or one or infinity. Proof We follow the proof given by M. S. Gowda in [1]. First, we remark that if LCP(A,q,R"} has a solution that is not isolated, then LCPiA,q,Rn^ has infinitely many solutions. The theorem is proved if we show that LCP[A,q,Rl) has at most one isolated solution. Suppose the contrary and let x, and u, be two distinct isolated solutions of LCPiA,q,R").

By

Theorem 4.61, both solutions are stable. Considering a matrix A of the form A - A + el sufficiently close to A, we have that LCPiA,q,R") must have solutions close to both x, and ut. Hence, LCP(A,q,R") must have at least two solutions which is impossible by Theorem 4.59 since A is a P-matrix. ■

120

EXISTENCE THEOREMS We now give a general existence theorem in Hilbert spaces also obtained by topological degree. Let (//,<,>) be a Hilbert space and K c H a closed convex cone. Suppose given the following operators Al,A2:H^> H, T:K -> H and L:K^>H. The following notions will be used. We say that Ai is positively homogeneous of order 1 if A2{tx) = tA2(x) for each x e H and fejR+\{0}, and we say that A\ is a-coercive if (Ax(x),x) > a\x\ for each x e H. We say that a mapping/ H —> H is strongly continuous if, for every sequence {;en}ngJV c / / which converges weakly to x* we have that

M*»)}„ejv

is n o r m c o n v e r

g e n t toJO).

The next result is due to (Goeleven, D., V. H. Nguyen and M. Thera [1]) Theorem 4.62 Let (//,<,>) be a Hilbert space, K a H a closed convex cone and g eH an arbitrary fixed element. Suppose given the operators A j, A2:H^>Hand T, L -.K^Hsuch that: 1) A] is linear continuous and a-coercive, 2) Ai is strongly continuous and positively homogeneous of order 1, 3) L is strongly continuous and positively homogeneous of order 1, 4) T is strongly continuous and positive homogeneous of order p > 1, 5) (T(x),x) > 0, for each x e K \ {0}. If there exists x0 e K such that (g,x0) > 0, then for each X e R, the nonlinear complementarity problem ECP^A} + A2-AL + T-g;K)

has a

solution X'(Z) such that x*(A) * 0. We remark that problem ECP(AX + A2 - AL + T - g; K) has interesting applications in Elasticity Theory. Let K a R" be a closed convex cone, D c R" an open bounded set and S,T:R" -> R" two continuous mappings. Suppose that S[D) C K. If A is a non-empty subset of R" we put A* = {v e R" | (x, v) > 0 for all x e A]. We

121

CONES AND COMPLEMENTARITY PROBLEMS know, by Proposition 4.40, that the ICP(T,S,K) S-Vl(T,S).S).

is equivalent to the

Considering the Generalized Normal Map h defined by:

h(x) = S(x) - PK[S(x) - T(x)] for all x e R", we can show that a vector xt e D is a solution of the equation h(x) = 0

(4.54)

if and only if xt is a solution of the variational inequality S-Vl(T,S) (Pang, J. S. and J. C. Yao [1]). The Generalized Normal Map when S is the identity is not exactly the Normal Map considered by S. M. Robinson [1]. We have the following result about the problem ICP(T,S,K) defined in subsection (VI) of section 4 of this chapter. Theorem 4.63 Let K c R" be a closed convex cone, D a R" an open bounded set and S,T:R" -> R" two continuous mappings such that S{D)(zK. If there exists a continuous mapping f from R" into itself such that: 1) deg(xF,D,0) is defined and nonzero where T(x) = S{x) -PK[S{x) - f(x)} for every x e R", 2) x ecD{\S~\K)

and/x>0 imply T{x) + nf{x) )*{K ^(K-S(x))\

then the problem ICP(T,S,K) has a solution x,

eD.

Proof To prove this theorem it is sufficient to show that equation

S(*)-P4s(*)-7t*)] SW-^[S(*)-7{*)] = = 00

(4.55)

has a solution x, in D. Assume that S(x) - PK[S(x) - r(x)] has no zero in D.

Consider the

homotopy H(t,x) defined by: H(t,x) = S{x) - PK[S(x) - tT(x) - (l - t)f{x)]. 122

EXISTENCE THEOREMS From assumption (1) we have that the degree of H(0,) at 0 with respect to D is defined and it is nonzero. Also, we have that H(\,x) = S(x) - PK[S(x) - T(x)] for all x e D. The theorem will be proved (considering the properties of the topological degree) if we show that 0 £H(t,cD) for all t e ]0,1[. Indeed, if we suppose that H(t,x) = 0 for some t e ]0,1[ and x e dD, then for this x we have S{x) eJTand (y S(x),tT[x) + (1 - t)f(x)) > 0 for all y e K. (v - S(x),tT(x) (4.56) (4.56) Dividing by t > 0 in (4.56) we obtain a contradiction of assumption (2) with

„-!=!..

Theorem 4.63 is a particular case of Theorem 3.1 proved in (Pang, J. S. and J. C. Yao [1]). A more interesting relation between the topological degree and the Complementarity Theory is now established by the concept of exceptional family of elements for a mapping introduced recently by G. Isac, B. Bulavski and V. Kalashnikov in [1]. Consider again the space R" endowed with the Euclidean structure. First, consider the closed convex cone K= R" Let f:R" -» R" be a continuous map. Definition 4.21 We say that a set of points I xr J

r>0

c R" is an exceptional

family of elements for f if pcf -» +oo as r -> +oo and for each r > 0 there exists&> 0such that: r i) fi(xrx) ) = = -pr -prx!,ifxi>0.

n) y;.(^)>o,z/^r = o. Let K c R" be a closed pointed convex cone. We denote by K° the polar of

123

CONES AND COMPLEMENTARITY PROBLEMS K, that is if0 = - K'. By Moreau's Theorem [Theorem 2.23] we have that each vector x e R" has a unique representation of the form z = z+- z~, where z+ = Pdz) and z~ = -P. (z). [We remark that - z~ is the orthogonal complement to z+]. Definition 4.22 Let/: R" —> R" be a continuous map. We say that a set of points \xr\ c R" is an exceptional family of elements for f, {with respect r>0

r

to K) if (x )

-> +00 as r -> +oo and for each r > 0 the point

f

flixj)

belongs to the open ray 6\,[x) ',sr\ = <>> = (.xr] +/zs r |//> 0>, where

■r-W'-WWe remark that Definition 4.21 and Definition 4.22 can be unified by the following definition (Kalashnikov, V. V. [1]). We say that \zr\ cK is an exceptional family ofelements for fif' |z' -> +oo as r -> +oo and for every r > 0 f/zere exute /4 > 0 SWC/J f/za? i*/ie vector 5r =J(zr) + firz' satisfies the following properties: 1) sr eK', 2) (sr,zr) = 0.

For example, Definition 4.22 implies the last definition if we take z' = (jcr) for any r > 0. Consider now two continuous functions from i?" into itself and the ordering defined by R". Definition 4.23 We say that a set of points \xr\

r>0

c R" is an exceptional

family of elements for the couple (f,g) if\xr\ -> +oo as r -» +oo, £ ( / ) > Qfor each r> 0 and there exists //, > 0 such that for i = l,2,...,n, we have:

124

EXISTENCE THEOREMS rr),ifg (xrr)>0, fi(xr) = -M,&-M,g,(x -M f^x^-Mrg^x'pfg^X), rgi(x ),ifgi(x i i) f,(xr)

r

r ii) flx ii) flxr)>0,if )>0,if gi(xgi)(x

r

) == 0.0.

The concept of exceptional family of elements for a function was recently introduced by G. Isac, B. Bulavski and B. Kalashnikov in [1] and the importance of this concept is supported by the following theorems. Theorem 4.64 For any continuous functions f:R"—>R", either a solution for the complementarity problem

there exists

, -, find x, e R"+ such that N ECP(f,R"++):\ ):{ , . . . .. V >[f(xt)eR"+and{xt,f(xt)) / ( * ) > «= 0 or an exceptional family of elements for f. Proof We know by Proposition 4.20 that the solvability of the problem ECPif,R") is equivalent to the problem to find a fixed point for the mapping XF(JC) = P „ - / j ^ . ( * ) j ;

x eRn.

Hence, we consider the equa-

tion x = ^(x) or x-P (x) = 0 /K-w)+*-y*)=° f(PK(x)) Since lRn,R")

+

sn+

57 (4.57)

(4- >

is a Hilbert lattice, we know that -fL(*) = x+ and because

x - x+ = -x~, the equation (4.57) becomes f(x+)-x~ x = 0. 0.

(4.58)

Let us set F(x) = f(x+\-x~. x . Then we conclude that our problem is to solve the equation 125

CONES AND COMPLEMENTARITY PROBLEMS + F(x) f(x+)-x-=0. H*)=.= f(x )-x-=0.

(4.59)

Now, we examine problem (4.59). It is obvious that the mapping F:R" -»ff" is continuous. Let {Sr}r>0 be the family of spheres of radius r: Sr = \x^R {xeRnn\\x\ \\\x\\ = r} r\

(4.60)

and Br the open ball or radius r, i. e. n Br = [xeR {xeR"\\\x\\
(4.61)

We consider the homotopy between the identity mapping / and F defined by H(x,t) (l-t)F(x); H{x,t) = tx + {\-t)F{x);

0
(4.62)

and we apply Theorem 4.58 [Poincare-Bohl] with y = 0 and D = Br. We have H{x,t) = tx {l-t)f(x++)-{l-t)x)-(l-t)x-(i-O/l{l-t)f(x H(x,t) = tx + + + + = t(x t(x + (l-t)f(x = + x-) x-) + + (l-t)f(x )-X- )-x-

= = £c

+

-f

4.63)

tx+++{l-t)f(x++)~x-. tx +{l-t)f(x )~x-.

Two cases are possible: i) There exists an r > 0 such that 0 gH(x,t),xeS r ,te[0,l]. rem 4.58 implies deg(F,JBrr,0) ,0) = deg(/, Br ,0). ,0). deg(F, deg(/,5

Then Theo(4.64)

It is well known that deg(7, Bn0) = 1 (cf. (Lloyd, N. G. [1]) and (Rothe , E. H. [1]). Hence, deg(F,B„0) = 1 too. We have that the ball Br, contains at least one solution to the equation F(x) = 0 [cf. Kronecker's Theorem in

126

EXISTENCE THEOREMS (Lloyd, N. G. [1], Theorem 2.11)], Therefore, the problem ECP[f,R^

has

a solution. ii) For each r > 0 there exist a point ur eSr and a scalar tr e [0,1 [ such that H(ur,tr) = 0. 0. We remark that L r

i|2

(4.65)

= (u* - u~ ,u* - u~\ = L*

+ L~

= r1.

If fr = 0, then ur solves equation (4.59), which implies again that the problem ECP(f,R^ has a solution. Otherwise, tr > 0, and then (4.63) and (4.65) yield + tru;+{l-t tX+(\-t u-r r)f(U;) r)f[u r) = u;

(4.66)

From (4.66) we have

(l-*,)/,(<) (l-f r )/,M ==-f,(i£) ~tr(<)(,r «/(*4>o J/"M<>° and

+ - -)., */(«,),. < 0 . (l-r r )/(« r ) = (Mr),' (i-^)4<)=K */*W^°-

(4-67) (4.68)

Now we put xr - ur+ and we rearrange (4.67) and (4.68) as follows:

4*1=- \-t -xU tr— X

1 ri r

ifx!>0 0

(4.69)

and

4 x 1 = 7 ir 7i ( < ) , - 0 ' r

z/

<=0-

(4 70) (4.70)

-

We take // r = —— and we note that (4.69) and (4.70) represent relation (i) I

—

ir

127

CONES AND COMPLEMENTARITY PROBLEMS and (ii) from Definition 4.21. In order to demonstrate that \xr\

r>0

is an

r

exceptional family of elements, we must show that \\x \\ —> +00 when r -» +QO . On the contrary, we suppose the set |w*| to be bounded. In r>0

2 11 -11 n iTTii^ +oo which means that the this case, we have that II -II / 2r || +|| ►>+00

this case, we have that " r = V ~ K ~~ > which means that the right-hand side of (4.66) is unbounded. On the other hand, the left-hand side of (4.66) is bounded since the set {<}, | u+r r>0 j is supposed to be bounded and/is a continuous function. This contradiction completes the proof. ■ Theorem 4.65 Iff : R" -> R" is a continuous function and K cz R" is a closed pointed convex cone, then there exists either a solution for the problem ECP(f,K) or an exceptional family of elements for f Proof By Proposition 4.20 and Moreau's Theorem [Theorem 2.23] we have that the solvability of the problem ECP(fK) is equivalent to the solvability of the equation + F{x) )-x-=0, F{x) ==f(x+f(x )-x-=0,

(4.71)

where x+ = Pg(x) and x~ = - PK<> (x). Repeating now the proof of Theorem 4.64 exactly, we obtain that either there exists a solution for the problem ECP(f,K), or for each r > 0 there exists a point ur eSr and a scalar tr e]0,l[ such that the equality tru* + (l - tr)f\u*J = u~ is true. Dividing both sides of that equality by (1 - tr) and rearranging, one yields the relation ; K U;+ U; u; f(<)-~ -^-7r^<=«; = ^r +- jbr( -U^ /(<)=j~"7 -"-)' u

which means that f\u+r) which means that f\u+r)

128 128

-

^\ur'■>sr) ■ Th e fact that ||rr|| -> +00 as r -> +00 is 6[ur> sr) ■ Th e fact that ||rr|| -> +00 as r -> +00 is

e e

(4 72)

EXISTENCE THEOREMS established in the same way as in the proof of Theorem 4.64. Thus {xr \

r>0'

,

where xr =urr is an exceptional family of elements for/(with respect to K) and this completes the proof. ■ Theorem 4.66 Let f, g : R" —> R" be two continuous functions. If the following assumptions are satisfied: 1) there exists b eR" such that g(x) = 0 ifx = b, 2) g maps a neighborhood of the point b homeomorphically onto a neighborhood of the origin, then there exists either a solution of the problem ICP(f,g,R")

or an

exceptional family of elements for the couple (fg). Proof We consider the following equation with respect to the variable (z,x)eR"xR":
-f/w-*1.

F(z,x)== J) W ' + ==00. F(z,x) F{z,x)= 0.. + y ,g(*)-z \g[x)-z ) \g{x)-z )

(4.73)

fl

The problem ICP(f,g,R") is equivalent to the solvability of equation (4.73). Indeed, if (z,x) solves (4.73), then x is a solution of the problem ICPif,g,R"\. Conversely, ifx is a solution of the problemi /CPICfif,g,R") then (z, x) is a solution of (4.73) where

[&(*). */,'a(*)>o,

z/=*-/xW» »/a(*) == 0o Z/=*-/x(*)» */&(*) z = l,2,...,«. The mapping F(z,x) is clearly continuous over R2". Let Sr be a (2« - 1) dimensional sphere: Sr = l(z,x) sR2n\\^z,x-b)\\-*; = r\ and Br an open ball

129

CONES AND COMPLEMENTARITY PROBLEMS of radius r, i. e., Br = l(z,x) e R2n | |(z,x - b)\\ < r | . Furthermore, we cons(z truct a homotopy H(z,x,t) of the mappings F(z,x) and G(z,x) = \

\

A*))

in the

standard way: H(z, tG{z,.x) t)F(z, x) x) {z,3x, H(z, x, t) t) = =■tG{z, tG{z, x) + + (l{\- t)F(z, _'tz -(l-r)z_'tztz-+ {\-t)f{x)-{\-t)z{\-t)f{x)-{\-t)z+

''

+ Jg{x)+{l-t)g{x)-{l-t)z\ Jg{x)+{l-t)g{x)-{l-t)z\i-ty) ■ J

= =

V + ( i - * ) / (( ** )) -- ^^

-ty

+
Hence, we have (tz++(l-t)f(x)-z-)' ) - * ■

H(z,x,t) =

•

'

U(*)-(i-0*

J

(4.74)

Two cases are possible: A) there exists an r > 0 such that H{z,x,t) * 0 for all (z,x) e Sr and t e [0,1]. In this case, the Poincare-Bohl Theorem implies the equality deg(F, B„0) = deg(G, B„0). Since |deg(G, 5 r ,0)| = 1, we have that deg(F, fir,0) ± 1. As above, we conclude that the ball Br contains at least one solution of the equation F(z,x) = 0 and the solvability of the problem ICP(f,g,Rn+) is proved. B) For r > 0 there exists a point \zr ,xr\ e Sr and a scalar tr e [0,1 [ such that H(zr,xr,tr)

= 0.

(4.75)

We remark that

"»)f ! ■Kt- =\\z;( \) ;( \\x'-bf=r\

hr,x'-b)2 IIV

130

r

/ 2n

II Hn l f r \\„

+ z

II

+

ll„

(4.76)

EXISTENCE THEOREMS If tr = 0, then (z,,/) solves the equation (4.73) and hence, xr solves the problem 7CP(/,g,J?;). Otherwise, i. e. if tr > 0, from (4.74) and (4.75), we obtain r r tX+{l-tr)f(x tX+{i-t ) )=z;,= z;, (4.77) r)f(x and

. 4*1

z+r=^-L. i->/ \-t,

(4.78)

Substituting expression (4.78) for z* into (4.77), yields

^-g(x')+{i-tr)f{x') !/(*') = z; which implies for i = 1,2, ...,n

-^4, rM»o 4*1=1 ^ ' \(z-\ (4 JKrh (1-0' i/bl^o-

(4.79) (4 ?9) '

=

Taking fir = lxr\

r>0

Jr.—j we note that (4.79) guarantees the family of points

(i-0 2

to be exceptional if p |-> +oo when r -> +oo. To prove the last

relation we suppose on the contrary that the family \xr\

r>0

has a finite

cluster point x . We remark that the corresponding cluster point t cannot be equal to 1 (otherwise (4.76) and (4.77) contradict each other). Then the continuity of functions / and g together with (4.77) and (4.78) imply the boundedness of the family {zr}r>0 , which again contradicts (4.76) as r-H-oo. This completes the proof. ■ 131

CONES AND COMPLEMENTARITY PROBLEMS The concept of exceptional family of elements for a continuous function opens up a new research direction in the Complementarity Theory, since, when a continuous function from R" into itself is without exceptional family of elements we have that the Explicit Complementarity Problem associated to this function has a solution. Hence, an important problem now, is to find tests or an algorithm which can be used to decide when a given function is without exceptional families of elements. In the paper (Isac, G., V. Bulavski and V. Kalashnikov [1]), is shown that any coercive function is without exceptional family of elements and there are also given examples of functions which are not coercive but which are without exceptional families of elements. It seems that several classes of functions used in the study of Complementarity Problems are without exceptional families of elements.

5. Some special problems in complementarity theory In section 4 we studied several existence theorems. Now, in this section, we present other problems which are necessary to be studied when the Complementarity Theory is applied. • Boundedness of the solution-set Let [H, <, >) be a Hilbert space and K a H a closed convex cone. Given a mapping / : JT -> //, we consider the Explicit Complementarity Problem associated t o / and the cone K: [find xn e K such that ECP(U f K)-\ ' '■{f{x0)eK'and{x0,f{x0)) / ( * ) ) = 0. The feasible set is ? = ix e K\f(x) eK*\ and denote by S the set of solutions. In the paper (Isac, G. [7]) it is remarked that the set ? can be unbounded and because S e ?, it is possible to have that the set S is unbounded. Because the problem ECP(f,K)K) can be the mathematical 132

SPECIAL PROBLEMS IN COMPLEMENTARITY THEORY model for some equilibrium problems, it is important to know when the set 5 is a bounded set. Another reason to study the boundedness of S is the fact that some recent numerical methods for solving the problem ECP(f, K) are based on global optimization. The boundedness of the set S was studied in the paper (Isac, G. [7]) when/ has the form f(x) -T{x) + b, where T G L(H) and beH. Using the numerical range of the operator T, in the paper (Isac, G. [7]), is computed the radius for ball containing the solution-set S. For the nonlinear case, only the following results are known. Let T : K ->• H be an operator not necessarily linear and S: K -> H a nonlinear operator. We say that T is homogeneous of degree p > 0 if T[Xx) = A?T(x) for every x G K and every X e R+. Let (p : R+ —> R+ be a mapping such that qj(t) > 0 for every t > y, where y e R+. We denote by a>(T) - UT(X),XS)\X eJT,|[JC|| = l] and we say that T is K-range bounded if fi)(7) is a bounded subset of R. If T is JST-range bounded then MK{f) = sup 0 and Jf-range bounded, then for every x €K\{0} we have i»,(7)||x||/M"1 < {T[x),x) < MK(f)\x\p*x. Assuming T homogeneous of degree p> 0 andA-range bounded and S ^-asymptotically bounded (see Definition 4.3), we consider the following explicit complementarity problem: find x* e K such that t ECP(T + S,K):- T(x,) + S(xt)eKK\ and

(T(xt) + S(xt),x,)

= 0.

Theorem 5.1 Let (H, <,>) be a Hilbert space and K c / / a closed convex cone. Suppose that T : K —» H is not a necessarily linear operator, homogeneous of degree p > 0 and S: K-> Ha p-asymptotically bounded nongeneous of degree p > 0 and S: K-> Ha (p-asymptotically bounded non133

CONES AND COMPLEMENTARITY PROBLEMS linear operator. If, the following assumptions are satisfied: 1) T is K-range bounded and M^T) < 0,

then then the the solution-set solution-set of of ECP(T ECP(T + + S, S, K) K) is is bounded. bounded. Proof Indeed, for every x e K \ {0} such that ||x|| > r, we have (T(x) + S{x),x) = {T{x),x) + (S(x),x) < MK(T)\\x\r+\\S(x)\\\\x\\ <M*(r)|Mr^to|H

(ll+ll

<MK(T)\\x\r (T)\\x\r\c<44\\x\\ +c<44\\x\\

(5.1)

(5>1)

^M,(r)Hr+c^W)]|4

Since from assumption (1), we have -M^T) > 0, we obtain, using assump tion (2) that there exists a> r such that c^Q|x|) /|x|f < -MK(f)(T) for every x e K \ {0} such that |x| > a. (The number r is the same used in Definition 4.3). Hence, from (5.1), we deduce that every x e JST\ {0} such that |x| >a is infeasible for problem ECP(T + S, K). Thus, we have that, if JC SK \{0} is a solution of the problem ECP(T + S, K), it is necessary to have \x\ < a (since every solution is feasible), and the theorem is proved. ■ Theorem 5.2 Let (//,<,>) be a Hilbert space and K a H a closed convex cone. Suppose that T : K -» H is not a necessarily linear operator, homogeneous of degree p > 0 and S : K -» H a nonlinear operator (p-asymptotically bounded. If, the following assumptions are satisfied: 1) T is K-range bounded and mg(T) > 0, 2) '

rtWfi

lira ■ ^!LJl = o lim II II H — \\x\\p

2) lira ^-SL = o 134 134

SPECIAL PROBLEMS IN COMPLEMENTARITY THEORY then, the solution-set of ECP{T+ S, K) is bounded. Proof Indeed, using assumption (1), for every x e K \ {0} such that r < \\x\\, we have l {T(x) (T(x) ++ S(x),x) S(x),x)>m > mK(T)\\x\r' - -c44\\x\\ ^(1*1 K(T)\\x\r

= ^(r)iHr-^(wji4 ^(r)|Hr-^(W]l4

(5 2)

'

Since mg(T) > 0, we obtain from assumption (2), that there exists a > r such that

v v ^(IHI) " < mJT) \\4P

for every x eK\{0} such that a < be . From the last

inequality and (5.2), we deduce that every x e l \ { 0 } satisfying o < | x | cannot be a solution for the problem ECP(T + S,K) and hence every solution x e K \ {0} of this problem must satisfy ||x|| < a. ■ To finish this subject, we remark that the study of the boundedness of the solution-set of the problem ECP(f,K) is an interesting open problem. • Solution which is the least element of the feasible set Consider the problem ECP(f,K) where K is closed convex cone in a Banach space E and f:K->E*■E" is an arbitrary mapping. The feasible set of this problem is ? = lx e K\ f{x) G K* I. The following problem is important in Economics and in Engineering (Cottle, R. W., J.S. Pang and R E. Stone [1]) and (Isac, G. [12]). Under what conditions does the problem ECP(f,K) have a solution which is the least element of the feasible set? (The least element is considered with respect to the ordering defined by the cone K). Several results about this problem, considered in the linear case are presented in the book (Cottle, R W., J.S. Pang and R. E. Stone [1]) and some results for the nonlinear case in the book (Isac, G. [12]). Now, we consider the case of the Implicit Generalized Order Complementarity Problem. Let E(T) be a locally 135

CONES AND COMPLEMENTARITY PROBLEMS convex space ordered by a regular, pointed, closed convex cone, K c E. Suppose, that with respect to the ordering defined by K, the space £ is a vector lattice. Suppose also, that the mappings Tt:E -> E ,(i= l,2,....m) are given. Consider the problem IGOCp{{f}"=v K\ , where fi(x) = x-Ti(x), (/= l,2,....m) and define H(x) = v(T1(x),.,.,Tm(x)) and G{x) = v(0,i/(x)). The feasible set of the problem IGOCPUf}™^, K\ is in this case the set D={x e K | H(x) < x}. In the paper (Isac, G. and M. Kostreva [1]), it is proved that the problem IGOCPl {/)}",, K\ has a solution if and only if the mapping G has a fixed point in K. Let A c E be a non-empty subset. Definition 5.1 fFe say that T : A —> E is an {sm)-compact operator on A if and only if all sequences of the form T(Xi)>T{x2)>->T{xm)>-,VmeN,xmeAeN,xm <=A contain a r -convergent subsequence. Examples: 1) If T[A) is a sequentially compact set, then T: A -> E is an (sTw)-compact operator. 2) If T : A -» E is such that T\A) is a bounded set and A" is a completely regular cone, or if 7L4) is an order bounded set and K is a regular cone, then T is (sw)-compact. Definition 5.2 Let ¥: E -> E be a mapping. We say that T : A -> E is *Fisotone if: 1) there exists (i + XP)~ and is isotone, 2) T+ *Fis isotone on A. 136

SPECIAL PROBLEMS IN COMPLEMENTARITY THEORY Examples 1) If H s isotone, then T is IP-isotone with *F{x) = 0 for all x e E. 2) The fianction fix) = sin x, with x e R is not isotone, but there exists a constant a > 1, such that/is ¥^isotone, where ^(x) = ax, for all x e R. The following fixed point theorem is necessary. Theorem 5.3 [Isac] Let E(T) be a metrizable locally convex space ordered by a normal closed convex cone K czE. Let A czE be a closed subset and let T : A —> A be an {sm)-compact isotone operator not necessarily continuous. If there exists xQ e A such that T(XQ) < xo, then T has a fixed point in A. Proof For a proof of this theorem one can see (Isac, G. [2]). ■ Theorem 5.4 Let E(f) be a metrizable locally convex space ordered by a normal closed convex cone KaE. Suppose that E is a vector lattice. If the following assumptions are satisfied: 1) G is W-isotone,

2) (/+< (i+vy^G+yjw^K, 3) (i + ¥ ) " (G + VP) is (sm)-compact on intervals, 4) there exists x0 e D = j x e K\ H(x) < x\ such that ix e K\ X < x0 and G(x) = x) is non-empty and compact, then D has a least element x** which is also a least element of the set m M= \x eK\ G(x) = x\ and a solution of the problem IGOCPl{fA*.)" ,K\, •Ii=\'

Proof Let D, = ix e K\ X < x0 and G[x) = x) and for each x« e D* con sider the non-empty closed set = [JC| 0 < x < xt]. Let / = \x\,xl,...,x"\

DXt - [0, x, ] 0 fl A ,

where

[0,x»]0

be an arbitrary finite subset of D*.

We have 137

CONES AND COMPLEMENTARITY PROBLEMS f)D^r». * H *<*> ,w hwhere e r e D £; =£[ = o , [o,x. x ; ] orr]omD. A .

(5.3)

r=\

Indeed, for each r = 1, 2, ...., n, consider xr, e l ^ and denote x00 = A xr.. Since x00 < x.r, for each r = 1, 2,....,w, we have ( 7/ ++TT))--''((G G++TT))((A A::0000)) << (( / + T T )) - 1 ( G + ^ ^ )) ( X ;; ) = X;,

for each r = 1, 2,.., n, which implies (/ + ¥) _ 1 (G + ^ ( J C 0 0 ) < x00. Applying Theorem 5.3 to the mapping (7 + ¥)"'(G + ¥ ) and the set A = [o,x°°]0 we obtain an element z. e [0,x001 such that G(z.) = z.. Hence, z» < x00 < xt < xQ, for each r - \,2,..,n, z. e f t and z. eflT^, which proves (5.3). Then, since Z). is a compact set, we have

flA.^^.

x.eD. i.eft

(5.4)

If x», e P | D^ , then x„ e £>. and x„ < x. for each x, e D». So, x„ is the least element of D*. But xtt is also the least element of D. Indeed, if x„ is not the least element of D, there exists x <=D such that either (5.5) or (5.6) below hold: x< x„ , (5.5) x and x„ are not comparable.

(5.6)

Applying Theorem 5.3 to the mapping (/ + VF)~1(G + VF) and the set A = [o,x = x A X 0 \ , we obtain an element x eA such that G(x) = x,

138

SPECIAL PROBLEMS IN COMPLEMENTARITY THEORY 0 < x < x0, ? e A . Hence, we have xtt < x < x = x AX 0 , which contradicts (5.5) and (5.6). Consequently, x** is the least element of D. Obviously, since Ma D, we have that x.. is the least element of M. The proof is now completed, since x** is a fixed point of G. ■ • The cardinality of the solution-set The cardinality of the solution-set is close to uniqueness of solution. Generally, the uniqueness has been studied only for the Linear Complementarity Problem [see (Cottle, R. W., J. S. Pang and R. E. Stone [1]) and its references]. Some results about the cardinality of solution-set for the Linear Complementarity Problem are proved in (Gowda, M. S. [1]) and for the Nonlinear Complementarity Problem in (Kojima, M. and R. Saigal [1], [2]), (Isac, G. [12]). The study of the cardinality of solution-set and in particular the uniqueness is an interesting open problem in Complementarity Theory.

• Nonexistence of solution For some practical problems and for some numerical methods, it is important to have some nonexistence tests. Not many results are known about this problem. We note that few results about this problem are proved in (Yao, J. C. [1]). • Sensitivity analysis When the Complementarity Theory is applied to the study of some problems in Elasticity, Engineering or in Economics, it is important to study the dependence of solution-set of the perturbations applied to the function. Not many results are known about this problem. • Nonlinear complementarity and quasi-equilibria Some new results establishing interesting relations between complementarity problems and quasi-equilibria have been recently obtained in (Noor, M. A. and W. Oettli [1]).

139

CONES AND COMPLEMENTARITY PROBLEMS 6. Complementarity and fixed points The Fixed Point Theory is currently used in Complementarity Theory. Many existence theorems in Complementarity Theory are obtained using relations between complementarity problems and some fixed point theorems, see for example (Isac, G. [12]), (Cottle, R W., J. S. Pang and R. E. Stone [1]) and (Isac, G. [6], [16]). In addition, when we apply Tihonov's regularization to the study of the Linear Complementarity Problem in Hilbert space, we use some fixed point theorems (cf. Isac, G. [13]). Finally, several numerical methods for solving linear or nonlinear complementarity problems are based on fixed point theorems. In this section, we will show another useful aspect, studied by G. Isac [6], [4], [12], which is the fact that Complementarity Theory can be applied to prove new fixed point theorems. The study of fixed points on cones for nonlinear mappings is an important domain of research in Nonlinear Analysis (Krasnoselskii, M. A. [1]), (Amann,H. [1]). Let (H, <, >) be a Hilbert space, KczHa closed convex cone and/: K-*H a mapping. Consider the Explicit Complementarity Problem

((

ECP(f,K):-

find x, E K such that

'to) = 0.

f{x.)eK' K' aand(xt,f(x.)) ,f(xt))

Theorem 6.1 Let (H,<,>) be a Hilbert space and K cH a closed convex cone. Iff: K -> H has the form fix) = x -g(x), where g : K^>K, then x, is a solution of the problem ECP(fK) if and only ifxt is a fixed point ofg. Proof If xt is a fixed point ofg, t h e n / * , ) = 0, and we have that x. is a solution of problem ECP(fK). Conversely, if we suppose that x, is a solution of problem ECP(fK) then we also have that

140

COMPLEMENTARITY AND FIXED POINTS

i

xt eK and ((xx - xx*, )^ > 0, for every x eeK. K. t , / /((xx, ,) ^

(6.1)

SinceX**) = x* ~ g{x*) and g(x») e K, we deduce that (g(x*) - xt, xt - g(xt)) > 0, which implies 0 < (x, - g(x,), x, - g(xt)) < 0, that is g(xt) = xt. U Considering cp-asymptotically bounded operators, G. Isac proved in paper [6] the following fixed point theorem, [for cp-asymptotically bounded operators see Definition 4.3 and the paper (Weber, V. H. [1])]. Theorem 6.2 Let (//,<,>) be a Hilbert space ordered by a Galerkin cone K(Kn)ni_N. Let T : K —» K be a mapping satisfying the following assumptions: 1) 7X0) ^ 0, 2) T is (ws)-compact, 3) T is (p-asymptotically bounded, where lim (pit) exists and it is different from + co, then Thas a fixed point x* e JST\{0}. Proof The initial proof is in (Isac, G. [6]). A new proof is given in (Jachymski, J. [1]). ■ The next result is a generalization of the classical Hartman Stampacchia Theorem. Theorem 6.3 [Hartman-Stampacchia] Let E be a locally convex space, E* its topological dual and let C be a compact convex set in E. Iff: C —> E is a continuous mapping, then there exists x. e C such that (x - xt ,f{x„)) > 0 for every x e C.

141

CONES AND COMPLEMENTARITY PROBLEMS Proof A proof is in (Holmes, R. B. [1]). ■ Let K a Hbe a closed convex cone and/: K->H. Definition 6.1 We say that f satisfies condition (M) if there exists an element wo e K and a real number r > ||«0|| such that \x - UQ,f(x)j > 0 for all x e Kwith \\x\\ = r. We remark that condition (A/) was considered by J. J. More for the cone /?" in (More, J. J [1]). Proposition 6.4 Let (//,<,>) be a Hilbert space, K czH a locally compact convex cone and f : K —> H a mapping. Iff is continuous and satisfies condition (M), then the problem ECP(f, K) has a solution x* such that \x,\ 0, for for all all xeK (x - x, ,/(*.)) > 0, xeK r. r.

(6.2)

We have two situations: a) |x»| < r. IfxeK, then there exists X e ]0,1[ sufficiently small such that w = Ax + (1 - X)x, sKr, and, from (6.2) we have (w~x*,f(x,)) (w~x*,f(x,))

= = A(x-x.,f(x.)) A(x-x.,f(x.)) •(*)} >>0,0,

that is (x-x,,f(xt)); * » > 0 for all x eK, which implies that x, is a solution of the problem NCPtf K).

142

COMPLEMENTARITY AND FIXED POINTS b) \\x.\\ = r. In this case we have (x. - w0,/(;c.)) > 0 (from condition (M)), and, since (x - x, ,/(*.)) ^ 0 for all x e Jf„ we obtain

(x-u0,f(xt))=(x-xt,f(x,))+(xt-u0,f(xt))>0,,f{xt))>0, = 0, that is we have (x-M (x - w00,/(^))>0,foralljcGJC ,/(*»)) > 0, for all x e Kr .r.

(6.3)

If x e K, then there is a /I e]0,l[ such that v = /bc + (l-/l)w 0 eKr (since |a 0 | < r). If we put x = v in (6.3) we have /l(;c - w 0 ,/(*,)) > 0, that is (x-u (x - 0M , 0 ,/(JC»)) f(x,)) > 0, 0, for for all all x e K. K.

(6.4)

Since |M0| < r, from (6.2), we have (M w 00--** „ / ( * , ) ) > 0 .

(6.5)

Now, from (6.4) and (6.5), we deduce (x-JC»,/(JC»)) > 0, for all x e K, that is x» is a solution of the problem ECP(f, K) with |x*| < r. ■ Taking % = 0 a n d / = / - 71, we get, using Theorem 6.1, the following statement. Corollary 6.5 IfKczH H is a locally compact convex cone, T : K -*K a continuous operator and there exists a number r > 0 such that (T(x),x) < \\x\ for every x e Kwith \\x\\ = r, then Thas a fixed point in K. Remark 6.1 Corollary 6.5 can be considered as a variant of Altaian's Fixed Point Theorem (Altaian, M. [1]) for cones.

143

CONES AND COMPLEMENTARITY PROBLEMS We now introduce a generalization for Galerkin cones of condition (M). Definition 6.2 Let K(K\'"/neAf„ be a Galerkin cone in H and f ' : K ->H a mapping. We say that f satisfies condition (GM) if there exist a bounded sequence of positive numbers {rrnfneN Cand a sequence {u„}neN cK such n} that for every n e Nwe have r u 0i) rn>l n>\un\\> n\> ii) (x-u (oc-un,f(x))>0, n,f(x))>0,

for all x e Kn with \x\ - rn.

We introduced condition (GM) to have more flexibility for condition (ii) on every cone K„. Since Kn can be constructed by the finite element method, it is important to verify condition (ii) independently on every cone Kn and not on the cone K. The main result of this section is the following theorem. Theorem 6.6 [Isac] Let (//,<,>) be a Hilbert space and K(Kn)n^N a Galerkin cone in H. Suppose given two continuous operators S,T: K -> H such that S is bounding, T is compact, and (S + T)(K) c K. If the following assumptions are satisfied: 1) I -S satisfies condition (S)+,(or the more general condition (5) ) 2) I -S - Tsatisfies condition (GM), then S + Thas a fixed point in K. Proof The theorem will be proved if we show that problem -T,K) has a solution (see Theorem 6.1). Since I - S - T ECP(l-S-T,K) satisfies condition (GM), then from Proposition 6.4 we have that for every n e N the problem ECP(l-S-T,KnT,K ) n) has a solution x„ e K„ such that |x„| being bounded and I —S a. bounded operator, we

144

COMPLEMENTARITY AND FIXED POINT

WL*

have that {(l-S)(x„)}n

eN

is norm bounded and from reflexivity we obtain

that {x„}neN

has a subsequence, denoted again by {x„}n£N such that

|(/-5')(x n )]

is weakly convergent to an element u e H. Since T\s a

">nn€N

compact compact operator, operator, considering considering aa subsequence, subsequence, we we may may also also suppose suppose that that [r(xnn>)n&N )| is norm convergent to an element v e H. Because x„ is a solution of the problem NCP(l -S-T,Kn),we ' « ) > w e have (x„, xn - S(x„) - T(xn)) = 0, for every n e N,

(6.6)

(x„, xx„n - S(xn)) = (xn, T(xn)), for every n e N.

(6.7)

that is

From (6.7), we deduce lim(xnn,x,xnn-S(x„))=lim(x lim(x - S(x„)) = lim(x )) n)) = (x(x n,T(xnn,T(x t,v). t,v). n-»oo*

n-froo'

(6.8)

Let {Pn}neN be a sequence of projections such that for every n e N, P„ is a >n<=N projection onto ^T„, and for every x e K, lim Pn(x) = x. n—*ao

We set xn = Pn{xt). Since, for every n e N, x„ solves the problem ECP(l-S-T,KnT,K ) n) which is equivalent to a variational inequality and since denoting zn = xn +1 1 + — \xn, we have that z„ e ^„ (for every n e N) h 1 + (

^"'

we obtain, 0<(z 0<(z„-*„,*,, - % ) - % ) ) n-xn,xn-S(xn)-T(x„)) = {x„ +x„ln,xnn=

S(xnn)-

T(xnn))

= (x„,x„ -S(x„)-

T(xn)) + -(xn,xn ^n

-S(xn)-

T(x % ) n>))

145

CONES AND COMPLEMENTARITY PROBLEMS = — \x„ >{x xnn,x„-S(xn)-T(xn)), which implies (xn,T(xn)) < (x„,xn - S(xn)). By computing the limit in the last inequality, we deduce (x,,v)<(x (x.,v)<(x„u). t,u). .«>.

(6.9)

From (6.8) and (6.9), we obtain lim(xn,xn-S(xn)) {*.))< (xt,u). — - 1

n->oo '

Since (I - S) satisfies condition (S)+, we have that {*„} convergent subsequence \xn\

'*) keN

We denote again lxn\

"*) k&N

N

has a norm

(Condition (S)+ implies condition (S)+).

by {x„}neN

and x, = \xmxn. n—>oo

The proof is

complete if we show that x« is a solution of the problem NCP[l -S -S-T,K). Indeed, let z e if be an arbitrary element. If we denote z„ = Pn[z), we have lim(zn -xn) = z-x,. Since z„ e K„ for every nsN ■N and x„ solves the »l->00

problem ECP{l -S -T,Kn),v/ewe obtain (using again that ECP{l -S-T,Kn,*») ) is equivalent to a variational inequality) (zn-xn,xn-S(xn)-T(xn))>0. T(xn))>0.

(6.10)

Taking the limit in (6.10) as n tends to +a>, we obtain by the continuity of S and Tthat (z-x■t,xJC+, t- X* S(x,)- T[x,)) > 0 for all z e K, which implies that x, solves the problem ECP(l -S-T,K).*) and the theorem is proved. ■ We say that T: H —> H is a cp-contraction (in Boyd and Wong's sense [14]) if there is a mapping
146

COMPLEMENTARITY AND FIXED POINT In (Isac, G. [16]), it is proved that if T : H -> H is a (p-contraction with cp continuous, then I-T satisfies condition(S)+. Corollary 6.7 Let K[Kn")w=N )n^N be a Galerkin cone in the Hilbert space (H,<,>) and S,T: K —> H two continuous operators such that S is a (^contraction with

) be a Hilbert space and JSr(JSr„) ( a Galerkin (Kn)neN * cone in H. If S,T: K -> H are continuous, (S+f)(K) C K, T is compact and S is a contraction orq>-contraction with

0 such that (x,S(x) + T(x)j < \\x\\ , for every xe K with \x\ = r , then S + Thas a fixed point in K. We recall that in a Hilbert space a mapping T is a pseudo-contractive mapping if and only if I — T is monotone. In this sense, we say that S:K-> K is p-pseudo-contractive if / - S is strongly p-monotone. We know that a strongly p-monotone mapping satisfies condition (S)+ (Isac, G. [16]). Corollary 6.9 Let (//,<,>) be a Hilbert space and K(K n) isJV a Galerkin (E»)neN cone in H. Let S,T: K -> H be two continuous operators such that (S + T)(K) cK,Tis' is compact and S is p-pseudo-contractive and bounded. If I —S - T satisfies condition (GM), then S + Thas a fixed point in K. We denote by " <" the ordering defined by K in H Corollary 6.10 Let (H,<,>) be a Hilbert space and K(Kn")neN )neN a Galerkin cone inH. Suppose, given two continuous operators SQ,T:K^>H,>H,i such that

147

CONES AND COMPLEMENTARITY PROBLEMS S0 is bounded, T is compact, and S0 (x) < T(x) + x for all x e K. If the following assumptions are satisfied: 1) So satisfies condition (S)+, 2) So - T satisfies condition (GM), then there exists an element x* e K such that S0 (x*) = T(x-). Proof We apply Theorem 6.6 withS = I-S0. ■ Some similar fixed point theorems for multivalued mappings are proved in [74]. Remark 6.2 Theorem 6.6 is valid if we replace in assumption (1) condition (S)+by{S)\. Another subject about Complementarity Theory not presented in this chap ter, but completely open to new and interesting investigations, is the Complementarity Problem associated with vector valuedfunctions. This kind of Complementarity Problem has interesting applications to Pareto Optimi zation. The reader can obtain new informations about this subject in the papers of Chen, G. Y. and Y. Q. Yang [1] and their references, Giannessi, F. [1], Isac, G. [12]. The paper of Hadjisawas, N., D. Krawaritis, G. Pandelidis and I. Polyrakis [1] has of applications to the study of the Com plementarity Problem for vector valued functions.

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165

CHAPTER 2

Metrics on Convex Cones

1. Introduction One of the most interesting and specific instruments of investigating a large class of nonlinear problems on convex cones, is based on Hilbert's projective metric, and its variants. In 1895, David Hilbert introduced a metric in an early paper on the foundations of geometry (Hilbert, D. [1]). This metric, reconsidered in 1903 (Hilbert, D. [2]), is now called Hilbert's projective metric. Special cases of Hilbert's metric occur in the earlier work of Cayley, Beltrami and Klein (Klein, F. [1]). In his papers, Hilbert constructed a model for a metric hyperbolic geometry in which there are three noncolinear points forming a triangle with the length of one side equal to the sum of the lengths of the other two sides. The metric defined by Hilbert was, for a long period of time, neglected, until 1957 when Garrett Birkhoff [1] and H. Samelson [1], independently remarked on the impor tance and the depth of Hilbert's idea. In 1907, O. Perron proved the follo wing result which is now considered a classical theorem (Perron, O. [1]). We consider the space Rm as an ordered vector space, with the ordering defined by the closed pointed convex cone R™. Theorem [Perron] If A is positive linear transformation from m Rmm into itself, i.e., Ax > Ofor any x > 0, then there exists anx0> 0 (xQ e R+ ) such A"x x x>0 that n->oo lim« jr = -f%, *™\\A"xl \x \' for any x > 0 ■ Q

"-*00 JA"xj p 0 |

In 1908 and 1909, G. Frobenius reconsidered the problem studied by 167

METRICS ON CONVEX CONES Perron, but with respect to primitive non-negative m x /w-matrices (Frobenius, G.[l], [2]). We recall that a non-negative m x wi-matrix, is said to be primitive if there exists a natural number r, such that Ar > 0. The result obtained by Frobenius, is the following. Theorem [Perron-Frobenius] If A is amx m-primitive matrix, then there exists a vector xo {unique up to proportionality) such that Ax0 = AQKQ, for A"x x xeR?\{0}. some AQ > 0, andfurthermore, n—>oo lim j, « = j-^n for all xe R™\ {0}. ■

i-||^|

Po||-

*^"P**| Foil

The Perron-Frobenius Theorem is of central importance in economics (Fujimoto, T.[l], [2]]), (Morishima, M. [1]), (Morishima, M. and T. Fujimoto [1]), (Oshime, Y. [1]),( Solow, R. M. and P. A. Samuelson [1]) and (Thorlund-Petersen, L. [1]), in population biology (Golubitsky, M. E. B. Keller and M. Rothschild [1]) and (Nussbaum, R. D. [1], [2]), and in other sciences (Borwein, J. M., A. S. Lewis and R. D. Nussbaum [1]) and (Nussbaum, R. D. [1], [2]). Since its discovery, this theorem has been extended in many ways. The first extension of the Perron-Frobenius Theorem to infinite dimension, is obtained by R. Jentzsch in 1921 and it is the generalization to linear integral operators (Jentzsch, R. [1]). Nonlinear extensions of the Perron-Frobenius Theorem have been given in 1953 (Solow, R. M. and P. A. Samuelson [1]), in 1964 (Morishima, M. [1]), in 1974 (Morishima, M. and T. Fujimoto [1]), in 1979 and in 1980 (Fujimoto, T. [1], [2]), etc. In 1957, Garrett Birkhoff showed that the Perron-Frobenius Theorem for non-negative matrices and Jentzsch's Theorem for integral operators with positive kernel, could both be proved by either an application of the Banach contraction principle or by Picard's fixed point theorem, in suitable metric space, where the metric is a projective metric, similar to projective Hilbert's metric (Birkhoff, G. [1]). At about the same time, H. Samelson used the Hilbert's metric in finite dimension to give a simple proof for Perron's Theorem (Samelson, H. [1]). 168

INTRODUCTION After 1960, G. Birkhoff and L. Kotin used in several papers the projective metric to prove the existence and uniqueness of solutions to certain type of systems of linear differential equations (Birkhoff, G. [2], [3], [4]) and (Birkoff, G. and L. Kotin [1], [2], [3]). After 1970, P. J. Bushell, in several papers, applied the projective metric to the study of some nonlinear problems (Bushell, P. J. [l]-[7]). For example, he showed that a positive homogeneous operator of degree a, is an a-contraction with respect to Hilbert's projective metric, and consequently, various existence theorems can be proven using the contraction mapping theorem. Working in this sense, P. J. Bushel obtained interesting existence theorems for some kinds of Volterra and Fredholm nonlinear integral equations. The existence theorems for nonlinear integral equations, in particular for Hammerstein equations, obtained by A. J. B. Potter, in the papers (Potter, A. J. [l]-[3]), are also based on the Hilbert's projective metric. The projective metrics also have other kinds of applications, as for example, in the study of some problems in dynamic programming (Bellman, R. and T. A. Brown [1]), in the study of some operatorial equations in C -algebras (Danes, J. [1]), in the study of some iterative processes (Nussbaum, R. D. [1], [2]) and also in the study of the classical DAD problem. [We recall this problem . Given an rcx«-matrix A with nonnegative entries, the DAD problem is to know when it is possible to find diagonal matrices Dx and D2, with positive diagonal entries, such that the scaled matrix DXAD2 is doubly stochastic, (i.e. having row and column sums 1)]. This problem is an important special case of a large class of matrix scaling problems, with numerous applications in diverse fields. (Borwein, J. M., A. S. Lewis and R. D. Nussbaum [1]), (Nussbaum, R. D. [1], [2]). The Hilbert's projective metric and its generalizations, as for example, the generalizations proposed in (Thompson, A. C. [1]), (Stetsenko, V. Ya. and B. Imomnazarov [1]), (Turinici, M. [1]) etc., represent a useful and powerful mathematical instrument, in order to study some nonlinear problems on convex cones. The principal aim of this chapter, is to introduce this special domain of nonlinear analysis to the reader. This chapter can also be considered as a complementary part of Nussbaum's memoirs (Nussbaum, R. D. [1], [2]). 169

METRICS ON CONVEX CONES

2. Hilbert's projective metric In this section we will describe Hilbert's projective metric and we will study its essential properties. Let IE,I I) be a Banach space and Ka E a closed pointed convex cone, that is, K is a closed subset of £ satisfying the following properties: l)K+KcK;c t f ; 2) X K c K for all X e R+ and 3)Kf](-K)={0}.{0}. Define the ordering induced by K in the usual way; x < y if and only if JC>0}. In this case the ordered Banach v - x e K. Clearly K = ix sE\x>0\. space (JE,| \\,K) is Archimedean, that is for x, y e E, n x < y for all n e N imply x < 0. Also, because K is closed, we have that the intersection of K with every straight line is closed. Definition 2.1 We say that a function p : K x K -> J? U{°°} is a projective metric on K if for all x, y, z e Kwe have: ;; = p(v,x), 1) /?(x,v) p(x,y) = p(y,x), 2> p{x,^)0, 3) p[x,y)>0, 4) p{x,y) for some X A eR+\ {0}. p(x,y) = 0, if and only if x = Ay Xyfor Remark 2.1 If p is a projective metric on K, then p(x,y) is constant on rays, that is p(Ax,py) for all A, p > >00.. p(A,x,ny) p(x,y),foT&\\X, (2.1) ■My) = = p(x,y), Indeed, we have Indeed, we have ffyx,ny) ,My)*< p(Ax,x) + p(x>py) - p(x,/i v) < p(x, v) + >o(v,/i v) = p(x, v), ffyx,ny) < p(Ax,x) + p(x>py) - p(x,/i v) < p(x,y) + p(y,p v) = p(x, v), and and 170

HILBERT'S PROTECTIVE METRIC p(x,y) < fty) + p(tiy, y) == p(x,My)^ p(x, ^y) < p(x, X x) + p(A x, //v) p(x,y) < p(x, p(x,{iy)+p(My,y) P(X,AX)+p(Ax,{iy) = p(Ax,juy). For every xy e Kwe define M[x,y) | a > 0| x < ay] M(x,y) = inf mf{a>0\x
(2.2)

m(x,y) = sup{/3>0\x>/3y}. m{x,y) sup{/? > 0| x > /3y\ .

(2.3)

and

When the set | a > 0 | ; c < a v } is empty we put M(x,y) = oo. We have m(x,y) > 0 and we can show that rr{x,y) =

1 M(y,x) M(y,x)'

(2.4)

The next result gives a relation between m(x,y) and M(x,y). Proposition 2.1 For every x, y & K\ {0} such that M(x, y) is finite we have m[x,y)y <x< M(x,y)y. m[x,y)y<x<M(x,y)y.

(2.5)

Proof From (2.2) it follows that for each n e Nwe have x < M(x, V) + — v n\ which implies that n\x - M(x,y) v] < v. Applying Archimede's axiom, we deduce x - M(x,y)y < 0. From the definition of m(x,y) and using again the fact that the ordering is Archimedean, we obtain x - m[x,y)y > 0 and therefore the proposition is proved. ■ 171

METRICS ON CONVEX CONES Proposition 2.2 For all x,yeK\{0}\{0} we have 0<m (x,y) < M(x,y) < oo. Proof From (2.3) we deduce m{x,y) > 0. Suppose that for some x,yeK\{0} we have mix,y) m{x,y)>> M(x,y) (2.6) In this case we have -[M{x,y)-m{x,y)]y>0. -[M{x,y)-m(x,y)]y>0.y > 0 . Because we also have [M{x,y)-m{x,y)]y [M(x, V) - m{x,y)]y = [M(x,y)y-x] [M(x,y)y - x] + [x -[x-m(x,y)y]>0, m(x,y)y] > 0, we deduce \M(x,y)-m(x, m(x,_v)]y = 0, and since M(x,y) - m(x, v) * 0 we have v = 0, which is impossible since by assumption y e K\ {0}. ■ Definition 2.2 Hilbert'sprojective metric dh (•,•) is defined in K\ {0} by: i) <4(0,0) dh(0,0) = = 0, ii) dh(x,0) = dh(0y) (S>y)==+oo, +oo,

i W ,v)=ln[^\ _m[x,y) _ Remark 2.2 From (2.4) it follows that d4(x,v)=ln[M(^v)M(y,x)]. h(x,y) = ln[M(x, v)M(y,x)]. Proposition 2.3 77ze function dh is a projection metric in the sense of Definition 2.1. Proof By Remark 2.2 and because M{x,y) < M(x,z)-M(z,y)w) it follows that the function dh satisfies properties (1) and (2) of Definition 2.1. If dh(x,y) = +oo, we obviously have dh(x,y) > 0. If dh(x,y) < +oo and M(x,y) < +oo, then by Proposition 2.2 we have m(xy) < M(x,y), with equality if and only if x = m (x,y)y. Therefore, we

172

HILBERT'S PROJECTIVE METRIC remark that properties (3) and (4) of Definition 2.1 are also satisfied for the function df and the proposition is proved. ■ Remark 2.3 Note that m(x,y) = 0 if and only if x - Xy g K for all X > 0. Because K is a convex cone, it is equivalent to say that the line from y to x leaves K at x. Thus, we have that dh(x,y) < +oo, if and only if x and y are interior to the intersection of the line through them, with K. It is interesting to consider the following geometrical interpretation of the numbers m = m(x, y), M = M(x,y) and df,(x,y) proposed in (Kohlberg, E. and J. W. Pratt [1]). This geometrical interpretation is the following. Suppose that x and y are two elements in the relative interior of K. Replacing x by Ax for a sufficiently large X > 0, we obtain that: in the twodimensional subspace generated by x and y, the line through x and y leaves K at two points, a and b. By the definition of m(x,y), the point x- my is obtained by moving from JC in the - v direction, until the nonnegativity constraint is violated. By a similar reason, the point, yyy

1

xx :is obtained by moving from y in the

M M -x direction until the nonnegativity constraint, used in the definition of M(x,y) is violated. We have the following figure:

ax , %,xb By similar triangles, we have that m = = and M=, M = , which implies ay yb ay yb

173

METRICS ON CONVEX CONES .. s, , M(x,y) , ay oyxb xb drf„x,y = IInn —vM ^' = ll nn ^—==. . h(x,y) ) = m[x,y)v] ax TW(X, ax yb yb

,„ m (2.7)

Hence, from the (2.7), we have that dh{x,y) is the logarithm of the cross ratio of (a,x,y,b). We recall that the cross ratio of any four points {a,x,y,b) lying in that order on a straight line in any linear space, is defined as , ,.. ay xb %{a,x,y,b) = ., *(«,x,*A) = = ax S^ 4yb ax yb

(2.8)

where ay,x6,ax and vi are distances along this line. The cross ratio is an important notion in projective geometry and its fundamental property, is the fact that it is invariant under projection (Bussemann, H. and P. J. Kelly [1]). Proposition 2.4 If x, y, z are arbitrary elements in K and z is a positive combination ofx andy, then dh(x,y) = dh(x,z) + dh(z,y). Proof Because for A,// > 0 we have dh(Ax, {Ax,ny) ny) = dh{x,y),

(2.9)

it is sufficient to consider only convex combinations. In this case using formula (2.7) we have

dhh(x,z) + d E ••^5 42 ]1 ==4J ^S 441 | " =d^y) dh(x,y). h(z,y)==II nnf a dkfey) ■ axzb az yb J axzb ax ax yb yo * Recall that a function

0 with the property A + n = 1. 174

HILBERT'S PROJECTIVE METRICS Proposition 2.5 Hilbert 's projective metric is a quasiconvex function in each of its argument. Proof Let x,y,z e K be arbitrary elements and X./J. such that A,ju > 0 and X + H=\. We must show that dh{Xx (Zx + juz,y) juz,y)<max{d < max{d h(x,y),d h(x,y),d h{z,y)}. h(z,y)}.

(2.10)

Note that, to show inequality (2.10), it is sufficient (because of formula (2.9)) to show that ddh X+ \2 X+ Z y 2Z'y)

\2 2 ' )

max d y x d yz y d z y ~~max{d{^x*( ' )'>^ )' ' )}>>( ' )}■ ■

Indeed, we have

41*4^)

jA^( l 1 \^ ~)^ Jj ) " + m(z,^) "> 2 J m[x,y) + m[z,y) \2

=

max{dh(x,y),dh(z,y)}.

m In some particular cases, it is easy to compute the dh(x,y). In this sense we consider the following examples: Example 2.1 Let£ = .R" and K=R". Rn+.

If x,y e Int(JT) then we have

ft

M(x,y) = max< —'- >, and

' UJ

175

METRICS ON CONVEX CONES

fei U-J

m[x,y) = nix,y) = min< min<—' —->'-> 1

which imply that

UJ

tel-

dhh(x,y) ]nmzx.\-LLj d {x,y) == \nmaz\-L-\. \. U \ U [y [yiXj iXj\

Example 2.2 Let E be the real Banach space of real n x n symmetric matrices and K the cone of positive semidefinite matrices in E. In this case the Int(K), is the set of positive definite matrices in E. It was shown in AM(B-U) (Bushell, P. J. [4]) that for A,B e Int{K), dh{A,B) = In—} - ' where 1 1 K(B~ A) ■

XiJJD) and XJJ)) denotes the maximum and minimum eigenvalues of D, respectively. Example 2.3 Let E = R" and K = Rn+. In (M. Golubitsky, E. B. Keeler and M. Rothschild [1]) is used the following method to compute dh(x,y) forx,y eInt(K). Denoting by A(x,y) = i(a,b) eR + \ {0} | x < ay and ay < bx\ , we have that dd

hix>y) =

fmin{ln(ft)|(a,A) fmin{ta(ft)|(a,A) eA{x,y)} eA{x,y)} ifA{x,y)* ifA{x,y)* [+oo ifif A(x,y)- [+00 A(x,y)-

The main property of Hilbert's projective metric is the fact that it contracts a wide class of mappings as explained below. Let [E,\ |) be a Banach space ordered by a closed pointed convex cone K c E. Elements JC and v belonging to K \{0}, are said to be linked if and only if there exist finite, strictly positive real numbers X and n with x < X y and y < ju x [or equivalently, if and only if there exist finite, strictly positive real numbers a and J3 with ay < x < fiy ]. This is an equivalent relation which splits K into a

176

HILBERT'S PROJECTIVE METRICS set of mutually exclusive constituents (components). For the next results, wesetJT + =iir\{0}. We say that a linear mapping T: E -> E is positive if T(K+) c K+. For every (x,y) € ExK+we define: sup(x / v) = inf [/I Ay) \X\Ixx < E is a positive mapping, then |inf(x/v) by definition the protective diameter ofT is A(r) = A{T) = sup{d sup{dhh(T{x),T{y))\{x,y) (T{x),T{y))\{x,y) eK eK++ xx K K++]] and for every (x,v) e EX K+ the oscillation ofx andy is O,SC(JC I/ v) osc(x v) == sup(x sup(x // y) y) -- inf inf(x (x I/ y). 7).

In (Bauer, F. L. [1]) is showed that 0 < osc(x/y) < +00 for all (x,y) e E xK+. Note that the oscillations are well defined since, a Banach space, ordered by a pointed closed convex cone, is an Archimedean space. If mf(x/y) = + 00 then ny < x for all n e N and contrary to hypothesis y < 0. Hence, we have -00 < m£{x/y) < +00 and by a similar argument we deduce that -00 < sup(x/y) < +00. Considering again a positive linear mapping T:E->E,vtQ define , . [\osc(T(x) I T(y))\ . . N(T) = sup^ — ± - Y — p ^ 0< oscixI v < +00,(x,y) jI cwc(;c/}>) osc{xly)

]

<EE sExKA

jj

and

177

METRICS ON CONVEX CONES k{T) = sup{^(r(x)/ sup{dh(T{x)I T(y))/dh{x,y)\0y)\ h{x,y)

< +»,(*, +*,(*, v) €K++ x K K++]. ).

The number N(T) is named the Hoph oscillation ratio (Hopf, E. [1]), while the number UT) is named the Birkhoff contraction ratio (Birkhoff, G. [1]). Relations between N(T) and k(T) are given in the following results. Proposition 2.6 IfT: E-> E is a positive linear mapping then : 1) 0 < inf(x/v) < mf(T(x)/T(y) < sup(71(x)/71(y)) < sup(x,v), for all (x,y) €€ K+ (x,y) K+ xx K+, K +,

£

2) N(T) = tanh(±A{T)j A(r) Proof An elementary proof of this result is in (Bauer, F. L.[l]). ■ Proposition 2.7 IfT :E-*Eis

a positive linear mapping then N(T) < k(T).

Proof The proof is based on a simple limiting argument. For more details see (Ostrowski, A. M. [1]). ■ The next result was proved for the first time in (Bushell, P. J. [3]).

fT.E^E is apositive linear mapping then k(T) < N(T). Proposition 2.8 IfT:E^>Eisa Proof Let (x, y) e K+ x K+ be an arbitrary element, such that 0<^(x,y)<+oo. 0. By Proposition 2.6 and from the definition of Hilbert's projective metric dh{^) we have 0 < inf(x / v) < inf(7T(x) / 7{v)) 71^)) < sup(T(x) s u p ^ x ) I/ T\y)) <sup(x/y) < S U p ( x / v ) <+oo <+Q0 and 0 < inf( v / x) < inf(7T( mf(T{ v) / ^x)) < sup(T{y) snp(T\y) I T\x)) < sup( v / x) < +00 178

HELBERT'S PROTECTIVE METRICS As it is proved in (Bauer, F. I. [1]) we have for every a > 0, sup([« y] I y) = aasup(x sup(x /I y) +1, +1, sup([axx + y\l

(2.11)

inf([ax inf ([a x ++ y]/y) v] / y) == aamf(x/y) inf(x I y) ++1l..

(2.12)

and

in4]

Hence, from (2.4) and the inequality [sup( r(y) / T[x)) - inf(j{y) I 7T(J<:))] < JV(r)[sup(>> / x) - mf{y I x)] it follows that l [{inf(T( [ { i n fXt)/T(y))}^ / r ^ --{' SW - f(T(x)/T(y))}-' su^)/^))}"1'

))}"']

[<'

^
]

Replacing JC by ax + y (a>0) and using (2.12) and (2.11) we deduce that

[suP(r(x)/r(v))-inf(7t^)/r(v))] T{ v)) + l][a inf(r(x) / T(y)) [a sup(r(x) / 71( r(>>)) +1]

^ N(T)\sup(x/y)-mf(x/y)} iy(r)rsup(x/>.)-inf(w^]

> f o^r a ; 0^ ,-,foror>0. [a sup(x / y) +1] [a inf(x / >>) y) +1]' +1]'

^r

Integrating with respect to a (over [0, a]) we obtain

gsup 4\ain£(T(x)/T(y)) f5f51 )+1+ 1l\ * "W\amf(x/y) l i ^ ^+ \_" |ainf(7{x)/7T(>>)) + lJ

v ; v ;

Lainf(x/y) + l_

and computing the limit a s a ^ +co it follows dhA c/ (T{x),T{y))
(2.13)

179

METRICS ON CONVEX CONES Obviously from (2.13) we have that HT) < N(T). ■ We have also the following classical result. Theorem 2.9 [Birkhoff]

Let (E\ J) be a Banach space, ordered by a

closed pointed convex cone K cz E. If T : E —» E is a positive linear mapping then:

Q'

1) dh(T{x),T{y)) < tanhQA{f)Jdh{x,y),forfort all (x,y) e K+ x K+, 2) if T has finite diameter with respect to dh(,\ then T is a contraction with respect to the projective metric dh{,) in each constituent of the quasimetric space (K+, dh). Every constituent is invariant under T. Proof Part (1) of the theorem is a consequence of Propositions 2.6-2.8,

G

since k(T) = N(T) = tanhl — A(f) I. If x andy are two arbitrary elements in the same constituent, then so are T\x) and T(y), and thus, part (2) of the theorem follows. ■ Remark 2.4 Theorem 2.9 is also valid in a locally convex space ordered by a closed pointed convex cone. Now, we consider the case of nonlinear mappings. Suppose that (E,\ |) is a Banach space, ordered by a closed pointed convex cone K
HILBERT'S PROJECTIVE METRICS remark that it is sufficient to show that dh(x,y) is finite for all x,y e Int(K). Indeed, let s > 0 be an arbitrary real number such that B(x,s) c K and £

£

IWI

IN

B(y,£) c K. Then x ~j-rty ^ 0 and v - -r—rx > 0 and hence £E Ibdl m[x,y) < M(x,y) £ is a mapping (not necessarily linear), we say that T is no«negative if 7T(jr) c JT and if T [lnt(^0] c Int(K), we say that T is positive. Let r : E —> £, then T is said to be monotone increasing ii x < y implies 7\x) < TXy). If r is positive, the projective diameter A(7) of T is defined by A(r) = sup{^(7tx),T(y))\ x,y €Int{K)} and the contraction ratio k(T) of T is defined by k{T) = M{X\ dh(T(x),T(y)) < Mh{x,y), for all x,y eInt{K)}. Also, in this case, for every x,y e Int(K), we define osc(x/y) =M(x,y) - m(x,y). If T is positive and if T(fa) = Xp T(x) for all x e Int(K) and X > 0, we say that T is positive homogeneous of degree p in Int(K), (p > 0). Theorem 2.11 Let T : E —> E be a monotone increasing positive mapping which is positive homogeneous of degree p with respect to Int(K). Then the contraction ratio k(T) does not exceed p. Proof Since we know that m (x,y)y <x< M(x,y)y, we obtain 181

METRICS ON CONVEX CONES P P [m(x,y)} T{y). [m{x,y)]PT(y)
(2.14)

From (2.14) and the definition of mQ (resp. MQ) we have

:[*MT

[m(x,y)]P<m(T{x),T{y))<M(T{x),T{y))<[M( <m(T{X),T{y))<M(T{x),T{y))<[M(XX,y)} ,y)}PP

H*.^)]'

which implies that k( T) < p. ■ The following result is an interesting consequence of Theorems 2.9 and 2.10. Theorem 2.12 Let (E,\\ |) be a Banach space, ordered by a closed pointed convex cone KaE, such that Int{K) is non-empty and S=(lnt(K)f]S(0,l),dh*) ) is a complete metric space. Let T : E -» E be a mapping. 1) If T is a monotone increasing mapping and positive homogeneous of degree p[0 0 as in the proof of Theorem 2.10, such that B(x,s) a K and B(y,e) a K, then we have £

S

x > Tr-j,y and y > J - J ^ which imply that x and y are linked. We consider

Ml

HI

T(x) the mapping f(x) = ,,_) L which is the composition of a strict contraction

II Wll (see Theorems 2.9 and 2.77) and a normalizing isometry (because the (see Theorems 2.9 and 2.11) and a normalizing isometry (because the property dh(Jbc,yy) = dh(x,y) for all X,n> 0 and x,y e Int(K)). Applying the Banach contraction mapping theorem, we obtain a unique element u e S 182

HILBERT'S PROTECTIVE METRIC such that f(u) = u.UT is in the case 1) then the element JC» = \\T(u)\\m~p)-u is such that T\x*) = xt. If Tis in the case 2) we have 1\u) = \\T(u)\\-u, that is u is an eigenvector in Int{K). Finally, we can show that u is unique in £. ■ Suppose again that (Z?,| ||j is a Banach space ordered by a closed pointed convex cone K c E such that Int(K) is non-empty. For r e JR+\{0} we define Sr = \x eInt{K)| \x\ = A . We can show that for all r e /?+\{0}, (£r,dh) is a metric space. Suppose that T : E —> E is a positive nonlinear mapping. The contraction ratio, k(T,£r) of the positive mapping T restricted to Sr is HT.Sr) = inf{^ inf[X > > 0| 0|ddhh(T[x),T{y)) (T[x),T{y)) < <M Mhh(x,y) (x,y) for for all all x,y x,y

eS eSrr}.}.

Let T : E -» E be a positive mapping and let a e R+. We say that T is a-corccave (resp. a-convex) if and only if T(Xx) > A?T(x) (resp. T{Xx) < XaT{x)) for all x e Int(K) and A e ]0,1]. Example 2.4 Let Q be a closed bounded subset of if" and let E = $Q.) (the space of continuous real-valued functions defined on Q endowed with the supremum norm). Let K a 0Q) be the cone of non-negative functions on Q. In this case, Int(K) is the set of strictly positive functions on Q, hence, Int{K) is non-empty. Let h Q x Q - > i?+\{0} be a continuous function and define JL:E^E by Ax{s)= J" j*x(s)= J k(s,t)x(t)dju{t), Jn

ssQ.,

where // is Lebesgue measure on R". J is a continuous linear mapping from £ into £ such that J>{Int{K))<^ Int(K). L e t / : Q x R+\{0} -> tf+\{0} be a continuous mapping. Using the mapping/ we define ? : Int(K) -> 7«?(A)

183

METRICS ON CONVEX CONES by 9x(s) =M*(s)) for all s e Q. If/fax) >f{s,y) or f(s,x) y and/5,^;) > ffex);,x) orfa,Ax) < Xaf{s,x) for all (s,x) e Q x R+\{0} and X e ]0,1], then "? is increasing a-concave or decreasing a-convex. The Hammerstein operator T = ^°^ maps Int(K) into itself and is a-concave or a-convex. Before we can get the next result, we note that T is a-concave (resp. a-convex) if and only if T(ji x) < /fT(x) (resp. T(ji x) > ^1\x)) for all x e Int{K) and // > 1. Theorem 2.13 Let \E,\ |) be a Banach space ordered by a closed pointed convex cone K czE such that Int(K) is non-empty. Assume that the norm of E is monotonic {i.e., 0 < x < y implies \\x\\ < \\y\\). Let T : E —» E be a positive mapping and a e R+. If either T is increasing a-concave or decreasing (-a)-convex, then k (T,£r) < afor all r e /?+\{0}. Proof We give the proof in the case T is decreasing and (-a)-convex. The proof in the other case is similar. Suppose that r > 0 and x,y e £r are two arbitrary elements. Since we know that 0 < m(x,y)y <x< M(x,y)y we have 0 < m(x,y) < 1 < M(x,y) (because the norm || || is monotonic on K). Applying the fact that T is decreasing and (-a)-convex, we obtain a a [M(x,y)]'aaT{y) [M(x,y)]T{y) < T[M{x,y)y] < T{x) < T[rr{x,y)y\ < [m{x,y)\ [m{x,y)\ T[y) T[y)

r.and m(T(x),T(y)) > [M(x,y)]~ , which

Thus, M(T{x),T{y))<[m{x,y)]'a

a

imply that ,

,

\M(T{x),T{y))]^

\(MY]

,

d =l n ^ - ^ ^ j < l n i y ^ad Jf(x,y), h{T{x),T{y)) ^4r ,7tv)) = W h^adfcy). dh(T(x),T{y)) - In! ^ ^ J * ^ { - j and the theorem is therefore proven. ■ and the theorem is therefore proven. ■ 184 184

HILBERT'S PROTECTIVE METRIC Corollary 2.14 If the assumptions of Theorem 2.13 are satisfied with a < 1 and in addition (£ndh) is a complete metric space, then there exists an elementxt e Sr and a real number X*> 0 such that T\x*) =Xt x*.

rTix) v

Proof Consider the mapping/: Sr -> Sr, defined by f(x) = ,,

A for all

I Wll

x € Sr- By using the properties of Hilbert's projective metric, and applying Theorem 2.13, we obtain dh(f(x),f(y)) for all \y)h= dh(T(x),T(y)) 0). It follows that ifX\ > A^ then, x\ < x2. Proof Let T be is decreasing and (-a)-convex. If we suppose, for contra diction, that xi i X2, then we have M(xix2) > 1. Thus, using the fact that T is decreasing and (-a)-convex, we get: xXlx = = (\IX{)T{x{) (l/wOd) >> (i/A,)r (1/A,)r [M(xhhxx22)x )X22]\ > {\IX{)[M(x {\IM)[M(xhhxx22)rT{x )rT{x22))

= Q,{\2IX )x22rx ya2x. 2. 2lX x)[M{x x)[M(xhhxx22)x

Since X2 < X\ and M(xi,x2) > 1, we have (X21 X^M{x^ I x2)]

(2.15)

< 1.

By applying Tto both sides of inequality (2.15) and using the (-a)-convexity of T, we obtain

185

METRICS ON CONVEX CONES X X a xx<(X x, <(X2IX 2 IX x) X~)

a

\M{xx lx2)Y X2

which implies that M(X M(xXx,X,x2)<(X 2/X2X) 2)<(X

a [M(x ,x 2 )] a .. IAt)~a[M(x ux21)]

Since Af(x,,x 2 )>l, it follows that XX<X2, which is a contradiction. When T is a-concave and increasing, the proof is identical to the proof of Theorem 6.1 presented in (Krasnoselskii, M. A. [1]). ■ We say that a positive operator T: E->E, is strictly (-l)-convex if for each x e Int(K) and for each X, where 0 < X < 1, there exists a number, rj{x,X) > 0, such that

^T{Xx)<{\-rj)X^ ^ ( l - ^ r ' lT(x). ^x).

(2.16)

An operator, T : E -> E, is strictly (-l)-convex if and only if, for each x e Int(K) and X > 1 there exists a number t],(x,X) TJ,(X, X) > 0, such that T(Xx)>(l + T],)X-l-T(x). For a positive operator which is strictly (-l)-convex and decreasing we have a result similar to that of the Theorem 2.15. Theorem 2.16 Let \E,\ |) be a Banach space, ordered by a closed pointed convex cone, K c E, such that Int{K), is non-empty. Let T : E -> E be a positive operator which is strictly (-l)-convex and decreasing. Letxx andxi be eigenvectors of T corresponding to eigenvalues Xx and X2 respectively (xh x2 e Int{K); XhXi > 0). It follows that if Xx > X% then x, < x2. Proof Suppose that x, i x2. Then, by definition of M(xx,x2), we have M(xhx2) > 1 and

186

HILBERT'S PROTECTIVE METRIC

J_

*i = [jjT(xr)Hs [j-fiM^Xih] i

i-

( 1l \ 1+

! 1+ *M*i.*2)]~ !%) MiH( i+ M x MiH( *M*i.*2)]~ %) -h-( ^)[ ^' 2)r%) (A ") (Pi }

1

= = -~ -~ (\+V*\M{x (\+V*\M{xxx,x ,x22)] )] (x (x22), ), where 77, > 0. Suppose then, that where 77, > 0. Suppose then, that

©

« = [^-](l + /7.)[M(x1^2)]-1
= T(xx) < 7l(ai 2 ) < (l - Tj)a~1T(x2) = (l - 77)" A2x2, where 77 > 0.

Hence, it follows that /L,x, < or_1/l2^2, which implies that 1

x ,, << l| -X

\M(xx,x22)jC )x2.. jAf(jCj,JC

U+17J

By definition of M(xi,x2y, however, it follows that we have M(x\,xz) ^

(

1 "l

M(xx,X2), \l + TJtJ

which is impossible. (AS]

M(XX,X2)) M*i,*

2 . Thus, we find that, we must have a > 1. We get —H > — \AXJ \+T]t Because to the fact M(xhx2) > 1 and T is decreasing, it follows that

K >(l + T] )[M(x x )]~ T(x )

T[x2)>l[M(xl,x2)x2]

t

u 2

1

2

and hence,

187

METRICS ON CONVEX CONES M Mix (xi'xi):il x ) —1v + -77.2' > 1. This implies that X2 > 1, which is impossible. By this 1+7* contradiction, we deduce that we must have xi < x-i. ■ contradiction, we deduce that we must have x\ < x-i. ■

Theorem 2.17 Let \E\ J) be a Banach space, ordered by a closed pointed convex cone, KaE, such that Int(K) is non-empty. Assume that the norm of E is monotonic. Let T : E -> E be a positive, strictly (-l)-convex decreasing operator. Then, T is a strictly non expansive operator on £rfor all r e R+\ {0} (i.e., dh(T(x), T{y)) < dh(x,y) for allx,y e £ ) . Proof For every x,y e Sr, since 0 < m(x,y)y < x< M(x,y)y and T is decreasing we have 7[M(;c,>')>'] < T(x) < T\m(x,y)y]. By applying the monotonicity of the norm, we get 0 < m(x,y) < 1 < M(x,y). Also, because T is strictly (-l)-convex, there exist elements rj and rj, > 0, such that (1 + ^\M{x,y)\lT\y) T[y) rj)[m{x,y)\ll{y). Vr(y)
and

rr{T{x),T{y))>{\++11 m{T{x),T(y))>(l r?,lM(x,y)]-\ ,lM{x,y)\\ which imply that

,

»f. d

d^), TOO)**4w" 4^vf:11 4^41= 4(IM,IM) v!':11<<4^41 - * M^• ■

_m[x,y)(\ + + T] T]tt)j)j \m[x,y)j lm[x,y)(\ \m[x,y)j ■ The following example is to remark on the importance of Theorem 2.17. The following example is to remark on the importance of Theorem 2.17. Let Q a R" be a compact set and £ : Q x Q -> /?+ \ {0} and / : Qx (R+ \ {0}) -> R+ \ {0} continuous functions. 188

HILBERT'S PROTECTIVE METRIC Consider the Banach space, E = $(Q), endowed with supremum norm, and ordered by the closed pointed convex cone, K = {w eeE\ £ | u(x) > Ofor all x e f i ) . In this case, because Int(K) = j u e E | u(x) > 0 for all x e QJ, we find that Int(K) is non-empty. Consider the operator T:E^yE

defined by

T{u){x)= r( \ak{x,y)f(y,u{y))dy, ^ ( x . y ) / ^ , ^ ) ) ^ , x eQ. M )(x)= ^ ia

If we denote T v x

d ){ {*) ) == \Qkix>yHy) >yHy)4y>

*eQ

and Tff(u)(x) = f(x,u(xj), f(x,u(x)), ),

X xeCl, €xe&,

we can remark that Tc[lnt(K)] c Int(K), Tf[lnt(K)] c Int(K) and T =7>2}. If a = mini min{£(x,}')| k[x,y)\x,y jc,;y e Q x Qj and b = max{A:(;»c, max|A:(;»c,j>)| j>)| x,y e Q x Q> then we can show that dh(Tc(u), Tc(vj) < adh(u,v)

for all u,v eInt{K)

where

MM! <11..

tanh-^ a = tanh 2 2

189

METRICS ON CONVEX CONES Indeed, we have 0 < a < Hx,y) (y)dy ajnv{y)dy
for for every every vv ee K. K. If u(y) Q, it If u(y) = = 11 for for every every yy ee fi, it follows follows from from (2.17) (2.17) that that

(2.17) (2.17)

~b~ dhh(T (Tc(v),u)
{\W :

im

k(Tcc)) = £(r £(7/ = tanh tanh-A(T -A(7/
HILBERT'S PROJECTIVE METRIC (I) IfE = R" andK=

Rl = {(xux2,...,xB)\

xt > 0,1 V~y\= -ytf f *\Y\M{x,y)-m{x,y)} l*->h (*«->v)

.1=1 ..1=1 1=1

J

(S Li=l li=l

J

[M(x,y) - m(x,y)] < < [exp{c/ [exp{dhA(x,y)} l]m(x,y), < [M(x,y) (x,y)} -- l]m(x,y), which implies |\\x-y\\< x - y | | <eexp{d x p {h(x,y)}-\. ^(x,y)}-l.

(2.19)

Moreover, if we denote w = (l,l,..,l) then we have M(x,y) = maxjl + {x M(x,y) (JC,.t --y)/y y,)(/ v,-11 11 < iz << n) n) < < 11 ++ ||x |JC-y\\ -y||//w(;c,y), I m[x,y),

(2.20)

and if ||x - y\\ < m[y, u) we can show that m[x,y)>\-\x-y\l m[y, u). m[x,y)>\-\x-y\lm[y>u).

(2.21)

From (2.20) and (2.21) it follows that if ||x - y\\ < m{y, u) then 1-dhh(x, \x -- y\y\\>>m(y,u) \\x m(y,u) tanhj tanhj-d {x,y) v) \.\. [2

(2.22)

The completeness of S follows from (2.19) and (2.22). ■

191

METRICS ON CONVEX CONES (II) IfE =$[0,l],fl) andK= {x&E\ x(i) > Ofor every t e [0,1]} then S is complete. Proof We use the same arguments, as in the proof of case (I), but now we consider that u(t) = 1 for every t e [0,1]. ■ (III) IfE = Mnxn(R), is the space of real n x n-matrices with norm \\A\\ = sup I \\Ax\\\ \\X\\ = l | , and ifK is the cone of real positive semi-definite symmetric matrices in E, then Int{E) is the cone of real positive definite symmetric matrices in E, and therefore £ is a complete metric space. Proof The proof is based on some arguments presented in (Bushell, P. J. [2] and [4]). ■ We consider now a general completeness test for the pseudo-metric space (K,dh), proven for the first time in (Krause, U. [1]). Let E be a real vector space and K 1. We note that < and <• both define a partial order in E. On the set of real numbers, R, we use the usual order. By R+ we denote the set of all non-negative real numbers. For x, y e K\ {0}, define Mx.y) ~ sup{/l € R+ | Ax
192

HILBERT'S PROJECTIVE METRIC Let u e K be an element. We say that a sequence {xn}neN c K is guided by u if there exists a null sequence of real numbers {s„}neN, £„ > 0 for all n e N, such that u < xn < xn+m< e„w + x„ for all m,ne N. Theorem 2.18 The pseudo-metric space (K,dh) is dh-complete, if and only if, every guided sequence has a supremum in K with respect to <*. Proof Suppose thatKis ^-complete. Consider a sequence {xn}neN^

c

c f

guided by u e K. We may assume that w * 0. It follows that 1 Mxn+m, xn)> for all m,n G N. Note that Mx„, xn+m) > 1, since the l + e„ sequence is increasing. Hence, we have dh(xn,xn+m)<\n(\ + sn) for all m,n e N and {x„}neN is a c4-Cauchy sequence. Thus, {x„}neN converges with respect to dh to some x e K\ {0}. The sequences defined by Mx ,xm) and A(xm x') are increasing and decreasing, respectively. The sequence Mx',xm) is upper-bounded by (1 + s{)Mx,x{) < + oo, which we can show using xm < xm+i< (1 + E\)x\. We find that the limit 8 = lim Mx*,xm) < +oo *

m-»+oo

exists. From the convergence of {x„\neN N to x with respect to df, we have exists. From the convergence of {xn]neN to x with respect to df, we have I 2 >1(JC„ , x ) • Mx ,x„) > — for « > «o

Mjc„, x)- Mx ,x„) > — for « > «o

J_

which follows that Mxm x)> — > 0 for n > n0. Thus, p= lim A(xn,x*) 25

n-*+ao

1 exists and p > 0. We show that —> — x* is a supremum of {^„}neJV in A" with P

respect to the partial ordering <*. Indeed, from the definition of A(-,-) we have Mxn, x*)x„ < rx* for any arbitrary r > 1 and « e N. This implies that

T

xn <» — JC* for all n e N. \p) 193

METRICS ON CONVEX CONES Assume now that x„ <* y for all n e N and some v e K. From the definition of A(v), we have X(x*,xn)x* < rxn for all n e N, r > 1 arbitrary. This yields X(x',x„)x* 1. Since {x"IneN n} converges to x*, we have

lim /i(xn,;c j-Alx ,x\\- = 1 and, therefore,

H->-H»L

(4

5-p = 1 or S = —. This yields — — ;\x* < r 3 vforforallall r > 1 or — \x*<,y. p \p) \p) Conversely, suppose that every guided sequence in K has a <»-supremum in K. Let {x„} nEJV cif b e a Cauchy sequence with respect to d^. We can show that there exists a subsequence uk = x„(k), such that

1

A 4), is Kuk+\>»k)'Kuk>uk+i)>Y^zr -for tokeN. dh(uk+uuk)
Hence, there exists ak > 0 and /?* > 0 such that a 4 w t < uk+l,/3kuk+l < uk and

#*/%> _JI F - Consider the sequence {^t}AeAr defined by induction as ' 1 + 4-* X X\ = \ and Xk+\ = —- for all A: > 1. Let {vk}kfLN bebe the sequence defined as a

k

Vk = XkUk for all k e N. The sequence { v ^ is is guided by Vj. Indeed, from the properties of {« t L ^, it follows that v* < vk+1 and vi+1 < (1 + 4 ~*)v**)v* vk <

Hence, n n

v

*

+

i -

v

i =

I

I' n

(i>

and, therefore, v„ v +1 < I - v jv, for all n eN. --(!) ' From this, we obtain for arbitrary m,n e N,

194

<(Hj ' . * ©

{vk+\~vk)<] \4~\ ±

v

*+i

HILBERT'S PROTECTIVE METRIC

v

n+m n+mI

I(

n+m-\

1

i

r o fn+m-\ n+m-\

^\

"2{ * Vv 4=n k=n

I/

-v„= I ( v( 4A4 ++ 1II -- vVv 4i4))) << S Z 444"*v <^ ^| - ' v4i4<

V v n + m—- vVp ==

V

+

k=n k=n

k=n

V ^ 4X~** v, £ l

and hence vn+m -v„ <2-4""v 1 . If we put £„=2-4~" we deduce that is {v*)teAf is aa;s e ( l u e n c e guided by vi. Therefore by assumption, {v t }. „ has ■N >keN a <*-supremum v e Jf. The theorem will be proved if we show that {vk}keN converges to v , with respect to dh- For, in that case, {«i} ieAr also converges to v* with respect to dh and since {uk}k£N is a subsequence of the keN dh-Cauchy sequence {x„}„^N > we obtain that {x„} converges to v* with neJVconverges ieJV respect to dh. Since v is a <*-supremum of {v„ } neA ,, we have v„ < rv* for all r > 1. Hence Mymv') > - for all r > 1, that is Mynv) > 1 for all n e N. It r suffices, therefore, to show that for a given 0 < e < 1 there exists ne e N such that A( v*, v„) > 1 - e for all n > ne We know that n vn+m-v-v^^-vj < lUr Vj) for all /«,« e. N. Select an n such that 2-4""< s for n £ all n > A?£. Since v; < v„ we have that vn+m < (l + f)v„ for all n>rie,me

N.

Choose some fixed n > rig. Since v, < v„ for 1 < i < n, we conclude that v* < (1 + s)v„ for all k e N. This, in particular, means that vk <» (l + f)v„ for all k e N.

Since v is a <*-supremum of {v t } t € ; v , it follows that

e—i

v* <» (l + s)vn, that is v* < (l + e)rvn for arbitrary r>\. Thus, ^(v*,v„) > f - Y — J

f

1 or all r > 1, that is A(v*,v„) > — l+e

A + sJ 1 ,>l-e, >\-e, we obtain that iX{V\V v ' . vHj)>\-E > 1 - £ for all n > ne and the \+£ theorem is proved. ■ Since

From Theorem 2.18, we can deduce various criteria for (^-completeness. We say, that K is complete for the relative uniform convergence, if for every sequence {*„}neJV <= K satisfying 195

METRICS ON CONVEX CONES -snnuu < -E < xxn+m -xnn< < eennu, u, for for all all m,n m,n ee AN^ n+m -x with u e K and {en)IneN being a null sequence, there exist an x* e K and a null sequence {Sn}n^N, such that ~8nu < x„ - x. < S„u, for all n e N. Corollary 2.19 IfK is complete for the relative uniform convergence, then K is dh -complete. Proof Suppose that {x„}neA, c: K is a sequence guided by an element u e K. By the completeness for the relative uniform convergence, there exist x> e K and a null sequence {^„}neA, such that -Snu <xp-x,< ^sn 8nu. Since u < x„ and {*„}„6A, is increasing, it is implied that x„ <* x. for all n e N. Furthermore, if xn <*y e K for all n € N, then x. < (1 + S„)ry for all r > 1 and all n e N. Hence, x* < ry for all r > 1, that is, x. <. y. Thus, x. is a <»-supremum for {*„}neA, and by Theorem 2.18 we see that JT is ^-com plete. ■ Corollary 2.20 Let E(x) be a locally convex space and K c E a pointed convex cone. If the following three assumptions are satisfied: 1) K + K czK,{K, being the v-closureofK), 2) every increasing Cauchy sequence in K converges in K (with respect to T) 3) every order interval in K is r-bounded, then K is dh-complete. Proof Suppose that {*„}n6A, is a sequence in K guided by an element u e K, then 0 < xn+m -x„< s„u, for all m,n e N, with en > 0 converging to 0. Suppose that the topology ris defined by the family of seminorms {p,} ie/ For every seminorm ph there exists by assumption (3) a constant a, such that Pi{xn+m -x„) < en ■al., for all m,n e N. Hence, {x„}n<_N is a Cauchy sequence for x. Since {x„}neN is increasing, then by assumption (2) it con-

196

HILBERT'S PROTECTIVE METRIC verges to some x, e K. The Corollary will be proved if we show that x, is a <» -supremum for {x„}neN.r " Indeed, since for fixed n, um = xn+m - x„ defines an increasing Cauchy sequence in K, we must have x, -x„ e K, considering assumption (2). Hence x <* JC. for all n e N. Suppose that y G K is an element such that x„ <* y for all n e N. Let r > 1 and choose some £• > 0 such that r > 1 + s. Then xn < (r - e) y or {r - s)y -x„ e K for all n e N. Thus, it follows that, and hence, (r - s)y - x* e K. Applying now assumption (1), we have that ey + (r - s)y - x, e K, that is xt < ry. Thus, x, <» y and xt is a <*-supremum for [xn} . ■ neN

From Corollary 2.20 we obtain the following result. Corollary 2.21 Let E(T) be a locally convex space, and K aE a pointed closed convex cone. IfKis normal, and sequentially complete for T, then it is dh-complete. Let (is,|| I), be a Banach space and K czE, a closed pointed convex cone. Denote again by < the ordering defined by K.. If u e K is an arbitrary element, such that || u\\ = 1, we define Ku = ix eK\dh(x,u) <+oo| and £u = ix e Ku | |JC|| = l|. We can show that (Ku ,dy) is a.pseudo-metric space and (£„ ,dh) is a metric space. For this, it is sufficient to remark that if x,y e Ku then 0 < dh{x,y) (x,y) < dh(x, (x,u) u) + + ddhh(y, (y, u)< u)< + + co. co.

(2.23) (2.23)

Indeed, (2.23) follows from m{x,u) i x _ x —-.——y<m(x,u)u<x K

M{y,uY

'

197

METRICS ON CONVEX CONES and M(x,u)u

x<M(x,u)u<

;' )y.

m[y,u) Theorem 2.22 If the norm of E is monotonic, that is 0 < x < y implies \\x\\< \\y\\, then (Swdh) is a complete metric space. Proof For every x,y e £u we have 0 < m(x,y) m{x,y) < < 11 <<M(x,y) M(x, v)<< +oo. +oo.

(2.24) (2.24)

Indeed, since 0 < m{x,y)y < x, we deduce /w(x,v)|[y|| < ||x||, which implies that m(x,y) < 1. Similarly we can show, that 1 < M(x,y). Thus x -y < [M(x,y) - m(x,y)]y and finally ||x -y\\ < | M(x,y) - m{x,y) m(x,y) |\ <<exp{d exp{dh{x,y)} h(x,y)} - - 1.1.

(2.25) (2.25)

Suppose that {x„}neN, is a Cauchy sequence in (Swdh). From (2.25) we know that {*„}neJV is a Cauchy sequence in E and hence converges in norm to an element x,e K with || x. \\ = 1. The inequality dh(xn,u)
+ dh(xm,u) implies that

M(x. ,U) f,v " ' < a < +oo for n > m. m(xn,u)

Thus, we obtain 1 a-'<m(x a" <m(xn,u)
Therefore, for n > m we get a xu<xlu<x a' n n
HILBERT'S PROJECTIVE METRIC and by computing the limit as n -»+oo, we deduce that dh(x+,u) [xt,u) < In a 2 < +00. Similarly, we can show that dh{xn,xm)<€ X * implies that the inequality <2e iand hence, {x„}n£N is convergent in {Swdh). ■ dh(x.,xn)<2s Remark 2.5 If Int(K) is non-empty and u e Int(K) then Ku = Int(K) and £U = S. Applications We now present some applications based on the results presented in this section. Application 2.1 Let (E,\ |) be a Banach space, K c E a closed pointed convex cone a n d / : K —>K a mapping. We recall that/is monotone increasing, when 0 < x < y, implies that fix) < fiy), and that / is positive homogeneous of degree p,(p eR+ \ {0}) if 0 < x and X e R+ \ {0} imply

Ate) = Xpfix).

Theorem 2.23 Let \E,\ ||J be a Banach space ordered by a closed pointed convex cone KczE. Assume that the norm is monotonic. Letf: K —> K be a mapping which is monotone increasing and positive homogeneous of degree p with 0 < p < 1. If there exists u e K with \\u\\ = 1, such that dh{u,fiuj) < +00, then f has a unique fixed point x, e Ku and 1

[*W«MF

1

Proof Ifx € £uthen m[x,u)u<x<

1

")]F

-u<x <[M( [AW(/(M),W)]I-P U)\-P U< xt t <[M(/(H),«)]I-P <[M(/(M),M)]I-P

• u.

M(x,u)u

and hence

199

METRICS ON CONVEX CONES

r*tM(f{u),U)U.

[m,{x,u)]Pm(f(u),u)u < f{x) < [M(x,U)] [M{x,U)]P Therefore,

dh(f(x),u) (f(x) ,u)
(2.26)

Furthermore, by applying/to (2.26), we obtain dh(f(x),f(yf)(y)Y< pdh{x,y).

f(x)

Let ¥(*) = ,, -. '., for every x e £u. This well defined mapping, VF, is the V \x)\ composition of a strict contraction and a normalizing isometry and it maps BM in itself. Because {£wdh) is a complete metric space, then by the Banach contraction principle there is a unique element, v € £u, such that ^(v) = v. I

We can show that, xt = |/(x)||i-p • v is the unique fixed point o f / i n Ku. Since xt - f(x.) < f[M(x„u)u] < [M(x.,u)]P M\M(f(u),u)u implies M(xt,u)<[M(x„u)]P)]'M. M(f(u),u), the conclusion of the theorem follows. ■ Consider the integral equation

0-

y{t) =loIslr^Gisly^Yds, y{t)=\ s\n[^G{s)[y{s)}pds,t>0, t > 0, where 1) G is continuous in [0,+oo[ and

200

(2.27) (2.27)

HILBERT'S PROJECTIVE METRIC 2) 0 < a < G(0 0, and K be the cone of non-negative functions in E. Let

i;

f(y)(t)=Jlslr(^G(s)[y(S)Yds y(s)]pds

(0

and suppose that/? e]0,l[. 2

2

d——1 (¥) 1 u. ?)'■u
Let u(t) = (t/p)^; (t/p)\-P\ then p2a\—-j

2

From Theorem 2.23, equation (2.27) has a unique solution, y(t), such that a ^ ^^i, i-PJ 1i

2

^ 1

22

\1

,

_

22 1.,!=?,

?#-

- " ^ < ^ ) < ^ 1 Z Z J ^ •f '', forO<^<+oo. forO
Application 2.2 Let 7x(f) = j £(f,.s):x:( .?)<&, where £(-,-) is positive and continuous in the unit square 0
(2.28)

for every positive JC(-) continuous in [0,1]. If we consider the function u(i) = 1 in 0 < t < 1, then using (2.28) we can show that

201

METRICS ON CONVEX CONES

a

dh{Tx,u) {Tx,u)<\J^
o

and therefore ,A(r) <21n[ — I <+oo. Applying Theorem 2.12 we obtain that T has a unique positive continuous eigenfunction (associated with a positive eigenvalue). Application 2.3 Consider the positive solutions of the eigenvalue problem k x u AM{X) = j$nk{x,y)f(y,u{y))dy, x e Qx Mx) = n ( >y)f{y> (y))

eQ

(2.29)

where Q c R" is a compact subset and where i : Q x f i - > R + \{0} and / : Q x (R+ \ {0}) —> i?+ \ {0} are continuous. Suppose that for each x,fix,u) is strictly (-l)-convex and decreasing as a function of u. Consider the operator T associated with equation (2.29) and defined and studied in the example presented after Theorem 2.17, where the space E and the cone K are also defined. In this case, the metric space (£r,d^) is complete and the

Ttx) _3i

operator ¥ : £ r -» Sr, defined by ^(x) = ,,_\ L, has a unique fixed point in

«^)ir

Sr. Thus, for each r > 0 there exists A, > 0 and ur e Sr such that T\ur) = XrUr Hence, problem (2.29) is solved. Application 2.4 The next result can be used to solve some nonlinear integral equations. Theorem 2.23 Let \E,\ ||J be a Banach space ordered by a closed pointed convex cone KczE. Suppose that Int(K) is non-empty. Let f: Int{K) -> Int(K) be a monotone decreasing mapping, which is positive homogeneous of degree -p(0
202

HILBERT'S PROTECTIVE METRIC S=(Jnt(K)f]S(0,l),di,) is complete, then there exists a unique x, e Int(K) such thatf(x,) = x,. Proof The mapping / is a p-contraction. Indeed, we know that for all x,y e Int(K) we have m(x,y)y <x< m{x,y)y

M(x,y)y

which implies that f[m[x,y)y]>f{x)>f[M{x,y)y\. f[m[x,y)y]>f{x)>f[M{x,y)y\.

M

Thus, using the homogeneity of/we see that p lm[x,y)]f(y)>f(x)>[M(x,y)]-pf(y).)Vf(y)[m{x,y)Yf{y)>f{x)>[M{x,y)Yf{y).

Applying the definitions of m and M, we obtain M{f(y),f(x))<[M(x,y)]P and

y)Y

P P m{f(y),f(x))>[m(x,y)]:,y)Ym(f(y),f(x))>[m(x,y)] .

It follows that

m^(y)hMf(yim^{^§_ ^(/w,/w)=*(/w./(4=4^i. ^4^41 -MM. |_ m[x,y) J L m[x,y) _

Consider now the mapping *F : S-» 5 defined by

203

METRICS ON CONVEX CONES

w= V(x)-.

* ^H f o r a l l x e 5 ' We have

dhh(V(x),V(y)) d (V(x),V(y)) = d
{\\A4'\\f(y)\\r

Applying the Banach's contraction principle to W and £, we obtain a fixed point y, of T. Thus, we have ¥(>-.) = >>* or/[>>,) = Ay,, where A= \\f[yt)\\ is strictly positive. If we set x, = 2

i1 1+p

-y,, the theorem is proved. ■

Note that Theorem 2.23, can be used in particular, to solve the integral equation k{t,s)[y{s)]'PpdM{s), y{t)=jnk{t,s)[y{s)]-

dp(s), t teCl,

where Q is a compact topological space, p. a positive Borel measure, and k a continuous function defined on fi x Q. In the paper (Potter, A. J. B. [1]) the case/? =1 is also considered. Application 2.5 In this application, we give a generalization (important for Economics) of the Perron-Frobenius Theorem for mappings, without additivity. L e t / : R™ -».ft™ be a continuous mapping. We say that/is homogeneous of degree 1 if f{Xx) = Afix) for all A > 0 and x e R™ If for some positive integer r, x > y implies / r(x) >f(y) then, we say that/is primitive. (The exponent r is in the sense of composition of functions). Lemma 2.24 Let (X,d) be a compact metric space and let/:.letf:X->Xbea continuous mapping such that for some positive integer r, d(fr(x),f(y)) » ) < d(x,y)for all x *y. Then, fhas a unique fixed point, x», andf(x) -» x, for allx e X.

204

HUBERT'S PROJECTIVE METRIC Proof Suppose, r = 1. The mapping/has at most one fixed point, because if x and y are fixed points, then d(f{x)j{y)) = d(x,y), which implies that x = y. The Lemma will be proved, if for any x e X, the limit of every convergent subsequence of/"(;t), is a fixed point. Let f"k(x) ->y. Since d(f(x),f"+1(x)\'(*)) is a decreasing sequence of numbers, it has a limit Ob, which is also the limit of every subsequence. It follows that t+1 d(y,f{y)) = Hm lim <*(/*(x),/* d(fn>{x),f"*++1 \x)) (x),/"* + 22(x)) rf(y,/W) (*)) = «o a0 = = « £ 4/"Yxmd(f^\x),f^ {x)) k-*co

k-*x>

d(f(y),f(f(y))), H/W,/(/W))> =

which implies thaty =fiy). Suppose r > 1. In this case, for ally = 0,...,r - 1 and for any x e X, we have that fm+J{x)

= (fr)"(fJ{x))W) converges to x„ the fixed point o f / ' . Be

cause d(xt,f(x.)}

= d{f(xt),fr(f(xtyj\' ( * ) ) ) <
)).

r

we obtain that

J[x,) = x* and also that/ (x) -> *». ■ Theorem 2.25 Iff : U™ —> R™ is a continuous mapping which is homogeneous of degree 1 and primitive, then: 1) there exists xt > 0, unique up to proportionality, such thatj[xt) = A„ x* for some A* > 0,

2)

mrM'

forxeR:x{0}* + m \ { o } .

-

Proof Since/is continuous and primitive, we can show that (1) follows from (2). Thus, by setting S = U e Rm+ \ \x\ = lj and defining Y : S -* S as

fix)

^ M = I, , vii, it is sufficient to show that

"«/wir

205

METRICS ON CONVEX CONES

{

there exists there exists x, x, eS eS such suchthat that n W(x)->x.forallxeS. V (x)->x.forallxeS. eS.

(2-30) '

For this, we will show that

WwwHw U(V(x),V'(y))
(231)

[for all x*y,x,y .}»<eint(S).

(We denoted by int(S) the relative interior of S). Indeed, if x < Ay then by homogeneity and primitivity, we see that f(x) < Afr(y). This implies that M(r(x),r{y))<M{x,y) M(fr{x),r{y))<M{x,y)l**y) and hence that

/'Ml Mi.j') •

M(V(x),V(y))<\^M(x,y).

X IV (WIL Iv )|J 'Jlrwllj

Similarly, we can show that

Y(y) 1

K*'}') n{n*),ny))> fM»(x>y)v) ■ . * ) ■

' IV J/'W WJ .

Applying the definition of dh, we obtain dJx¥r(x),x¥(y))
(2.31) is thus proved. The proof will be finished, if we show that (2.30) is true. Let D = Vr(S). Since xVr(x)>0 whenever x > 0,(because / is primitive), we have that D c int(S) and Hilbert's metric is defined on D, and induces the ordinary topology. Because *¥r is continuous on S, (since / is continuous and

206

HILBERT'S PROTECTIVE METRIC / r ( x ) | > 0 for x e S), D is compact. Applying (2.31) and Lemma 2.24, we obtain that ¥"(*) —> x* for all x e D and hence for all x e S. ■ Theorem 2.25 was generalized by U. Krause. (See Krause, U. [1], [2]). The reader can find other interesting applications of Hilbert's projective metric in the papers cited in the references of this chapter. We finish this section with some comments about the position of Hilbert's projective metric in the class of all projective metrics defined on a convex cone. Let [E,\ I] be a Banach space ordered by a closed pointed convex cone K c E such that Int(K) is non-empty. If / : K -> K is positive, that is fJnt{K)) c; Int(K), we recall that the contraction ratio kd (/) of/is defined by kdh{f) = inf{A| dh{f{x),f{y))

< Mh(x,y) for all x,y eInt{K)} .

In 1973 P. J. Bushell proved in the paper (Bushell, P. J. [2]) the following result. Theorem 2.26 Let q> be a differentiable real valued function in [0, +oo[ such that: 1) Ofor t e [0, +oo[ and Ofor t e [0, +oo[ and

0, and 3) (p(s + t)<
[ (M(x,y))~ is iis a pseudo-metric in Int(K), I Vm{x,y)J_

then, it follows that d (x,y) = q> In —. ' .

and ifkp is the associated contraction ratio, then k^f) > k(f). This result imposed the problem of studying the position of Hilbert's projective metric in the class of projective metrics defined on a convex cone. Let E be a real vector space and K c E a pointed convex cone. Let/: E -> E be a mapping such that f{K) c K. In this case, we say that/is 207

METRICS ON CONVEX CONES non-negative. If d is a protective metric on K, we define the contraction ratio off with respect to d by kd{f) = inf {A 14/(x),/(j,)) < M{x,y) for all x,y e K} . If kjf) < 1, we say that/is a strict contraction with respect to d. Suppose that E is the Euclidean space Rm. In this case, if K a R™ is a closed pointed convex cone, we write x > 0 if x is in the relative interior of A". We say that a mapping/: JT -> Rm is positive if fix) > 0 for all x eK\ {0}. Theorem 2.27 Let < be the ordering defined by a closed pointed convex cone K a Rm Then every positive linear mapping is a strict contraction with respect to dh. Conversely, if d is a projective metric on K, such that every positive linear transformation is a strict contraction with respect to d, then there exists a continuous strictly increasing function q>:R+ -> R+, such that d(x,y) = cp{dh,{x,y)),for all x,y > 0. Furthermore, the contraction ratios satisfy kd ( / ) < kJf),for every positive linear mapping f Note that Theorem 2.27, was proved for the first time by E. Kohlberg and J. W. Pratt in the paper (Kohlberg, E. and J. W. Pratt [1]). In the same paper, a generalization of Theorem 2.27 is given for locally convex spaces. The proof of Theorem 2.27, is long, and it is based on several technical results.

3. Thompson's metric Thompson's metric is another powerful and interesting tool for investigating nonlinear operators on cones. As we discussed in section 2 of this chapter, Hilbert's projective metric is well known, but Thompson's metric has the advantage of not being restricted to a section of the cone when completeness is indispensable for solving problems. In this section we will present Thompson's metric, and some of its applications. 208

THOMPSON'S METRIC We shall note that Thompson's metric was inspired by Hilbert's projective metric. In 1963, A. C. Thompson defined this metric on a convex cone in a Banach space. The definition of this metric, considering a convex cone in a locally convex space is the following: Let E(f) be a locally convex space, and K cz E, a closed pointed convex cone. We recall that the cone K is normal, if and only if, there exists a family of seminorms {p a } ^> which generates the topology x and such that 0 < x ] <{X- l)[pjx) +pjy)] < 2m(X - 1), which implies that Pcfc -y) =PcLx -y + u-u)
■

Elements x mdy belonging to K where x and y are not both zero, are said to be linked if and only if there exist finite (strictly positive) real numbers X and ju, such that x < X y and y < [i x. We can show that this is an 209

METRICS ON CONVEX CONES equivalence relation, which splits K into a set of mutually exclusive constituents (the equivalence classes). Let Cx be the equivalence class of an arbitrary element x eK\{0}.(0). We can show that Cx has the following properties: 1) y,z e Cx=>y + z e Cx, 2) y e Cx and a > 0 => ay eCx, 3 ) 0 ? C x. Let x, e JC\ {0} be an arbitrary but fixed element, and C^ be its equivalence class. If x,y e Cx% we define a(x, v) = >] and p\x,y) = P= inf{//| v < ju*} Remark 3.1 1) Since K is supposed to be closed, we have x < ay and v < fix. 2)a*0and/?*0. Indeed, if a = 0 or ft = 0 it follows that x = v = 0, which is impossible because x,y e Cx<. Let d,: Cx, x Cx% -+R be the function defined by dt{x,y) = ln[max(a(x,y),/3(x,y))\

'))]

(3.1)

Lemma 3.2 For every x. e K\ {0}, {CXt ,dt) is a metric space. Proof It is sufficient to show that d, is a distance. Indeed, from (3.1) we have that d,(x,y) = d,(y,x), for every x,y e Ca. lfx=y then from (3.1) and the definitions of a(x,y) and p\x,y), we have that dfcy) = 0. Conversely, if dfcx.y) = 0, then max{a(x,y),p\x,y)} = 1, which implies that a = 1 or p= 1. In this case we see that x < y and >> < JC. Because K is pointed, we deduce

210

THOMPSON'S METRIC that x = y. If x *y then, a > 1 or fi > 1, and it follows that dfcy) > 0. Consider three arbitrary elements x,y,z e CXt. In this case: x < ay ; y < pxx x< a2z; z< fax z < a?y ; y fii, we get dt(x,y) - lna, < ln(a 2 « 3 ) = lnor2 + lna 3 < ln|max(a 2 y5 2 )| + ln|max(a 3 , J32)] = dt (x, z) + dt (z, y). We obtain a similar conclusion, if we suppose that Pi > a.\. ■ The distance d, is named the Thompson's metric. The most important result about this metric is the following theorem. Theorem 3.3 IfK is a closed pointed convex cone, sequentially complete in a locally convex space, E(T), then for every x, e if\ {0}, the equivalence class CJ( is a complete metric space with respect to the metric d,. Proof Since by Lemma 3.2 we know that d, is a distance, it is sufficient to show that the metric space ( C ^ , d,) is complete with respect to dt, Let {xn}n€N c Cx% be a Cauchy sequence with respect to d,. For allp,q e N n<=N we define aa

m pq

> l} >***,} = inf ' l\x\x {A pp-*

=

mf

X

X

i) The sequence {JC„}neJV eJV is bounded with respect to topology r. Indeed, since {xn}neN neN is a Cauchy sequence with respect to dh there exists n0 e N such that dt{xp,xq}<

1 for all p,q > n0, which implies that

max(a w ,a 9 p ) < exp(l) for all p.q > n0.

211

METRICS ON CONVEX CONES In particular, we have apno < exp(l) for all/? > n0. Hence, xp < expO)*^ < 3 * ^ . pa(xp)

<3pa(Xno).

Applying the seminorm pa we obtain

If we define M = max{/?a msx{pa(xl),..,pa[x^pa[xa^,

we have thatpjx„) < Mfor all n e N. Because pa is an arbitrary seminorm of the family ( p a } a e ^ , we conclude that {x„}ni_N is r-bounded. ii) The sequence {*„}„eA, is r-Cauchy. Let/? a be an arbitrary seminorm of the family {pa}aejf lo€/# ■ If £> 0 is given, there exists Se>0,

such that exp(£ £ )
f €

, where M0 =3M. Since

M/

{xn} n r f , is a Cauchy sequence with respect to d,, it follows that there exists ne e N such that d,{Xp,xq) < Se for all p,q > ne . This implies that m ^

w

f. + 4

£

, a J < <1 l ++^,andthat, p <[l + ^-]x 9 ;, g <[l + ^ J ~W0' ~M0.

V

e -1 1 = e, for sMp,q > n^ By Lemma 3.1, we have pa[xp - xq\ < 3M 1 + \ M0 J Thus, we deduce that {x„}n<_N is a r-Cauchy sequence.

Since K is

sequentially complete, there exists x0 = (r) - lim xn. n—»oo

iii) xo w f/ie /zwzY of the sequence \xn IneN }

wzY/j respect to dt.

Let £•> 0 be given. Since {x„}n6W is a J r Cauchy sequence, d^Xp,xq) < sfor p and q sufficiently large. For such/? and q, xp < exp(£)x? and xq < exp(£)xp. Because K is r-closed, it follows that xp < exp(spco and XQ < exp(s)xp for every p sufficiently large. Therefore x0 E CXt and d^x^xo) < s for p suffi ciently large. Thus the theorem is proved. ■ The next result is a consequence of the proof of Theorem 3.3.

212

THOMPSON'S METRIC Suppose that (is,| ||j is a Banach space, and K c E is a closed pointed convex cone. Suppose also that the norm is monotonic (i.e. 0 < x
|lx|| ^:[MI). Proposition 3.4 Let C be an equivalence class of Kand {x„}ni.N, a Cauchy sequence in C with respect to dt. Then, {*„}neJV is a Cauchy sequence in norm. Moreover, lim dt (xn, x,) = 0 implies lim \xn - xJ = 0. n—>°o n—>°o

n—»oo «-»oo

'

The next result gives a supplementary information about convergence. Proposition 3.5 Suppose that Int (K) is non-empty. If \xn}IneN is a sequence in Int(K), such that, lim|jcn -x*| = 0 where x* e Int{K), then «->oo «->0O

limdt(xn,x*) = 0,

n->oo n-»oo

Proof Since x, e Int{K), there exists S> 0 such that \y e is I | y - J C , | | < £ } CJK". If we put e„ = \\ x» -x„\\, we may suppose that £„ * 0 for all «. We have x, ± d(x, -xn)/

en e K, which implies

sn-5 s„+S £„-S £ +$ -x, <xn.<■f. —H nn x*. ——.—x, -^-x,<x -^-x,<x <-^—x <-^—x tt 8 8 nn For an n sufficiently large, such that 8— sn > 0 , we have

f^" + S , - 8 ) 8 'Ssno-sj . ) v. 0 V o

dfi?,(x„,x,)
and by computing the limit as n - * » we obtain dfe„, xt) -> 0. ■ Theorem 3.3, was proved for the first time, by A. C. Thompson [1] and /Vo-

213

METRIC ON CONVEX CONES position 3.5 by Y. Z. Chen [1]. Applications We now present a few applications ofThompson's metric. In doing so, we hope to stimulate some interest on this subject. Application 3.1 Let (E,\ |) be a Banach space ordered by a closed pointed convex cone. Suppose that the norm is monotonic. Theorem 3.6 Suppose that K is normal and complete. If for the mapping f:E^E,-+E, the following assumptions are satisfied: 1) there exists r, 0
Let \E,\ |) be a Banach space ordered by a closed

pointed convex cone KczE. Suppose that K is normal, Int(K) is non-empty and the norm is monotone. In Chapter 1 (Def 4.14), we introduced the notion of heterotonic operator. We consider again this notion. Let D be a non-empty subset of E a n d / : D-^Ea mapping. We recall that/ is said to 214

THOMPSON'S METRIC be heterotonic operator on the set D, if and only if, there exists an operator T.DxD^E) - > £ ssuch that: 1) 7T(;c,x)=y(;c)forall;ceA 2) T(x,y) is monotone increasing on D with respect to x for any y, 3) T\x,y) is monotone decreasing on D with respect to y for any x, We will apply the following classical result. Theorem 3.7 [Generalized Contraction Principle] Let (X,p) be a com plete metric space andf: X-^Xa mapping such that p(ftx)M) * L(a,fi)p(x,y), (x,y eX;a< i*Ap(x,y) < p), where 0 < L(a,p) < 1 for 0 < a C be a heterotonic operator. Suppose that for each [a,b] a ]0,1[, there exists a positive number L(a,b) e ]0,1[ such that > ,£(«■*). l\tx,rxx)- U > tL("'b) T■ T(x,x) = f(x) for allxeCandt * e | e [a,b]. (3.2) (3.2) Then, f has exactly one fixed point, xt e C, and for any point xo e Int(K), T\XQ,XQ)

-^>xt =fixt) asn^> +oo.

Proof Let x,y G C. We pick two real numbers a and b, such that a, b e ] 0,1 [ and dt(x,y) e [ - l n 6 , - l n a ] . We can select such numbers by applying the definition and the properties of d,. Without loss of generality, assume

a{x,y) > p\x,y), i.e., d{x,y) = ln(a(x,y)). By assumption, -J. < a(x,y) < - . 6 b a From (3.2) we obtain

215

METRIC ON CONVEX CONES l f(x) = T(x,x) > T\fi{x,yY T\fi{x,yyly,a(x,y)y\ y,a(x,y)y\>y)y > T^a{x,y) T\CC{X,y)~lly,a{x,y)y} y,a{x,y)y}

I**)

L b) > > a(x,y)a(x,y)-L^^b)

-L(a,b)

b) ■ = a(x,y)-^' ■ T(y,y) T(y,y) = a(x,y)-^'b)-f(y) -f(y)

HaJ,)

which implies that, /?(/(*), f{y)) ^ a(x,yp""bb' which implies that, /?(/(*), f(y)) < a(x,yp"' '

f and and

l 1 (x,y)x >T a\ f[y) > f[y) = = T{y,y) T{y,y) > > T^a(x,y)T^a(x,y)-lx,p\x,y)x\ x,p\x,y)x\ > T^a(x,y)T^a{x,yylx,a(x,y)x x,a(x,y)x

-L{a,b)

b) > a{x,yyL(a'b)-T{x,x) \x,x)■ = a(x,y)-^'

which implies that a[f{x),f[y))

■ f{x)

\L(a,b) (a

< a(x,y)

' .

Therefore, dt(f(x),f(y))\y)) < lr^a{x,y)NL{a£(«.*)! ^ = L{a,b)dt(x,y). We know from the Generalized Contraction Principle, that / has a unique fixed point x» € C. Let x0 e C and x0 ^ x*. Denote 8n = dt{f(x0),x,\.

*\n

Since / is a generalized contraction of C, it follows that {8„}n^N is a decreasing sequence. If 8„ -> 8* > 0, then 8* < 8n < dfeo, x,) for all n e N. Again, because / is a generalized contraction, there exists a positive constant, 6 < 1, which depends on 8, and d,(xo, *»), such that, Sn = 4(/"(* 0 )>*.) =S 9dt(fn-\x0),x.) xA for for all n e N. It follows, that 8, < 8n < #V,(x 0 ,x.) -» 0 as n -> +oo. That is, however a contradiction. We conclude, therefore that \S„ } „ —» 0 as w -» +oo. Hence, |/"(x 0 ) - *. -» 0 as H -> +oo (by Proposition 3.4.) ■ Theorem 3.9 Let C be a component of K and f: C -> C a heterotonic operator. Suppose that there exists a lower semicontinuous function
THOMPSON'S METRIC npcf'x) >
sCandts t e ]0,1[.

(3.3)

Then,/has exactly one fixed point x* e C and for any point, x0 e Int{K), f(x0) -> xtasn-+ +00. Proof It is sufficient to show that condition (3.3) implies condition (3.2) of Theorem 3.8. p(,) >g,p(0 Note that where wherelog,tXo%,,p{,)-T{x,x)>tL(a'b) -f(x). The theorem now follows from Theorem 3.8. ■ The following is an application of Theorem 3.8 to the study of the Hammerstein equation. Let Q c R" be a compact non-empty set. Consider the Hammerstein equation x(t)= | Q X(t. s)f(s, x(s))ds, where X e 0& x Q) is positive, x e 0£i) and the real valued function f[s,x) is such that/*,*) = F(s,x,x). Suppose that F(s,x,y) is a real valued function, monotone increasing in x and monotone decreasing in y. Consider the convex cone K = [x e 0 for all s e 0\. The func tion F(s,x,y) is such that the following properties are satisfied: 1) F(s,x(s),y(s)) > 0 for all x,y e <%Q), F(s>x>y) 2) Fy>x'y' is monotone decreasing in x and monotone increasing in y, G(s,x,y) where G is a positive continuous function on Q x ^(Q)x ^(Q),

f

G l^ A * ^ ) , ^ 1 ^ ) ) GQ >x, 3) \>a(X) = mm\ , \ \ . \ ' - |\ss eQ >A, and X e ]0,1[. G(5,45),4s)j j G(.s,x(.s),;t(.s))

217

METRIC ON CONVEX CONES In this case the operator T defined by

L M

T{x,y){t) F{ \s,x{s),. HS))ds k&;; t e Q n*>y)(t)== lnnX(t,s)F(s,x{s),y( is such that T:Int(K)xM(K)^>Int(K) T:Int(K)xInt(K)^>Int(K) Int{« <*): <*)

w;

r(x,x)(0 = and T(x,x)(t)=

r(^)) IIn \ X{t,s)f{s,x(s))ds. n

For every A X e ]0,1[,

I

ii1 T[Ax,r li x)= (jk.r *) [Ax,r x)=

l 1 K(S))C J * !tM) Z:(t,s)F(s,Ax(s),r x(s))ds s . ^ i:(s),. ) , / ! 'x(.s))a Jn

r

Wx(4

--L^^SS$)^ '^ :

*'. I L :(M) Q

^(^(^r's^))

Gi[s,Ax(s),A~lx(s))

^F(s,x(s),x(s))

<')■

^I - L ^ ^ M M ^ ^ " ^ >

•n

r(M)-

G(S,X(S),X(S))

x(s),

(,))d

j Z(t,s)F(s,x(s),x(s))ds (5,x(; w> I.a :(*,*) = X{t,s)f(s,x{s))ds (s,x(s))< = a a(X) >

n

Jo

n

Hence, we can apply Theorem 3.9 to obtain an element x. e Int(K) such that x.{t)=SnX{t,s)f(s,x.(s))ds. t'.*) (s, *,(.?))<; Application 3.3 The next result, is based on Thompson's metric and on Caristi-Kirk's fixed point theorem (see Chapter 4). Let {E,\ I) be a Banach space and K c E a closed pointed convex cone. Suppose that the norm is monotonic. Theorem 3.10 Let C be a component of K and f: C -> C a mapping. Suppose that there exists a function R+, such that: 1)

218

THOMPSON'S METRIC J[x) >, *{/(*)) / / ; x-x, , / oforr allx e C. 2)x>flx)>
(3.4)

/WW)^1, «(x,/(x))< and

g(jO *
gfc)X) > ,L
* ( / ((**)))T) "^ ' Applying the definition of G?, we obtain that i ^(x,/(x))
p(x)

K/M)!_

==In l np(x)-In ^ x ) - l n #>(/(*)) ^/(x))

i

and by Caristi-Kirk's fixed point theorem we know that there exists a point x, e Csuchthat/(x*)= x,. **• ■ Application 3.4 The following is an application to the study of the Explicit Complementarity Problem which was studied in Chapter 1. Let (H,<,>) be a Hilbert space ordered by an isotone projection cone (see Chapter 1). If K c H is an isotone projection cone, we denote by PK the metric projection onto K. The following result was proved for the first time in (Isac, G. and B. Nemeth [1]). Theorem 3.11 Let (i/,<,>) be a Hilbert space ordered by an isotone pro-

219

METRICS ON CONVEX CONES jection cone K a H. If the mapping h : K -¥■ H satisfies the following three properties: 1) h is isotone with respect to the ordering defined by K, 2) there exist pi,p2 e]0,l[ such that h{Ax) < Aph(x) for every x e K and A € R+, where p =pt if A < 1 andp =p2 if A >1, 3) there exist x0 eK \ {0} and A, p. > 0 such that px0< Pg(h(x0)) < Axo, then, ECP{I - h,K) has a solution x, that is unique in C^ and the sequence, with xx P X {xn}„eN "}neN defined bk? y xxn++\ = ^{H*"))' K{K »))' (» 66 ^0 ^0 with i' arbitrary arbitrary in in C^, C^, isis PK(> (% :.)). (» convergent to x„. Moreover, FnLsJV {xn}neN is convergent to x, with respect to the norm of H and n

PL dt(xn,x.)< ,xt)<—°—d t(x0d,x t(x {)0,Xl) ~1-A) 1-Po where pQ = max(/?,,/>2). Proof We recall from Chapter 1 that problem ECP(I - h,K) has a solution, if and only if, the mapping T{x) = PK(h(xf) has a fixed point. Since K is an isotone projection cone, we obtain [applying assumption (1)] that T is isotone with respect to the ordering defined by K. Using assumptions (2) and (3), we can show that T is a p0 contraction with respect to d, of the component C . We can also show that T{ CJQ ) c C . This theorem is a consequence of Banach's contraction principle and of the fact that every Cauchy sequence with respect to d, is a norm Cauchy sequence. ■

4. Working with two cones In this section, we will introduce another metric on a convex cone. To introduce this new metric, we will work with two cones and we will use an idea similar to Thompson's idea. The results presented in this section, are 220

WORKING WITH TWO CONES based on the paper by V. Ya. Stetsenko and B. Imomnazarov [1]. The idea to use two cones KczK\ has an interesting consequence, in that by this way we obtain not only the existence of a solution for a nonlinear equation, but also a solution belonging to a smaller cone. If the cone K, is composed by some particular elements, we obtain by this method a solution satisfying some particular properties. For example, this is the case when K\ is the cone of continuous positive real valued functions defined on a compact space, and K is the subcone of K\ formed by the monotone increasing functions. Let \E,\ I) be a Banach space and K, K\ a E two pointed closed convex cones such that K a K\. We will denote the ordering defined in E by jr(resp.JTi)by<(resp<:). Definition 4.1 We say that K is Ky-normal if there exists a > 0 such that for all x eK, y s K\ with x + y G K it follows that a max{||x|,||>|} < \x + y\. The following result is known (see V. Ya. Stetsenko and B. Imomnazarov [I])Proposition 4.1 The cone K is K\-normal if and only if there exists a constant p> 0 such that for allx,y e K, 0 < x // > 0, such that x < Ay andy < ux. (0 (i) When x and v are linked we also say that JC and y are connected. The binary relation, x % y o x and y are linked, is an equivalence relation on K. If x eK\ {0} we denote its equivalence class with respect to ^ b y Ci£x). Proposition 4.2 Let Kbea K\-normal convex cone in E. If for x,y s Kwe

221

METRICS ON CONVEX CONES have x < Ay,y< Ax, \\x\\ < Mand ||y|| < M, then |JC -y\\ < M{\ + 2p)(A -1), where p is the normality constant. Proof Since x -y < (X - l)y andy -x < (A. - l)x, it follows that - (X - l)x<x -y a n d - (A - \\x<x -y < (A - \)y. 0) 0) (i) For this, we deduce 0 <(x - y) + (A - l)x <(A - l)y + (A - l)x and by Proposition 4.1 we obtain \\x-y + (A- l)x\\ < /fl(A - l)y| + p - \)x\ < 2pM{A -1). On the other hand, we know that | | x - > ; | | - | P - l ) x | | < | | x - y + (A-l)x|| which implies that II* -\HI - ll(^ - 1WI + 2pM(k - 1) < M{\ + 2p)(A -1). Thus, the proposition has been proven. ■ For every x, y e K, we define a(x,y) = = inf<X inf< A x x<Xy\, < Ay >, and

fi{x, MU /3{x,y) = mfU y) = and we set

222

ySAx\ yZAxi

WORKING WITH TWO CONES dt(x,y) =

\r[max{a(x,y),fi(x,y)}].

Theorem 4.3 Let K be a K\-normal convex cone in E. IfCk is an arbitrary component ofK, then (C*, d*) is a metric space. Proof By the definition of d* we see that dt(x,y) = lr[max{a(x,y),0(x, lr[max{a(x,y),j3(x,y)}] v)}] = ln[max{/3(x,y),a(x, v)}] = = dJy,x), = dt(y,x), for all x,y eCkk. It is easy to conclude that d*(x,y) = 0, if and only if, x = y. Let x *y, (x,y e Ct}. Then, one of the numbers a(x,y), p\x,y) will be larger than unity. Indeed, if we assume the opposite, we find using the definitions of a and J3, that XQ < 1, JIQ < 1, such that, x<X0y and y<ju0x0. This implies that x < X 0/u0x < x, and that x = X 0ju0x when XQ/JO < 1 which is impossible. We now show that d* verifies the triangle axiom. For this, we suppose that xy^z e Ch are arbitrary elements and that X\,fj\, (i = 1,2,3) are the smallest of the numbers which satisfy the relationships: x<X,y;

y< u,x; x<X0z; 2 z< unx;2 z<X,yv and

0) ^ " o r 1

(i)

(i)^ (i)

y
'or3

We can suppose that X\> p.\. Then, dt(x,y) = InXx< l n ^ / l ^ ) = \nX2 + ln^ 3 < ln[max{^2,//2}] + ln[max{23,yu3}] = dt(x,z) + d,(z,y). If we suppose that H\ > X\ we get the same conclusion. Therefore the theorem is proved. ■

223

METRICS ON CONVEX CONES Theorem 4.4 Let (E,\ ||) be a Banach space and K, Kx czE closed pointed convex cones. IfK is K\-normal, then for every component Ck ofK, {Ck,d*) is a complete metric space. Proof Considering Theorem 4.3, it is sufficient to show that (Ck,dt) is com-plete. Let {xn}n£N c Q be a Cauchy sequence. We set <*„ = >nfU xp £Axq j , (p,q = 1,2,...). First, we will show that \x„} ., is norm bounded. Since [xn ] 7o

=

^o(^)

suc

i s a Cauchy sequence, then for a given s > 0, we can find h that d,(xp,xq) < s for &\\p,q > no- If we consider s= 1, we

obtain d,(xp,xq) < 1 for allp,q > n0. This implies that maxja^ja^l <e, for all p,q > n0. In particular, aprlo < e for all p > n0 so xp< ex < 3x Because K is K\ -normal, we have \xp < 3/dpc^ , for all p > «0- If we denote M0 = max|||x1|,...,||^no|,3p||xnj)||, then ||;x:n|| < M0 for all n eN.

Hence,

(x„ ) ., is norm bounded. We will now show that {x„}n<_N is a Cauchy sequence with respect to the norm of E. Let s> 0 be a given number. Then there exists a number S> 0 (<S=<5(£)) such that 1 l + * es' <<1 M00{l (l + + 2^)' M 2p)i"' Using again the fact that {x„}ni_N is a Cauchy sequence, we can find a natural number nx = nx(S), such that d,(xp,xq) < S, for a\\p,q >nh From this we deduce,

224

WORKING WITH TWO CONES a ^ <es pq pq qp w

s <\ + 11 M00(l(l + + 2p) 2p)" M

Thus, £g < 1 1] + ■ *,; 3 '>$> 2p) (i{) M0(l + 2p)p'

Xp

ss 1+1 X *'${M0{l M0(l + + 2p) 2p)\PXp

H

By Proposition 4.2 we have £

|k-*J<M 0 (l + 2p) >) 1 + M (l + 2p)- - 1. 0

=

£,

for all p,q > ri\, that is \\xp- xq\\ < e for all p,q > n\. This means, that {*«}«£# l& a Cauchy sequence with respect to the norm. Since E is a Banach space and K is closed, there exists an element xt e K such that ||x„- x, \\ -» 0 when n -> +00. The theorem will be proved if we show that {x„}n£N converges to xt by the metric d* in C*, and that xt e C*. Indeed, because \xn } is a Cauchy sequence with respect to d,, d,(xp,xq) < e for all sufficiently large p and q which implies that x pp <e"x JC„ < e"xaqqa;; (\) (\)

x<e x??a <exeDexppD (( ii ))

(4.1)

On the other hand, \\xq- x, || -» 0 as q -> +00 and x* eK. Passing to the limit in (4.1) when q -> +00, we obtain

xxp$eW, e% 4/x*1 x*.-t/(f}Xp

(4-2) (4.2)

for a sufficiently large/?. Since e > 0 is an arbitrary number, it follows that the sequence {x„}neN converges to xt with respect to the metric dt. The theorem is proved. ■ 225

METRICS ON CONVEX CONES Corollary 4.5 1) Every Cauchy sequence with respect to the distance d. in an arbitrary component ofK, is at the same time a Cauchy sequence with respect to the norm of space E. 2)If {x„}neN is a Cauchy sequence with respect to d., then its limit with respect to the metric d>, and that with respect to the norm of the space E coincide. Application We now give as an application of metric d,, a fixed point theorem. Theorem 4.5 Let (E,\\ |) be a Banach space andK,K^E K^E E closed pointed convex cones. IfK is Krnormal andf: E^Eis : is an operator, such that the 'f:E->E following assumptions are satisfied: l)j{K)czK, /*)££ 2) there exists p e [0,1[ such that for x,y e K satisfying x
z_ n

< ,x.)nn,x,)<-—d.(x ,x,)<——d.(x ,x:) -d.< (^O'^lj i) d.,{xd,(x nd,(x 00,x

\-p

p"

J

-Mxo-*i)

M ii) \\x„- x.\\ < M(k M(*0Xl Xl + + 2p) 2p) e'-r ee^M^'0-X]) -l,foreveryn>koeN, where, h G N,where, --l,foreveryn>koeN, 1

M lc M(k

0) = P \ o)

226

s-

i+fc, l+*n . 2e v >*<-.,,) pe

• |*Ao | and KQ IS an arbitrary natural number. •Ik!

WORKING WITH TWO CONES Proof We will show that /(C A (x 0 )) c Ck(x0). By assumption, x0 and/so) are linked. Lety be an element of Ck(x0). From assumptions (1) and (2), it follows ihatflxo) andy(y) are linked and by the property of transitivity of the connectivity relation, flx0), fly) and xQ belong to the component Ck(x0). Thus, / ( C A ( J C 0 ) ) C Ck(x0). The operator/is a contraction with respect to d, because d*(f(x), f(y)) < pd.(x,y) for allx.y e Ck(xQ). From Theorem 4.4, we know that \Ck(x0),dt j is a complete metric space, and the Banach's contraction principle is applicable. Hence,/has a fixed point x» e Q(x 0 ), which is unique in this component, and we conclude that and we conclude that (i) is also satisfied. To show estimation (it) we choose k0 e N. From conclusion (i) it follows that

A J* <&(* v ;c] <*Y— ,— ad,{xQ,xl) \-p which implies that xt< and and finally finally that that

«[

/o ——a - — A<*'(*0.*l) A(x ( x 00..*i) *i) x p

e~

J*

£—d.(x - d . ( j,x, j)c , ) £+

p , 11--" P < n e e> |xv + |
x,.

Q 0x

|X* || — /-'

.. ••||x r Ao, |. -*fc. ■

(4.3)

-|M. -IMI-

(4-4) (4-4)

By a similar argument, we can show that n n P ——dAxn,X\) Ulj Pc

K I < pl-pe,}-p >\W\±Pe

227

METRIC ON CONVEX CONES Inequality (if), is thus a consequence of (4.3), (4.4) and Proposition 4.2. ■ Remark 4.1 As shown in (Stetsenko, V. Ya. and B. Imonmazarov [1]), Theorem 4.5 can be used to study the solvability of the Urysohn integral equation. The results presented in this section, can be developed for cones in locally convex spaces.

5. Monotone semigroups and metrics on cones Some problems about Volterra functional equations can be solved using another special metric on cones. This metric can be defined by using a special monotone semigroup acting on the cone. The results presented in this section, are based on the papers of Turinici, M. [1], [2], [3], [4]. Let E(T) be a real locally convex space with the topology r defined by the family of seminorms Q = { A } I £ / • Let K c E be a pointed closed convex cone. Denote by < the ordering defined by K (i.e., x < v <=>y-x e K). Let S be a semigroup on K, that is S is a mapping (t,x) -» S(t,x) = S(t)x from R+xK into K, satisfying the following properties: Si) S(0)x =x,for S,) = x,for allx allx ee K, K, S52) S{t S(t + s)x = S(t)S(s)xfor all t,s e R R+ andx ee K. K. + andx We say that the semigroup S is monotone, if the following property is satisfied: 0s ' t > s imply implyx<S(s)y. * x<S(s)y. yj Remark 5.1 By Si) and S2) we can show that a semigroup S on K is 228

MONOTONE SEMIGROUPS AND METRIC ON CONES Archimedean, if and only if, 5/) x,y eKandx< S(t)yfor allt>0 imply xR+\J{+oa) {+00} d,:KxK->R+\J{-Hx>} defined by e^xiT. d^(x.y) x < ^S(t)y>}■ (x,y)eKxK. f{/>0|x<5' inf{ s(x,y) = mf{t>0\x<S(t)y\,

M

(5.1)

(It is understood that the infimum of the empty set is +00) Proposition 5.1 The couple (K,ds) is a generalized metric space. Proof First, we recall that a metric space is said to be generalized if its metric can take the value + 00. It is easy to remark that ds is symmetric, that is ds(x,y) = ds(y, x) for all x,y e K. If x = y, then by the definition of ds and by property (Si) we have ds(x,v) = 0. Suppose thatjcy e K are two elements, such that, ds(x,y) = 0. It follows by (S3) that x < S(t)y, y < S(t)x for all f > 0 and this givee [by property (S4)] ]hat x < v and v < x, which imply that x = v. The proposition will be proved, ,i we show that ds satisfies the triangle inequality. For this, we suppose that x,y,z e K are such that ds(x,y) < +00 and ds(y,z) < +000 Letting ds(x,y) < t, ds(y,z) < s be arbitrarily fixed, we obtain [by (5.1) combined with (S3)], x < S(t)y, y < S(s)z, y < S(t)z and z < S(s)y. Hence, by the semigroup condition (S2) it follows that x < S(t + s)z and z < S(t + s)x, which imply that ds(x,z)
METRIC ON CONVEX CONES To obtain the completeness of the metric space {K, ds), it is necessary to impose some special conditions on the cone K and to the semigroup S. In this sense, we suppose that the following conditions are satisfied: (Hi): E is a Q-sequentially complete locally convex space [i.e., any Q-Cauchy sequence in E is Q-convergeni). (H2): K is a Q-sequentially closed cone {i.e., any Q-convergent sequence in K has its limit in K, too) (H3): For any couple, i e I, b e K, there exists a function fb * R+ -> R+, such thatfiib(i) -> 0 as t -> 0 and pt[S(t)x -x]< fib{i) for all t el?+, x e K, andx < b. (H4): K is a Q-normal cone in the following sense: for any couple i e /, b e K there exists a positive number, kLb, such that 0 < x < y < b implies p,{x) < kiibp,{y). (H5): For any sequence {yn}neN <=•£ where {^j ij Q-convergent to an element y e K and any x eK, relation x < S(t)y„ for all n € N imply x < S(t)y. Under these assumptions, let a e K and r > 0 (r e R) be arbitrary fixed. Denote by K(a,r) the ds-closed ball with center a and radius r. Let x,y be two arbitrary elements in K{a,r). For simplicity, we set r = ds{x,y). By (S4), we obtain x < S(t)y and y < 5(r) x such that S(t)y = u + x and S(t)x = v+ y, for some u,v e K. Obviously, u + v = S(t)x -x + S(t)y -y, 0<x,y< S(r)a < S(3r)a < b where b = 2S(3r)a. Applying condition (H3), we obtain We also have

p,{u +v) < 2fb(t), ie I.

0<M,V<M + V<

S(t)x + S(t)y < S(2r)x + S{2r)y

and applying condition (H4), we deduce

230

MONOTONE SEMIGROUPS AND METRIC ON CONES p,( MM)<^,.( p,.( )<^.6AM(+v)<2^,. y;.AA(/). (/). M +v)<2^.i iy;. Now, we remark that x-y x- y = = S(t)y S(t)y-y-u -y-u implies -y)< i[S(t)y-y] p{S(t)y - y] +p +Pil(u), pPii(x {x-y)

Hence, we proved the

Proposition 5.2 Under assumptions (H3) and (H4) the evaluation all iel v) < (l + + 2K (d(d ie/ p{x2k^ )fitb Pi b)f s(x,y)),for i{x-y)<{\ bKb s{x,y)),for all

(5.2)

holds for any couple x,y e K{a,r) and any a e K, r>0, where b = 2S(3r)a. Let {xn}n£N be a ^-Cauchy sequence in K. Without loss of generality, suppose that {x„}neN c K(a,r) for some a e K and r > 0. From (5.2) it follows that {x„}neN is Q-convergent to JC*. Suppose s> 0 is arbitrarily fixed. There is, by the initial choice of our sequence, a natural number n(e) e N such that xm < S(s)xn and x„ < S{spc S(s)xmm,, for for all alln,m n, m>>n(e). n(e).

(5.3) (5.3)

Using assumption (H2) and passing to the limit as m -> +00 in the first relation of (5.3), we obtain that x* < S(s)x„ for all n > n(e). Using assumption (H5) and passing to the limit as m -> +00 in the second relation of (5.3) we deduce that x„ < S(e)xt for all n > n(s) and consequently we have d3(x„,xt) < e for all n > n(e), proving that {xn}neN is ^-convergent to x,. Thus we proved the following important result.

231

METRIC ON CONVEX CONES Theorem 5.3 If the assumptions (Hi)-(Hs) are satisfied, then the generalized metric space (K,ds) is complete. Applications Several interesting applications of the space (K,ds) in connection with the study of Volterra integral equations, are presented in the paper (Turinici, M. [4]). We will mention only a fixed point theorem. Let \E(T),Q= {Pi\ifLl\ be a locally convex space and K c E a closed pointed convex cone. Suppose that assumptions (Hi)-(H5) are satisfied. Let / : K -» K be a mapping. Let

R+ be an increasing function which satisfies the following property: 1) (r)]/(x) for all / e R+ and xeK. Theorem 5.4 Let (p : R+ -> R+ be an increasing Junction andf: K -» K a monotone mapping. If the following three assumptions are satisfied: 1) cp satisfies property (P), 2) fis cp-homogeneous, 3) the setK° ={xsK\x< S(a)f(x)and f(x) < S(a)x for some a > 0 j is non-empty, then there exists a couple of mappings ¥ : K° -» K° and 5: £° ->/?+, such that, for any x elf the following conclusions hold: i) the set K(x) = j v e K \ x < S(fi)y,y < S(fi)x for some fi>6\ is invariant under f and *f{x) is the unique fixed point off in K(x),

232

REFERENCES ii) the iterative process [fm{x):m e N] in K(x) is ds-convergent to d(x). Proof The proof can be found in the paper (Turinici, M. [4]) and is based on some special results proved in (Turinici, M.[2]) and (Matkowski, J. [1]).

■

We end this chapter with the remark that each of the results and mathemati cal techniques presented here, can become the starting point for new research.

6. References BAUER, F. L. 1. An elementary proof of the Hopf inequality for positive operators. Numer. Math. 7 (1965), 331-337. BELLMAN, R and T. A. BROWN 1. Projective metrics in dynamic programming. Bull. Amer. Math. Soc. 71 (1965), 773-775. BIRKHOFF, G. 1. Extensions of Jentsch's theorem. Trans. Amer. Math. Soc. 84 (1957), 219-227. 2. Lattices in applied mathematics. In: Lattice Theory. Proceed, of Symposia in Pure Math. Vol. II, Amer. Math.. Soc. (1961). 3. Uniformly semi-primitive multiplicative processes (I). Trans. Amer. Math. Soc. 104 (1962), 37-51. 4. Uniformly semi-primitive multiplicative processes (II). J. Math. Mech. 14(1965), 507-512. 233

METRIC ON CONVEX CONES BIRKHOFF, G. and L. KOTIN 1. Essentially positive systems of linear differential equations. Bull. Amer. Math. Soc. 71 (1965), 771-772. 2. Linear second order differential equations of positive type. J. d'Anal. Math. 18 (1967), 43-52. 3. Third order positive cyclic systems of linear differential equations. J. Diff.Eq. 5(1969), 182-196. BORWEIN, J. M., A. S. LEWIS and R. D. NUSSBAUM 1. Entropy minimization, DAD problems and doubly-stochastic kernels. Preprint, Faculty of Mathematics, University of Waterloo (1993). BUSEMANN, H. and P. J. KELLY 1. Projective Geometry and Projective Metric. New York (1953) BUSHELL, P. J. 1. On systems of linear ordinary differential equations and the uniqueness of monotone solutions. J. London Math. Soc. (2) 5 (1972), 235-239. 2. Hilbert 's metric and positive contraction mappings in Banach space. Arch. Rational Mech. Anal. 52 (1973), 330-338. 3. On the projective contraction ratio for positive linear mappings. J. London Math. Soc. 6 (2) (1973), 256-258. 4. On the solution of the matrix equation T'AT = A2. Linear Algebra Appl. 8 (1974), 465-469. 5. On solutions in a cone of ordinary differential equations in an ordered Banach space. J. Math. Anal. Appl. 49 (1975), 211-214. 6. On a class of Volterra and Fredholm non-linear integral equations. Math. Proc. Cambr. Phil. Soc. 79 (1976), 329-335. 7. The Cayley-Hilbert metric and projective operators. Linear Algebra Appl. 84 (1986), 271-280.

234

REFERENCES CHEN, Y. Z. 1. Thompson's metric and mixed monotone operators. J. Math. Anal Appl. 177 (1993), 31-37. DANES, J. 1. The Hilbert protective metric and equation in a C -algebra. Czechoslovak Math. J. 37 (112) (1987), 522-532. EDELSTEIN, M. 1. On fixed and periodic points under contractive mappings. J. London Math. Soc. 37 (1962), 74-79. EDWARDS, C. M. and M. A. GERZON 1. Monotone convergence in partially ordered vector spaces. Ann. Inst. H. Poincare XII4 (1970), 323-328. EISENACK, G. and CH. FENSKE 1. Fixpunktheorie. B. I.-Wissenschaftsverlag Mannheim (1978). FROBENIUS, G. 1. Uber matrizen aus positiven elementen, (I), S. B. Kg. Preuss. Akad. Berlin (1908), 471-476. 2. Uber matrizen aus positiven elementen, (II). S. B. Kg. Preuss. Akad. Berlin (1909), 514-518. FUJIMOTO, T. 1. Nonlinear generalization of the Frobenius theorem. J. Math. Economics 6 (1979), 17-21. 2. Addendum to nonlinear generalization of the Frobenius theorem. J. Math. Economics 7 (1980), 213-214. GOLUBITSKY, M., E. B. KELLER and M. ROTHSCHILD 1. Convergence of the age structure: Applications of the projective metric. Theoretical Population Biol. 7 (1975), 84-93.

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METRIC ON CONVEX CONES HILBERT, D. 1. Uber die gerade linie als kurzeste verbindung zweier punkte. Math. Ann. 46(1895), 91-96. 2. Neue begrundung der Bolya-Lobatschefskyschen-skyschen geometrie. Math. Ann. 57 (1903), 137-150. HOPF, F. 1. An inequality for positive linear integral operators. J. Math. Mech. 12 (1963), 683-692. ISAC, G. and A. B. NEMETH 1. Projection methods, isotone projection cones and the complementarity problem. J. Math. Anal. Appl. 153 Nr. 1 (1990), 258-275. JENTZSCH, R. 1. Uber integralgleichungen mitpositiven kern. J. Reine Angew. Math. 141 (1921), 235-244. KLEIN, F. 1. Uber die sogenannte nicht-euklidische geometrie. Math. Ann. 4 (1871), 573-625. KAWANO, N., T. KUSANO and M. NAITO 1. On the elliptic equation Aw = 0(;c)z/ in R2. Proc. Amer. Math. Soc. 93 (1985), 73-78 KOHLBERG, E. 1. The Perron-Frobenius theorem without additivity. I. Math. Economics 10 (1982), 299-303. KOHLBERG, E. and A. NEYMAN 1. On the relationship between convergence in Hilbert 's projective metric and convergence in direction. Harvard Business School, Working paper 79-54.

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REFERENCES KOHLBERG, E. and J. W. PRATT 1. The contraction mapping approach to the Perron-Frobenius theory: why Hilbert's metric? Math. Oper. Research 7 Nr. 2 (1982), 198-210. KRASNOSELSKII, M. A. 1. Positive Solutions of Operator Equations. Noordhoff (Groningen), (1964). KRASNOSELSKII, M. A. and P. P. ZABREIKO 1. Geometrical Methods of Nonlinear Analysis. Springer-Verlag. Berlin (1984). KRAUSE, U. 1. A nonlinear extension of the Birkhoff-Jentzsch theorem. J. Math. Anal. Appl. 144 (1986), 552-568. 2. Perron's stability theorem for nonlinear mappings. J. Math. Economics 15 (1986), 275-282. 3. Relative stability for ascending and positively homogeneous operators onBanach spaces. J. Math. Anal. Appl. 188 (1994), 182-203. KRAUSE, U. and R. D. NUSSBAUM 1. A limit set trichotomy for self-mappings of normal cones in Banach spaces. Non. Anal. Theory Meth. Appl. 20 Nr. 7 (1993), 855-870. KRIETE,H. 1. Internal completeness of cones. J. Math. Anal. Appl. 161 (1991), 545554. MATKOWSKI, J. 1. Fixed point theorems for mappings with a contractive iterate at a point . Proc. Amer. Math. Soc. 62 (1977), 344-348. MORISHIMA, M. 1. Equilibrium, Stability and Growth. Clarendon Press, Oxford (1964)

237

METRIC ON CONVEX CONES MORISHIMA, M. and T. FUJIMOTO 1. The Frobenins theorem its Solow-Samuelson extension and the KuhnTucker theorem. J. Math. Econom. 1 (1974), 199- 205. NUSSBAUM, R. D. 1. Hilbert's Projective Metric and Iterated Nonlinear Maps, (I). Memoirs Amer. Math. Soc. 391 (1988). 2. Iterated Nonlinear Maps and Hilbert's Projective Metric, (II). Memoirs Amer. Math. Soc. 401 (1989). OSfflME, Y. 1. An extension of Morishima 's nonlinear Perron-Frobenius theorem, J. Mat. Kyoto Univ. 23 (1983), 803-830. OSTROWSKI, A. M. 1. Positive Matrices and Functional Analysis. In: Recent Advances in Matrix Theory, (Madison), University of Wisconsin Press (1964). PERRON, O. 1. Zur theorie der matrices. Math. Ann. 64 (1907), 248-263. POTTER, A. J. B. 1. Existence theorem for a non-linear integral equation. J. London Math. Soc. (2)11(1975), 7-10. 2. Hilbert 's projective metric. Applied to a class of positive operators. In: Ordinary and Partial Differential Equations. Lecture Notes in Math. 564, Springer Verlag (1976), 377-382. 3. Applications of Hilbert's projective metric to certain classes of nonhomogeneous operators. Quart. J. Math. Oxford (2) 28 (1977), 93-99. SAMELSON, H. 1. On the Perron-Frobenius theorem Mich. Math. J. 4 (1957), 57-59.

238

REFERENCES SOLOW, R. M. and P. A. SAMUELSON 1. Balanced growth under constant returns to scale. Econometrica 21 (1953), 412-424. STETSENKO, V. YA. and B. IMOMNAZAROV 1. Existence of eigenvectors of nonlinear not completely continuous operators. Sibirskii Matem. Zhurnal 8 Nr. 1 (1967), 146-155 THOMPSON, A. C. 1. On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14 Nr. 3 (1963), 438-443. 2. Generalization of the Perron-Frobenius Theorem to Operators Mapping a Cone into itself. Ph. D. Thesis, University of Newcastle upon Tyne (1963). 3. On the eigenvectors of some not-necessarily linear transformations. Proc. London Math. Soc. (3) (1965), 577-598. THORLUND-PETERSEN, L. 1. Fixed point iterations and global stability in economics. Math. Oper. Research 10 Nr. 4 (1985), 642-649. TURINICI, M. 1. Projective metrics and nonlinear projective contractions. An. Stiint. Univ. Iasi 23 (1977), 271-280. 2. Sequentially iterative processes and applications to Volterra functional equations. Ann. Univ. Marie Curie-Skladowska Sect. A 32 (1978), 127-134. 3. Fixed points for a sum of projective contractions. Ann. Stiint. Univ. Iasi 25 (1979), 99-105. 4. Volterra functional equations via projective techniques. J. Math. Anal. Appl. 103 (1984), 211-229.

239

CHAPTER

3

Zero-Epi Mappings

1. Introduction In this chapter, we will focus on two goals. The first is to present, with minimally necessary details, the theory of zero-epi (shortly 0-epi) mappings and to show that, in many applications, this concept is an adequate substitute for degree theory. The concept of 0-epi mapping agrees with the concept of essential compact vector field (with respect to an open bounded subset introduced by A. Granas [1]). However, here, we present the theory of 0-epi mapping as defined and developed by the Italian School (Furi, M., M. Martelli and A. Vignoli [3], [2], [1]), (Furi, M., M. P. Pera and A. Vignoli [1]), (Furi, M. and M. P. Pera [1], [2], [3], [4]), (Furi, M. and A. Vignoli [1]), (Massabo, I., P. Nistri and M. P. Pera [1]), (Pera, M. P.[l], [2], [3]) and (Ize, J., I. Massabo, J. Pejsachowicz and A. Vignoli [1]) et al. The concept of 0-epi mapping can be considered as a refinement and generalization of topological degree. In fact, 0-epi mappings have properties such as existence, boundary dependence normalization, localization and homotopy invariance analogous to those of degree theory. Moreover, since 0-epi mappings may act between different spaces, in many cases, they can be used to study the solvability of boundary value problems where the classical degree theory is not applicable. Another fascinating aspect about the concept of 0-epi mapping is the fact that we can have a 0-epi mapping with respect to an open bounded set but with topological degree zero. To support this statement, we will show two examples: one in the finite dimensional case, and another in the infinite dimensional case. Finally, 0-epi mappings may also be viewed as a simple tool which helps the use of Schauder fixed point theorem (because the normalization property is a reformulating of this fundamental fixed point theorem) and 241

ZERO-EPI MAPPINGS they can be successfully combined with degree theory to obtain new and deeper results. We present in this chapter not only the concept of 0-epi mapping, but also some of its variants and refinements. The concept of O-epi-mapping with respect of a convex cone will be also given. We remark that when the map/ is a compact perturbation of the identity, t h e n / i s 0-epi, provided that the Leray-Schauder topological degree is defined and different from zero. Furthermore, monotone maps are 0-epi under suitable assumptions. In (Ize, J., I. Massabo, J. Pejsachowicz and A. Vignoli [1]), a generalization and refine-ment of the concept of 0-epi mapping is also presented and applied to the study of some nonlinear problems. The class of 0-epi mappings in the generalized sense (Ize, J., I. Massabo, J. Pejsachowicz and A. Vignoli [1]) contains many more types of mappings. In particular, it contains any class of mappings for which a classical degree theory satisfying the boundary dependence property under compact perturbations is defined, for example k-set-contractions, condensing vector fields, maps of type (S)+, ,4-proper mappings to name a few. Another interesting extension of the concept of 0-epi mapping to £-setcontractions is in (Tarafdar, E. U. and H. B. Thompson [1]). In several papers, such as (Furi, M., M. Martelli and A. Vignoli [1], [2], [3]), (Furi, M. and M. P. Pera [1] - [4]), (Furi, M. and A. Vignoli [1]), (Furi, M., M. P. Pera and A. Vignoli [1]), (Pera, M. P.[l] - [3]) and ( Ize, J., I. Massabo, J. Pejsachowicz and A. Vignoli [1]), the 0-epi mappings were used as the main mathematical instrument to study the solvability of several types of nonlinear problems and in the study of global branches of solutions of the problem: /(x,A) = 0, A e A where / : £ x A ^ F is a continuous mapping, E and F are Banach spaces and A is a "parameter space", which is a (not necessarily finite dimensional) Banach space. Finally, the second goal of this chapter is to show other interesting ways for applications of this simple, deep, and strong mathematical instrument. For example, we will apply this concept to the study of Nash equilibrium for a couple of mappings and to the study of

242

ZERO-EPI MAPPINGS ON BOUNDED SETS Complementarity Problems. In this way, we will manage to apply the concept of 0-epi mapping to optimization problems and to the study of equilibrium (in economical or physical sense). We hope, by this new approach, to stimulate the research in this area.

2. Zero-epi mappings on bounded sets Let (E,l I) and (F,|) |) be normed vector spaces. First, we recall some definitions. If Q c E is a non-empty subset, we denote by Q the closure of Q, and by 9Q the boundary of Q. Let Q cr E be a non-empty subset and f:Q-> F a mapping. We say that / i s compact if/(Ci) 1S contained in a compact subset of F, and we say t h a t / i s completely continuous i f / i s continuous and it maps bounded subsets of Q into relatively compact subsets of F. A continuous mapping f:E—>E is said to be a compact (resp. completely continuous) vector field if x -> x - f(x) is a compact (resp. completely continuous) map. The next theorem is a basic pre requisite for this section. Theorem 2.1 [Schauder] Let Q be a convex (not necessarily closed) subset of a normed space \E,\ ||)- Then each continuous compact mapping f:Q —> Q has at least one fixed point. Proof A proof of this classical theorem is presented in (Dugundji, J. and A. Granas [1]). ■ Recall furthermore that a continuous mapping / : Q -> F is proper iff~'(D) is compact for every compact subset D a F. We can show that, if/ is proper, then /(&) is closed. In this case, we h a v e / ( Q ) = / ( Q ) . We say that a topological space X(T) is normal if it is Hausdorff, and for all closed subsets A,BaX A,BczX such that Af\B = 0, there exist two open subsets [/and 243

ZERO EPI MAPPINGS V such that A c U, B a V and Uf]V = . It is well known that every normed vector space is normal. We recall the following classical result. Theorem 2.2 [Urysohn] A Hausdorff topological space X{r) is normal if and only if, for every two closed subsets A and B such that AC\B = 0, there exists a continuous function (p:X —> [0,l] such that (p[x) = 0 for every x e A and F is zero-epi {shortly 0-epi) if and only if 1) 0 if{dCl)(i.e.,fis O-admissible), 2) for any continuous compact mapping h:Q.—>F such that h(x) = 0for every x e dQ., the equation f(x) = h(x) has a solution in Q. Remark 2.1 If E = F and / : Q -» E is a continuous compact perturbation of the identity, then the above definition agrees with the definition of essential compact vector field (with respect to Q) as given by A. Granas [1]. Definition 2.2 If Q cz £ is an open bounded subset andp e F an arbitrary element, we say that a continuous mapping /f.Q^F : Q - > F is p-epi if and only if 1) p € f{dQ)(i.e., f"isp-admissible); 2) the mapping f-p> f-p defined by (f-p (f-p)(x) =fix) -p is 0-epi. p) {x)=fix) The fundamental properties of 0-epi mappings are similar to the properties of Brouwer's topological degree. We will now give the main properties of p-epi mappings. Existence (or solution) property If / : Q --^» FF is p-epi, then the equation f(x) = p has a solution in Q.

244

ZERO-EPI MAPPINGS ON BOUNDED SETS i: Q -> E is p-epi if and only if Normalization property The inclusion /: p e Q. Proof If we suppose that the inclusion i: Q —» E is p-epi, then by the existence property of p-epi-mapping we have that p e Q . Conversely, we suppose that p e Q. It is sufficient to suppose that 0 e Q and to show that the inclusion i :Q —> E is 0-epi. Indeed, let h:E —» E be a continuous and compact mapping such that h(x) = 0 for any x g O . Since 0 e Q, the equation i(x) = h(x) has a solution in Q if and only if the mapping h:E -> £ has a fixed point. But, since /z(£) is compact, applying Schauder's Fixed Point Theorem (Theorem 2.1), we have that h has a fixed point and the property is proved. ■ Localization property If f:£l -> F is 0-epi, Q , c Q is an open set and /~'(0) c Q,, then the restriction off of f to Q,, i.e. f

:Q, —> F is 0-epi.

Proof First we have 0 g / ( ^ Q , ) as Q, n. Suppose given a con tinuous compact mapping /z:Q, -> F such that h(x) = 0 for every x e dd\. Let h* be the extension of h to Q given by : , . .

h{x):

fO,i/xeQ\ni [/z(xj, z/ x e i i , . \h[x),ij

The mapping /z» is continuous and compact. By assumption, the equation f(x) = h,(x) has a solution x* e Q. Since / - 1 (0) a Q,, we must have that xt e Q,, and the property is proved. ■ Homotopy property Let f:Q-> f:Q -> F be 0-epi and let h:Qx [0,l] -> F be a continuous and compact mapping such that h(x,0) = Ofor any x e f l . If f(x) + h[x,t) 9* 0 for all x e dQ, andfor any t e[0,1], then the mapping 245

ZERO EPI MAPPINGS /(•) + h(-,l): Q -> F is 0-epi. Proof Let g: Q -» F be a continuous compact mapping such that g(x) = 0 for all x e 9Q. The set D = {x G Q| / ( * ) + h(x,t) = g(x) for some t e[0,l]| is closed since [0,1] is compact. By Urysohn's Lemma, there exists a continuous function [0,l] such that q>(x) = 1 for every x e Z> and q>(jc) = 0 for all x e dQ.. Consider the equation f(x) = g(x)-h(x,
(2.1)

and remark that the mapping h,: Q —» F defined by k(x) = g(x) -/*(*, p(x)) is continuous, compact and vanishes on dQ. Then, since / is 0-epi, there exists a solution x. of equation (2.1). We observe that x* e D and hence cp(x.) = 1. Obviously, f{x,) + h[x„,\) = g(x.), and the proof has been completed. ■ Boundary dependence property If / : Q -> F is 0-epi awrf g:Q -> F is a continuous compact mapping such that g(x) = 0 ^or all x e d Q, tfien / + g: Q -> F is 0-epi Proof The property is a consequence of Definition 2.1. ■ The next result establishes an interesting relation between the concepts of p-epi mapping and the Leray-Schauder degree (Rothe, H. [1]) and (Lloyd, G. [1]).

246

ZERO-EPI MAPPINGS ON BOUNDED SETS Theorem 2.3 Let \E,\ |) be a Banach space and Q <= E an open bounded set. If f: Q -> E is a p-admissible compact vector field and the LeraySchauder degree deg(/,Q,p) ^ 0, then f is p-epi. Proof Let g. Q —> E be a continuous compact mapping such that g(x) = 0 for all x e SQ. We remark that / - g is a compact vector field such that ( / - s){x) = /(■*•) f ° r a n x e ^ - By ^ e boundary dependence property of the Leray-Schauder degree, we have deg(f - g,Q.,p) = deg(f,Q,p) * 0. Hence, the equation f(x) - p = g[x) has a solution in Q, that is, / is p-epi.

■

The following example, recently published in (Ding, Z. [1]), shows that the concept of 0-epi mapping is more refined than the topological degree. Consider the space E = h and select positive real numbers a\, a^r, s, p and k such that re r t, rsIkIIk \-, A1 rm k 2 2 2 ^L, L > 1 , rs -a2 < p,a\ +a\< p2 a s>a,,-{p-a2)<-,-j-L>l, _ 2
€<-

-P-

1 5 l/2

4

Set Set

° I = {*=( X I.L^ 6 / 2|HI
tol2 == {x Q {x == (xn)nEN {Xn)nsNel2\\\4 el2\xl>al,x2>a2) Q2 = {x = (xn)nEN el2\xl>al,x2>a2) and define Q = Qi f| ^2- Note that Q is open and bounded. In the paper (Ding, Z. [1]), it is showed that the mapping / : Q -» l2 defined by

f(x) = ((*,0,0,...) /(x) o 2,)[o,^,^,..^ ^,^,...) £,0,0,...) + r ( * 2 - ar{x 2 ) 2(-a

247

ZERO-EPI MAPPINGS has the following properties: The mapping / - / i s 0-epi on Q and deg(7 -f - g, Q,0) = 0 for any compact continuous mapping g : Q-> l2 such that g(3Q) = {0}. This means that we can solve the equation (/ -f)(x) = g(x) in Q for any such mapping g, and this result does not follow from the direct application of degree theory to the mapping/ -f-g. Remark 2.2 It is important to also remark that there exist 0-epi mappings Q F is 0-epi and proper, then/ maps Q onto a neighborhood of the origin. Proof The theorem will be proved if we show that, if D0 is the connected component of F\f(dQ) F\f(Xl) containing the origin, then D0 c / ( Q ) . Indeed, since / is proper, we have that f(dQ) is closed and D0 is open, which implies that Do is path connected. Let x e D0 be an arbitrary element and ^:[0,l] -» D0 be continuous mapping such that cp(0) = 0 and cp(l) = x. The homotopy (x,t) -»,/(*) _ 9(0 implies that /(•) - #>(l):Q -» F is 0-epi and in particular we have that x e / ( Q). ■ Theorem 2.5 Let [E,\ ]|) and (F,\\ |) be normed vector spaces and Q c E an open bounded set. If / : Q -> E is continuous, injective, and proper, then f(Q) is open if and only iffisp-epiforanyp O). pee / ( Q iffisp-

248

ZERO-EPI MAPPINGS ON BOUNDED SETS Proof Supposep e / ( Q ) a n d / /?-epi. From Theorem 2.4, we have that /-/? maps Q onto a neighborhood of 0 and hence / Q ) is open. Conversely, suppose that fiQ.) is open. Since / is proper and injective, we have / is invertible on its image and/"7 is continuous. The theorem will be proved if we show that if 0 e f{&), then/is 0-epi. Consider a continuous compact mapping ft: Q -> F such that h(x) = 0 for all x s 8Q and define a mapping T-.F^Fby

! n W/"'(4 ^^( ) *w= |o [0

,ifytf{Cl). ,ifyef{Cl).

We have df(Q) = f(dQ) because /(Q) = / ( Q ) and / ( Q ) is open. Hence, *¥(y) = 0 for all v e
is a solution of the equation/*) = h(x) and the theorem is

Corollary 2.6 [Extension of the normalization property] Let [E,\ |) and \F,\ I] be normed vector spaces and Q ac E an open bounded set. Let f:Q->F

/ : « - > * ■

be continuous, injective, and proper. If f[Cl) is open, then/is

p-epi if and only ifp e f{&) ■ ■ We say that a set D c F is star-shaped with respect to the origin if Xy e D for every y e D and X e [0,1]. Theorem 2.7 Z,e/ (jB,| |) #«£? (F,\\ ||) 6e normed vector spaces and fic£ fic£ a« opew bounded set. If / : Q -> F is 0-epi and there exists a starshaped (with respect to the origin) subset DczF such that DC\f(dQ) = <j>, then the equation fix) = h(x) has a solution in Q for any continuous compact 249

ZERO-EPI MAPPINGS mapping /?:Q-»F satisfying the property h(dQ) a D. In particular,

£>c/Q). Proof Indeed, if x edCl and X e [0,1], we have Xh(x) e D and/*) g D, which imply that/*) * Xh(x) for all x <= 5Q and for every X e [0,1]. By the homotopy property, we deduce t h a t / - h is 0-epi. Consequently, we have thatX*) = h{x) for some xeQ. Let/? eZ) be an arbitrary element. By taking as h the constant mapping h{x) = p for every * e Q, we derive that p e fCi), and the theorem is proved. ■ An interesting problem is to ascertain under what conditions the property to be 0-epi, is conserved by the uniform convergence. The answer to this problem is given by the next result. Theorem 2.8 Let \E,\ ||J and \F,\ ||j be normed vector spaces and Q c E an open bounded set. Let {f„}neN be a sequence of 0-epi mappings from Q into F uniformly convergent tof. If f is 0-admissible and proper, thenf is 0-epi. Proof First we remark that / is continuous. Consider a continuous com pact mapping /i: Q -> F such that h{x) = 0 for all x e d Q. Let x„ be any so lution of the equation f„(x) = h(x) and set *n = f{xn)-f«{x forallall« n==1,2,.... 1,2,.... n=f(Xn)/ . (n*) , ) ==/ ( /(*„) * , ) - * (~h(x„) * « ) for Z

We can show that lim v„ = 0. Since f-h f-his is proper (because h is compact n—>oo

continuous), we have that {xn}neN has a cluster point x., and for this point we deduce^x,) = h(xt). ■ Theorem 2.9 [Perturbation Theorem for 0-epi Maps] Let (E,\\ |) and (F,| I) be normed vector spaces and Q^Ean 250

open bounded set.

ZERO-EPI MAPPINGS ON BOUNDED SETS / : Q - > F is a proper 0-epi mapping and fcQx[-l,l]->F If f:Q->F fcQx[-l,l]->F >F a continuous compact mapping such that h(x,0) = 0 for any x e Q , then there exists a real number e > 0 such that f(-)-h(-,X) f ()-*•>*) is 0-epi for every A satisfying \ X \ < &

Proof Considering the homotopy property, we observe that it is sufficient to show that there exists e > 0 such that f{x) * h[x,X) for all x e dQ and X e ]-s, E[. Suppose the contrary. In this case, there exists a sequence {{xn,An)}neN in dQ x [-1,1] such that jimA„ = 0 and f{xn) = h(xn,An). Since / is proper and h is compact, we obtain the existence of a cluster point x0 of the sequence {x„}neN, such that XQ e 9Q. Obviously, we have f(x0) = h(x0fi) which is a contradiction of the O-admissibility of / and thus the theorem is proved. ■ We now give some examples of 0-epi mappings. Examples 2.1 /. Let (E,\\ I) and (F,\\ |) be Banach spaces and L : E —> F a linear bounded operator. Suppose that L is surjective and dimKerL = n<+co. Also consider given an open bounded set Q c= E and a continuous mapping g.Q->R", such that g (x) ^ 0 for any x edQf)KerL. If there exists a n linear mapping J : R —» E such that Im J = KerL, and the Brouwer topological degree deg(g./)./-;(fi),0) ^ 0, then the mapping M:Q.-^ FxR" defined by M(x) = {Lx,g[xfj is 0-epi. Indeed, let T:F x R" -» E be defined by T(y,z) = Sy + Jz, where S: F-+E is any bounded linear right inverse of L. We can show that T is an iso morphism and M is 0-epi if and only if the composition / = MT from r\Q)

into FxR" is 0-epi. We have

251

ZERO-EPI MAPPINGS f(y,z) = (L{Sy + /(y,z) + Jz), Jz), g(Sy g(Sy + + Jz))=(y,g{Sy Jz))=(y,g{Sy ++ Jz)) Jz))

= (y,z)-(0,z-g{Sy {y,z)-{0,z-g{Sy + + Jz)). Jz)). Hence, / is of the form I -h , where h maps ^ ' ( Q ) into the finite dimen sional space {0} x R". Using the definition of the Leray-Schauder degree, we have 1 deg(/,rdeg(/,r-11(Q),0)=deg (Q),0)=deg If. y: fi

(0}x«"n V |{o}*«"

= =

,r\n)f\{o}xRnn),o\ ,r\n)f\{o}xR"),o\ I

/

11 (Q),0)*0, deg(g '0, < /,J- (Q) deg(gJ,J) 0)^0,

which implies that / is 0-epi. ■ II. Let (E,\\ I) and ( F , | fl) be Banach spaces and Q c E an open bounded set. Let L : E -» F be a bounded Fredholm operator of index 0 with dim KerL = n, and let h : Q-+F be a continuous compact mapping such that h (x) € ImLfor any x e d£l fl KerL. Suppose that l deg(QhJ,J-\n),0)*0, deg(QhJ,J{Q),0)*0, 0, where J: R" —>E, Q : F —> R" are linear continuous operators, such that Im J = KerL and KerQ = ImL. Then, there exists e> 0 such thatL -Ah is 0-epi for every X satisfying 0 < \X\ < s. Indeed, from Example I, we have that the operator M: M : D.Q.->lmLxRn, defined by M(x) = (Lx,Qh(x)), is 0-epi. Let 0 : F -> ImZ, be any continuous retraction. Since M is proper in Q, we obtain, by Theorem 2.9, that there exists e > 0 such that the mapping x -> (Lx - AOh[x),Qh[x)\ is 0-epi for any 0< |A,| < e. Let k : Q ->F be a continuous compact mapping such that k(x) = 0 for any x e 9Q. From the boundary dependence property for 0-epi mappings, we have that the mapping 252

ZERO-EPI MAPPINGS ON BOUNDED SETS x -> ->• (Lx [he - A,®(h(x) M>[h{x) ++ rrllk{x)),Q{h(x) k{x)),Q{h(x) ++ A~ A~llk(xjf) k(xjf) is 0-epi provided that 0< |A,| <s. Now, the result is true since the system

Lx = M&(h{x) Mdh{x) ++ZZ-xxk{x)) k{x)) i Q(h(x)++A-r\x)) Q(h(x) (x)) ==0 o

is equivalent to the equation Lx - Ah(x) = 0. ■ Remark 2.3 We recall that a linear operator L : E —> F is called a Fredholm operator if dxmKerL < + oo, ImZ, is closed, and codim ImZ < +00. The index of Z is the integer iL = dim KerL — codim ImZ-. It is also known that a bounded Fredholm operator is proper on bounded closed sets. For other results about Fredholm operators, the reader can consult (Larsen R.[l]), (Schechter, M.[l]). m.Let (E,\ I) and (F,\\ ||) be Banach spaces and Q F be a bounded Fredholm operator of index 0. Ifh : Q —► F is a continuous compact mapping and there exists a compact linear operator T: E —> F such that 1) Ker(L + T) = {0}, 2) Lx + Tx ^/l[h(x) + Tx]for onyx BdQ. and X e [0,1], then L -his 0-epi. Indeed, by the Fredholm alternative, we have ind(L + T) = ind(L) = 0 and, by assumption (1), we obtain that L + T is an isomorphism. Since 0 e Q, Corollary 2.6 implies that L + T is 0-epi on Q. The result is a consequence of the homotopy property of 0-epi mappings. Application It is well known that a classical result related to the existence

253

ZERO-EPI MAPPINGS of nontrivial solutions of nonlinear operator equations on infinite dimen sional Banach space is the following theorem. Theorem 2.10 [Birkhoff-Kellogg] Let (E,\ |) be an infinite dimensional Banach space and let 5(0,1) be the closed unit ball ofE. IfT: 5(0,l)->£ is a continuous and compact mapping such that there exists k > 0 satisfying \\T(x)\\ > k for allx e 0 such that X* = XJ\X').

Proof The proof can be found in the paper (Birkhoff, G. D. and O. D. Kellogg[l]). [See also (Krasnoselskii, M. A. [1])]. ■ Now, we give a generalization of the Birkhoff-Kellogg theorem which was proved for the first time by M. Furi and A. Vignoli in [1]. The proof is based on the concept of 0-epi mapping and on the following result: Lemma 2.11 Let \E,\ |) and (F,\ ||) be Banach spaces and ClcEan open bounded set. Let T . Q -> F be a 0-epi mapping andp e F \{0} an arbitrary element. V/T(Q) is bounded, then there exist r. > 0 and x. e9fi such that T{x*) = r*p. Proof Because T (Q) is bounded, there exists r > 0 such that T\x) - rp * 0 for all x e Q. Hence the map T - rp is not 0-epi. Consider the homotopy i / : f i x [ 0 , l ] - > F defined by H (x,X) =1\x) - Xrp. Since H(-,0) is 0-epi and H(x,\) is not 0-epi, by the homotopy property, we see that there exist x.e SQand n e ]0,1[, such that T{xt) = firp. Ifwenowsetr. =//r,the proof is completed. ■ Theorem 2.12 [Furi-Vignoli] Let (E,\ |) and (F,\\ ||) be Banach spaces and Q cE an open bounded set; F is supposed to be infinite dimensional. Let T: Q -> F be 0-epi and S : Q -> F continuous compact and satisfying

254

ZERO-EPI MAPPINGS ON THE WHOLE SPACE /or a// \\S(x)\\>/fc>0 >k>0for all xeSQ. x eSQ. IfT{Q) is bounded, then there exist A, > 0 and x. eSQ such that J\xt) = A, S(xt). Proof Denote by p the radial retraction of F \ {0} onto the boundary 65(0,1) °f the unit ball 5(0,1) of F. (We have p :F ^ 55(0,1)). Since S is compact and dim F = w, there exists p e F, with ||p|| = 1 and such that -p t(S(a)). We have that, for A> 0 large enough, the mapping T- X S is not 0-epi. Indeed, if this is not the case, by Lemma 2.11, there exist a sequence {*„}n6jv in SQ and a sequence {rn}nsN in tf+ such that

fil

)«z_n) I == (^]p, T{xn) - »S!fo) = rj?, for all ne JV, that is, % ^^--S(x N. % ) VwJk I.n Es AT. n Since {Tfo)},,^ is bounded, then

^TW) ™ ---► > •00

r

as n -> oo. Considering the n compactness of 5, we may assume that the sequence {s{xn)}ni_N converges to some element v e S(<Xl). From this, we note that there exists some P> 0 such that -fip = v, which implies, -p = ~ = p(v) e p f s f ^ j ) , which is impossible. ■

IHI

Remark 2.4 Since, as it is remarked in (Ize, I., I. Massabo, J. Pejsachowicz and A. Vignoli [1]), ^-proper mappings with nonzero degree are 0-epi, we have that Theorem 2.12 contains as a particular case the generalization of the Birkhoff- Kellogg's Theorem proved in (Fitzpatrick, P. M. and W. V Petryshyn [1]).

3. Zero-epi mappings on the whole space Let [E,\ I) and (F,\\ ||) be normed vector spaces. Suppose given a conti nuous mapping/: E ->Fand an arbitrary element/? e F

255

ZERO-EPI MAPPINGS Definition 3.1 We say that f bounded set in E.

is p-admissible if and only iff'

(p) is a

Definition 3.2 We say that the mapping f is p-epi with respect to E if f is p-epi on any bounded open set Q^>f~1(p), i.e. the restriction / ] 5 is p-epi in the sense of Definition 2.2 for any bounded open set Qr>/~ (/?) [or equivalently if the equation fix) - p — h{x) is solvable for any continuous compact mapping h with bounded support]. Remark 3.1 Concerning Definition 3.1, in view of the localization property for 0-epi mappings, we may restrict ourselves to sufficiently large open balls centered at the origin. Concerning 0-epi mappings with respect to the whole space E, we give only the specific properties. We will indicate the proofs only for the results which are not easy consequences of analogous results previously obtained in the context of 0-epi mappings with respect to bounded sets. Homotopy property Let (E,\\ |[j and (F,\\ ||J be normed vector spaces, f: E —» F a 0-epi mapping with respect to E and h : E x [0,1] - > F a completely continuous mapping such that h (x,0) = 0 for any x e E. If the set D = I x e E | /(x) + h(x, t) = 0 for some t e ]0,l]J is bounded, then fi-) + h(-, 1) is 0-epi with respect to E. ■ An outcome of homotopy property is the following result which is analogous to Theorem 2.7: Theorem 3.1 Let (E,\\ |) and (F,\\ |) be normed vector spaces, D c F a star-shaped set with respect to the origin and f: E —> F a 0-epi mapping with respect to E. Iff'(D) is bounded, then, the equation fix) = h(x) has a solution provided that h : E -> F is a completely continuous mapping with ImA c D. In particular, we have Im/z> D.

256

ZERO-EPI MAPPINGS ON THE WHOLE SPACE Proof The proof is similar to the proof of Theorem 2.7. ■ Corollary 3.2 Let (E,\ |) and (F,\ |) be a Banach spaces and L : E^tFa bounded linear operator. Then L is 0-epi with respect to E if and only if it is an isomorphism. Proof Indeed, if L is an isomorphism we can apply Corollary 2.6 since L is continuous, injective, proper, and open mapping. Conversely, if we suppose that L is 0-epi with respect to E, we have that L is one-to-one, since L is admissible (and KerL must be bounded). Since E and F are Banach spaces, it is sufficient to show that L is onto. Let pe F be an arbitrary element and consider the segment D = | / p | 0 < f < l } . The positive homogeneity of L implies that L~'(D) is bounded. Applying Theorem 3.I, we obtain that D a IvaL, and hence p slmZ,. ■ Proposition 3.3 Let \E,\ fl) and (F,\\ ||J be normedspaces andf: E ->F a continuous coercive mapping (i. e. lim / ( * ) = +°o). If f is 0-epi with |jr|-»W

"

respect to E, then the equation fix) =h(x) has a solution for any completely continuous mapping h : E ->F with bounded image. In particular, fis onto. Proof It is well known that the condition lim | / ( x ) | = +oo is equivalent to the fact that the inverse image under / of any bounded subset of F is bounded. The Proposition is proved if we remark that, taking any ball DczF centered at the origin and containing Imh we can apply Theorem 3.1.

m Remark 3.2 Note that under the assumptions of Proposition 3.3, the mapping / +h is 0-epi with respect to E for any completely continuous mapping h : E -» F having bounded image. In particular, we have that/is p-epi with respect to E for any p € F.

257

ZERO-EPI MAPPINGS Theorem 3.4

Let (E,\ ||) and (F,\ ||) be normed vector spaces and

f:E->Fa f:E Fa O-admissible mapping. Suppose that f is proper on bounded i —>r closed sets. Let {f n}n€fJ \fn) I neJV be a sequence ofO-epi mappings with respect to E,from E into F, converging uniformly to f on bounded subsets ofE. If the sets /„_1(0) are uniformly bounded, then f is 0-epi. Proof The proof is similar to the proof of Theorem 2.8. ■ Let (//,<,>) be a Hilbert space and /f.H^H : / / ^ / / a mapping. Recall that / is {f(x)-f(y),x-y) y)>0 for all monotone (in Minty-Browder's sense) if (f{x)-f{y),x-y)>0 x.y

G H.

-+H ia continuous Theorem 3.5 Let (//,<,>) be a Hilbert space and f-.H^H f-.H^H monotone operator which is proper on bounded closed sets. If (f{x),x) (f{x),x):)>0 >0 for x with |\x\| sufficiently large, then f is 0-epi with respect to E. Proof Consider for every n e N, where the mapping fn-H / „ : / f-> / - » H is defined by

/,(*) = Q)x + /(x). By a result proved in (Minty, G. J. [1]), we have that/, is a homeomorphism of// onto H. From Corollary 2.6, we have that/, is 0-epi with respect to E for every n G TV. The Theorem is now a consequence of Theorem 3.4. ■ It is also interesting to consider 0-epi mappings with respect to the whole space, acting on finite dimensional spaces. In this sense, we remark only the fact that in (Fun, M., M. Martelly and A. Vignoli [3]) is proved (using some special results on homotopy) that there are no 0-epi mappings n mm m f.R ^R R Hwith respect to Rn if n < m. Also, a O-admissible mapping f:R f.R"^R r->J f:R" -> R" is 0-epi with respect to R" if and only if deg(/,0) * 0 (deg(f,Q)

258

ZERO-EPI MAPPINGS ON THE WHOLE SPACE is the topological degree deg(/,fi,0), where Q is any bounded open set containing/-1(0)). Examples of 0-epi mappings with respect to the whole space are given in (Furi, M., M. Martelly and A. Vignoli [3]), (Furi, M. and M. P. Pera [1]), (Furi, M. and A. Vignoli [1]), (Massabo, L, P. Nistri and M. P. Pera [1], (Pera, M. P. [3]), (Furi, M. and M. P. Pera [4]), (Pera, M. P.[2]) and (Furi, M. and M. P. Pera [2], [3]). Let (j5,|| I) and \F,\ |) be Banach spaces and / : E -»F a continuous mapping. To investigate, under suitable assumptions on / the set/ _1 (0) of solutions of the equation/(*) = 0, and in particular to give conditions of/ which ensure that the set/ _ 1 (0) contains an unbounded connected component, it is necessary to introduce the notions of 0-regular and O-regularizable mappings. Definition 3.3 Let U a E be an open subset and /:£/-> f:U->F F a continuous mapping. We say that f is 0-regular (zero-regular), if and only if the following properties are satisfied: 1) /~'(0) is a bounded subset ofU, 2) for every completely continuous mapping h:U -> F with bounded support and such that h(x) = 0 for every x e d U, we have that the equation fix) = h(x) is solvable in U, 3) f is proper on bounded closed subsets of U. Remark 3.3 The mapping / is 0-regular, if and only if it is 0-epi and proper on bounded closed subsets of U. As for 0-epi mappings, we have for 0-regular mappings the following properties: Existence (or solution) property If f:U —» F is 0-regular, then the equation fix) = 0 has a solution in U.

259

ZERO-EPI MAPPINGS Localization property If f:U -> F is 0-regular and Vis an open subset of Ucontainingf~\0), then fv is 0-regular. Let f:U-+F f:U—>FF be a continuous mapping. We say that / is a regular embedding of U in F if / is a homeomorphism onto f{U ),f{U) is open in F, and f{ U) is closed in F. Also, we can show that a homeomorphism between U sndf{U ) is proper, if and only if f{U) is closed. Normalization property If f:U -» F is a regular embedding of U in F, then fis 0-regular, if and only if 0 e.f{U). Let f,g:U-±F :U->FF be continuous mappings. We say that a continuous map ping H:U x [0,1] -> F is a 0-homotopy joining/withg if: l)i/(-,0)=/and//(•,.!) = £, 2) the set D = [x eU \ H(x,X) = 0 for some A e[0,l]| is a bounded subset, oft/, 3) the mapping (x,X) -^H(x,0) - H(x,X) from U x [0,l] into F is completely continuous. Homotopy property If f,g:U -> F are 0-homotopic, then f and g are either both 0-regular or both not 0-regular. A consequence of the homotopy property is the following result: Theorem 3.6 Let (F,|| |) and ( F , | J) be Banach spaces and UcEan open subset. If f:U -> F is a proper 0-regular mapping, then f maps U onto a neighborhood of the origin. Moreover, fiJJ) contains the connected component ofF \f{dU). Proof The proof is similar to the proof of Theorem 2.4. ■ Proof The proof is similar to the proof of Theorem 2.4. ■ 260 260

ZERO-EPI MAPPINGS ON THE WHOLE SPACE Remark 3.4 Note that in (Furi, M. and M. P. Pera [2]) it is indicated that a linear bounded operator L : E -»F is 0-regular, if and only if, L is an isomorphism. Several applications of 0-regular mappings are given in (Furi, M. and M. P. Pera [1], [2]), (Furi, M. and A. Vignoli [1]), (Furi, M. and M. P. Pera [3], (Massabo, I., P. Nistri and M. P. Pera [1]) and (Pera, M. P. [2]). We now apply the concept of 0-regular mapping to the study of the structure of the set of eigenvectors of nonlinear equations involving compact perturbations of 0-epi mappings. Lemma 3.7 Let (X,d) be a compact metric space. If A and B are two closed disjoint subsets ofX then either: 1) there exists a connected subset of X intersecting both A andB, or 2) X is the union of two disjoint compact subsets XA and XB such that A czXA andB c l j . Proof The proof can be found in the book (Kuratowski, C. [1]) or in (Whyburn, G. T. [1]). Theorem 3.8 [Furi-Vignoli] Let (E,\\ ||j and (F,\\ |) be Banach spaces such that F is infinite dimensional. Suppose that f:E—>F is a 0-regular mapping sending bounded sets into bounded sets and h : E —» F a completely continuous mapping such that \\h (x)\\ > sfor some s > 0 and all x e E with sufficiently large norm. Then there exists an unbounded component of the set of eigenvectors D = \x <EE\f(x) = Ah(x) for some A>0\ _1 l starting from the boundary of the (bounded) set /-'(O) / - 1 (0) LM \Jh(#((),£■)). (B{0,s)).

Proof Consider two positive real numbers rj < r2. Using the O-regularity of

261

ZERO-EPI MAPPINGS / we can show that A = / _1 (0)UA _1 (5(0,f)) is bounded, and hence X = \x eZ)|r,
is also bounded.

Indeed, choose any sequence {x„}neN a X.

The set X is compact.

Since X
sequence {Xn}neN of non-negative real numbers such that f{x„) = Xnh[xn) for all n e N. Because / maps bounded sets into bounded sets, we have that/X) is bounded and because \\h(x)\\ > e for all x e X, we may assume {Xn) -> X, > 0. The complete continuity of h and the properness of/ imply that [x„}n&N has a subsequence convergent to some element x, e E. This implies f(x,) = X,h(x,),

that is x. £ D and r, < dist(x,,,4)
Hence, JO e X, and the compactness ofXis proved. Similarly, we can prove that any bounded and closed subset of D \ Int{A) is also compact. We now show that there exists a connected subset of X which intersects both X x {xeX\dist(x,A) Xn=\ = r^}l] and Xriri ={xeX\dist(x,A) = jx eX\dist(x,A) ==rr22).}. To show ri ={xeX\dist(x,A) 4='.j this, we suppose the contrary. By Lemma 3.7, the set X is the union of two disjoint and compact sets Kr^ and K such that X^ a K^ and X^ c K^.

Set 5S = dist(K dist(K rrK r2) > 0 and consider the open set U = U\\JU2, where rrK r)>0 U1l = Ix eE\dist(x,A).

Since

by the localization property, we obtain t h a t / i s 0-epi on U. On the other hand, dU^\D <5UC\D= =<j><j> and ||A(JC)|| \\h(x)\\ >> 88>>>080f> 0 for all x &3U which contradicts Theorem 2.12. Finally, given any integer n > 1, consider the set UD/"'(0),

Xn=<x eD — . By the above argument, we note that there exists a connected subset £„ of Xn having two points x„ ety„ such that dist(xn, A) = — and dist{y„ ,A) = n. The same arguments used above to show n that X is compact, can be further utilized to show that there exists a cluster point xt, edA of the sequence {x„}neN. We will prove that the component

262

ZERO-EPI MAPPINGS ON THE WHOLE SPACE £ of D\ Int(A) starting from xtt is unbounded. Since any bounded and closed subset of D\ Int(A) is compact, if we suppose that Z is bounded we can show, applying Lemma 3.7, that there exists an open and bounded subsets of E such that E c f and dt/ r\(D\Int(A)) = 0. Since x„ is a cluster point of {*„} ^ and lim|y B | =+QO , there exists k , which is impossible since d1/{\D\Int{A)) = . U Definition 3.4 Let \E,\ ||j and \F,\ ||j 6e Banach spaces, Ua E an open subset and /:£/-> f:U->F F a continuous mapping. If there exist a Banach space (G,\ [I and a continuous mapping 0:U —» G swc/z ?/zaf: /) sends bounded subsets ofU into bounded subsets ofG, 2) the mapping (/, Fx G defined by (/.<&)(*) = (/(*), {0}. We say that / is nontrivially O-regularizable if / is O-regularizable and not 0-epi. I f / i s nontrivially O-regularizable, then we must have G* {0}.

263

ZERO-EPI MAPPINGS Proposition 3.9 A O-regularizable mapping f:U^F f:U^F ► F is 0-regular if and only if f is O-admissible and proper on bounded closed subsets ofU. Proof Indeed, if / is 0-regular, then by definition, it is O-admissible and proper. Conversely, because / is supposed to be O-regularizable, there exists a mapping 0:U £>:£/ -> G such that (/,):£/-» F x C? is 0-regular. -+G Consider a compact mapping kU->F k:U -> F F with bounded support and such that k(x) = 0 for all x e dU. Obviously the equation (/(*),O(x)) = (k(x),0) is solvable in U, which implies in particular, that / i s 0-epi. ■ The following results are consequences of analogous results proved for 0-epi mappings: Theorem 3.10 [Perturbation Theorem] Let (E,\\ fj and (F,\ |) be Banach spaces and UcE an open bounded subset. If f:U^F f:U —>FF is a O-regularizable mapping and h:Ux[0,l]^>F h:U x [0,ll —» F a compact mapping such that h(x,0) = 0 for any x e U U , then there exists s > 0 such that /(■) - h(-,X) is O-regularizable for every \A\< e. ■

/0-*M

Theorem 3.11 [Continuation Principle] Banach spaces, U a E an open subset, mapping and h:U x [0,1] -> F a completely

Let (E,\\ |) and (F,\\ |) be f:U^>F f:U-*F F a O-regularizable continuous mapping such that

h(x,0) - Ofor any x e U . If there exists a mapping O: U —> G, where G is a Banach space, which 0-regularizes f such that the set l S,4 = {x e
= h{x,X) h(x,X) for some A X e[0,l]} e[0,l| =

is bounded and does not intersect dU, d U, thenf-h{-,\) O-regularizablebyby
264

ZERO-EPI MAPPINGS ON THE WHOLE SPACE The following result has interesting applications to the study of the solution set of some nonlinear equations (Furi, M. and M. P. Pera [1]): Theorem 3.12 Let \E,\ ||J and (F,\ ||J be Banach spaces and U a E an open subset. If f:U —> F is a mapping nontrivially O-regularizable by a mapping O: U -> G, where G is a Banach space, then there exists a connected component D off ~l(0) intersecting O_1(0) and satisfying at least one of the following two conditions: 1) D is unbounded, 2) D Pi dU is non-empty. Proof Denoting by S =/ _ 1 (0) and using the regularity, we remark that 50 = STIO^O) = (/,O) _1 (0,0) is compact subset of U. Two situations are possible: (I) the set [/is bounded and (II) the set [/is unbounded. I) The set U is bounded. For the moment, suppose that / is a proper mapping. We show that Sx = SC\dU is non-empty. Indeed, since G * {0}, there exists an element UQ e G such that u0 * 0. For each t e R, we put _ tu &(*) Si{x) = ((/(*)>^K*) o) ■ Since ®{u) is bounded in G, there exists t* 6 R / ( * ) . (x) - ft
265

ZERO-EPI MAPPINGS U ) such that So <^ A0, Si c Ai, A^V\Ax = <j) and4,lM=s. A0\jAl=S. Thus, Thus, there there exists an open neighborhood % ofA0 in [/such that % f] Al = . Therefore, as above, we can show that the equation X*) = 0 has a solution x, &d\ (Note that 1/0 => S0 and applying the localization property of 0-epi mappings, we obtain that (£0) is 0-regular onf 0 ). Since x* e S, we have

x, e#%r\s ed%r\s =(0%r\AQ)u(0ntr\A1)=+, which is a contradiction. Hence, we can find a connected component D of S which connects S0 and dU. Let us show now that the assumption "f is proper"is not necessary. Indeed, let G\ be a 1-codimensional subspace of G. We have (,,<£ ):C7 -> G, x /fcc G = G, ©Rx © Rx00,, where x0 € G, and = (<J5 Rx0. 15022):C/ The mapping /» = (/,,):U -> F x G, is O-regularizable. Moreover, because (/.,0) = (/,, 2 )-(O,0 2 ) with (/"*,©2) proper and (0,O2) is compact, we have that / . is a proper mapping. Hence, by the previous argument, there exists a connected component of /, -1 (0,0) intersecting both So and dU, with the same being true for / - 1 (0) 3 /»_1(0,0). II) The set U is unbounded. Let n0 eN be a natural number such that the open ball centered at the origin with radius n0 contains the set S0. By the part (I) and the localization property of 0-epi mappings we have that for any n> n0 there exists x„ e So and a connected component Dn
266

ZERO-EPI MAPPINGS ON THE WHOLE SPACE S f l ^ o * ^ thus, {*•} a n d S\J<3D0 a r e t w o c l o s e d disjoint subsets of Sf\ A which, by assumption, cannot be connected by a component of S. Hence, by Lemma 3.7, there exists an open neighborhood 1/ o of x* in DQ such that <5'fon<S' = ^. On the other hand, there exists m > n0 such that xm € % and B{0,m) z> Z)0. As a consequence we deduce that the connected set Dm intersects both % and E \ %. Then, Dm D ^fo * ^, which is impossible since Sf)d% = $ a n d the theorem is proved. ■ Similarly, the next theorem can also be used to obtain information on the structure of the solution set of a large variety of nonlinear equations. Theorem 3.13 Let (E,\ ||J and \F,\ |) be Banach spaces, Uc E an open subset andf: U —> F a continuous mapping. If there exists a continuous mapping O: £/ -» /? sending bounded sets into bounded sets such that the mapping {f,Q>):U -+FxR defined by (/,0)(*) = (f(x),0(x)) is Q-regular, then the equation fix) = 0 has a connected subset D of solution set intersecting the set O ~ (0) and such that each one of the sets D+ ={xe D | > 0} 0} anrfZT andD' ={*e ={xe D D || 0(x) 0(x) << 0} 0} is either unbounded or intersects d U. Proof

We introduce the following notations: S S= =ix[xeU\f{x) &U\f{x) ==Oj, 6\,

+ SSQ = = [x [x eU\ eU\ f(x) f(x) ==00and andO(x) O(x)==o}, o}, S S+==[x[xeU\ eU\f{x) f(x) ==0 0and and0>(JC) 0(x)> >o}o)

0

and S~ = ix e £7j/(*) = 0 and O(x) < OJ. Since the mapping (fi<S>) is and S =ix eU\f(x) = 0andO(x)<6\. Since the mapping (£<X>) is 0-regular, it follows that So is a non-empty, compact set and S0 C\dU = ^. O-regular, it follows that So is a non-empty, compact set and S0 C\dU = ^. We We will will show show that that S* S* and and S~ S~ are are non-empty. non-empty. Indeed, Indeed, for for every every nn ee N, N, let let 5(0, 5(0, n) n) be be the the ball ball centered centered at at the the origin origin with with radius radius n. n. Let Let nn00 ee N N be be such such B(0, H00). The boundedness of O implies the existence of a real that S00 a 5(0, U. constant M > 0 such that 0(x) * M for all x e 5(0, n0) fl £/. 267

ZERO-EPI MAPPINGS Consider the mapping g: [0,1] x B(0,n0)f]U ^ F x R defined by

g{t,x) == (f{x),®{x)-tM). g[t,x) (f{x),0{x)-tM).

There exists (tt,x ,x*) e]0,l]x e{B(0,n such that g(/„x.) = (0,0). 0)f\U) t)e]0,\]x^B(0,n 0)C\U) Indeed, if not, the homotopy and the localization properties for 0-regular mappings would imply the solvability in 5(0,«Q)f)U of the equation g(l, x) = (0,0), which is impossible since g(l,x) * (0,0) for all gjl, xxsB(0,n eB(0,n0)f]U. Thus,/x.) = 0, 0>(x,) = / . M > 0 and finally, we have 0)f]U. ST * <j>. Similarly, by the same method, but taking instead a constant M< 0, we obtain S ' * §. Consider now an extension of U adding two points denoted by -co and + co. We put U = U U{-°°} U{+co}, and we extend the topology of U to U. A subset Q of U is an open neighborhood of + oo (resp. of - oo) if +co e Q (resp. -co e Q), and the difference U \ Q is a closed subset of U whose intersection with j x e U\ ^(x) > Ol (resp. jx eC/ |
s;-=(s-ndu)\j{-K>}, s,'-=(s-rw)u{-w}, 268

ZERO-EPI MAPPINGS ON THE WHOLE SPACE observe that, since S0C\dU = <j>, it follows that S*f and S? are closed in if. We show now that there exists a connected subset D* of 5* \(s*f US*+) such that S,*~ f l c ^ D * ) * ^ and s; + D C / ^ J D * ) * <*>, where c/ .lD*j denotes the closure of D* in if, Suppose the contrary, that is such a connected set does not exist. Applying Lemma 3.7, we can find two disjoint and closed (thus, compact) subsets of S say A'~ and A*+ such that A*-^>S*f, A*+^ S*+ and A'-\JA*+ = S'. Hence, if being Hausdorff, there exist two disjoint and open subsets of if, U'~ and lf+ such that U'-z>A*-,U'+ 3 A'+. We put U~ = if' C\U and U+ = U'+ V[U. Note that, by definition, t/ - no _1 (]+oo,0[) and t/ + riO" 1 (]-oo,0[)are bounded. Moreover, U ~ and if are open in U, are disjoint and So is contained in U~\JU+. Hence, applying the localization property for 0-regular map pings, either the restriction of (f, O) to U~ or its restriction to U+ is 0-regular. Suppose that (f O) is 0-regular in U ~. An argument analogous to the one used above in proving the non-emptiness of .S^ implies now that there exists x* e BIT such that j{xt) = 0, 0(xt) > 0 (observe that t/"nO_1(]0,+oo[) is bounded). Thus, x, e S+ C\dU~ contradicts the fact that Sf]U is contained in the union of the disjoint open subsets U+ and U~. Therefore, there exists the connected subsets D as claimed. By denoting D = Ur\clu,(D*) , we have that D is connected and each of the sets _1 D~ == DD nO no-'Q-oo.oQ D (]-oo,0[) is either unbounded or D++ == DDD^QO.+ooQ, no-'Qo^Q, D~

intersects 8U. Moreover, since 0(Z)) is an interval containing the origin, we also have DC\ S0 & , and the theorem is proved. ■ From Theorem 3.13, we deduce a classical result proved in 1971 (Rabinowitz, P. H. [1]). Theorem 3.14 [Rabinowitz]

Let \E,j ||J be a Banach space and

T : [0,+oo[ x E —> E be a completely continuous mapping such that 269

ZERO-EPI MAPPINGS 7T(0,JC) = Ofor allx e E. Then the equation T(k,x) = x admits an unbounded connected set of solutions (Ajc) emanating from (0,0).

Proof Denote by f:R*E-+E- » £ an arbitrary completely continuous extension of T, for example taking f[A,x) = 0 for all X e ]-oo,0[ and d e E. Since the identity mapping on E x R is 0-regular and the set {(A,x) eRxE\£ (JC.A.) = r[f{X,x)fi)

for some r e[0,l]}

is the set {(0,0)}, it can be shown, applying the homotopy property for 0-regular mappings, that the mapping (X,x)&RxE^>[x-T{A,x),l} eRxE ?->► l ; c - 7(A,x)„ ( / l , ; t ) , / l )eExR «ExR eExR (X SX) € is 0-regular. Then, taking into account Theorem 3.13 E: = Rx E,F:= E, frXjc) = x- f(ljc) and ®(Xjc) = K the conclusions of the theorem ffllow.

■ Remark 3.5 In the proof of Theorem 3.14, we used the following form of the homotopy propertyjbr 0-regular mapping: "Let/: U-> F be 0--egular and let H : [0,1] x U -> F be completely continuous and such that 7/(0, x) = Ofor all x e U. If the set {xeU\f{x) + H{r,x) = 0 for some r e[0,l]| is bounded and

w

does not intersect dU, then the mapping/+ H(l,-) is 0-regular." Initially , P. H. Rabinowitz proved Theorem 3.14 using the topological degree. The proof of this theorem presented here was proposed in (Furi, M. and M. P. Pera [4]). From this proof, it follows that in fact Rabinowitz's theorem is a consequence of a Schauder's fixed point theorem. Other applications of 0-regular mapping are presented in (Furi, M. and M. P. Pera [4]), (Furi, M. and M. P. Pera [2], [3], [4]), (Massabo, I., P. Nistri and M. P. Pera [1]) and (Furi, M , M. P. Pera and A. Vignoli [1]).

270

ZERO-EPI MAPPINGS ON CONES 4. Zero-epi mappings on cones Zero-epi mappings can also be used to study the structure of the solution sets of nonlinear equations involving mappings acting on ordered Banach spaces. Several problems will be considered in this sense. For this, we must introduce the concept of zero-epi mapping with respect to convex cones. Let (E\\ \>K\) an(^ (^2'| l ' ^ ) be ordered Banach spaces. The ordering on E\ (resp. on E2) is defined by the pointed closed convex cone K\ c E\ (resp. by K2 c= E2). Let Q c K\ be a bounded non-empty subset. Suppose that Q is open with respect to K\. Denote by 5Q (resp. by Q,) the boundary (resp. the closure) of Q with respect to K\. Definition 4.1 We say that a continuous mapping f: Q —>E2is (Kx,K2)-0-epi, if and only if 1) 0 g/(*?Q) (i.e., f is ^-admissible with respect to K\), 2) for any continuous compact mapping h : Q —> K2 such that h(x) = Ofor all x e 3Q the equation fix) = h(x) has a solution in Q. More generally, we have the following definition as well: Let/? e K2 be an arbitrary element, where eventually/? = 0. Definition 4.2. We say that a continuous mapping f: Q -> E2 is \Kx,K^-p-epi, if and only if, the mapping f'—p defined by {f-p)(x)=j{x)-p, Cl, is (xM [Ki,K2)-0-epi.
If/is/7-admissible with respect to K\, that is p <£ fiXl) and the equation fix) = h(x) has a solution in Q for every continuous and compact mapping h : Q -> K2 such that h(x) =p for any x e dCl, then/is (K (KXu,K K2)-p-epi. ) -p-epi. IffIff is (JSri,JST2)-/>-epi, t h e n / - k is (jf1,jK"2)-/)-epi for every continuous and compact mapping k : Q -> K2 such that k(x) = 0 for every x e 5Q. The

271

ZERO-EPI MAPPINGS fundamental properties of 0-epi mappings with respect to convex cones are similar to the properties of 0-epi mappings on an arbitrary open bounded set. However, but the proofs contain some specific details. Existence property Iff: Q -> E2 is (Kx,K2)-p-epi where p e K2, then the equation fix) =p has a solution in Q. Proof The property is a consequence of Definition 4.2. ■ Normalization property The inclusion i: Q -> E\ is {Kx,K^-p-epi if and only ifp eii. Proof If the inclusion i: Q —> Ej is [Ki,K2)-p-epi then, by Definition 4.2, it follows that p e Q. Conversely, suppose that p e Q. It is sufficient to show that equation i(x) = h[x) has a solution in Q for every continuous and compact mapping h : Q -> K\, such that h(x) = p for every x e dQ. Consider the mapping h : K\ —> K\ defined by \h(x), if x ee Q ,./ ,_[h(x),if

*-w-'\p,[p,

if x xeQ. €£Q. Q.

Applying to h' the Schauder's fixed point theorem, we obtain a point x, € Kx such that h\x.)= x,. The definition of h' and the fact that/? e Q imply that xt must be in Q. It follows that the equation i(x) = h(x) has a solution in Q. ■ Localization property Iff-.Q -> E2 is a (Kx,K2)-p -epi mapping and there exist two open sets Qi and Q2 such that Q b Q2 c Q, Ci.x f] Q2 = , <™d f-\p) e Q, UQ 2 , then yj_ or / j - » {Kx,K2)-p-epl /-'(rfcn.uoj,

272

ZERO-EPI MAPPINGS ON CONES Proof Suppose the contrary, i.e. there exist two continuous and compact mappings fy:Q, -^JT2 (i = 1,2) such that h,{x) = 0 for every x e 8Q{ and f(x)-p*h,[x) f{x)-p* h;(x) every x e Q Consider the mapping h : Q->K2 defined by hi{x),if

xefi]

h[x) ^(x), z/if xx eQ. e Q22 A(x) -= •y/^(JC), 0, o,

!/i£fi$Q,un ! / i £ f i \ ( Q , U n 22 ). ).

The mapping h is continuous, compact and f(x) - p * h(x) for all x E Q, which is impossible since/is supposed to be [Ki,K2)-p-epi. ■ Q^>E Homotopy property Letf: Q^E andlet 2be2be (Kl,K2)-0-epi H : Q x [0,1] ->K2 be a continuous and compact mapping such that H(x,0) = 0 for any x e Q. If fix) ^ H(x,X) for all x e dQ. and for any X e[0,l], thenf-H(-,l) )-0-epi. 2)-0-epi. uKu2K f~H{;$ (-,1) is (K(K Proof Let k : Q —>K2 be a continuous and compact mapping such that Hx) = 0 for all x e 5Q. The set D=[x D = [x ee Q Q || //(x) ( * ) == i/(x, i/(x,X) A) ++ *(JC) k(x) /or some *2 6e[0,l]} [0,1]} is closed in Q and Z) (~}dQ = <> | . By Urisohn's Lemma, there exists a continuous function (x) = 0 for every x e 9Q. Consider the mapping /*♦ : Q —> K2 defined by ht(x) = H(x,®(xj) h,(x) H(x,®(x)) + k(x). The mapping h* is continuous, compact and h* (x) = 0 for any x e dQ.

273

ZERO-EPI MAPPINGS Then, since/is (KUK2)-0-epi, there exists x> e Qsuchthat f(xt) = H(xH(x.,®(x.))+k(x.). t,(x. ))+k(x.). Note that x.eD which gives f(x.)-H(x.,l) /f(x ( t*)-H(x.,l) ) H(x.,l) completed. ■

k(x ==k{x.) k(xtt)) and the proof is

Theorem 4.1 If f:Q -> E2 is (K],K2)-0-epi and there exits a star-shaped

l r\D)r\dn=t (with respect to the origin) subset D c K2 such that f~ (D)f)d£l = (j> (i.e.

j{x) t Dfor every x e dCi), then the equation fix) = h(x) has a solution in Q for any continuous compact mapping h : Q —>K2, such that h(dCl) c D. In particular, D czf{Q). Proof Let h : Q —>K2 be a continuous and compact mapping such that h(dQ) cz D. Consider the homotopy H:QQ xx[0,1] H: [0,1] -> -> KK22 defined defined by by H(x,X) H(x, X)== Ah(x). Xh(x). Since Xh(x) e D for all x e dQ. and X e [0,1], and b e c a u s e ^ ) ^ A it follows that f(x) * Ah(x) for all xe dQ. By the homotopy property, we deduce that the equation f{x) =h(x) has a solution in Q. Let p € D be an arbitrary element. By taking as mapping h the constant mapping h(x) = p for every JC e Q we obtain that/? e/(Q), and the theorem is proved. ■ Theorem 4.2 [Perturbation Theorem] If If f:Cl-*E / : Q - >2 £ 2 is proper and (Kx,K2)-0-epi and H:Q x [—1,1] —> K2, a continuous compact mapping such that H(x,0) = 0 for any x e Q, then there exists a real number e > 0 such thatf-H{-,X) is [KltK2)-0-epifor every Xsatisfying \X\ < s. Proof The proof is similar to the proof of Theorem 2.9. ■ Proof The proof is similar to the proof of Theorem 2.9. ■ 274 274

ZERO-EPI MAPPINGS ON CONES Recall that, if X(x) is a topological space, and AczXa non-empty subset, we say that A is a retract of X if there exists a continuous function r : X -+A such that r< =id\. It is known (Dugundji, J.[l]) that, in a Banach space E, any closed convex and non-empty subset is a retract of E. Let h:Q -> Jj^be a continuous and compact mapping such that h[x) * JC for every x e dQ. In this case, since K\ is a closed convex cone, and Q. K is a continuous compact mapping such that h(x) + p^x for every x e 9Q and i(h + p, Q.,K) * 0, then I -his (K,K)-p-epi. Proof Let k : Q. -> K be an arbitrary continuous and compact mapping such that k(x) = 0 for every x E 5Q. Let h* : Q -> K be the mapping defined by h*(x) = h[x) + p. Consider the compact homotopy H: Q x [0,1] -»Jf ->K defined by H(x, X) - h\x) + Ak(x). For all X e [0,1] we have 7/(x,A,) * x for every x e 9Q, and, by the property of invariance with respect to a homotopy of the topological index, we obtain i(H(-,l),Q,K)= i(H(-,0),Cl,K) = i(h*,Q,K) * 0 . This implies the

275

ZERO-EPI MAPPINGS existence of an element x, e Q such that xt = h (x,) + k(x*\ and hence id -h -p is (A,^)-0-epi. ■ The following notations are necessary for the next results. For any p > 0, we denote by: B(0,p) x e lE\|x| <)rU„ £(0,/>)rU„ *(0,p) = = I{xeE ?(O,P)- =[xe£,|H| x | |x|
s;=s(o,/>)nir s;=s(o,p)r\K1.l.

Note that Bp is open with respect to K\ and the closure of Bp with respect to Kj , denoted by B+p, is B(0,p)C\ K\, where 5(0,/?) is the closure of B(0,p) with respect to E\. Theorem 4.4 If f : Bp -> E2 is (Kh K2)-0-epi and h : B+ -> K2 is a continuous and compact mapping, such that Af(x) ^ h(x)for all x e S^ and X>\, thenf-h

is (Kh K2)-0-epi.

Proof Consider the compact homotopy H : Bp x [0,1] -> K2 defined by H{x,f) = th{x). Note that, because fix) ± th{x) for all x e Sp and te [0,1], and since all the assumptions of the homotopy property of 0-epi mappings with respect to convex cones are satisfied, the theorem follows. ■ Corollary 4.5 Ifh : Bp -» K\ is a continuous compact mapping such that h(x) * X xfor all x e S+p and X>1, then id - h is (Ki, K^-Q-epi.

The following theorems are generalizations of two classical results known The following theorems are generalizations of two classical results known

276 276

ZERO-EPI MAPPINGS ON CONES as the Krasnoselskii 's compression (resp. expansion) of a cone theorem (Krasnoselskii, M. A. [1]), (Amann, H. [1], [2]). Theorem 4.6 [On the Compression of a Cone][Pera] Letf: Bp -> E2 be a (Ky K2)-0-epi mapping and h : Bp -±K2a continuous compact mapping. Consider 0 < a< p. If the following assumptions are satisfied: 1) fflx) * h{x)for allx e Sp and A> 1, 2) there exists q e K% q > 0 such thatf(x) - h(x) * Xqfor all x e S* and X>0, 3) there exists 5 e R+ such that ||/(x)|| < 8 for all x e B+a, thenf-h is (Kh K2)-0-epi on the set Proof

Bp\B+.

Suppose by absurd that there exists a continuous and compact

mapping h0 : Bp\B+^>K2

such that h0 (x) = 0 for all x e Sp[jS* and

fix) - h(x) * h(£x) for all x e Bp\B*.

Consider the mapping h" : Bp -> K2

defined by

.(x) = \h(*). fc h\x)

+

+ + lho{x), ifxeB ifxeB p\B a p\Bt ffx^XK

h'(x) = {^> [0,

***W

ifxeB^. ifxeB*.

The mapping h is continuous and compact. With the assumptions of Theorem 4.4 being satisfied for/and h + h , we deduce t h a t / - (h + h ) is (K\, JT2)-0-epi on Bp. Now, considering the absurd assumption and the localization property, it follows t h a t / - ( / z + h ) is (K\, JK2)-0-epi on B*. Let Xy be a real number, such that S+M ..■■ -<X <X sup{||/z(x)| e B* >. X,x, where M = sup|||/z(x)| M4\ x•■*K)

IMI

jj jf ||

>{«; HI-

277

ZERO-EPI MAPPINGS Also consider the homotopy H:B+a x [0,l] -> JT2 defined by H(x,X) =A. Xxq. Since from assumption (2), we have/*) - h(x) * X Xxq for all x e S*, we deduce (applying again Theorem 4.4) that/-(/i + h ) - ^q is (JTi.J^-O- epi on £+. Hence, there exists x, e 5*such t h a t / * . ) = /z(x.) + A,ig, which implies ||/(x. )|| = |/I(JC. ) + A. , J > llj |?|| - ||/z(x. )||| > S. This is impossible con sidering assumption (3) and therefore the theorem is proved. ■ Theorem 4.7 [On the Expansion of a Cone][Pera] Let h: B+p -> JTi ie a continuous and compact mapping. Consider 0 < a < p. If the following assumptions are satisfied: 1) h(x) * Ax for all x e S+ and X> 1, 2J rtere exists p e. K\,p ?*0 such that x -h(x)* Xpfor allxe S+ andX> 0, then the mapping id - h is (Kh Ky)-0-epi on the set B*\ B*. Proof The proof is similar to the initial proof proposed by Krasnoselskii in [1] for his classical version of this result. Note that Krasnoselskii's proof is based on fixed point theory and not on topological degree. ■ Consider again the ordered Banach spaces (£,,| |, A",) and (E2,\

\,K2).

Definition 4.3 We say that a continuous mapping f: K\ —> E2 is (K\,K2)-p-epi on K\(p s K2) if and only if 1) f (p) is a bounded subset of K\ (i.e., f is p-admissible with respect to *), 2) f is (KhK2)-p-epi on every subset Q c K\ which is bounded, open with respect to K\ andf~x(p) c Q. Several results and applications associated with this concept are presented 278

ZERO-EPI MAPPINGS ON CONES in (Pera, M. P. [1]). The concept of regular mapping can be introduced also on a convex cone. This concept is useful in the study of the structure of the solution sets of nonlinear equations which involve mappings acting in ordered Banach spaces. Definition 4.4 We say that a continuous mapping f: U —>E2 defined on the closure of a relatively open subset UofKu is (KhK2)-0-regular (zeroregular) in U if and only if: 1) f~'(0) is a bounded subset ofU, 2) for any completely continuous mapping h: U ->K2 with bounded support contained in U, the equation fix) = h(x) has a solution in U, 3) f is proper on closed bounded subsets of U . The basic properties of 0-regular mappings with respect to cones are the following: Existence (or solution) property Iff: U -+E2 is (K^K^-O-regular, then the equation fix) = 0 has a solution in UcK\. ■ Localization property Iff: U -+E2 is (K\,Ki)-^-regular and U\ and U2 are open disjoint subsets of U such thatf~'(0) c U\ U Uz, then f, or fi_ is (KhKi)-0-regular. ■ Normalization property Let f: U ->K2 be continuous, one-to-one and proper. IffiU) is an open subset in K^ then f is (K^K^-Q-regular if and onlyifOefiU). In particular, the inclusion i : U —>K\ is (K\,Ki)-0-regular, if and only if 0 e U. m Homotopy property Letf: U ->E2is (KhKi)-§-regular. If H:U x [0,l] -^>K2 is completely continuous, H(x,0) = 0 for all x e U and the set

279

ZERO-EPI MAPPINGS D = [x e U\ f(x) = H(x,A) far some X e[0,l]} is a bounded subset ofU, then the mapping x -+fix) -H(x,\) is (KhKj)-0-regular. ■ The proofs of these properties are the same as in the general case. Definition 4.5 We say that a mapping/: E\ —> E2 satisfies the maximum principle iff~'(K2) c K\ {i.e., fix) e K2 implies x EKI). Theorem 4.8 Let (Elt\ ||, J5T,) and {E2,\ \,Kt\ be ordered Banach spaces and let f: E\ —> E2 be a 0-regular mapping. Iff satisfies the maximum principle, then its restriction to the cone K\, i. e., f , is (K\,Kz)-0-regular. Proof Let h : K\ —> K2 be a completely continuous mapping with bounded support. Since/is a O-admissible mapping on the whole space, there exists p > 0 such that / _ 1 (0)U supp(/z) c B(0,p). Applying Dugundji's Extension Theorem (J. Dugundji [1]) we obtain a continuous retraction r:B(0,p) -> B* . Consider the continuous compact mapping h*:B(0,p) —>K2, defined by h' = h or, and the set D = {* e B(0,p) | f(x) = Ah'(x) for some Ae[0,l]}. Because/satisfies the maximum principle, it follows that D c K\. Hence, D does not intersect the boundary of 5(0, p). By the localization property, f is 0-regular in B(0, p). Applying the homotopy property, it follows that h is 0-regular in 5(0, p). Thus, the equation fix) = h\x) is solvable in f-his / 5(0, p). Since any solution of the above equation belongs to 5*, we

280

ZERO-EPI MAPPINGS ON CONES deduce that the equation f(x) = h(x) is solvable in K\. (^.JQ-O-regular. ■

Hence, ff

u

is

Definition 4.6 Letf: U -^E2be a continuous mapping. We say that f is O-regularizable by the mapping E2 x E3 given by (/,0)(x) = (f(x),$>(x)) is (KhK2xR)-0-regulzr in U. Note that/is (ISTiyfiy-O-regular if and only if, it is O-regularizable by the trivial mapping O: U -> {0}. By an argument similar to the proof of Theorem 3.12, we obtain the following result: Theorem 4.9 If the mapping f: U -> E2 is O-regularizable by the mapping <£>: U —> E3 (where the convex cone K3 is nontrivial), then the equation J(x) = 0 has a connected component D of solution set intersecting the set DQ = I x e UI <&(*) = 0| and such that, either D intersects the (relative to K{) boundary ofU, or D is unbounded. ■ The next result, proved for the first time in (Furi, M , M. P. Pera and A. Vignoli [1]) is the variant for cones of the well-known Birkhoff-Kellogg Theorem. Theorem 4.10 Let U a K\ be a bounded open (relative to K{) subset and f:U^E22E2 a (K\,K2)-0-regular mapping such that f(U) is bounded. If f:U->E h : U ->K2 is a continuous compact mapping such that Wx)l >a>0 for allx e dU, then there exists X- > 0 andx* e dUsuch that f(x*) = Ath[xt). (Note that dU is the boundary of U relative to K\).

281

ZERO-EPI MAPPINGS Proof Considering the homotopy property for 0-regular mappings with respect to cones, we remark that the theorem will be proved if we show that there exists A,. > 0 such t h a t / - X»h is not (KhK2)-0-regalar. Denote by B2a the set {x e K21 || x\\ < a}. It is known (M. Furi, M. P. Pera and A. Vignoli [1], pg. 291) that there exists a retraction r : K2 -> K2\B^a such that x < r(x) for any x e K2. (Note that "<" is the ordering defined by K2). Consider the continuous compact mapping k : U -> K2 defined by k(x) = r(h(x)), Define M = sup{|/(jr)|| * el/). The number Mis well defined since fiJJ) is bounded. We can show that, for X> — , the equation f(x) = Xk[x) has a — M — M no no solution solution in in U U .. Thus, Thus, for for X X> >— — the the mapping/mapping/- Xk Xk is is not not (^i,JT (^i,JT22)-0-re)-0-reM aa gular in U. Hence, for X > M — , the mapping/- Xh is not (J5r1,Jr2)-0-regular a , the mapping/- Xh is not (J5r1,Jr2)-0-regular gular in U. Hence, for X > — M a in U. Indeed, if there exists X0> — M for w h i c h / - X$h is (Jfi.^-O-regular, in U. Indeed, if there exists X0> — a for w h i c h / - X$h is (A^iQ-u-regular, then, by considering the positiveahomotopy H:U x[0,l]-»J£2 defined by then, the positive x[0,l]-»J£2 by H(x,t)by= considering tX0(k(x) -h(x))(note thathomotopy k(x) > h(x)H:U for all x e U), defined we deduce, H(x,t) = the tX0(k(x) -h(x))(note all x e U), weindeduce, applying homotopy property,that t h ak(x) t / - >XQh(x) k isfor (JST^iQ-O-regular U. By \4 applying the homotopy property, t h a t / - XQ k is (JST^iQ-O-regular in U. By M M this contradiction we have that, taking an arbitrary X, > — , the mapping a ■ f- X* h is not (A"I,JSr2)-0-regular in U and the proof is finished. f- X* h is not (A"I,JSr2)-0-regular in U and the proof is finished. ■ The following result has interesting applications to the study of the solution set of nonlinear equations depending on a parameter n XsR XeR: X eR: ={x [x (xt) eee RR++ || x, x,. xt > >> 0,1 0,i == l,2,..,n} l,2,..,n} .. >-0,i=l,2,..,n). + = {* == (*,) (*,)

Theorem 4.11 Let (Eu\\ \\,KX) and (E2,\\ \\,K2) be ordered Banach spaces, L : E\ —> E2 a bounded linear isomorphism satisfying the maximum principle and h : R" x £, -> E2 a completely continuous mapping such that 282

ZERO-EPI MAPPINGS ON CONES hlR" x KA0(peR), H>0{M<=R), then the equation L(x) = h(X, x) has a connected component D of solution set contained in R" xiB* \Bp \ such that, either D is unbounded or it intersects the set [R" X Sp) U[R" X S * ) . Proof The theorem is a consequence of Theorem 4.9 if we show that the ( is O-regularizable in RR"xiB^\B R^x+X(B:\B;). {K\B;) p).

mapping (X,X)-^L(X)-UX,X) (/l,x)-> L(x) L{x)-h[A,x) hU,x) U,x)

Indeed, using assumption (1) and (Furi M., M. P. Pera and A. Vignoli [1], Remark (3.2), we have that the mapping yy¥{X,x) ¥(A,x) = (L{x)-h(X,x),X) (L(x)-h(A,x),X) is 0-regular in R"xB+.

We show that *¥ is 0-regular in R" x {B+0 \ Bp V In

fact, if this is not true, then there exists a compact mapping n K22xxxR" Rl ti = (B+a+p\B;) R"++ [K\B;) ti a f(A,,*,) t , * , ) :: R Rn+++xxx{^\B )^K2 ^K ti^fah,): n

R:*(B:\B;), (K\B;) with bounded support contained in R^x R" x IB* \ Bp I such that, Rn+x (K\B;\ (X, x) ee R" (B* V(X,x)x))*h\\,x) ^(X, * ti(X, x) for all (X,x) (B* \ \B+\. BJ\. Extend ti to R" x B+ by setting ti(X, x) = 0 for all (X, x) e R" x B+p . Note that *P - ti is 0-regular in R" x B+a and, by the localization property, is 0-regular in R"xBp.

Moreover, for any \i>0, the set

283

ZERO-EPI MAPPINGS i(A,x) V(A,x) - -ti{A,x) t(pq,0)for forsome somet te[0,l]} e[0,l]J i(X,x) e / ?f > B 2£+p| Iy{l,x) ti{l,x) ==t{jjq,0)

>,te[0,\],Z-h = o} = = {(A,x)|lW-WA,x) {{A,x)\L{x)-(h(A,x) + A,(A,x)) h,{A,x)) == t^{orsomete[0,l],A-h = 0} 2{A,x) 2{A,x) is bounded and does not intersect R>S Rn+ x+p5; ; ; (we used assumption (2)) and the fact that h' has bounded support contained in R:X(B;\B;\. R"+ x(B+ \B*\.

Hence,

by the homotopy property, the mapping T - ti - (ji qJS) is 0-regular in R"+ x £+ 0. This implies that the equation L(x) - h(0jc) = Mq 5 ; for all ju ft >> 0. + has a solution in BK p■. On the other hand, for // >

M== supkh(0,x)\\ supj|/<0 5 x)||xes;} ^ t ^ where M x SBJ\ the

m

above equation is not solvable in B+p. This contradiction shows that the mapping (A,x) -> L(x) - h(A,x) is O-regularizable in R" x (B+ \5+), and in R>(B;\B;), (/L,JC)-> L{x)-h(A,x) this way the theorem is proved. ■ Remark 4.1 In the proof of Theorem 4.11, "0-regular" means 0-regular with respect to appropriate cones.

5. Zero-epi families of mappings and optimization Two branches of modern mathematics which have been recently developed are namely optimization and nonlinear analysis. The results obtained in optimization use: a) differentiability or certain generalizations of differen tiability (directional derivative, subgradient, generalized subgradient, infragradient, etc.); b) convexity or certain generalizations of convexity; c) geometrical methods (cone of feasible directions, tangent cone, etc.) and d) multipliers. Meanwhile, nonlinear analysis has provided several strong and 284

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION specific methods for solving nonlinear problems. In particular, one such method is the application of zero-epi mappings. What appears to be quite interesting is the attempt to establish some relation between the concept of "zero-epi-mapping" and some problems in optimization. In this way, we can open a new direction of research and we can indicate another kind of application of zero-epi mappings. In this sense, we introduce the concept of zero-epi families of quasiconvex mappings, and we apply this concept to the study of Nash equilibrium points for a couple of mappings. The main results of quasiconvexity and the concept of quasi-subgradient as defined and studied in (Greenberg, H. J. and W. P. Pierskalla [1], [2]) and (Crouzeix, J. P. [1]) we will suppose as known. Let E\ and F\ (resp., E^ F2) be locally convex vector spaces such that ( E ^ F J J (resp., (E2,F2)2) is a duality with respect to the bilinear form (,)j (resp., (,)2). Consider the dual system (£j (is, xE2xE ,F12xF ,Fl2xF ), 2), defined by the

H*

bilinear form: ((x^xj^y^yj) ({xl,x2),(yl,y2)) = {x (xul,yv,), + 2(x ,y22,y)2.)2 . i)i+(x Set ro(.Ei) = [f:E1 -> ]-oo,+oo]| / is convex lower semicontinuous and/ *+co}; and, for e a c h / e T0(Ej), let (P) be the following minimization problem: (P):a = infinff(x). (P):a= inff(x). XEEI

It is well known that problem (P) can be transformed into a min-max problem by the following technique (Auslender, A [1]) and (Ekeland, I. and R. Temam [1]). Perturb problem (P) by a mapping q> e r o ( £ , xE2) such that (p(x,0) = f(x) and consider the following perturbed problem (Pu)' (pu): h(u)= inf inf
Generally, the convex mapping h is not lower semicontinuous, and, if we

285

ZERO-EPI MAPPINGS consider the polar mapping h' of h, we have h relation is valid:

< h, and the following

- a = -h(0) -h(0) < -h" -h" (0) == inf h'(v). ti{v). veF222 veF vef

If we denote by p = inf/z*(v), we want to know under what conditions VGF2

P = - a. In order to solve this problem, we consider the Lagrangian of problem (P) defined by cp, i.e. L(x,v) = sup((w,v) supl(w,v)2 -
The functional L is convex-concave, and, because P = inf sup L(x, V) and a = - sup inf L(x, L(x, v), veF

2xeE,

we have that fi= -a

lefi'^

if the following is true: L(x,v). I(x,v) inf sup i(x,v) L[x,v) = sup vinf L{x,v). e/r

v e

^i6£,

Jte£,

2

(5.1)

It is well known that (5.1) is true, if there exists a saddle point for L, that is a point (x,, v») such that L(x, v,) < L(xt, v,) < L(x.,v), for all (x,v) e £ , x F2. The next proposition is a classical result. Proposition 5.1 If(x», v.) is a saddle point of the Lagrangian L, then x* is a solution of problem (p) and v. is a solution of the following problem:

286

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION J3=\ p= inf h*» • V veF2 vsF2

fc*(v).

■ ■

From Proposition 5.1, it follows that certain optimization problem can be solved by the min-max technique. Many optimization problems can be transformed into min-max problems using other methods than the perturbation method, such as for example the decomposition method (Ekeland, I. and R. Temam [1]). Note that H. Amann studied multiple solutions of differential equations using the min-max approach [see the references of (Isac, G. [1])]. The importance of min-max approach in Pareto optimization and in game theory is also well known. Consider now the concept of Nash equilibrium point which is more general than the concept of saddle point. Nash equilibrium point plays a role of great importance in mathematical economy and in strategy theory. Let g:E: x F2 -».R be a convex mapping with respect to x eE\ and let h:Ex x F2 -> if be a convex mapping with respect to v e F2. Definition 5.1 We say that a point (x*,v») sEl x F2 is a Nash equilibrium pointfor the couple (g,h) if and only if: 1) g(x,,vt) < g(x,vt),for allx e Eu 2) h(x*, v,) < h(x*, v) ,for all v e F2. Remark 5.1 If L:EX x F2 ->R is a concave-convex functional and if we put: g(x,v) = -L(x,v), for all (x,v) e £ j x F2, h{x,v) - L(x,v), for all (x,v) e E\ x F2, then every Nash equilibrium point (x»,v*) eE{xF2 of the couple (g,h) is a saddle point for L. Now, we propose the following computation method for the Nash equilibrium point for a general couple (g,h). Note that (xt, v») is a Nash equili-

287

ZERO-EPI MAPPINGS brium point for the couple (g,h) if and only if

M ^ (x.,v.) ^ (*.,v.) ^ ^ ' "(*.')W(x.,v.) J iM where

°(*,v)(z>') = g{z>v) + *(*.'). for a11 fef) e £> x **

If we denote y¥x{Z) = <&X{Z + X), where X = (x,v), Z = (z,t) € E, x F2, then we have the following result: Proposition 5.2 The point X, = (x,, v,) is a Nash equilibrium point for the couple (g,h) if and only if ^¥ *. ((00 ) = min V^X.,{Y). (7). /fe£,xf E £ ] X /*2 2

(5.2)

■

Remark 5.2 If g is convex with respect to x, and h is convex with respect to v, then ^xiZ) is a convex mapping with respect to Z for each X. However, this result does not work for quasiconvex functions. The following results about quasiconvex mappings and quasisubgradients are useful. Let E, F be two locally convex spaces such that (E,F) is a duality defined by the bilinear form (,). If A e E is a convex set, then a mapping/ A -> R is called quasiconvex if /[^c, + (1 - A)x2] < max[/(x,),/(;c 2 )] for all X e [0,1] and xhx2 e A. Proposition 5.3 The mappingf. A -» R is quasiconvex, if and only if n

/ E M ^maxf/^,),..-,/(*„)] -i=i

288

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION n

for all At G [0,1] such that ^ / l , , = 1 andxt &A(i= \,2,...,ri). 1=1

Proof The proposition is a consequence of the following relation:

r A

I n

A,*, 1 1f 1*1

.1 =1 _i=l Li=l

JJ

f

x x

i =f u \+ 5KX ( 2-' 11^1

J

n

n

{

'A, K XKJ,1/Z■}J

A\j=2 yj=2 = 2 v^

M=2 •1=2 M=2

i1=2 =2 1=2

'

) _

XAxji ■ Aj'YiAi s I L'=2^ '=2 ' J_ n f

< m acx//(( **.,)),,// ^ <max

_j=2 V

,«

V ' Lu^i 1=2

x

*j/

■B

■

■ Remark 5.3 Quasiconvex mappings are very important in economics. For more details on quasiconvex mappings, the following references are recommended (Greenberg, H. J. and W. P. Pierskalla [1], [2]), (Crouzeix, J. P. [1]), (Mangasarian, O. L. [1]), (Martos, B. [1]) and (Avriel, M. [1]). The concept of quasi-subgradient was defined in R" by H. J. Greenberg and W. P. Pierskalla [1], [2]. We consider here the concept of quasi-subgradient in locally convex spaces. Consider again the dual system (E,F) and a quasiconvex mapping /: E-> R . Let XQ G E be an arbitrary point. The quasi-subgradient of/at x0 is the set d* f(x0) c F defined by

xX * e0=*f{x)>f(x0)\. If we put qQ [x ) = inff/(x) | (x - x0, x*) > o] X

q[x*} = inf x X\ sup|(x-x 0 ,x*)|/(x) < ^,J > 0 , JC

then we have the following result:

289

ZERO-EPI MAPPINGS

/w

,x x0,J > 03 = /(*o) o /(*o) x* eJf(x0)^mf[f{x)\(xt,x-x[X f(xQ)<*f(x =?c(**)qG(x). 0,)>0\ 0) = JC

Let df{x0) be the classical subgradient of/at x0, that is tf(x0) = jx* e F ( x * , x - x 0 ) < / ( x ) - / ( x 0 ) , f o r a l l x e £ J . The following result gives a relation between q and R is a function andx0 e E, then we have q qc (x ), then there exists x e E such thatj(x) < ft and (x-x {x-x0,x'}>0. 0,xj > 0 . It follows that sup {(x-x o ,x*)|/(x)0. Now we note that q(x ) < fj, which implies
and X
sup{(x*,x - x 0 )| / ( x ) < AJ < X - / ( x 0 ) < 0, X

which implies f(x0) = q[x'} < qG(x] < f(x0) and, consequently, x ' e d\f(xQ). ■ Proposition 5.6 0 e d*f(x0) <=>Xx0) = infftx). Proof The proposition is a consequence of the definition. ■ Remark 5.4 It is possible to have a mapping / and a point x0 such that

tf(xo) = 0butd'f(xo)*. 290

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION Example 5.1 Consider E = R and/: E ~» R given by

J-Vx, ifr/ x > 0 /(*)=H*' *-° /(*) = [+oo, if x < 0. v

' [+00, ifx<0. In this case we have d/0) = $ but ^ / ( 0 ) = {-k \ k e R, k > 0]. In this case we have dj(0) = § but ^ / ( O ) = {-£ | £ e J?, it > O}. Before the concept of 0-epi family of mappings is introduced, we recall Before the concept of O-epi of mappings introduced, we recall some notions and results from family the general topology.isLet Xbe a topological some notions and results from the general topology. Let Xbe a topological 1S space and let {4}, e ;if, Defora any family of subsets ofJ£ The family {4}, ^ e / ofx called locally finite if, for any x e X, there exists a neighborhood Vx of JC such that Ftfl 4 = ^ f° r a ^ ' G ^ \«/» where 7X c / is a finite set. A family {/■}., of numerical mappings defined onXis called a continuous partition of the unity if, for every i e I,f is continuous, positive and for every x e X, the family {/X*)}ie/ is summable in R and £/■(*) = 1. A partition of the unity is locally finite. A topological eIiei is locally finite if (supp(/))J. ^ t y {fi] {fi))•<=! Hausdorff space X is said to be paracompact if, for all open covering of X, there exists a finer locally finite open covering. It is known, that a metrizable topological space is a paracompact space. The following theo rem is a classical result: or a a paracompact paracompact space space X, Theorem 5.7 For all open covering {4}, /. ffor e7 {4},, liel there exists a continuous locally finite partition of the unity {/,} . , which

is also subordinate to {4j,- /

zw

the following sense: for all j e J, there

exists ij e I such that suppf/^j c ^ , Proof The proof of this result can be found in (Bourbaki, N. [1]). ■ Also, we will use the following result: 291

ZERO-EPI MAPPINGS Theorem 5.8 [Dugundji] Let X be a metric space and let A a X be a closed subset. Iff: A —> E (locally convex space) is a continuous mapping, then there exists a continuous mapping / " : X-> E such that f'A = f and f(X)czconv(f(A)). Proof The proof of this fundamental result is in (Dugundji, J. [1]). ■ Zero-epi families of quasiconvex mappings in R" The following lemma is a fundamental result for our construction: Lemma 5.9 Let X, Y be two subsets of R", where Y is convex. Suppose thatfX xY —>R verifies the following assumptions: 1) X"vV) is upper semicontinuous for every v e Y, 2) fix,-) is quasiconvex for every x e X. Then, for every a e R, such that,

supinf/tay)
Y Y x*xy* x<=xy* x^xy^Y

(5.3)

there exists a continuous mapping p : X—» Y such that fx,p(xf) < 0 for all xeX. xeX. Proof Consider, on X, the induced topology. By (5.3), we obtain that, for every x e X, there exists yx e Y such that^x, yx) < x) is upper A*, y*) < a.a. Since J(-y A-y*) semicontinuous, there exists a neighborhood Vx of x such thatfx'yj a, JV&) <<«. for all x' e Vx. The paracompactness of A' implies that there exists a locally finite continuous partition of unity {^,} i e / and a function i - » x, such that supp(^) c VXj. Clearly, the mapping/? defined by p(x) = £ (Pi{x)yXi, for all x e X t—i <=/ is a continuous mapping. For each x e X, let Ix be the subset of I, defined 292

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION by i elx <=> x e supp(^). Then, from the quasiconvexity offix, •), we have

/(*>p(*))=/U,I>.(*K - m f c {/( jc ^x / )} < «• V V

I,, --Eee///

//

, € ,<e'x e X

X

Since supp(^) c Fx , it follows f(x,p(x)) f{x,p[x)^<msodf\x,y < max{/(x, x\\ vX()} < a and the >'e/x

lemma is proved. ■ Theorem 5.10 Z,e/ {/*} gS ^ e a family of real quasiconvex and lower semicontinuous mappings defined on R". The set t> c: R" is an open bounded set. Suppose that the mapping x-^fx{y) is a continuous mapping on dVfor every y e R" . If the following assumptions are satisfied: 1) ^fx{0) * <j> for allxed

V,

2) 0 * tffx (0), for allxed V, then, there exist a continuous mappingp: dV —>Rn \{0} and a number 8 >0 such that fx{p(xf) < fx{0) - e for allx e dV. Proof By assumption (2) we have /,(0)> fx(0) > inf fx(y) for all JC x ee 8t>. 5Z>. yeR" ysR"

Let *P be the mapping defined on 8 V by

V(x) = fM fx(0)-mffx(y)>0. yeR"~ yeR

Since 8 V is a compact set and ¥ is a lower semicontinuous mapping, there x ¥(x0)== inf ¥(*). exists JC0 e 8 V such that ^(XQ) Tfx). x&%> xtdD

If we choose 0 < e < ^(XQ), then we have f/,(0) > inf inf ffxx(y) ++ e, e, for for all all xx ee 58 Z T> > x(0) > ye«"

and, consequently, 293

ZERO-EPI MAPPINGS

supMnifti) (fx{y)-fxfM) {0))<-<-s. £.

xzdpyeR"

Then, from Lemma 5.9, we ascertain that there exists a continuous mapping

p-.d'D^R" l" such that fx(p(x)) < fJO) - e for all x e d V, and the proof is therefore complete. We remark that we must have p : d V -> R" \{0}. ■ Consider now an open bounded set "D c R" and a family {fx}xfg

of

quasiconvex mappings on R" depending on x eV . Definition 5.2

The family {/.} {fx}x<*^e Z > is called 0-epi if there exist a 1

continuous mappingp : dZ> —> R"\ {0} and a number e>0 such that

W*))

m

fhx(p(x))
forallxEdZ? x e dV

(5.4)

and p is 0-epi, where p:T> ->R" is a continuous extension ofp to "D. Remark 5.5 The existence of the mapping p in Definition 5.2 is assured by Theorem 5.8. The concept of 0-epi family is well defined if we show that it is independent from the mapping p and the extension p. Indeed, let p be a mapping which verifies (5.4) and which has two continuous extensions px,p2toV such that p, is a 0-epi mapping. In this case, we have A and the = |*

^|*

mapping p2 - px has the following properties: i) (p2 - P\ X*) = 0 f° r all x € 9 Z>, ii) p2 - pl is a compact mapping on V . Then the boundary dependence property of 0-epi mappings implies that the mapping p2 = px + (p 2 - /?,) is a 0-epi mapping. Suppose now that px,p2:dt> -> R" \ {0} are two continuous mappings verifying (5.4). We can

294

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION suppose that s is the same for pj and p2. Let p2 [resp. px ] be a continuous extension of p2 [resp. px ] to V , where /?2 is supposed to be 0-epi. Consider the mapping

x pt(x) = tpl(x) + (l-t)p)M all x e V and fe[0,l]; 2(x) ) for all* using the quasiconvexity of£ for all x e d V, we obtain /,(?,(*)) ( A (*))} < /,(/>,(*)) < max{/ max{fx(JtPl((x]),f < /,(0) /,(0) -- ** for for all* all *ee [0,1] [0,1] (5.5) (5.5) A (x)),/ x(pJ[ 2{x))} But, from (5.5), we have 0€pidD),foTa\\t<= 0 * /?,(dZ>), for all t e [0,1]. [0,1 ].

(5.6) (5.6)

We set A(JC,0 = f(p, h(x,t) t(px - p^22\x), Xx), for all x e V and t e [0,1],

and we remark that the mapping h is a compact mapping on V x [0,1] and i £ h(x,Q) = 0 for all x e V . Sincep, •ep, ,p,^t[p =~t[p t[ + p2, from (5.6), we have 1-p 2] l-p 2] t[pl-&]+ p2(x) {x) + t{p t(p\x-p -p22\x) \x)

= pp22(x) (x) + h(x,t) * 0 ]for all x e dV and t e [0,1],

and, using the homotopy invariance property of the 0-epi mappings, we obtain that pl=p A =2/+(p Wl-p A2) "A) == A+a(-,l):Z>->*" p2+h{;l):V-+R" is 0-epi mapping. Therefore, Definition 5.2 is well posed. Zero-epi families of quasiconvex mappings in Banach spaces Lemma 5.11 Let X be a subset of a Banach space (Y, || ||) and let

295

ZERO-EPI MAPPINGS f-.XxY^Rt Rl be a mapping f:XxY—>Rbea assumptions: mapping which which verifies verifies the the following following assumptions: 1) fix,x -y) is upper semicontinuous relatively to xfor every >yeY, y e Y, 2) fix, •) is quasiconvex for every x s X. Then, for every a eR such that supinf f(x,y) < a, x*xy*Y

(5.7)

there exists a continuous mapping p : X -> Y such that fix, x -p (x)) < a for allx sX Proof Since Y is a vector space, the following relation is true: supinf f(x,y) = supinf f(x,x-y). (x,x-y)y) xeXyz*' x<=xy*Y xexy*r

(5.8)

From (5.7), using also (5.8), we have that, for every x e X, there e x i s t s ^ such that fix.x -yx) < a. As fix.x -yx) is upper semicontinuous, there exists a neighborhood Vx of x (in the topological space X) such that fix \x' -yx) < a, for all x' e Vx. Because A!" is a paracompact space, there exists a locally finite continuous partition of unity {#>,},.6/ and a function i —> xt such that supp (
/<*) =iel!><(*K is a continuous mapping. Clearly, the mapping fix, x-y) is quasiconvex with respect t o y . Indeed, for every y , , y2 € Yand every XhX2 e [0,1] such that X] +A2 = 1, we have f(x,x - [Xxyx + X2 v 2 ]) = f{x,X (x,X -yxx)) + X2(x - y2)) f(x,X x(x x{x-y y{x-y

296

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION <max{f(x,x-yiyi),f(x,x-y <max{f(x,x),f(x,x-y )}. 2(x,x-y 2)}. 2)}. Since |sup(p,-)} is a locally finite family, it follows that, for each x e X, {sup(p,)L/ lie/ the set Ix <= / defined by

/ , = {ie/|xesupp(p,)cF,} {ie/|xesupp(p,)cF,} is a finite set. Using quasiconvexity, it follows that

f(x,x-p(xj) /f(x,x-p(x))=/\x,x-Y,(x)) =f = / x, x - X
ieK

and the proof is thus complete. ■ Remark 5.6 If the set X considered in Lemma 5.11 is a compact set or a weakly compact set and if fix,x-y) is a weakly upper semicontinuous mapping with respect to x for every y e Y, then we can prove that the mapping p defined in the proof of Lemma 5.11 is a compact mapping (it is a mapping of locally finite rank). Therefore, in this case, / —p is a compact vector field. If £ is a Banach space and t> a E is a subset, then V denotes the closure of V with respect to the norm and "D a denotes the closure of 2? with respect to the weak topology; then we have V c= V a. But, if "D is convex, then V =Va. IfdZ> {resp. dat>] denotes the boundary of Z? with respect to the norm topology [resp, the weak topology], then we have

ff [\^XzV {\{E\V)(£\z>)ff =a =d affz>. z>. V. dv=v n#> cp'n(£\p)' n#>cz>Ti( dV=V

a

a

a

Theorem 5.12 Let \fx } =„ be a family of real quasiconvex mappings de-

297

ZERO-EPI MAPPINGS fined on a reflexive Banach space E. The setVczE is an open bounded set. Suppose that the mapping
1) ^/x(0) ^f *,x(0)^^,forallxe^V, xe^V, 2)

0ecrfx{0),forallxe<7V, X BcPf, OeJf, M

p:dV^>E and a number ss>0 then, there exist a continuous mappingp: dV-^Eanda > 0 such that: i) x -p(x) * 0, for allx&d V, ii) f/x,(x-p(x)) V. xe 0ZJ. ( * - / < (*)) * ) ) < fx(0) - e, for allxsd Proof Using a property of the quasi-subgradient and since 0 £^/ J (0)for

w

all x e ^ P , we have fx(0) > inf /xW. fx(y). for all xe^Z>. i e ^ . _ye£"

_ye£

Let ¥ be the

mapping defined on d°V by ¥(*) = fx{0) - inf fx{y) > 0. yeE

The mapping ¥ is weakly lower semicontinuous, and, because E is a reflexive Banach space and V is a bounded set, we obtain, using the variant of Weierstrass' Theorem for lower semicontinuous functions (Vainberg, M. M. [1]), that there exists x0 e d" V such that yiJ(x0)= inf T(x). If we choose 0< e < ¥(x 0 ), then we have fx(6)-

inf fx(y)

> e for all x eff"D

and, consequently, sup inf (fx(y) - fx(0)\ < s. But, from Lemma 5.11, we x^daT>y^E

obtain that there exists a continuous mapping p: daV -> E such that obtain that there exists a continuous mapping p: dcTV -> E such that fx(x - p(x)) < fx(Q) - e, for all x e daV =3 . fx(x - p(x)) < fx(0) - s, for all x e daV =3 dV. Since e > 0, we have, using (ii), that (i) is also true, and the proof is complete. ■ Consider now an open bounded * c <

2QO

and I* { / , } , , beafc-nilyof

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION quasiconvex mappings on the Banach space E depending on x e P f f . Definition 5.3 The family {fx}

_, is called 0-epi if there exists a com-

pact continuous mappingp: d^V —> E and a number s> 0 such that: 1) x -p{x) * 0,for allx e 8° draz>, V, 2) f{x-p{xj) E is a compact continuous extension ofp to "D°'. Remark 5.7 The existence of the mapping p in Definition 5.3 is ensured by Theorem 5.8. The concept of 0-epi family in the case of Banach spaces is well defined if we prove that it is independent of the mapping p and the extension p. Indeed, let p be a compact continuous mapping which verifies (1) and (2) and which has two compact continuous extensions p\, p2 : Va ^E, where / - p \ is a 0-epi mapping . In this case, we have 1 Al e°i> „ ==Pi\

„»

g"7>

which implies that (/ - p 2 \ x ) -(I-p\\x) (I - (/ - p ^ xI*)) = p\(x) p\(x) -p -p22{x) {x) = = 0, 0, for for all all xed^V. xed^V. Since the mapping (/ — p2) - (/ - p\ ) is compact and (/ - /?j) - [(/ - p2 ) - (/ - px)] = / - p2, we can use the boundary dependence property of 0-epi mappings and we obtain that I - p2 is a 0-epi mapping. Suppose now that p\ and p2 are two compact continuous mappings which verify (1) and (2). We can suppose that s is the same for Px and/?2. Let p2 [resp. /?,] be a compact continuous extension ofp2 [resp.pi] to T>a,

299

ZERO-EPI MAPPINGS where I - p2 is supposed to be 0-epi. Consider the mapping p,(x) = tpx(x) + (1 - 0 p2(x), for all x e Va and t e [0,1], and, using the quasiconvexity of/i for all x & dat>, we obtain fx(x - pt(x)) < max{fx(x - p,{x)),fx(x

- p2(x))}

(5.9)

But, from (5.9), we have Oe(/- j p,)(^Z>),forall?e [0,1].

(5.10)

We put h(x,t) = -[t(px -£>)(*)]> for all x e Va and t e [0,1] and we observe that the mapping li is a compact mapping on Va x [0,1] and h(x,0) = 0, for all x e Va. From (5.10), we have (/ - p 2 \ x ) + h{x,t) = (/ -p 2 ){x) - t(px -p 2 ){x) = [I -(tp x + (1 - t)p2)\x) for all x e T>a. Using the homotopy property of the 0-epi mappings, we obtain that (/ - px) = (/ - px) + /j(-,l):Z>CT -> E is a 0-epi mapping. Therefore, Definition 5.3 is well posed. ■ Zero-epi families of quasiconvex mappings and localization of Nash equilibrium points Theorem 5.13 Let {fx}x€^ be a family of real, proper quasiconvex and lower semicontinuous functions defined in R". The set "D c R" is an open bounded set. Suppose that the mapping x ->f(y) is a continuous mapping on V for every y e R" and 0 id 'fi(0) for all x edV. Suppose that 300

ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION d"fx{Q) is non-empty for all x ^dV. If {fx}xeS isisaaO-epi 0-epi family, then there xsVV exists XQ e T> such that*o fX()(0) = inf x (y). f:* 0 f00 yeR" Proof Suppose the contrary, that is to say: {/,} {/.} xeV* is a 0-epi family, but

J*

/_(0) > inf f(y), (y), for all x e V. ' x \ yeR"

Since 0 £ inf fx(y)

for all

yeR"

x&V.

Let *F be the function defined on Z? by ¥(x) = / , ( 0 ) - i n f / , ( j O > 0 . yeR"

Because Z> is a topological compact space and ¥ is a lower semicontinuous mapping there exists x« e V such that ¥(;c,)= jceZ? inf ¥(*)>(). If we choose 0 < s < ^ ( x . ) , then we have fx(0)>

inf f (y) + s for all

yt=R" x y eR"

x € V and, consequently, sup inf {My)(fx{y)-f x{0))<-e. fM)<-£gD yeR" yeR xev x

From Lemma 5.9, we note that there exists a continuous mapping p:V *->n\{0} p:V^R fx(p(xj))-* {0} such that /,(/<*)) « * ) ) '< quently, 0 g pip), which is impossible because {/-) {fx}xeS is a 0-epi family ' xeZ> and the existence property of 0-epi mappings is contradicted. Therefore, the proof is complete. ■

301

ZERO-EPI MAPPINGS A similar result is valid in reflexive Banach spaces. Theorem 5.14 Let [E,\\ \\j be a reflexive Banach space andV czEan open bounded set. Let {/-}I j e P " be a family of real proper and quasiconvex

mappings defined on E such that the mapping cp{x,y) = fx(y) is weakly CT continuous. Suppose that d fx(0) *

. V \f\ fxeV* ^a is a 0-epi family, then there exists x0 e Z> such that

A0(0) = yeE inf/,*o0(y)W. Proof As in the proof of Theorem 5.13, we suppose the contrary, that is to say: {/x} =0 is a 0-epi family, but fx(0) > inf fx(y) for all x e V. Since 0 td'fJSi) for all xed^V,

it follows that,/fO) > inf fx(y) for all

Let *F be the mapping defined on 0

■ o-

yeE yeE

xeV".

>y ¥(*) = ./ r ( 0 ) - iyeEn f / r ( v ) > 0 .
Z> is a weakly compact Since *F is a lower semicontinuous mapping and V v set in E, there exists .x* €e V " ssuch that *(*•)= i^to ■ (see M. M. :s; JC» x^D"

Vainberg [1]). If we choose 0 < s < ^(x,), /,(0) > inf f(y) yeE~ yeE

and, consequently

then we have

+s f
sup inf(fx(y) -fj0))<-e. (o)) <-£. e.

r f S ' )

1

€

a &V Z> cr

Because the mapping

^

<&(x,x - y) = fx(x - y) - fx(0) is is weakly upper semicontinuous with respect to x for every y, we can use Lemma 5.11, to show that there exists a compact continuous mapping p:Va a -> £ such that ffX A - p(jc)) /,(0) 2? . < ,A (o) -- 8 £ for all x ee © A 0(xX-pi <*)) <

302

ZERO-EPI MAPPINGS AND OPTIMIZATION Consequently, 0 e{l-p){z>a\,^') which is impossible because {fx}xeSc I 9

is a

0-epi family and the existence property of O-epi mappings is contradicted. Therefore, the theorem is proved. ■ As an application, consider two reflexive Banach spaces E and V and two mappings, h,g : E x V->R. We put X¥X{Z) = a E x V such that s a am 1) ^x^x^S" ily of quasiconvex mappings, {**)Sxe5a * ffamily x 2) q>(X,Y) = ¥x(Y) is a weakly continuous mapping, 3) 0 £ ^ ^ ( O ) *(/>,for all X e d°V, then, if fi?x}xefi'7 *s a ®~eP*family, fomily, there exists at least one Nash equilibrium point in "Dfor the couple (g,h). ■ The Corollary suggests an algorithm to localize Nash equilibrium points. Starting from an open bounded subset Z>0aEx V, we decide if ^>X} X^D" >*€£><,"

M

is a 0-epi family using CFVQ . If yes, we reduce 2?o to a smaller subset Z?; a 1S e am is aa ^~ O-epi (for example by subdivision) and we decide if {^x]x&v P' ffamily ^y X&V? using m) < s for e > 0 acceptable and {^x}x&Z ^~ePifamily. fe™dlyx^ Remark 5.8 The assumption 0 g ^ ^ ( 0 ) means that the Nash equilibrium points are not on the boundary eft) because Proposition 5.6 implies that 303

ZERO-EPI MAPPINGS 0 e d*tf^xip) <» ^ ( y ) ^ ^ 0 ( 0 ) for all Y e £ xK Remark 5.9 It is possible to substitute in our theory in R" the quasisubgradient d* by the generalized differential defined in (Mirica, S. [1]) or by the value set for a distribution in the White's sense (White, R. E. [1]). Remark 5.10 Since the topological degree can be used to decide if a mapping is 0-epi with respect to a given set, one could recommended, concerning the estimation of the topological degree the papers (Erdelsky, P. J. [1]), (O'Neil, T. and J. W. Thomas [1]), (Peitgen, H. O. and M. Prufer [1], (Priifer, M. and H. W. Siegberg [1], [2]), (Rabinowitz, P. H. [1]) and (Siegberg, H. W. [1]). Open problem It is interesting to extend the concept of 0-epi family of mappings to the mappings which are not necessarily quasiconvex.

6. Zero-epi mappings and complementarity problems In applying the concept of 0-epi mapping, another possibility is the study of complementarity problems. This section will initiate and consider this new applications. About complementarity problems, consider as known the notions, the results, and the notations introduced and studied in Chapter 1. The subject being completely open to new research, we present now only a few results. Let (H,<,>) be a Hilbert space and let K c H be a closed convex cone. Denote by K' the dual of K. lfPK denotes the projection onto K, that is, for every x € H, Pg(x) is the unique element satisfying, \\x-PK{x)\\ = rm^\x-y\\, then we have the following classical result: 304

ZERO-EPI MAPPINGS AND COMPLEMENTARITY For every element x G H , P^x) is characterized by the following properties: (PK{x) x,y) >0, > 0, for allyeK ally eK (x) --x,y)

(6.1)

and (PK{x)-x,PK(x))

= 0. 0.

(6.2) (6.2)

Suppose given two mappings/ g: H -> H. The Implicit Complementarity Problem associated to the mapping/g and to the cone K is find xt G H

such that

ICP{f,g,K):\g(x G K, f(x*) G K and and ICP{f,g,K):- • g(x eK* tg(x*) t) eK,f(x t) {g(xt),f(x {g(x 0. t))t)) = 0. t),f(x Consider the mapping > 00/or for all all xx GG ^A]j . . Theorem 6.1 Lef (//,<,>) fee a Hilbert space and K czH a closed pointed convex cone. Let f and g be two completely continuous mappings from H into itself. Suppose there exists a completely continuous mapping
305

ZERO-EPI MAPPINGS fi>> 0 and x &dQ.C\g~*{K), wehave have 2) for every fi edQOg l(K), we

f{x) to the problem ICP(f,g,K) and x. e Q. Proof As we remarked above, the theorem will be proved if we show that the equation O(x) = 0 (6.3) has a solution x> in Q, where i/ A: [0,l]->ff defined by H(x,X)-H(x,(S) for for all all(x (x,X) Q xx [0,1]. [0,1]. h(x,X) = H(x,X)-H(x,0) f/l) ee Q By assumptions and using the properties of PK, we have that h is continuous and compact. Moreover, h(x,0) =/f(x,0)-//(x,0) = 0 for all JC e Q. We show that *¥(x) + h(x,X) * 0 for all x e 5Q and X e [0,1]. Indeed, if JC e 9Q and X = 1, then Y(x) + V(x) + h(x,l) h(x,l) = = V(x) V(x) ++ (D(x) (D(x) --¥¥((JJCC)) ==
ZERO-EPI MAPPINGS AND COMPLEMENTARITY x g( o) = PPK[S(XO) - C01 - ^oM^o)] s(*o) K[g{xo) ~ *o/(*o) *o/(*o)" ^oM*o)] •

(6.4) (6-4)

Using (6.1) and (6.2), we obtain g(x e *K,, *(*>) 0) £ K*o) " ^o/(*o) *o/(*o) - 1(11 "- ^oM*o)],y) (g(*o) - [«(*o) ^oM*o)].^) 2= ^ 0, for allv all y e K,

(6.5) (6.6)

and

((g(*o) ^ o j- -[g(*o) W - -V*o/(*o) ^ o ) - -^0--^*oM ) ^ x)o)]» ] . ^s( x)o)) ) == 0°•

(6.7)

From (6.5) and (6.6) and (6.7), we have

l o r a11 eK (y -- g(*o). g(*o)>*o/(*o) + (1 -- *oM*o)) all y eK. (y *o/"(*o)+( ^oMxo)) *- 0,°' ffor y -

(6.8)

Dividing (6.8) by Xo > 0 and considering (6.5), we have a contradiction of 1—X assumption (2) with // = l - A p . Hence, we have that ^(x) + h(x, X) * 0 for /i 0 all x e dQ. and A, e [0,1]. Applying now the homotopy property of 0-epi mappings, we obtain that vP(-) + /z(-,l):Q—> H is 0-epi, which implies that the mapping Y(x) + h{x,\) = *P(x) + 0(x) - 4/(x) = O(JC) has a zero in Q. But, considering our assumption about O, we have a contradiction. From this contradiction, we have that 0(x) = 0 must have a solution x* e Q. ■ Remark 6.1 Theorem 6.1 is at the same time a refinement and a generalization of Theorem 4.63 [Chapter 1] proved for the first time in (Pang, J. S.andJ.C.Yao[l]). Recall that, if in problem ICP(f,g,K) the mapping g is g(x) = x for all x e H,

307

ZERO-EPI MAPPINGS then we obtain the Explicit Complementarity Problem (shortly ECP{f, K)), which is, (find x, ee K K such such that that find x, ECP{f,K):f(x.)eK* and(x.,f(x )) = 0. f{x.)eK*and(x.,f{x ('t)) t (*)) = 0.

{

Because the identity mapping on H is not completely continuous when H is infinite dimensional, it is impossible to obtain from Theorem 6.1 an existence theorem for the explicit complementarity problem. However, if (H,<,>) is the Euclidean space (R",<,>), (*",<•>) we have the following result: Corollary 6.2 Let (Rn,<,>) be the Euclidean space, K a R" a pointed closed convex cone and f: R" —> R" a continuous mapping. Suppose that there exist a continuous mapping

R" defined by *P(x) = x - PK[x - 0 andx € dClflK, we have f(x) + ju) be a Hilbert space and K c H a closed pointed convex cone. Let f and g be two completely continuous mappings from H into itself. Suppose there exist a completely continuous mapping tpfrom H onto itself a vector u0 e K and a bounded open set Q c H such that: 1) the mapping *P from D. into H defined by V{x\ ¥(*) = =
(l-A)[^x)^(x)^[if-^(x)f. {\-X)[
ZERO-EPI MAPPINGS AND COMPLEMENTARITY Then, the problem ICP(f,g,K) has a solution x* E Q. Proof Considering again the mapping (x) = 0 has a solution in Q. Assume that <S>(x) has no zero in Q. Define the mapping H(x,X) = Xg{x) - PK[A(g{x) - /(*)) + (1 - X)uQ] + (1 - X)(p(x) for all x e Q ancUs [0,1]. We have H{x,0) = (p(x)-u0 = V(x) and H{x,l) = g{x) -PK[g{x) - f{x)] = (D(x) for all x e Q. Consider the mapping h : Q x [0,1] -»//defined by h(x, X) = H(x, X) - H(x, 0) for all (x, X) e Q x [0,1 ]. We show that ^(x) + h(x,X) * 0 for all x e SO and X e [0,1]. As in the proof of Theorem 6.1, we have that ¥(*) + h(x,X) * 0 for all JC € 5Q and A.= 0 or X = 1. Suppose that ¥(*) + h(x, X) = 0 for some x e SQ and X e ]0,1 [. Denote this x by x0 and this A, by X,0. Thus, we have x

¥(x0)+h(x0,X0) = 0.

309

ZERO-EPI MAPPINGS Since xiJ(x0)+h(x0,X0) = H(x0,X0), we have H(x0,Xo) = 0 which implies, ^og(*o)+ (1 - ^oM*o) = ^ o W * o ) - /(*o)) + ( 1 " ^o)"o] or /(*o)) + (1 ~ *o)»o] • M-~ /{**))

X *x PK[MSM *x,0 (*o) = PK[MS( O)

Using (6.1) and 6.2), we obtain

M'o)**

(6.9) (6-9)

(kXo(x {x00)-[A00g{x00(*o) )-A0Qf(x f(x00)) (*o)-+ (\-A (\-A00)i0)u (1-A )uQ0],y)>0,for ],y)>0, for ally eK eK and

(6.10) (6.10)

(^W-M«o)-vW+(i-^H].*;i,W) = 0 -

(6.11) 6n

(-)

From (6.9), (6.10) and (6.11), we have kXa(x0)sK and

K

(6.12) (6.12)

(v - ^ ( x 0 ) , ( l - ^0)[> e JT. (6.13) (6.13)

From (6.13), we deduce that

(i-^)[^*o)-*o]+M*o)4*-*4*o)f, which is a contradiction of assumption (2). Hence, we have that T(x) + h(x,A) * 0 for all x e dQ and X e [0,1]. The conclusion of the theorem is obtained by a similar argument as in the proof of Theorem 6.1.

■ If we take the function (p to be g in Theorem 6.3, we obtain the following result:

310

ZERO EPI MAPPINGS AND COMPLEMENTARITY Corollary 6.4 Let (H,<,>) be a Hilbert space and K c Ha closed pointed convex cone. Let f and g be two completely continuous mappings from H into itself. Suppose there exists a vector u0 e K and a bounded open set QczHsuch that: 1) the mapping g(x) - u0 is 0-epi on Q., 2) for every p. > 0 andx e SQflg^W, we have f{x) + p\g{x)-u ]] i[K-g[x)]\ f(x) /{g{x)-u [g{x) -00"<J .]•[■e[K-g{x)]\

Then the problem ICP(f,g,K) has a solution x* e Q. ■ Remark 6.2 If all the assumptions of Corollary 6.4 are satisfied without assumption (2), then the conclusion of this corollary is true if the following assumption is satisfied: 3) (g{x) - uQ,f{x)) > 0 for all x € S Q f V W Indeed, if (3) is satisfied, then, for any p. > 0 and x e dQ(]g '{K) we have g(x)* w0 since g(x) -"o is 0-epi on Q and x [ju (u0-g(x),f(x)+p\g(x)-u «[si* 0)) o])"o]> < 0, which implies assumption (2) of Corollary 6.4. Proposition 6.5 Let (H,<,>) be a Hilbert space and K c H a closed pointed convex cone. Let f and g be two completely continuous mappings from H into itself. Suppose there exists a vector v0 eg~l(K) and a bounded open set Q c H such that: 1) the mapping g(x) - g{vo) is 0-epi on Q, 2) for every x e d£lC\g~ (K) we have: V \\g{x)~g(v 0)||^p\\x-v01,withp>0, 0

w-«(v )i

H) (/(*)-/(v 0 ),g{x)-g(v 0 )> >c\\x- v 0 f, withc> 0, 3) c\\x - v 0 | > p | / ( v 0 | . for allx e 8QC\g''(K). Then the problem ICP(f,g,K) has a solution xt e Q. Proof The proposition is a consequence of Corollary 6.4 and Remark 6.2, 311

ZERO-EPI MAPPINGS since from our assumptions, we have (denoting u0 = g(v0)): (*(*) - «b,/(*)) = (g{x) - g(v0),f(x) - /(v„) + /(v 0 )> = («(*)" g(v 0 ),/W - /(v 0 )> + (g{x) ~ «(v 0 ),/(v 0 )) > c||x - v 0 f - p\x - v 0 ||/(v 0 )| > 0,/or all x e < ? n f V ( * ) .

■

The concept of 0-epi mapping with respect to a convex cone can also be applied to the study of solvability of complementarity problems. If (H,<,>) is a Hilbert space and K c H a closed pointed convex cone, denote, as in section 4, B+p = \x e K\ \x\ < p\

and S* = \x e K\ \X\ = p\

for any real number p > 0. Theorem 6.6 Lef (//,<,>) be a Hilbert space and K c / / a closed pointed convex cone having a compact base andf: K —> H a continuous mapping. If there exists p > 0 such that px +J(x) g K for all p > 0 andx e S/, then the problem ECP(fK) has a solution x» such that \\x.\\ < p. Proof Note that the mapping h : B+p -> K defined by h(x) = PK [x -fix)] for all x s Bp is continuous and compact. We show that h(x) * Ax for all x € Sp+ and X > 1. Indeed, if we assume that h(x0) = XQX0 for some x0 e Sp+ and XQ > 0, then we have /l0x0 = PK[x0 - / ( ^ : 0 ) 1 , which implies (using formula (6.1)), ( V o - [^o - f{xo)\y)

> 0, for all v e ^ .

(6.14)

If in (6.14) denote po = Ag - 1, we have that //QX,, + / ( J C 0 ) eZ* , which is impossible. Applying now Corollary 4.5, we obtain that the mapping x - h(x) is (K,K) -0-epi on B+ . Hence, the equation x - PK[x - f(x)] = 0

312

ZERO EPI MAPPINGS AND COMPLEMENTARITY has a solution in B+p , which implies that the problem ECP(f,K) has a solution x- such that || x*\\ < p. ■ From an argument similar to the proof of Theorem 6.6, we also have the following result: Theorem 6.7 Let (H,<,>) be a Hilbert space, K cz H a closed pointed convex cone, andf: K —> H a mapping such that fix) = x + (p(x) with

0 such that Ax + (fix) i. K* for all x e Sp+ and X>\, then the problem ECPif.K) has a solution x* such that \\ x*\\ < p. ■ Now, consider again the Implicit Complementarity Problem. Theorem 6.8 Let (//,<,>) be a Hilbert space, K czH a closed convex cone andf, g:K—>H two completely continuous mappings. Suppose that there exist 0 < a 0, 3) there exists q e K\ {0} such that f (x) -Aq g K' for all x e S^ and A>0. Then the problem ICP(f,K) has a solution x* such that a< \\ x*\\ < p. Proof Define h : B+-> K by h(x) = PK [g(x) -fix)] for all x e B+p . From the assumptions, we have that h is continuous and compact. For all JC e Sp+ and X > 1, we have Xg(x) ■* h(x). Indeed, if there exist x0 e S/ and XQ > 1 such that Xog(x$) * h(xo), then, by formula (6.1) we have ( M * o ) - [ « ( * o ) - / ( * o ) ] . > ) * 0 ^ a l l v e K, which implies pog(x) +fix) e K* where po = AQ -1. We obtained a contra313

ZERO-EPI MAPPINGS diction of assumption (2). Thus, Ag(x) # h(x) for all x e Sp+ and A > 1. We now show that g(x) - h(x) * Aq for all x e Sa+ and A > 0. Indeed, similarly to the above, supposing that there exist x0 e Sj and ^> > 0 such that g(xo)-h(xo)=Aoq, we reach (using again (6.1)) a contradiction of assumption (3). Finally, since g is completely continuous, there exists 8 > 0 such that ||g(r)|| ^ 5 for all x e B* . Hence, all the assumptions of Theorem 4.6 are satisfied, and, from the cited theorem, we obtain that g -his (K,K)-0-epi on Bp+\ B* , and the conclusion of the theorem follows. ■ The next result is an interesting existence theorem for Complementarity Problems depending of a multiparameter: Theorem 6.9 Let (//,<,>) be a Hilbert space, K c H a closed pointed convex cone, g : H -> H a bounded linear isomorphism satisfying the maximum principle, andf: R" x H ->H a mapping such that the mapping 0. Then, the problem ICP[f(A,-),g[-),K^ has a connected component D of solutions set contained in R" x IB* \B* J such that either D is unbounded or it intersects the set (R" X S*\\J[R"

+ X

S*).

Proof If we consider L = g and the mapping h(te) = PK [g(x) -fite)] for all (te) e R" x H, we can show with a similar 314

ZERO-EPI MAPPINGS AND k-SET CONTRACTIONS argument, as in the proof of Theorem 6.8, that all the assumptions of Theorem 4.11 are satisfied. Hence, the theorem is a consequence of Theorem 4.11. ■ Remark 6.3 Theorem 6.9 has interesting applications to the study of the post equilibrium state of a thin elastic plate (as in the model presented in Chapter 1) if we set in Theorem 6.9 g(x) = x for all x e H and M*)

=

* - t,A,Li(x) + C(x) + W(x),vrhsKLi(i= 0'=l,2,..,n) 0 = l,2,..,n) i=i

are linear self-adjoint and completely continuous operators and C, 5R are nonlinear, completely continuous operators. The mapping j{h,x) can be considered as a generalization of Von Karman's operator used in elasticity theory. We conclude this chapter with the observation that the theory of 0-epi mappings must be developed. For many practical problems, it is important to have efficient tests which can be used to decide when a mapping is 0-(or p-)-epi with respect to a given set. If we compare the topological degree with the concept of 0-epi mapping, we remark, from the practical point of view, the simplicity and the elegance of this new concept.

7. Zero-epi mappings and &-set contractions Of particular interest, is the variant of the concept of zero-epi mapping, which can be obtained by replacing in the definition of zero-epi mapping, the compact mappings by A>set contractions. This development was presented for the first time in (Tarafdar, E. U. and H. B. Thompson [1]). Let (E,\\ |D be a Banach space and D c E a bounded subset. The measure ofnoncompactness of the set D is defined by a(-D) = inf \e> 0| D can be covered by a finite number of sets of diameter less than e }. The measure ofnoncompactness has the following properties: 315

ZERO-EPI MAPPINGS 1. a (D) = 0 if and only if D is relatively compact, 2. A c A implies a ( A ) ^ a (A), 3. a(D) = a(D), where D denotes the closure of A 4. a(Dl U A ) = max{a(D,),a(A)}, 5. «[co( D)] = a(D), where co(D) denotes the convex hull of A and 6. a(Dx +D2)< a(Dt) + a(D2). The concept of measure ofnoncompactness was introduced in (Kuratowski, C. [2]). For the proof reference can be made to (Lloyd, G. [1] and (Banas, J. andK. Goebel [1]). Let (A|| 1) and i.F,\ ||) be Banach spaces a n d / : £ - > F a continuous map ping. We recall that/ is said to be a k-set contraction if for each bounded subset DciE, «(/(D)) < ka{D), where k > 0. We know that the concept of zero-epi mapping is strongly based on Schauder 's Fixed Point Theorem and on Urysohn's Lemma . To introduce the concept of (/?,&)-epi mapping we must replace Schauder's Fixed Point Theorem by Darbo's Fixed Point Theorem. Theorem [Darbo] If \E,\ ||1 is a Banach space and D cz E is a closed bounded convex set, then any k-set contraction/: D —> D with k e [0,1[ has afixedpoint. Proof The proof can be found in the paper (Darbo, G. [1]). ■ Let ( A 1 I) and ( F , | ||) be Banach spaces and Q c £ a n open bounded subset of E. We recall that a continuous mapping /f:Q^F : Q - > F is said to be ^-admissible (p-admissible) if 0 if{dQ) (jp g=/( > FF is (Q,k)-epi if for each k-set contraction h:Q->F, with h(x)=0 on dCl the equation

316

ZERO-EPI MAPPINGS AND k-SET CONTRACTIONS y(x)=/j(^) has a solution in Q. Similarly, we say that a p-admissible mapping f:Q—>F is (p,k)-epi if the mapping/—p defined by ■p,x e Q (f-p)(x)=Ax)-p,xeU

is (0,k)-epi. As in the case of 0-epi mappings, we can prove the following basic properties for (p,k)-epi mappings: Existence (or solution) property If f.Q. —> F is a (p,k)-mapping, then the equation f(x) =p has a solution in Q. Normalization property The inclusion mapping i:Q—>E is (p,k)-epifor k e[0,l[ if and only if p e Q. Localization property If / : Q —> F is (0,k)-epi andf (0) is contained in an open set Qi c Q, thenf restricted to Q.\ is also (0,k)-epi. Homotopy property Let f:Q -» F be (0,k)-epi and h :[0,l] x Q -» F be an /3-set contraction with 0 < /? < k < 1 such that h(0jc) = Ofor all x e Q. Iff(x) + h[t,x) ^0 for allx € 9Q and for all t e [0,1], then / ( • ) + /J(1,-):

Q - > F is(0,k-

p) -epi.

Boundary dependence property Let f:Q.^>F / : f l - > F be (0,k)-epi and g: Q -> F be an /3-set contraction with 0 < /3 < k < 1 and g{x) = 0 for all x E 8Q. Then, f+g:Q^>Fis(0,k- {0,k-frp)-epi. The proof of these properties are in (Tarafdar, E. U. and H. B. Thompson [1]). The authors of this paper follow (modulo some technical results) the proofs of these properties for 0-epi mappings. We note that, since every

317

ZERO-EPI MAPPINGS compact mapping is a k-set contraction, it follows that every (0,£)-epi mapping is 0-epi and every (p,k)-epi mapping isp-epi. The class of (p,0)-epi mappings is strictly larger than that of (p,k)-ep[ mappings, but the importance of (p,k)-epi mappings is supported by the fact that a (p,k)-epi mapping is more solvable at the point p than a p-epi mapping. To conclude , we remark that several of the results presented in this chapter can be considered for k-set contractions, and from the practical point of view, this is an interesting aspect.

8. References AMANN,H. 1. Lectures on some fixed point theorems, IMP A, Rio de Janeiro (1974). 2. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev., 18 (1976), 620 -709. AUSLENDER, A. 1. Problemes de Minmax via I 'Analyse Convex et les Inegalites Variationnelles: Theorie et Algorithmes. Springer-Verlag, New York, (1972). AVRIEL, M. 1. Nonlinear Programming: Analysis and Methods. Prentice-Hall, Englewood Cliffs, New Jersey (1976). BANAS, J. and K. GOEBEL 1. Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics Nr. 60, Marcel Dekker Inc. New York, Basel (1980). BIRKHOFF, G. D. and O. D. KELLOGG 1. Invariant points injunction spaces. Trans. Amer. Math. Soc. 23 (1922), 96-115.

318

REFERENCES BOURBAKI, N. 1. Topologie Generate. Chap. 9, Hermann Paris, France (1971). CROUZEIX, J. P. 1. Contributions a I'Etude des Fonctions Quasiconvexes. Universite Clermont-Ferrand, These (1977). DARBO, G. 1. Punti uniti in transformazioni a codominio non compato. Rend. Sem. Mat. Univ. Padova 25 (1955), 84-92. DING, Z. 1. An infinite dimensional 0-epi mapping with degree zero. J. Math. Anal. Appl. 199 (1996), 458-468. DING, Z. and A. G. KARTSATOS 1. P-regular mapping and alternative results for perturbations of maceretive operators in Banach spaces. (Preprint, 1996). DUGUNDJI, J. 1. An extension ofTietze's theorem. Pacific J. Math., 1 (1951), 353-367. DUGUNJI, J. and A. GRANAS 1. Fixed Point Theory. PWN-Polish Scientific Publishers, Warszawe (1982). EDELSON, A. L. and M. P. PERA 1. A note on nonlinear problems depending on infinite dimensional parameters. Nonlinear Anal. Theory Meth. Appl., 6 (1982), 1185-1191. EKELAND, I. and R. TEMAM 1. Convex Analysis and Variational Amsterdam, Holland (1976).

Problems.

North-Holland,

319

ZERO-EPI MAPPINGS ERDELSKY, P. J. 1. Computing the Brower degree. Mathematics of Comput. 27 (1973), 133-137. FITZPATPJCK, P. M. and W. V. PETRYSHYN 1. On the nonlinear eigenvalue problem T\u) = X C(u), involving noncompact abstract and differential operators. Boll. Un. Mat. Ital. 15-B (1978), 80-107. FURJ, M., M. MARTELLI and A. VIGNOLI 1. Stable-solvable operators in Banach spaces. Atti Accad. Naz. Lincei Rend. 1(1976), 21-26. 2. Contributions to the spectral theory for nonlinear operators in Banach spaces. Annali Mat. Pura Appl. 118 (1978), 229-294. 3. On the solvability of nonlinear operators equations in normed spaces. Annali Mat. Pura Appl., 124 (1980), 321-343. FURI, M. and M. P. PERA 1. On the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces. Atti Accad. Naz. Lincei Rend., 67, 1-2(1979), 31-38. 2. An elementary approach to boundary value problems at resonance. Nonlinear Anal. Theory, Meth. Appl., 4, Nr. 6 (1980), 1081-1089. 3. On unbounded branches of solutions for nonlinear operator equations in the nonbifurcation case. Boll. Un. Mat. Ital. 1-B (1982), 919-930. 4. Co-bifurcating branches of solutions for nonlinear eigenvalue problems in Banach spaces. Annali Mat. Pura Appl., 135 (1983), 119132. FURI, M. and A. VIGNOLI 1. Unbounded nontrivial branches of eigenfunctions for nonlinear equations. Nonlinar Anal., Theory, Meth. Appl. 6 Nr. 11 (1982), 1267-1270.

320

REFERENCES FURI, M , M. P. PERA and A. VIGNOLI 1. Components of positive solutions for nonlinear equations with several parameters. Boll. Un. Mat. Ital., Serie VI, Vol. 1-C Nr. 1 (1982) 285302. GRANAS, A. 1. The theory of compact vector fields and some applications to the theory of functional spaces. Rozprawy Matematyczne, Warszawa, 30 (1962). GREENBERG, H. J. and W. P. PIERSKALLA 1. Review of quasiconvex functions. Oper. Research 19 (1971), 15531570. 2. Quasiconjugate functions and surrogate duality. Cahiers du Centre d'Etude de Recherche Operationelle Vol. 15 (1973), 437-448. ISAC, G. 1. Q-epi families of mappings, topological degree and optimization. J. Opt. Theory Appl. 42 Nr. 1 (1984), 51-75. IZE, J, I. MASSABO, J. PEJSACHOWICZ and A. VIGNOLI 1. Structure and dimension of global branches of solutions to multiparameter nonlinear equations. Trans. Amer. Math. Soc. 201 Nr. 2 (1985), 383-436. KRASNOSELSKII, M. A. 1. Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands (1964). KURATOWSKI, C. 1. Topology. Vol. 2, Academic Press, New York (1968). 2. Sur les espaces complets. Fund. Math. 15 (1930), 301-319. LARSEN, R. 1. Functional Analysis: An Introduction. Dekker(1973)

321

ZERO-EPI MAPPINGS LLOYD, G. 1. Degree Theory. Cambridge Tracts in Mathematics Nr. 73, (1978). MANGASARIAN, O. L. 1. Non Linear Programming. Mc. Graw-Hill, New York (1969). MARTOS, B. 1. Nonlinear programming. Theory and Methods. North Holland Publishing Company (1975). MASSABO, I., P. NISTRI and M. P. PERA 1. A result on the existence of infinitely many solutions of a nonlinear elliptic boundary value problem at resonance. Boll. Un. Math. Ital., (5) 17-A (1980), 523-530. MINTY, G. J. 1. Monotone (nonlinear) operators in Hilbert space. Duke Math. J., 29 (1962), 341-346. MIRICA, S. 1. A note on the generalized differentiability of mappings. Nonlinear Anal., Theory, Meth. Appl. 4 (1980), 567-575. O'NEIL, T. and J. W. THOMAS 1. The calculation of the topological degree by quadrature. SIAM J. Num. Anal. 12 (1975), 673-680. PEITGEN, H. O. and M. PRUFER 1. The Leray-Schauder continuation method as a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems. In: Functional Differential Equations and Approximation of Fixed Points (Eds. H. O. Peitgen and H. O. Walther), Springer-Verlag, New York, (1979).

322

REFERENCES PERA, M. P. 1. Sulla risolubilite di equazioni non lineari in spazi di Banach ordinati. Bool. Un. Mat. Ital., 17-B (1980), 1063-1075. 2. A topological methodfor solving nonlinear equations in Banach spaces and some related results on the structure of the solution sets, Rend. Sem. Mat. Univers. Politecn., Torino, 41 Nr. 3 (1983), 9-30. 3. Unbounded components of solutions of nonlinear equations at resonance, with applications to elliptic boundary value problems, Boll. Un. Mat. Ital. 2-B (1983), 469-481. PRUFER, M. and H. W. SIEGBERG 1. On computation aspects of degree in R". In: Functional Differential Equations and Approximation of Fixed Points (Eds.: H. O. Peitgen and H. O. Walther), Springer-Verlag, New, (1979). 2. Complementarity pivoting and the Hopf degree theorem, J. Math. Anal. Appl. 84 (1981), 133 149. RABINOWITZ, P. H. 1. Some global results for nonlinear eigenvalue problems. J. Func. Anal. 7 (1971), 487-513. ROTHE, E. H. 1. Introduction to Various Aspects of Degree Theory in Banach Space. AMS, Mathematical Surveys and Monographs Nr. 23, Providence Rhode Island (1986). SCHECHTER, M. 1. Basic theory ofFredholm operators. Ann. Scuola Norm. Sup. Pisa, 21 (1967), 361-380. SIEGBERG, H. W. 1. Abbildungsgrade in Analysis and Topology. University of Bonn, Diplomarbeit(1977).

323

ZERO-EPI MAPPINGS STENGER, F. 1. Computing the topological degree of a mapping in S". Numerische Mathematik, 25 (1975), 23-28. STYNES, M. J. 1. An Algorithm for the Numerical Calculation of the Degree of a Mapping. Oregon State University, Ph.D. Thesis (1977). TARAFDAR, E. U. and H. B. THOMPSON 1. On the solvability of nonlinear, noncompact operator equations. J. Austral. Math. Soc. (Serie A) 43 (1987), 103-126. VAINBERG, M. M. 1. Variational Methods for the Study of Nonlinear Operators. HoldenDay, San Francisco, California (1964). WHITE, R. E. 1. Value set at x e Q for an arbitrary distribution with applications to local extrema off<= C(Q) and a maximum principle for ordinary differential equations. SIAM J. Math. Anal. 11 (1980), 61-72. WHYBURN G. T. 1. Topological Analysis. Princeton Univ. Press (1958).

324

CHAPTER

4

Variational Principles 1. Introduction The study of variational principles forms a very important part of mathematics. Euclid was the first to ask the fundamental question : find the shortest line which may be drawn from a point to a line. Then, Apollonius of Perga, in his fifth book on conies, posed the problem concerning the determination of the shortest line which may be drawn from a point to a given conic section. Thus, it may be seen that a sort of theory of maxima and minima was known long before the discovery of differential calculus. Furthermore, one could say that efforts to develop this theory influenced the discovery of calculus. However, the first general methods of investigation on solutions of extremal problems were created about 300 years ago, at the time of the formation of mathematical analysis. Fermat, for example, after making numerous restorations of two books of Apollonius, often cites this old geometer in his "method for determining maximum and minimum" (1638). The later is, in some instances so closely related to the calculus that Lagrange, Laplace, Fourier, and others wished to consider Fermat as the discoverer of calculus. This is some thing he probably would have done had he started from a somewhat more general point of view, as was in fact done by Newton (Opuscula Newtoni, I, 86-88) . Descartes had already remarked, in a letter written on 1st of March, 1638, that Fermat's rule for finding maxima and minima was imperfect, and we know that many imperfections still existed for a long time after the invention of the calculus by Newton. Weierstrass, in his lectures at the University of Berlin, explained that, to a considerable degree, these inaccuracies are due to one of the greatest mathematicians, Lagrange, and that they have been diffused in the French school by Bertrand, Serret, and others. It was L. Euler who started a new field in mathematics, nowadays known as Variational 325

VARIATIONAL PRINCIPLES Analysis. L. Euler deduced the first general rule, in 1744. This rule is now known as Euler's differential equation for the characterization of the maximizing or minimizing arcs. Much of the terminology used in Variational Analysis was introduced shortly thereafter by J. L. Lagrange. One of the earliest papers along this line was Hilbert's famous address, in which he posed 23 problems. Among these, the 19th problem was to prove that the regular solutions of any analytic regular variational problem are analytic. As well, the 23rd problem was, vaguely to develop a theory of Variational Analysis. The first question which led to modern theory of Variational Analysis was Dirichlet's problem, that is, the problem of determining a function 0(x,y) whose values are given on the boundary of a Jordan region A and for which the integral '' d<5> d<S>

D[0]=^|VO|Wv,VcD^£,^] V dx ' dy) assumes its minimum value. It was one of Riemann's tasks to solve the Dirichlet's problem, although C.F.Gauss and W. T. Thompson had started to understand the above problem. The solution of Dirichlet's problem was considered as an obvious question at that time because it was considered the positive character of D[Q>] as an obvious argument for the existence of a solution. B. Riemann, while a student attending Dirichlet's lectures, had been very much fascinated by the above argument and, used it without proof under the name "Dirichlet's principle". However, after some years .Weierstrass criticized Dirichlet's principle and the question of determining a harmonic function that assumed assigned values on the boundary of a Jordan region A became a very difficult research problem. Weiersstrass published his objection in 1869. Afterwards, H. Poincare and H. Schwartz, as well as others, tried to clarify Weierstrass' criticism. Fifty years after B. Riemann, D. Hilbert, in a very famous publication at the International Congress of Mathematicians in 1900 answered some of Riemann's theorems by proving directly that the corresponding problem had a solution. More open questions came up at this place by trying to understand the non-regular integrals. These include some 326

INTRODUCTION of the most important and interesting integrals such as the Jacobi least action integral in the three body problem of celestial mechanics. C. Caratheodory has devised a quite new approach to a considerable portion of the theory of Variational Analysis. In his theory, the extremals appear as so-called curves of quickest descent. Leibniz himself remarked the deepness of variational principles when he said that "our world is the best possible since its laws can be described by extremal principles " (Rassias, Th. M. [1], [2], [3]). In our time, a new variational principle was discovered in 1972 by I. Ekeland in [1], [2]. This principle has significant applications in optimization, optimal control theory, game theory, and in the study of dynamical systems (Ekeland, I. [3]), (Ekeland, I. [Notes by S. Terracini] [1]). It is may be of interest to mention that Ekeland's principle, together with its various equivalent forms, has found important applications amongst others in several subjects of nonlinear analysis such as fixed point theory, the geometry of Banach spaces, the study of the normal solvability of nonlinear operators to mention just a few. In many practical problems, we need to minimize a lower semicontinuous function / on a complete metric space. If E is a Hausdorff topological space, we say that a function f:E—> /?(J{+°°} is lower semicontinuous if, for every AsR, the set \x € £|/(x) > X ] is open. If the space E is compact, there will always be a minimizer, that is, some point x, such that /(**) . However, there will always be a sequence \x„} in E such that lim f(xn) = inf£ / , and hence, when inf£ / is finite,

M

neN

we have that, for every 8 > 0, there is some xe such that: 327

VARIATIONAL PRINCIPLES f(x£)< f(x) + s, for all xeE.

(1.1)

An implication of the variational principle discovered by I. Ekeland is that (1.1) can be satisfied even when E is not compact, provided that some assumptions are satisfied. Hence, the principal aim of this chapter is to present all different forms of Ekeland's variational principle and their main generalizations as well as applications in nonlinear analysis. It is well known that this subject, by its implications, is now an important domain of nonlinear analysis. In this sense, it appears sufficient to cite the necessity to study the drop theorem, the petal theorem, and other similar results. Our point of view is to present Ekeland's principle as a particular case of the problem concerning the search of finding a stationary point for a generalized dynamical system defined on a complete metric space. In this way, we hope to stimulate the researches about the study of stationary points of generalized dynamical systems.

2. Preliminaries We denote by R the field of real numbers and by R the extended real axis, i.e. R = /fUj-^+oo} ■ The empty set will be denoted by . N is the set of natural numbers. I. Topological spaces A topological space is a pair (£,r), consisting of a set E and a topology x defined on E. A topology T is a system of subsets U ofE, called open sets (relative to t), satisfying the following properties: 1) the set E itself and the empty set belong to T, 2) arbitrary unions and finite intersections of open sets belong to x. In the sequel we denote a topological space by £(r). The metric space is a topological space of a special kind. A metric space (E,d) is a set E endowed with a distance d, that is a single-valued real function defined on E x E and satisfying the following properties:

328

PRELIMINARIES >00 for all x,ysE, 1) d(x,y) d(x,y)>0 > 2) d{x,y) d(x,y)-0= 00 if and only if x = y, 3) d(x,y) = d[y,x) for all JC,^ eE , 4) d(x,y) < d(x,z) + d{z,y) for all x,y,z e E. In this case, a set U c E is open if and only if, for every x0 e U, there exists r>0 such that B(x0,r) = ix eE\d(x0,x)< r\ c U. Let (E,d) be a metric space. A sequence [xn] in E is convergent if there exists some element a e E so that, for every s > 0, there exists nE e N such that, for all n>He,*/(*„,a) < £ . The point a is uniquely defined and is called the limit of the sequence \xn) . A sequence \xn} in E is said to be a Cauchy sequence if it satisfies the Cauchy criterion, that is, for every e > 0 there exists n£ e N such that d[xn,xm)<e for all n,m>nE. Any convergent sequence is Cauchy but the converse is not always true. The converse is true if the sequence of the space satisfies some special assumptions. We say that the metric space (E,d) is complete if every Cauchy sequence in E converges to an element in E. For complete metric spaces we have the following classical result: We recall that if D is a non-empty subset of E, the diameter of D is defined by diam[D) = swpid(x,y)\x,y <ED) . The following is a fundamental result.

w

Theorem 2.1 [Cantor] Let (E,d) be a complete metric space and \Fn]'n<=N a sequence of closed subsets ofE. If the following assumptions are satisfied: l)Fn+l
00

then there exists x0 G Esuch that f] Fn = {x0} . n=\

We now recall the fundamental notion of compactness.

329

VARIATIONAL PRINCIPLES Let (E,d) be a metric space and let ft be a collection of open sets in E. We say that the collection "U is an open cover of a subset D of E (D may be equal to E) if D is contained in the union of the sets of ft. . The set D is said to be compact in £ if every open cover of D has a finite subcover, that is, if n

there exists a finite number neN such that Z) c U t/,, £/, e # • 1=1

w

It can be shown that a set D c £ is compact if and only if every sequence \xn }nsN in D has a subsequence which converges to an element in D. A topological vector space is a real vector space E endowed with a topology T for which the vectorial operations: (*,>>)-» x + v and (a,x)—» ax are continuous for every (jc,v)eisx.E and (a,x) eRx E. Let us consider real vector spaces. Topological vector spaces of particular importance are the normed vector spaces. A normed vector space is a vector space E equipped with a norm, that is a function ||-||fromE into R which satisfies the following properties:

1.

HI\\x\\>OforallxeE, p>_0

2. |In JC |I== 00 / / and only if x = 0, 3. I ax I = \a\ ■ \\x\\ for all a eR and x e E , 4. I x + y I < |x| + \\y\\ for all x,y eE. Every normed space is a metric space, since the mapping d:Ex E -> R+ defined by d(x, v) = ||Jt- y \ is a distance on E. The converse is not always true. We denote a normed space by [E,\ ||). Let E(x) be a topological vector space. The set E*=£ (E,R) of continuous linear functionals defined on E with values in R is called the dual ofE. If (£,|| I) is a normed space, this is equivalent to saying that a linear functional

330

PRELIMINARIES / is continuous, if and only if there exists a constant M > 0, such that \f(x)\ < M\\x\\ for all x in E. We define, in this case, the norm of / e E', denoted by ||/||,, in the following way: |/||, = inf {M > 0| |/(x)| < M\\x\\, for all x e £:} #}

\A4.. sup = SUD sup >-fj+= sup /f ~

X

xeE,x#(. ie£,j*0

Ml

\\X\\

iJ : =

vMlJ

xeE,x*Q 0 " \\\X\\J x<=E,x*Q

/o(4

sup |/(x)|. sup

xeE,\x\=i xe£,|:c|=l1

A topological vector space E is said to be locally convex if there is a neighborhood base at 0 consisting of convex sets. A seminorm on E is a map p from £ into R satisfying the following properties: ^ ) > 0 for forallxeE, all x EE, 1) p(x)>0 p{Ax) = = |/l|.p(x) \A\p{x) for all X eR and x Ge E, 2)i /j(ybc) p{x + + y) y)
M

II. Convex sets Let E(x) be a topological vector space. A subset D of E is said to be convex if Ax + (l- X)y is in D whenever x,y e D and X e [0,1]. This implies that a convex set contains the closed line segment joining any two of its points. Half-spaces are examples of convex sets. For any x* € E*,x* * 0 (E* the dual of E), and any a e R , the sets:

331

VARIATIONAL PRINCIPLES H„ = \xeE\(x',x) a} ;are called closed half-spaces. The sets H* /£ = = jx {xe£|(x\x)aj eE\(x*txj > a) are called open half-spaces. These sets would determine the same region in the space E if x* and a were replaced by Ax and Xa, (for any X > 0) respectively. Therefore, these half-spaces depend only on the hyperplane a}. Ha = j{xe£|(x*,x) x e £ | ( / , i ) = ffj. A closed hyperplane H in E is called a supporting hyperplane of a set D a E at a point x0 in the boundary of D if x0 is in H and £) lies in one of the two closed half-spaces determined by H. This ensures the existence of a nontrivial continuous linear functional x* e £*such that sup(x*,x) = \x',x0). We say that xQ is a supporting point of D. It is easy to verify that the class of convex sets of £ is closed under the operation of intersection. The intersection of all the convex sets containing a set M c E is called the convex hull of M and is denoted by conv (M). It is a convex set and also the smallest one containing M. If D c E is convex and its interior is non-empty, we say that D is a convex body. If D e E is a convex body, then every boundary point of D is contained in at least one supporting hyperplane of A and also D is the intersection of the closed half-spaces which contain D and are determined by the supporting hyperplanes of D. III. Convex functions Let E(i) be a topological vector space and let/be a function from E into R . We say that / is a convex Junction, if, for all xux2 e E and X e [0,l], one has: f(Xxx]+{l-X)x /(A r r ( l - A ) x2)<Xf(x + ({l-X)f{x ! - A ) / ( x 22 ), ), [), ) + 2)<;i/(*

332

(2.1) (2.1)

PRELIMINARIES whenever the right-hand side is well defined. The effective domain of/ is the set defined by domf = {xe E\f{x) < +00}. The function/ is said to be proper if f(x) >-oo, for all x e E and f{x) <+oo for at least one x e E. The epigraph o f / i s the set epif = \{x,t) e E x R\f(x) < t\. We can prove that/is a convex function if and only if its epigraph is a convex set. We say that/is concave if the function - / i s convex. The function/is called strictly convex, if inequality (2.1) is strict for all X e]0,l[and xl,x2 edomf with x{ * x2. One can establish a correspondence between convex functions and convex sets by associating to each convex set D a E its indicator Junction SD which is defined by

JO

ifxeD, lfxeD

[+00

if xifxtD. £D. ifxtD. ifxtD.

M*)={° l+oo ' l+oo [+00 DK DK DK

'

'

We can show that the set D c E is convex if and only if the function 8D is convex. The support function of a convex set D is the real function 5 D defined on£* by 8'D\x*\ = sup|(x*,x)|x eD>. Theorem 2.2 Let E(v) be a topological vector space and let f:E —> J?U{+°°} be a proper convex function on E. Then, the function f is continuous on the interior of the dom f if and only iff is bounded from above on a neighborhood of an interior point of the domf. TV. Lower IV. Lower semicontinuous semicontinuous functions functions Let E(x) be a topological vector space and let / be a function from E into R. We say that / is lower semicontinuous at a point xQ eE if and only if, given any e > 0, there exists r or all all y «eU. a neighborhood U of x0 such that f(x00)-e
liminf/(x)= *->*0

sup inf/(x), U^U(x0)xeU

333

VARIATIONAL PRINCIPLES where &(x0) is a base of the neighborhoods of x0 in E(x). A function/from E into R is said to be upper semicontinuous at a point x0 if the function - / is lower semicontinuous at x0. Theorem 2.3 Let E(r) be a topological vector space and let f be a function from E into R . The following properties are equivalent: 1) fis lower semicontinuous on E, 2) for every X eR, the level sets \x e E\f(x) < X\ are closed, 3) for every A eR, the sets [x e E\f{x) > A] are open, 4) the epigraph off is a closed set in E x R. The next theorem provides an essential property of lower semicontinuous functions.. Theorem 2.4 Iff is a lower semicontinuous function on a compact subset D of a topological space E(r), then there is an x0 e E such that f[x0) = inf f{x) and hence infD / = f(x0) > -oo. xeD

If f is an arbitrary real function defined on E(x), there exists a greatest lower semicontinuous function (not necessarily finite) majored b y / , namely the function whose epigraph is the closure in E x R of the epigraph off. Such a function is called the lower semicontinuous hull off. Iff is a convex function, then the closure off which is denoted by elf is by definition the lower semicontinuous hull off, that is:

(c//X*) =

fli, Clim inf / ( v ) , if f(x) > -oo for every x e E, I[-oo -oo

ifif ff takes takesvalue value-- oo. oo.

The closure off is a convex function. The function/is said to be closed if c / / = / a n d hence, for a proper convex function, closeness and lower semicontinuity are equivalent. As a consequence of this fact, we have that

334

PRELIMINARIES epi(cl f) = epi f V. Set-valued mappings A set-valued mapping (or point-to-set mapping) T from a set E into a set F is a mapping which associates a subset of F to each point of E. We say that a set-valued mapping T from E into F is closed at a point x0 e E if, for each pair of sequences \xn) N^Ec£ and

f*«L*

Wi«y WfieJV

cCFi ?

sS uU cChh t h a t X

X y

*" " ~* ""*■ ' °'O ' »^ "

£e rF X

(W" )

ff o r aa11 U

°

""

£e

^^

aainl dd y

*» " ~* ~**<>' ^0'

w ee W

have that jy00 e r(x 0 ). VI. Weak topologies in a topological vector space Let E(x) be a topological vector space and E be its topological dual. We denote by <■,•> a duality between E and E .The topology on E whose neighborhood base X of zero is the family of sets1 UU '. ..X,e . =-\x{X eE eE \\(X'>(x*,x) )\ <<£e,i '1 ==1\,...,n\ ' — M} for Xu some e > 0 and some finite set of continuous linear functionals **,..., x* € E* is called the weak topology. When E is considered endowed with the weak topology, we denote it by E(w). Every neighborhood of zero of the form U . . i s open with respect to the topology r. Every set, open with respect to the weak topology is open with respect to the topology i, but the converse may not be true. The weak topology is weaker than the original one. Since every weakly closed set in E is closed in the norm topology, it follows that every weakly continuous function on E is continuous with respect to the topology r. The weak topology on E is the weakest topology on E with respect to which E* is the dual space of E, that is the weakest topology for which every continuous linear functional with respect to the strong topology is also continuous with respect to the weak topology. Therefore, a linear functional on a normed linear space is continuous if and only if it is weakly continuous. In particular, we have that every closed hyperplane and every closed half-space in E is weakly closed. Since a closed convex set DcE is the intersection of all closed halfspaces containing D and the weak topology on Ex R is the same as the product topology on Ex R determined by the weak topology on E and the 335

VARIATIONAL PRINCIPLES usual topology on if, we obtain respectively that any closed convex set D is weakly closed and any convex lower semicontinuous function is weakly lower semicontinuous. The weak topology determines an important kind of convergence in E, called weak convergence in contrast to strong convergence which is determined by the topology r(or by the norm topology if £ is a normed space). Theorem 2.5 A sequence KLtf {x„}neN tn in Ca topological vector space E is

weakly convergent to x0e E if and only if the numerical sequence Ux* ,xn)\ |(x*,x y|

neN

converges to to \x*,x \x*,x00)) for for every every x* x* eeE*'. E*'. converges

Proposition 2.6 Let E be a topological vector space. The following properties are true: 1) if a sequence {x „ of elements in E converges strongly to x0eE, n} \X»)neti then it also converges weakly to xo, 2) the bounded and weakly bounded sets are the same in E, 3) a weakly convergent sequence is bounded. The dual space E of E endowed with the strong topology can be regarded as an original space with its own dual space. In this case, we can introduce in E a weak topology by defining a neighborhood base of zero, using the values of functionals on the space E . Thus, we consider the neighborhood base of zero, the family of sets of the form:

U.. U»

={x* eE*\ (x*\**) «..g=\x*eE*\ (x**,x*) <e,i <£-,/ = l,...,m ,n\ for some 8 >0, > 0, = !,... l,...,n\ **

**

jp • *

and some finite set of continuous linear functionals x",...,x" e E". The topology generated by this base is called the weak topology on E*. The weak topology is the weakest topology on E* such that all functionals in E" are continuous. It is easy to show that the space E can be identified with a subspace of its bidual E . In general, E * is larger than E.

336

PRELIMINARIES We say that E is semireflexive if E = is*and if the evaluation map is an isomorphism of £ onto the strong bidual \E*A , we say that E is reflexive. For example, a Hilbert space is a reflexive space. We can also define another topology on E by using the family of sets of the form: UX xx ..,x X,e es U

v

n

xv..,xn,e

= <x*e£* = lx* eE* (x^xj

<e,i <s,i = !,...,«> \,...,n\ for some e > 0 and some finite set of

continuous linear functionals xx,...,xn e E, as the neighborhood base of zero. This topology is called the weak star topology and is denoted by w . The weak star topology is weaker than the weak topology on E*. The two topologies will be the same if and only if E is reflexive. Therefore, one has three different topologies on E and thus three separate notions of convergence. Furthermore, it is easy to show that strong convergence implies weak convergence and weak convergence implies weak star convergence. The definition of strong topology is the following: Let (E,F)be a duality of vector spaces. The strong topology on E with respect to the duality (E,F) is the topology of uniform convergence on the sets of the family of all weakly bounded subsets of F. Remark 2.1 In general, w -convergence does not imply w-convergence in E. Since the weak topology has fewer open sets that the initial topology T (or the strong topology), it follows that it has more compact sets, which is an important fact for existence problems. Regarding the weak topology, one has the following fundamental theorem. Theorem 2.7 [Alaoglu-Bourbaki] The closed unit ball of the dual of a normed vector space E is w -compact. This theorem has an important indirect role in optimization problems as well as in analysis. Corollary 2.8 Z,ef(.E,|| ||) be a normed vector space. E is reflexive if and

337

VARIATIONAL PRINCIPLES only if the closed unit ball B = [x e E\ ||x|| < l} is weakly compact. Corollary 2.9 If If E is reflexive, then every bounded sequence in E has a weakly convergent subsequence. VII. Conjugate functions Let £ be a topological vector space . We suppose that E' * {0}; for example, we can suppose that E is a locally convex space. For any extended real-valued function f:E -» R, the conjugate function of / i s the function / * : £ - » / ? defined by = SUp{(x*,Jc)-/(x)|^ sup{(x',x)-f(x)\x €€ E) E] £■}for for all all x* x* G sE'. E' /V) = f'(x')

It thus follows that the conjugate function of the indicator function SD of a convex set Da E is exactly the support function of D. Indeed, S'D(x*) = supjuc*,x)-5 D (x)\x eE) = supju[:*,x)| supjuc*,xM JC eD>for eD|for all x' e E' E'. The conjugate of / ' , that is the function / " o n E defined by: /**(**) =sup|(x,je*)-/*(jc*jx* =sup|(x,x*)-/*(je*jpc* sE > forallxeis, forallxe£, is called the biconjugate of/ Theorem 2.10 Let f be any extended real valued function on the topological vector space E(r). Then the following properties are true: 1) the functions f andf are always convex and lower semicontinuous in the weak star topology ofE and in the weak topology ofE, respectively, 2) f=f if and only iff is convex and lower semicontinuous on E. Corollary 2.11 A proper function is convex and lower semicontinuous on a Corollary 2.11 A proper function is convex and lower semicontinuous on a 338 338

PRELIMINARIES topological vector space E(T) if and only if it is the supremum of a family of affine continuous functions. VIII. Subdifferentiability Subdifferentiability L e t / b e a convex function from a topological VHL vector space E to (-oo,+oo]. The subdifferential o f / i s (in general) the multivalued mapping df\E -» E defined at every point x € E by:

df{x) = (x eE*\f{y)-f{x) eE*\f{y) -f(x)>(x*,y-xjforallyeE\ df{x) df{x) = [x eE*\f{y)-f{x) eE'\f(y)-f(x) > (x\y-x) (x\y-x) for (x*,y-x) for ally ally EE}. eE]. eE}. We say that x is a subgradient o f / a t x if x e df{x), and t h a t / i s subdifferentiable at x if df{x) ^ . I f / i s subdifferentiable, we have df adorn f. We can show that dfix) is always a w -closed convex subset of E*. Remark 2.2 If / is a proper convex function on E, then the (global) minimum of/over E is attained at a point x in E if and only if 0 e df{x). Proposition 2.12 Let E be a topological vector space and

/:£->i?U{+°o} f:E -» J?U{+°°} ,be a proper convex function. Then x' e dfix) if and only if /f{x) (*) + + //*(**) * ( * * ) ==\x*,x). (**»*)■

If in addition, f is lower semicontinuous then

x'edflx) if and only>ifx€.df*[x'}. if x edf'ix*). Proposition 2.13 2.13.. Let Ebea topological vector space and f:E —> /?U{+°°} be a proper convex function. Iff is continuous and finite at a point XQ in E, then f is subdifferentiable at this point and df(x) ^ for allx in the interior ofdomf. Theorem 2.14 Let Ebea topological vector space, fandg be some convex functions on E and X a (strictly) positive real number. Assume that: 1) d(Xf\x\ I) d{Xf){x) = Xdf{x) Xdf{x)forallxEE, for allx e E, 2) d{f + g)(x) c df{x) 3f{x) + + dg(x) dg{x) for for all all xx ee E, E,

339

VARIATIONAL PRINCIPLES 3) fis continuous at some point ' x0 edomff\domg edomfr\domg T\domg . Then, for all x edomff)domg = dom[f + g),we have the formula +

*(/ *X*) = */(*)+ **(*)•

IX. Gateaux differentiability differentiability L e t / be a real valued function defined on a topological vector space. The Gateaux differential generalizes the concept of directional derivative, which is well known in finite dimensional spaces. We say that fis Gateaux differentiable at a point point x, x, wmcn which belongs oeiong to some f zz -» - > /Jf'(x,z) ' (vx , z ) :== ,.f(x±*tM open set in the domain off,/ i if lim— lim— '

tt

— *-*-■1is a linear

continuous functional with respect to z on E. If/is Gateaux differentiable at x, we denote by f'(x) or Vf{x) the element of £'defined by (V/(x),z) = f'{x,z)

for all z in E.

The definition of Gateaux differential does not require a norm to be defined on E. Thus, its properties are not easily related to continuity. When £ is a normed vector space, a more satisfactory definition is given by the Frechet diffrential. We say that / is Frechet differentiable at a point x if there exists a continuous linear function, denoted by f'(x), such that

S5g[/(x+*)->,W-(^(*W]-0If/is Frechet differentiable at JC, then the Gateaux differential exists at x and the differentials are equal. Any one of these definitions gives a single possible value for the differential, hence denoted by f'(x) or Vf(x) and called the gradient or the derivative of/at x Suppose that E is a normed vector space. We say that/is continuously Gateaux differentiable at x if, on a neighborhood of x, the Gateaux derivative exists and is continuous as a mapping from E to E* with 340

PRELIMINARIES respect to the norm topology. A function/which is Gateaux continuously differentiable atx is also Frechet differentiable at*. Proposition 2.15 If f:E^> tf R\J{+^} U{+°°} is convex and Gateaux differentiable at x in some open subset of E, then it is subdifferentiable at x and df(x) = {V/"(x)} ■ Conversely, iff is continuous, finite, and has only one subgradient at x, then f is Gateaux differentiable at x and

*/(*Hv/(x)}.

X. The generalized gradient We recall in this section the concept of generalized gradient as defined in (Clarke, F. [2]). Let [E,\\ ||J be a Banach space and let D be a subset of E. A function f:D->-+RR is said to satisfy a Lipschitz condition near x (of rank kx) if there exists a neighborhood of x, denoted for example by Ur such that: | / U ) ~ /(*a)| * *xlh - xXi\\ A forallxx,x2

e U x.

Let f:E->R ->/? 1be Lipschitz near a given point x, and let v be any other vector in E. The generalized directional derivative of/at x in the direction v, denoted by f°(x;v), is defined as follows: ,„„/(v + / v ) - / ( v ) / ^ v H= ilimsup— m s u p 7 ^ - ^—, ,

/°M=

y-*x do do iio

f

where v is a vector in £ and t is a positive scalar. —>RRi be Lipschitz of rank k near x e X. Then Proposition 2.16 Let f:X -^ the following properties hold: 1) the function v -> f°(x;v) is finite, positively homogeneous, and subadditive on E and satisfies |/°(x; v)| ^ k\\v\\,

341

VARIATIONAL PRINCIPLES 2) f°{x;v) is upper semicontinuous as a function of(x,v) and as a function ofv is Lipschitz of rank k on E,

= (~/)V,v). 3) /°U-v) /°U-v)=(-/)>;v). The generalized gradient o f / a t x, denoted by dfix), is the subset of E* defined as follows: t df(x) (x\v) for all v inE) inE\. mE\. df[x) = = \x eE'\f°(x;v)>(x\v) eE {x;v)>(x ,v) forallv jorallv inh\.

We denote by |pr*|„ the norm in E" which is defined as follows: be* I =supjuc pc*l = =supl(x supjuc ,v)|v ,v)|v G£,||V| << 1>. 1>. > | VG£,||V| G£,| Theorem 2.17 Letf be Lipschitz of rank k near x e E. Then the following statements are true: 1) df{x) df{x\ is a non-empty convex w'-compact subset of E* and \\x* forevery every x x
edf(x)\. e
Theorem 2.18 Let f:E -» ->J? R be Lipschitz near x and admits a Gateaux (orHadamard, of strict, or Frechet) derivative f'(x), then f'(x) edf{x). f'(x),thenf'{x)ec>f{x) Proposition 2.19 1) If f:E->R X 6 E and x is a local -+R is Lipschitzean near a point xeE minimum, then.00 e€
342

CRITICAL POINTS FOR DYNAMICAL SYSTEMS Theorem 2.20 [Lebourg's Mean-Value Theorem] Let x andy be points in a Banach space E and suppose that f: E —> R is Lipschitz on an open set f-E^-R containing the line segment [x,y]. Then, there exists a point z e ]x,y[ such that f(y)-f(x)e(df{z),y-x). f(y)-f(x)e(df{z),y-x).

3. Critical points for dynamical systems Let E(x) be a topological space. We say that a set-valued mapping T:E->E is a dynamical system (in the generalized sense) if, for every x e E,T{x) is a non-empty set. Definition 3.1 A point x* e E is called a stationary point for the dynamical systemiTifr{x Tifr[xtt)) = = {x \x*}. t\. We recall that x0 e E is a fixed point for rifx r if x00er(x e T[x00). ). A stationary point is a fixed point, but the converse is not always true. To justify Definition 3.1, we consider the concept of a dynamical system in the classical sense (cf. Isac, G. [3]). Let X{r) be a topological space and i7r:XxR^ :IxJf-> X X a mapping. We say that (X(T),K) is a dynamical system if the following conditions are satisfied: Sj) Si) n(x,6) = x for all xxee X, 52) 7i\7i:(x,t^,t^\ S2) 7c\7i;(x,t\),t2\==n{x,t n{x,tx x +t +t2)2) for forall allxxeeXXand andtlt,1 eR, x, 212eR, 53) ;ris a continuous mapping. S3) If {X{T),TI) is a dynamical system in the classical sense and, for every xe X, we denote r(x) T(x) = - ift(x,t)\ (the trajectory of x mX), then the stationary U(x,t)\, lleR point given in Definition 3.1 is exactly the stationary point used in the classical theory of dynamical systems (cf. Isac, G. [3]).

343

VARIATIONAL PRINCIPLES The concept of dynamical system defined here in a generalized sense is used in game theory, in mathematical modeling, in economics, and in Pareto optimization amongst other fields. We begin this section with a few examples. Example 3.1 Let (E,d) be a metric space and / a function from E )R =/?U{-oo,+oo} into/? = •RU{-°°,+00} .• We say that the petal from x to f is the set rPJPJ(x) d{y,x)
+

d{y,x)
If we consider E = - R and d(x,y) = \y — x\, we have If we consider E - R and d(x,y) = \y - x\, we have

In this examples we have TPf{x) = A. The set-valued mapping x -» TPj{x) is the petal dynamical system denned by/on E. Example 3.2 Let (E,d) be a metric space and X c E be a non-empty complete subset. The flower-petal Pr\a,b) associated with y e]0,+oo [ and points a,b e E is the set Pr(a,b) = [x e £ \yd(a,x) + d(x,b)
344

CRITICAL POINTS FOR DYNAMICAL SYSTEMS

This form has been computed in (Penot, J. P. [2]). We remark that the flower-petal Pr{a,b) is exactly TPj(a), where d is replaced by yd and /(•) = d(;b). We now define the set-valued mapping TP{r,br.x) = Pr{x,b)r\X, for every x e X, where b is a fixed element in E\X. The set-valued mapping TP{y,b:x) is a dynamical system since, for everyxeX, xeTPP\y,b:x). \y,b:x). ixeX, xeT Example 3.3 Let ((ii,| £ , | |) 1be a normed vector space. The drop associated with a point a e £ and a convex subset B of E forms the convex hull of {a}{jB i.e. D(a,B) = {a + t(b-a)\b eB,t <=[0,1]\. eft,!]}. Let X be a complete subset of E and let B be the closed ball with center bgX and radius r < d(b,X). The set-valued mapping

r^)M=^M)nx, is a dynamical system defined for every x e X. Example 3.4 Let (E,K) (£,.£) be an ordered topological vector space, where K is a closed convex cone. Let X
345

VARIATIONAL PRINCIPLES dynamical system rK{x) is important in the study of conical support points as well as in Pareto optimization (cf. Isac, G. [3], [4],[5] ). Example 3.5 Let E(r) be a locally convex space with the topology r defined by the family of seminorms \Pa}aeA> an<* let G c £ be a nonempty subset. For XQ e E, we define the set valued mapping rr*j J0 (') O =\(p ={(Paaa{x ={(p {x0Q0-g))_\geG -g)) (p (Paa(x0-g))aeA *t}, ,-g)) a£A\geGJ g e G .and (pa{x0-g)) eeA
o-*)L>

which is in turn important in the study of the best vectorial approximation in locally convex spaces (cf. Isac, G. and V. Postolica [1]), (Bacopoulos, A., G. Godini and I. Singer [1]). We will now study the existence of stationary points for a generated dynamical system. In this spirit, we will prove some general existence theorems. Theorem 3.1 [Dancs-Hegedus-Medvegyev] [Dancs-Hegedus-Medvegyev] Let (E,d) be a complete metric space and T:F -> 2E a dynamical system. Additionally, let the following assumptions be satisfied: 1) Y(x) Y{x) is a closed set for allx e E, 2) x E £ T(x) for allx £e E, 3) y e T(x) implies T(y) c T(x)for all x,y 6 E, 4) /) for for every sequence \x {x„} „ in E (defined (defined bv by simple iterations. iterations, i.e. m\ nn€N €^ Gl\xn)foralln=l,2,...), Xn+lZllxJ, ,2,...), we have that lim d(xnn, xn+,) = 0. n—►<» n—>oo

we have that lim d(xn, xn+,) = 0. 346 346

CRITICAL POINTS FOR DYNAMICAL SYSTEMS Then F has a stationary point. Moreover, for every x e E, there is a stationary point of r in r(ic). Proof. We can suppose that d is bounded on E, since otherwise we could d consider the equivalent distance d'= y—j and remark that, if assumption (4) \ +d where satisfied for d, then it would also be satisfied for d'. If D c E is a non empty subset, we denote its diameter by 8(D) . Let x0 be an arbitrary element in E . Using the properties of T, we can construct a sequence \x„\ „, of elements of E, such that \Xnl„<=N'

s(r(xA) s(r(xn,)) x^=x ,xn n er(x„_,) er(i„.,) and d(x d{xn,xn,x > V V ""'" x,=x x, =xQQ0,x,x„ )>^—^ x)>n_nx_ 2

1i

2 33 44 ^l-j; T1 ; nw== 22,3,4,.... >> >> >— >—

2""

From assumptions (3) and (4), we have r\x„_,) rt*„J 2 rr\x„) t * J and £(r(x„))-> £(r(xM)) -> 0 as ; «-»+<». Since all of the assumptions of Theorem 2.1 are satisfied both for th exists x» e E such that E and the family of setss {r\xn)}J n e W ,' there

M^L*'

PI r(x„) = {x*}. From condition (2), we deduce that x» is a fixed point for

n=l

T, and, from condition (3), we obtain that r(x,)cnr(x r\jc.)e fir(x„n)={x4, ) = {*.}> that is n=l

T\Xt) = \x,}. The proof is thus finished. If we take x, = x, in the proof we obtain the last conclusion of the theorem. ■ Remark 3.1 3.1 Theorem 3.1 is true if condition (2) is replaced by the following condition: (2): T as set-valued mapping, is closed at every point x e E. Let £(T) be a topological space and D c: E be a non-empty set. Consider a dynamical system T:D —> 2D. We say that T is reflexive on D if x e T(x), for each x e D, antisymmetric on D if xj = x2 whenever x2 e T(xi) and xx e T(x2), and transitive on D if x3 e T(xi) whenever x2 e T(xi) and x3 e r(x 2 ). If x e D, we denote by 347

VARIATIONAL PRINCIPLES T e (x) the complement of r(x) with respect to E. T is said to be closedvalued if T(x) c D is closed in E for every x e D. Definition 3.2 We say that D is F-semicompact if r is closed-valued and c every open cover ofD, of the form {T j r (x ( xa)\x xa &D,a e A) has a finite suba ) |aeD,aeA} cover. Every compact subset of E is T-semicompact but the converse is not always true. Semicompactness was defined in (Corley, H. W. [1]) and used to study the existence of Pareto optima. Corley proved the following theorem. Theorem 3.2 Let T:D->2D be a closed-valued, reflexive, antisymmetric, and transitive dynamical system. If D is F-semicompact, then T has a critical point. Proof We define a partial order on D by x, < x2 if and only if x2 e r(x,). Proof! The ordered space (A<) is inductively ordered. Indeed, suppose, on the contrary, that D is not inductively ordered. Then there exists a totally ordered set T = {xa \a s A} in D which is not upper bounded in D. Thus, fl r(xa) = 0; otherwise, any element of this intersection would be an aeA

upper bound of T in D. Hence for any x e D, there exists xa e T such that c x£r(xa). Hence {r | r c(x ( xa)\asA} a ) | a e ^ j forms an open cover of D. Since D is c T-semicompact, D has a finite subcover, denoted by |r {r(x,)|/ (x 1 .)|/ = = l,2,..,«|, l,2,..,«| where JCI < x-i <...< x„. But, from transitivity, we have r (\ x f ) c r ( x M ) , i = 2,..,n, which implies that Dc:rc{x„). Thus P ( J C „ ) C D C , which is a contradiction since xn e T\x„). Hence, D is inductively ordered and, from Zorn 's lemma, it has a maximal element x. with the property that x er(x») implies x = x., that is r(x») = {x.}. ■

348

CRITICAL POINTS FOR DYNAMICAL SYSTEMS If E and F are topological spaces, we say that f:E->F / : £ - > F is a proper l mapping if and only if , for any compact set DcF, (D) is D c F , the set f~ rl{o) compact in E. Proper mappings are systematically used in nonlinear analysis. G. Isac proved the following theorem. DczE Theorem 3.3 Let (E,d) be a complete metric space and D a E be a nonD 3tY:D^2 empty closed set. Also, let Y:D—> 2Di be a generalized dynamical system g:£>-»£>, and g:D-> Dbe a continuous mapping. Finally let the following assumptions be satisfied: 1) the graph of Tis closed, 2) T(X) It*) is boundedfor every x ee D, 3) g(x) e er(x) T(x) for every xxeD, e D, 4) g is a proper mapping, C D 5) there exist 0
Then, there exists x* eD such that T(x»)= {#(**)} • Proof. Let {^„}neA,be the sequence given in assumption (5). Using assump tions (3) and (5), we deduce that |g(x„+1) - g(x„)\ ^ k"diamT{x0) for every

W*»)L*

« = 0,1,2,.., and, by induction, we obtain that {g{x„)}n£N

ls a

l«eW

Cauchy

sequence in E. Since £ is a complete metric space, the sequenceS [g(x )j {g(nX»)} neJV nsN is convergent. Because g is a proper mapping, the sequence {*„}neA, has a convergent subsequence denoted again by ixn)

. If *♦ = limx„, then, H-MO

g(xtt)eY(x )eT(xt).t). from the continuity of g and assumption (1), g(x

Since

limc/r'<37«r(^ limdiamT(xn)-0, n) = 0, we obtain that T(x») = {#(*.)}, and the theorem is

n->oo

proved. ■ The next corollary, that has been obtained by H. W. Corley, is an extension

349

VARIATIONAL PRINCIPLES of Banach's contraction theorem for multivalued mappings. Corollary 3.4 Let (E,d) be a complete metric space and D czEa non-empty closed set. Let F:.D—»2fl be a reflexive dynamical system such that, for every xeD, I\x) is bounded. If there exists 0 < & < 1 , x0 eD and xn+l er(xn), for every n=0,l,2,..., for which dimT{xn+x)
such that x0 co

metric space, there exists by Theorem 2.1, x, eD such that p | r(x„) = {x,}. We therefore have r(jc.) = {x.}. Indeed, we suppose that nr(*„H*.} neN neN

there exists x e T(x*). We have x = x. + k, for some k e K, which implies x - x , eK. Since x t eT(x n ) for all n e N, we have x, - x „ eK for all

350

CRITICAL POINTS FOR DYNAMICAL SYSTEMS n e N and finally, x = x, + kx = xn + k2 + &,, kx, ifor some kuk2 e K. Hence, xxe\e P)r(jc /1 ), which implies x = xt. \l[x„),v n<=N neJV

In particular, we also havei xx,t sx0 +K,i.e.x0 0 such thatp < Aq. The base 5 is Hausdorff if Ker(5) = {x e E\p[x) = 0,for allpeB}

= {0}.

Let (E, Spec E) and (F, Spec F) be two locally convex spaces, Mbe a subset of E, a n d / : M-> F be a mapping. We say that/is closed if, for every net {x,-}. g/ cM such that {x^.^-txeM , -+xe M and |/(jc,). e / | ^ ^ G F , we have

f(x) = y. Theorem 3.6 Let (E, Spec E) be a locally convex space, \Pa}aeA be a base for the Spec E, and M M ccz E be a non-empty set. The dynamical system 351

VARIATIONAL PRINCIPLES r:E->2E has a stationary point if and only if there exist a locally convex space (F, Spec F), a base {<7/?L B f°r tne Spec F, a complete subset 'pen

MQQM, f:M00^F,->F, a function 9afi-f(Mo)^K (pap:f[MQ) -> R+ for every M M, a M. ca function f:M 0QM, couple (a,fi) e AxB, and two constants ca,cp > 0 such that: 1) T\x) rtx) c Y{x) <= M M00,for ,for all allxx ee M M00,,

2) fis closed and /(M 0 ) is complete, 3) q>

f =Ho> «o> ^ f o r e^v eerryy ^ioc,p)eAxA, \4fi)fieB \P'>JaeA>J and Mfi {P*}**A>f H*> (\<X,P)£AXA, >fi)e e>Ax. A, cCa=c= pC=l =\= l and eB eB = st\ f\\ Conversely, t^mn-xTawoalXT u / o suppose o n n n n c o that +l-»of- all o i l the +riO conditions ^ A n / i i f i / x n o of r*iYthe +l"i*» theorem +H a/vr'ai-M are oro <PaB — 0)we satisfied. Since E is supposed to be Hausdorff and M0 is complete, we deduce that M0 is closed. The set-valued mapping rrU\M: M :M ) 0- >-»MM o 0 isis aa S

=I

tOT

a

p

dynamical system on M0 and, obviously a critical point of l\ Mo is a critical point of r . We now define on M0 the binary relation xWy o pa(x (x --y),c y),cppqJf(x) qpp(f(x) (f(x) xfty & < => max{c max[caPa (x -y),c aaPa pq

-- /(v))} f(y))\

aS(f(x))-
for all [a,p) e A x B. We can show that the relation 9? is an ordering on and{q M0 (using the assumption that {{pPaa}aeA andfi}^\
fieB

denote this relation by " < ". Let {jc,}.e/ be a totally ordered subset of M0. We can suppose that (/,<) is also a totally ordered set such that x, <* Xj if and only if i <* j . If i <j, then Xj <* Xj, and the definition of the ordering <* implies that

352

CRITICAL POINTS FOR DYNAMICAL SYSTEMS C cc

xXXX X xx aPa{Xxil ~ Xj) <Paf}{f( i)) <Pa/>(f{ j)). aPa(*l j) <Pap{f( i)) <Pap(f{ j)). aPa(*l -X j) *<

j))aPa( i)) ~~-Vccfi 9*[f\ afi(f(

which gives that the net {^a/?(/(*,))j.

MA*,))},J(for a and |3 fixed) is a decreasing

net in R R++.. Hence, there exists5 rra/3apeU \<pap(/(x,))| i rap. eR++such SI that latl^i/UJ)}.^^^. Let e > 0 and {a,B) e AXB. There exists i0 e / such that, for all i>i0, we have r rr aB ap aff ^ ^9 ^ <Pa8 (/(*«))ap ^^rrap aB++ + mm{c mm[c mm(c ,c,c,Cg)-£. ■ e. aa a a9p(/(*«))^ ap (/(*«)) fip))-e. Then, for all j^i^j^-fZ- IQ, we have xXX X X X x X x C C*Pa( c*P CaPa( ~j,~X j)^ J j)^ ) )<

<mm(c ,cBpp)-e
Xj j)) ■<Pa/>(f{

) is a base for the Spec E (resp. the Spec F), we 'ULM^L)"

that {pa}a&A (resp.jg^} 'aeA

> SeB

obtain that {*,}1/6/ is a Cauchy net in 16/ is a Cauchy net in M0 and {f{xs)} J,W flMo). Since MQ and

f{M00)) are complete and closed, there exist f{M

x e M 0 a n d yy ee // (( M M 00 )) such that limx,=3c and lim/(x,) = y. ISj

f{x) = y since the graph of/is closed.

We have

fei

Because of the fact i > i0, we have <paJ3 (/(*,)) < ra/8+ min(ca ,cp) ■ s and (paP is lower semicontinuous. Hence, we deduce (p <Pap ra/?ap+ min(caa a,c ,cfi)fi)•••seefor forevery everyese>-»>0, 0, ap {fix)) (Pap \J{fix)) W) <s^rra/3 ++min(c min\c .c,,; ior every u,

353

VARIATIONAL PRINCIPLES which implies, (pap{f{x)) < rap. If ij € / satisfy i <j, then we have: xx x x X C i-i ^^9aB\f\ Pa([* Xj) < * ))-9aAf\ <Paf)( /( (/A)^aAf\^)-ra^ ( */ )/ ) )~- 9afi *PaP CaPa ~- Xj] (pafl * ( / t(f( *x/j))^<Pa0 ))5ft* aPA

C fix,) RQA (A*t) cp1p(/U) C/rfj

({A/ xM "^, t)Y)raft>

XX A) < (p„R\ f(x,)) r n. -A fix ~ A J)) j)) *~
(3.1) (3.1) (3.2) (3-2)

Computing the limits in (3.1) and (3.2) with respect toy and using the fact thatp a , qp are continuous, we obtain. r x I\ccaaaPa {Xi -*)& <papP ((A / (x*i)), ) ) ~-~ rV ap /(f( (X*t)) .i)) ) ) "~~ PafiA*), 9afiA*\x), \c Pa -~x)< A & <Pap afi *-* Vafi <Pafi((A <PafiA Pa U <Pa Vafi

(f{x,) -~ f(x)) (papap(/(*,)) - ~(pap9f{x). [cpqp (Ax,) fix)) <

354

VARIANTS OF EKELAND'S PRINCIPLE 4. Variants of Ekeland's variational principle We present in this section as an application of Theorem 3.1, the fundamental variational principle obtained in 1972 by Ekeland as well as several of its variants (Danes, J. [2]), (Ekeland, I. [3]), (Georgiev, P. [1]), (Penot, J. P. [2]). Throughout this section, (E,d) will denote a complete metric space. We say that a mapping / from E into /?U{+°°] is not improper if, it is not identically equal to +oo . Theorem 4.1. [Basic Ekeland's Principle] Let f:E -> R[J{+co} be a -» /?U{+°o} lower semicontinuous, bounded from below, and not improper, function. There is a point x* in the space E such that the inequality d\ x*, x) > /f[x.) d\x*,x) ( x . ) - /(x) f(x) is satisfiedfor allx e6 E\ {x*}.

(4.1)

Proof. Let TPJ{x) = {v €€E\f{y) E\f[y) + d{x,y) < < /(*)} f{x)} =={yzE\d{x,y) < f(x) f(x) -- f{y)} f{y)} + d(x,y) [y eE\d{x,y) < be the petal dynamical system defined in Section 3 of this chapter. Since/ is lower semicontinuous, rpj-(x) is closed for every x e E. Condition (2) of Theorem 3.1 is evidently satisfied. Let>> eYPj(x) and consider zeT z^^pj{y)j(y). PJP{y).

be an arbitrary element

We have d{x,y)
and

d{y,z)
which implies thatit d(jc,z)
355

VARIATIONAL PRINCIPLES We can apply Theorem 3.1 if we remark that condition (4) of this theorem is also satisfied. Indeed, taking the inequalities x x 4xxn-u n-uxn) n) ^ /(*»-i) --/ f( ^ /(*»-i) ( * «„))

ffor or

»» == 22,3.... '3--

and summing them up by parts, we obtain 00

Zd(x !Ld(x I. n_n_ux„)<+<x, l,xn)<+co,

n=2

using the boundedness of / from below. Therefore, we have that lim d(xn, xn_x) = 0, which means that condition (4) of Theorem 3.1 is n-»+oo

satisfied. Now by applying the cited theorem, we obtain that the dynamical system TP , has a stationary point x, e E. Evidently, for this element x„ inequality (4.1) is satisfied. ■ The basic Ekeland's variational principle can be expressed in the following form. Theorem 4.2 [Ekeland's Principle-Petal Form]. If f:E -> /?U{+«>} is lower semicontinuous, boundedfrom below, and not improper, then the petal dynamical system YPf defined on E has a stationary point. Theorem 4.3 Letf:E ->» /fUf+oo} Let f:E — /?U{+°o| be lower semicontinuous, bounded from below, and not improper. If a is an arbitrary point in E, then, there exists a oint xt in E such that the theft point following inequalities are satisfied: (i)) d(a.x.)< d{a,x.)
356

VARIANTS OF EKELAND'S PRINCIPLE and hence the metric space (D,d) is complete. By applying Theorem 4.1 for the space (D,d) and the mapping f\ we D, /In. obtain a point x* satisfying the inequalities: d(a,xt)
(4.2) (4.2)

and d(x., x) > f(xt) - f(x),for d(x,,x)>/(*♦)f(x), for all allx allxeD\{x x eD\{x,}. t).

(4.3) (4.3)

We will now show that inequality (4.3) is true for all x e E \ [xt }. Indeed, if x sE\D and (4.3) is not true, we have d(xtt,x)
(4.4) (4.4)

From (4.4) and (i), and using addition (that is using the triangle inequality for d), we get d(a,x) d{a,x) -» R\j{+<x>) R U{+oo} is lower semicontinuous, bounded from below, and not improper, then, for any a e E, the petal dynamical system TPj has a stationary point x, which is an element of the petal F TPJ P f (a). Theorem 4.5 [Altered Ekeland's Principle! Principle] Let Letf:E f:E -> -+R\J{+ao} J?U(+ool be lower semicontinuous, bounded from below, and not improper. Then, for every y>0 and any a e E, there exists xt e E such that the following inequalities nequaliti are satisfied: (i) f(x.)
eE\{xtt).). eE\{x

357 357

VARIATIONAL PRINCIPLES Proof. The proof is similar to the proof of Theorem 4.3 but considering the Proof.' set D=\xeE\f(x) Dyr={xeE\f{x) = lx eE\f(x) + +rd(a,x)Zf(aj\ yd(a,x)Y ■ • [Ekcland's e-Principle] Let f:E f:E->-> J?U{-H»} Theorem 4.6 [Ekeland's flU{+<»} be lower semicontinuous, bounded from below, and not improper. Then, for every e > 0 and a e E such that f(a)< f(a) < inf inf f(x)+ f(x) +£,e, there exists anx£e E such that: (i) r0/(x f{x£e)s)+^( f(x + aj ed(a,x ed{a,x ^) 0 0 /f{x ) -sd\x> { *xs) A )>f°rr f{(x*e)~ f\xx)>f°

allxeE\ {xEeE}}. allxeE\{x allxeE\{x \.

Proof We take y = e > 0 in Theorem 4.5, and an element a e E such that / ( a ) < inf/(*) + f{a)<mff{x) + £• e. M ■ ;ce£ xeE

Remark 4.1. Formula (i) splits into the following inequalities: f{x / ( *£)
(4.5)

d(x ,)<1. d{xee,a)<\.

(4.6)

nxe

From (4.5) and (4.6), we obtain that JCE improves a from the point of view of minimization of/and that it is located in a neighborhood of a. Remark 4.2 Formula (ii) states that the downward slope of/in xz is smaller than e, that is: (f{xeE)-f(x) )-f(x) ) )nh supmax ^—,0 <s sup max / * gT 7- / U <s. xTx <s. sup max\\ d\dUr) <s ***s i e) ^—,0H s -

j7U)-/u) 1

If JCE is a minimum of/, then its downward slope vanishes. In particular, if £ is a Banach space and / is differentiable, then, from (ii), we have

358

VARIANTS OF EKELAND'S PRINCIPLE < s,s,sh s being small, which implies that, the derivative of/at x is Y^ *~ E almost flat. II

I,

V

/ (x )

II

E

£

Remark 4.3 If we replace the distance d with anyr d'=d'=XdU>0) Xd (X > 0) and take 1

X=—i=, d dd(x,y) ( x , j H= xjx--^ )y\), , A— i—, we obtain (if £ is a normed space and

and

d(x„,a\<4e d{x d(xes,a)<4s ,a)<4e

(4.7)

|/'(x )| .
(4.8)

£

£

which imply the following result. If {x„}neA, is a minimizing sequence for/ (that is, lim/(x„) = i n f £ / ) , then there exists another sequence {y„}netf N «—>oo

such that: f{y / ( ^n)
n-x»

liml/^Jl^O. Hk'U)L'=°-

n—KON

"ft

(4.9) (4.10) (4.10)

4 ii[ ((4.11) -)

Theorem 4.7 [Ekeland's Principle-Strong Form] Z,ef Let f:E-> f:E -*R\J{+<x>} R\J\+x>} be lower semicontinuous, bounded from below, and not improper. Ifs > 0 is an arbitrary real number and a e E is such that + £, /f(a)
(4.12) (4.12)

then, for an arbitrary X > 0, there exists a point Xx e E such that the following inequalities hold:

(0 /(**)*/(«) f{xx)
359

VARIATIONAL PRINCIPLES (n)0 (if)

d(xx,a)
(iii) f(x f(xxx)J) < <
(4.12) we have a point x\ such that x /f{*x f a ))< <
which is exactly (iii), and jd{a,x -d{a,x,)
(4.13)

From (4.13) we immediately have (i). Inequality (4.12) implies that f(a) - f(xx)*e,t and using (4.13), we deduce inequality (ii). ■ f(a)-f(x <£, x)<s, As a consequence of Theorem 4.7, we can refine Remark 4.3 of Theorem 4.6. Suppose that (jE,|| |] is a Banach space and/is a Gateaux differentiable function on E. We denote the Gateaux derivative of/ b y / ' . From Theorem 4.7, we obtain that the equations /(*»)= inf/ and / ' ( * , ) = 0 can be satisfied up to any prescribed approximation. Indeed, considering the distance defined by the norm | ||, we have from from inequality (iii) of Theorem 4.7: f\xx) ^-M\ 'W < tx), for te [0,1]. [0,1]. / f a ) ~- jM 7'W / ff axa ++ <*), forall allre re f{*x) < /f\x f[*x +tx), tx). for all re [0,1]. Letting t -> 0+, we have -jt\x\ -~t\x\ < \\f(x / ' fxa),x). ) , x ) . Now, taking the infimum of

360

VARIANTS OF EKELAND'S PRINCIPLE both sides over all x e E with1 |x|| If we take |JC|| == 1, 1, we we deduce deduce // '' (( xx jj ^^ T~r- 1 X= -fs, we obtain the following result. X=4s, Corollary 4.8 IZ,ef(is,|| e f ( £ , | |) be a Banach space and f:E->R\J{+oo} f:E-> R\J{+co] aalolower semicontinuous function, bounded from below and Gateaux differentiable. If s> 0 is an arbitrary real number and x£ is an element in E such thattf(x f\xEE)<mff ) < inf / ++ e, e,t then there exists a point x*E e E such that: x&E xsE xeE

(0 A*.) f{x f(x *)* e

(it)

Ee

\\xl-x \\x' e-xe\\<4£,

(iu)\fixl)\<47. mWAxDk^. If(E,\\ |) is a Banach space and /f:E^R\J{+oo} : £ -> -RU{+°°} is a lower Corollary 4.9 If{E,\\ semicontinuous function, Gateaux differentiable on E, and bounded from n-^+00, we below, then there exists a sequence \x such that, when n-^+00, n)neN \Xni 'ne/V

have : (0 /(*„)-> /(*»)-> (i) / U J - > i ninf f F £ //\, ((<0/'(*„)-> » ) / ' (n^)^0[inE\. ) ^ 0 (0(fo£*). f»£*). (it)f'{x

It is of some interest to mention a different form of Theorem 4.7 which may be useful in optimization. Supposing that \E,\ |] is a Banach space and /f:E^>R\J{+«>} : £ —> RU{+°°} is a convex function satisfying all the assumptions of Theorem 4.7, we obtain a point

= («J0

xAeE

such that f(xxx)
exists an element x% which minimizes the function *x- -> > / (f[x) * ) +H—\x j l k ix -—x\. *|A

Using subdifferential calculus, we have that 0 0edf(x &df(x^) x)

H—B*, +^B* where B* A

is the closed unit ball of E*. Hence, we have the following result: 361

VARIATIONAL PRINCIPLES Theorem 4.10 Let {E,\\ [E,\ |) ||) be a Banach space and f:E -> i?U{+°o} i?U{+°o} be a /:£->■ lower semicontinuous, proper, convex, bounded ounded from fro below function. Let f(x )<M f + £e > 0 and x e v X> £ E f. Then, for any > 0 and x£e e dom f such that f(x£)<mfinf + £. e. X >0,0, Ef £ / + < ? / ( J £ )\ such that: there exists a point x'e e domf and a point ** x* e&df\x\

(0 (0

), £)
(iii)\x\ .<^. v (SO 1*11 •*-■ ' II

WE'

x

V

' II ll£- ^ From Theorem 4.6, we immediately deduce another form of Ekeland's principle which is currently used in some applied problems. Theorem 4.11 [Ekeland's Principle-Weak Form] Let Let f:E-> f:E^R\J{+oo} R(J{+ao\ be a lower semicontinuous, bounded from below, and not improper function. Then, for every e > 0, there exists xe e E such that: (j) f(x ) 6 £d(x,x£££),forallx t££)) < In the following, theorem we will show that Ekeland's principle has some deep relations with fixed point theory. The part of the contraction mapping principle which concerns the existence of a fixed point can be obtained from Theorem 4.6. Theorem 4.12 [Banach's Fixed Point Theorem] Let (E,d) be a complete metric space and T: E^>Ebea contraction (i.e. d(7\x),T\y)) < kd(x,y) with 0 < k < 1 for all x,y e E). Then Thas a fixed point. Proof We define fix) = d(x,T(x)), for all x e E, and we apply Theorem 4.6 f{a)
362

VARIANTS OF EKELAND'S PRINCIPLE Indeed, if xe is not a fixed point for T ,we put x = 71(xe) in (ii), and we obtain d(x£,T(x£))-(1

- k)d(x£E,T(x££))
T(T(x£))) < kd(x££,T{x ,T(x£)),

that is k < k, which is impossible. ■ Theorem 4.13 [Caristi-Kirk's Fixed Point Theorem] Let (E,d) be a complete metric space and f:E -> /?U{+°°} be a lower semicontinuous /:£->i?U{+°o} function, bounded from below, and not improper. Let T : E -> E be a setvalued mapping such that ), for all xxeE f{y) < f(x)-d{x,y),forall f(y) f(x)-d{x,y),forall xeE andy andy ee 7t*). 71*)7(x).

(4.14) (4.14)

Then there existsxt e E such that xt e T(xt). Proof Applying Theorem 4.11 with e= 1, we find x, e E such that /(JC«)< f(x) + d(x, d{x,x,),forallx eE\{x*}. f{xt )
(4.15) (4.15)

We claim that xt e 7\xt). Indeed , if this were not the case, then, for all y 6 T\xt), we would have y*x*. In this case, from (4.14) and (4.15), we would get f(v)< t)-d(x f(x.\tt,y) f{y)
VARIATIONAL PRINCIPLES evolves to a stationary point, no matter what the initial conditions (such as initial position and velocity) are. A dissipative system may have many stationary points which are called stable equilibria. Knowing the equilibria points, the dynamics of the system can be derived. If, at the initial time, the system is at an equilibrium with a null speed, then it will not change and the motion will remain at equilibrium. If the system initially either has a velocity or is not at equilibrium, then the motion will start but will decrease progressively to reach equilibrium. The pendulum with a rigid stick illustrates this principle. The decreasing of the motion dissipates the energy of the system (for example amongst others in the form of heat) until it is reduced to zero and the motion becomes stationary. Hence, the term is dissipative. A dynamical system T:E -> E in the sense of definition given in Section 3 is dissipative with respect to a function f:E —> [0,+oo) (/"is not identically equal to + oo) ifif ^ f{y)
(4.16) (4.16)

Condition (4.16) states that^y) less that/*) unless x equals v. We /U>) is strictly stri say that a sequence" x-XQ,X],X2) ,x x , ,x„,... defined so that x^ e l{x„), for all n, 5>Xj ,X2 5 , , X X , . . . ( X J, ■.•.■■. * ■*« *00 ' - u * 1 ' 2%2 ^2 2 '>' si>■*«? tha the set \xn\n sN\ is a trajectory of is a motion starting at x0. We We say that this motion (n denotes successive steps or times). If (4.16) is satisfied, the energy decreases with rate nn

Zfeh/M^ /Wi/fcii)>j 4*«'*«+i)

until the system reaches a stationary point. (We supposed xn * xn+i for every n). The idea expressed in Theorem 4.1 is that, in a complete metric space, a dynamical system which is dissipative with respect to some lower semicontinuous positive function/(not identically +oo) has a stationary 364

THE DROP THEOREM point. Furthermore, the idea expressed in Theorem 4.2 is that, for every lower semicontinuous, bounded from below, and not improper, mapping f:E—> JRU{+°°}, the petal dynamical system TPj is dissipative and has a stationary point.

5. The drop theorem The Drop Theorem is a geometric result discovered in 1972 by J. Danes [1]. It has interesting applications in functional analysis and in the geometry of Banach spaces. In this section we will prove that Ekeland's principle implies the Drop Theorem. Since the Drop Theorem is a generalization of Bishop-Phelps Theorem about conical support points, we will start with this result and we will then show that Bishop-Phelps theorem is also a consequence of Ekeland's principle. Let [ £ , | I) be a normed space and D c E be a non-empty set. Given a convex cone KaE,we KaE,we SE say that a point xQ e D is a ^-support point for D if [K+x0](]D={x0}. If x0 E D is a AT-support point, the interior of K is non-empty, and D is convex, then x0 is a support point for D in the classical sense, i.e. there exists a continuous linear functional h on E, (not identically equal to zero), such that h[x0) - suph[D). Indeed, since [A" + jt0]n£>={*o}> w e obtain, using a classical separation theorem, that there exists a non-trivial linear functional heE*, with suph{D)<mf[K + x0). But, since x0 belongs to both sets D and K + x0, we can show that sup//(£>) = h(x0) = inf(A" + x0). In 1963, E. Bishop and R. R. Phelps [1] studied the existence of a ^-support point for a convex set with respect to the particular cone K(f, p) associated to a functional/and to a number p > 0. For / e E* with ||/|| = 1 and p > 0,

365

VARIATIONAL PRINCIPLES we denote

K{f,p) {xsE\\\x\\
Evidently, K(f, p) is a closed convex cone. If 0 1, we have that K(f, p) has a non-empty interior. Indeed, let x0 e E be an element such that |jc0| = land — 1 ). Let P be a real number such that 1 0 such that 1 < 1 + s < B . We define the sets:

= {x [x^E\B 1, D has a K(f,p)

-support point.

Proof. We consider the complete metric space

A—d , where d is the

distance defined by the norm of E. We remark that it is possible to apply

1 1. Theorem 4.1 (or 4.2) to the mapping - / a n d to the metric space ( D,—d V p J We hence obtain that the restriction, with respect to D, of the dinamical system TP_f, that is TP_f(x) = TP_f(x)f]D forallx e D, has a stationary point v e D. We have:

366

THE DROP THEOREM {v} = pjrPJ (v)r[D=DpL eE-f(y) < -/(v) -/(v) -- II || yy -- v||l v||l {v)nD=DrdyeE-f{y)l-f{v)-±\\y-v\\l = Dniv sE-f{y) < {v}=r {v)nD D^eE-f(y)<-f{v)-±\\y-v\\l = JDD{>' = Z ) n Dr\{yeE\\\y-v\\ Df]{y = =^x++ +vv,x v,xeE\ , x eeE\ £ | ||x| \\x\\ << p/(x)} < pf{x)) pf{x)) = Df)\y < pf(x)\ Dflfj |J\\x\\ C|

= = \K(f,p) [K{f,p)++v]nD, v]nD, =[^(/,p)+v]nA and the theorem is proved. ■ Remark 5.1. A generalization of Theorem 5.1 can be found in (Isac, G. [1], [2]). Similarly, we can prove the following variant of the Bishop-Phelps Lemma. Theorem 5.2 Let \E,\ ||j be a normed vector space and DczEbe a complet subset. If f e E is bounded on D and | / | = 1, then, for every p > 1 and every x e D, there exists a K{f,p)-support

point x, of D such that

x,eK(f,p) + x,eK(f,p) + x. x. Let (is,I I) be a Banach space. Given r > 0 and xo e E, we denote by B(xQ,r) the closed ball of center XQ and radius r. For every x e E such that 0 < r < \x - x0L the drop associated to x and B(x0,r) is by definition the set D(x,B(x0,r)) = convex} U % / ) ) = {x + t(b-x)\t t(b-x)\t e[0,l], e[0,1], b6 66 % % %// )) )) .. Since B(x0,r) is closed and bounded and {x) is compact, we have that D(;c,J?(x0,r)) is a closed set in E. Let (is,| Theorem 5.3 [Danes's Drop Theorem] lef (E,\\ |) be a Banach space, C

367

VARIATIONAL PRINCIPLES nE\C, e>0, be a non-empty closed proper subset ofE, z0 be a point in E \ C, e> 0, and 0 0, there exists x.x, ee Xp\PJx Xf\Pr(xQ0,b) ,b) \ < s — r.) such that P [x,,b)nx = {*.})• (satisfying (satisfying in in particular particular d{x,,x^) d{x,,x^) d(x,,x )J)si such that Pyy [[x,,b)nx x., b) f IX = = {*.})• [ x, ]). 0)< , ) << y

Proof We apply Theorem 4.5 [Altered Ekeland's Principle] considering as complete metric space the set X and the mapping f{x) = d(x,b) defined for every x e X. The function / i s continuous and bounded from below by r. Hence, using Theorem 4.5, we can find x, e X such that f(x < f(x) + + yvy d(x,,x) d(x,, x) x)for for each eachxxx eX\{x \x.} d(x,, for ee XXeX\{x \\t}[x.} f(x»)
xx xx f(x.)
(5.1) (5.1) (5.2)

From (5.1), we have that, for each x eX\{x,}, eX\{x,},(. \, one one gets gets xx €g PPjx.,b) and, r(x.,b) and,

368

THE DROP THEOREM from (5.2), we deduce tnat that it yd{x yd(x yd(xtt,x )<s-d(x,,b)<s-r,. xt,b)<s-r,. t,x00)<s-d(x t,b)<s-r,. ,x 0)<s-a\xt,b)<s-r,. <s-r,. proved. ■

r

.1The theorem is he theorei

Theorem 5.4 can be formulated using the dynamical system TTPP(y,b:x) (r,b:x) defined in Section 3. We have the following result: Theorem 5.5 .Let X be a complete subset of the metric space (E,d). Suppose x0 e X and b e.E\X, with 0 < r < d{b, X), and denote s = d(b,x0 ). Then, for each y> 0, the dynamical system YP(y,b:x) has a stationary point x0 such that xt erp[y,b:x (y,b:x0). Theorem 5.3 implies that, for a given z0 and r, there exists a "supporting D[x0,B(z drop" of C of the particular form Z)(JC ,r)) for a suitable xo. From this 0 ,5(z00,r) remark, we have that Theorem 5.3 is similar to Bishop-Phelps Lemma, in the sense that the support cone is replaced by the support drop. We will prove a form of Theorem 5.3 which is very close to Theorem 5.2 and has Theorem 5.3 as a consequence. Theorem 5.6 [The Drop Theorem] Let C be a complete subset of a normed vector space [E\ |) and xo be a point in C. Let b e E\C be an arbitrary point and B be a closed ball with center b and radius r such that 0 < r < d(b,C). Then, there exists xt ee. CDD[x CC\L){x0,B) 0,B) \with the property that D(x*,B)r\C={xt}. E(x.,B)r\C={x.}. etal Theorem Theorem ] tali taki taking Proof We apply the Theorem. 5.4 [The Flower-Petal d-r orm ^ ith and d ry==^d-r ww with X = C[)D(x CC\D(x0,B), 0,B), the distance defined by the norm dd + + rr Xf]Prr(x (x ,b) ,b) and d = d(b,C). We obtain an element x, such that xx,t e XC\P XnPr(x000,b) a d-r t-r d-r tPr(xvt,b)f]X = {xA. id we have ve. — —-< < —-, ,b)r\X ave / l {x J t}. We put t:= d(x,b) > d, and < ' d+r t+r + rr t t+r + dd + d+r d +t-rr tt+r + r' which implies (using the convexity and the relations ons r < t, y< y< < ) tha that 1t ' tt + + rr i

369

VARIATIONAL PRINCIPLES L)(x ,B)
and an

^P =x{x.}, c / r'(x.,b)C\X .}> r(i,)i)ni={ and the proof is thereby finished. ■ We can formulate another form of Theorem 5.6 using the dynamical system rTD( D, B) s j defined in Section 3.

■Wl

Theorem 5.7 Let C be a complete subset of a normed vector space [E,\ |) and x0 be a point C. Let b e E\C be an arbitrary element and r be a real number such that 0 < r < d(b, C). IfB is the closed ball with center b and radius r, then the drop dynamical system TD< B) has a critical point x, such thatxtereT thatx, D(B)nl{x0). Jxn). Let (E,\ I) be a Banach space and B a convex subset of E. The drop associated with a point a e E and the set B is the convex hull of {a}U#, that is L\a,B) = {to [ta + (1 (l - r)6| t)b\ *t e[0,l],Z> D(a,B) e[0,lU efl}. efl}. If A, 5 c £ , w e denote d(A,B) = inf(\\a-b\\\a UA, inf (\\a - b\\\ aeeA,b A,b€€BB}. } . For Foran anarbitrary arbitrary non-empty set A c E, we denote by dA the boundary of A. Theorem 5.8 [Danes's ineorem s.s [Lianes s Generalized oeneranzea Drop urop Theorem] theorem] Let Let A A be be aa nonnonempty and closed subset ofE, and B be a closed bounded and convex empty and closed subset ofE, and B be a closed bounded and convex subset subset of of E E such such that that d(A,B) d(A,B) >0. >0. Then, Then, for for every every xx00 ee A, A, there there exists exists aa point point xt eeAC\D(x AC\D(x ,B) such that AC\D{X.,B) AC\D{X„B) = {X,}. x, Af)D(x0,B) AC\D{x ,B) {x }. 0 t t Froot It B is an empty set, then there is nothing to prove, since, in this Proof If B is an empty set, then there is nothing to prove, since, in this 370 370

THE DROP THEOREM case, x,=x x, = 0x. 0. Let Let d= d= d( d( A,B). A,B). We We will will first first prove prove the the following following property: property:

((

For every e>0 and every A'c A'a A,A*<j> A, A'* <j> there exists a eA' sA' such that diam diam(D(a, (D(a, [D(a, B) fl A'} A') < e.

Indeed, denoting M = diam(B) + d(A',B) + l for

(5.3)

«M£4

Selo, S e ( o ,mini m i n j — ,,1> ,l>\, ll] ,

tK f<

~

tS_ 5_ tS S_ 8 d{A',B)~ d(A\B)~~d~~d d~ d

and ii IIn

7II IIii /i MI 77iiii *,$ . T
So, we obtain that diam[D{a,B)V\A diarr{D{a,B)V\A;;) <e, and property (5.3) is proved. an( ^ Now, using property (5.3), we can construct two sequences { 4 },-«=# and {a,}ieAr., such that, fa,}. dianLi\<\, diamiAA <—,

(5.4)

a, G e 4-i> 4_,,

(5.5) (5-5)

4 = A, =A D,(14_, f M - i whereD whereDtt ==D(a,,5),v4 D(at,B),A00 ==D{xD{x 0,A)f)A. 0,A)r\A.

(5.6)

371

VARIATIONAL PRINCIPLES By Cantor's Theorem [Theorem 2.1], we obtain that there exists x, such

fU=M-

that f| 4 = {x.}. Denoting D = E\ x,, Bj, we will now prove that 1=1

i=i

00

f]D D. C\Dti = D.

(5.7)

i=i

CO

Since x, e D, we have D c A for every i e N, and, hence, Da fl A. Let i=i

w h e r e < e[0,l], b BB xJC ee flfl AA -■ Then T^ 11 x*==f,a, '.a. ++(l( 1-~f,)fy, 0 ^ > where '.( e[°'1]> 6, i e 5- . i=i

!

Since

I0-1]

is

kU'

compact, the sequence {f,}' l e / V has a convergent subsequence {^ j'*eAT l '*J*eJV Let / = lim ?, . We face two situations: t-wo

i) If t = 1. In this case, \\x - a, II ==f(1 IIJC 1 -- 1t,t )|k )|L - 6, b, I < (1 - 1f,, )diam(D,) -» 0, IIII '* Vv I i/ / '' ''** IIII \\ '* l\\/ll '*'* '*'* IIII \\ 'k'*I / which implies that x = x,

sD.

ii) If t < 1. Then, there exists n0 e N such that 1 - tt > 0 for every k > n0, and we can write i. = x h —a,- -» x t x. € 5 , o, a, = —a,- -> x . <=B,

* *

1-r 1-r,.

1-f, 1-f,

\-t \-t

" "

x

\-t \-t t

since B is a closed set. Denoting b = j~--j—x,, we deduce that x = tx* + (1 - t)b e D. Since A <= A-i <= ^ ^ o . 4 for i > 2, from (5.6), we have 44=An4-i=AnA-in4-2=An4-2=-^n^nzx^^)=Anv4 = 0.^)4-1 = Dir\Di-ir\Ai_2=Dir\Ai^=-DiC\Ar\D(x<),A) = Dir\AJ1 and, using (5.7) ,we get Df]A Df] A = I fi A V JDu^ = fi(fi(AM A H A) = fl H 4A,, =={*}. {*.}. ■■ N=l

372

/

1=1

1=1

THE DROP THEOREM The point x, in Theorem 5.8 can be chosen from A in such a way that every minimizing sequence \xn]'neJV <=L)(a,B),d[xn,A) —> 0 must converge to a. In this sense, we present the following theorem which has been proved in (Georgiev. P. [1]). Theorem 5.9 [Strong Drop Theorem] Let A and B be two closed nonempty subsets ofE. The set B is supposed to be convex, bounded and such that d = d{A,B) >0. Let B£={xe E\d{x,B) <s). Then, for every s e(0,d) and every z e E such that D{z,B£}[\A^, there exists a point xt eAf]L){z,B£) such that AflD(xt,B) = [xt}, and {xn}IneN -* x, whenever {xn}neN a D(xt,B) and d(xn,A)-+Q. UneN

Proof We apply Theorem 5.8 on yiU{z} andB£, and we obtain a point xt eD(z,B£)f)A such that AC\D{xt,Be) = {*,}. Let {xn}

N

C £ ) ( X , , J B ) , X„ # X .

and for every n e N, and d(xd(x K,A)-K). n,A)->0.

Then, we have x„ = tnxt +(l-tn)bn,

for some t„ e[0,l),bn sB.

There

exists yne A such that \xn - v„|| —> 0. Let ** e[*„,^]n^(x.,J?.) e[x 4x„,y ]ndD(x„Be) and - ~^~ *. a n dcnc:= . andc : ~^~ = - ^n:=-^---^-x — - f - x .xt.t. n,ynn]r\dD(x„B 1

l

n

1

l

n

If c„ e mtB£, for some n e N, then we can show that zn

e'mtD^x,,B£^,

which is a contradiction. Hence, we have d\cn,B) > s for every n e N, and we deduce

Ik-^hti-OK-^Nti-O^k.^^ti-O^.

»

which finallv imnlies 1\-t- 1n). . < <

x — z \\ £

—> 0. ->o.

373

VARIATIONAL PRINCIPLES re \\x \\x„ x.\==(l (l- -tj$b tn)\b tn)diamD(xt,B) So, we have x.\\ < <(l (-l -tn)diamL)(x 0,0,and the t,B) ->-> n - -x,\\ n -n -x\ theorem is proved. ■ We now consider a generalization of Bishop-Phelps Lemma. If [E,j |) is a Banach space, B c E is a closed convex subset such that 0 £ B, and e > 0 is a real number, we denote:

BeEe

={xeE\d(x,B)<e),

= {X ={xeE\d(x,B)<e\, €eE\d(x,B)<e\,

b\A&R {Ab\AeR K(B) = {l={Ab\AeR +,beB), +,beB), K{Bte)) = {Ab\A eR++,beB ,b eB e(0,d(0,B)), K(B {Ab\AeR E} E} if e e(0,d{0,B)), K(x,B) = \z ■eE\z K{x,B) sE\z= ==x x++A(b-x),A A(b-x),A eR sR++,b ,b e5J eB\ SB\ if xX ifxtB. £B. We can show that K(B) and K(BC) are closed convex cones pointed to the origin and K(x,B) is a closed convex cone pointed to the point x. Theorem 5.10 [Phelps] Let [E,\ |) be a Banach space and B c E a bounded closed, convex, and non-empty subset such that 0 € B. Let A c E be an arbitrary closed subset. If for some z e E,AC\[K(B) + z] is bounded and non-empty, then the set A[\K{B) + z\ contains a K(B)-supportpoint for A. Proof. We must show that there exists an element x, eAC\[K(B) + z\ such

that ,4n[*(6)+ *.] = {*.}■ We obtain this fact as a consequence of Theorem 5.8. Indeed, since J f l ^ ^ + z] is bounded, there exists r > 0 such that AV\[K{B) + z\czB(z;r). 2r Let 'o - : ,t0 B\ and D = z + t0B. We have d{z,D) = tQd{0,B) = 2r and d{ 0,5) d(D,AP{K(B) + z\)>r. Thus, we can apply Theorem 5.8 for D and Af][K{B) + z] (we remark that AC{K{ B) + Z] is closed). We obtain a point 374

THE DROP THEOREM xt ee AC\[K{B) Xt Af\[K{B) + z] such that Af][K(B) {xt}. AfiK(B) + z]riD{a,D) {xt}. z]nD{a,D) = {xA.

(5.8) (5.8)

We now prove that (5.9) (5.9)

x0 + K{B)cK(x.,D). K(B)^K{xt,D). K(B)^K[xt,D). Indeed, we have x, - z = txbx, for some t\ > 0, 61 b\ ee B5 and and htld(0,B)

4M = \\x. | * -4 . < 44 \\xt -z\\ <
Consider an arbitrary element* e x,+K(b) and f3 = £0 - t i . Then x - x» = t2 b2, for some t2 > 0, and b2 e B. f

2 We denote 6b33 =- --LLJ--L ++-*-*■ ^LiGJB e ,B, £ and, if we put s = —, we get

tft

'n

'3

xx = = x» x» ++ rt226b22 =-xx» +5/st336b22 =x» =x» +s(t +s(t00bb33 -- ttxxbbxx))=x* =x» ++s(/s[t - x»); x»); t + 0Z>3 0b3++z zK(x*, therefore x e K[x t, D). Because B(z;r)f]D = , we have 5B(z;r)f]K(x ( z ; r ) f l 4 t^,D) ,D) = = J B(z,r)ni3(^,D) B{z,r)nD{xt,D) and, from from (5.9), we obtain + xxtt]f]B{z;r) [[K(B) 45) + ] D 5(z; r) ac K(x #(x., £>) D 5(z; r)== B(z;r) 5(z; r) Df]D(x £>(**,t,D). £>). t,D)f]B(z;r) Since £(5) K(B) is a convex cone and x, x„ eK[B) + z,we have K(B) + £(5) + x, xt C:K[B) C:K[B) + + Z, Z, and, and, from from (5.8), (5.8), we we can can deduce deduce [K(B) + xXt]f]A z]nAr\B{z;r) t]f]A = [K(B) + Xt]f][K(B) + z]r]Ar\B{z;r)

375

VARIATIONAL PRINCIPLES cB{z;r)r\D(x.,D)r\[K(B) cB{z;r)r\D(x„D)r\[K(B) + z\nA=\x. z]r\A={x.}, cB{z;r)r\D> ~}A={x.},, r\D(x„D)r\[K(B)+z\r\; cB{z;r)C\D(x.,D)r\[K(B) cB[z;r)r\D{x.,D)\}[K{B) + z\\]A={x.\, and the theorem is proved. ■ The next result was also proved in (Georgiev, P. [1]). Theorem 5.11 [Strong Phelps Lemma] Z,ef(£,| |) be a Banach space and B ac E a bounded, closed, convex, and non-empty subset such that 0 £g B. 5. Let A c E be an arbitrary closed subset. If e e [0,d\0,B)) and AV[K(B U non-empty and boundedfor some z eE, then there exists A n[ K(BCe)) ++zz] \ Z5 a K(B)-supportpoint x, for A such that x, e ^fl[^(5 \x„}n}neNiis x,eAf)[K{B z\and{x £) £ ) + z] and K(B) ++ x, and d(x d{x„n, A),A)^0. convergent to x,, whenever {x„ \x„}}n£NN c K{B) -> 0. Proof. First of all, we remark that Bte and K(Bet)) are closed and, applying z sucn f)\K(B that Theorem 5. JO, we K{BEEE))) + Theorem 5.10, we obtain obtain aa point point xxtt ee A A f]| \ \\K[B + z\ z\\ such such that that ^n[^(B £ ) + x,| = {x,}. K(B K(BEEe),), ), we we for tor every every

have have nn ee

A[\K(B£) t) Ar[K(B N i\ and ana

Because of the fact that K(B) is a subcone of + x.] X.] = {x.}. {X.}. Let {xn}neN d(x , A) -» 0. Then, a\x n,A)->{). men, xxn n

n

ttnn > A such such that that > 0,b 0,bnn ee B. B. There There exists exists yynn eE A

ca K(B) *x„ K{B) ++ x.,xx.,x n n*x.. = x,+/ x, + tnobn tor for some some = n n

c = c5 L=^-^-. : i £ \\x„-y„\\^0. Letz |kx-nv- vJ J- ->>00 .. Letz„ Letz„ ]r)d(K(B n e[x n =^^-. e [ xn,y„n,]f]^{K(B ^ ] D 4 4££)5 , ) ++ x,)andset ^ ) a n d s e t c„ n n

If cm e6 \n\B intBe for some m, then zm E mt[K[B int(A^5c£)) + x,), JC,), which is a contradic d{c a?(cnnfl,, B) 5) > e, for every n e N, and we have tion. Therefore d(c be — zz_ bc_

2 k -*J='-II --*J='-II «l='-II 66- - c J * £ '«' ' « ' hh ee nnccee r- * \\_n—«y^ g€ -->>_ o. 0

So, we get \\x ||xnnn-z-znn\\n | = fJfc tnn\\bnnn||0, nn-diam(B[J{0})->0, complete. ■

and the proof is

In all the forms of the Drop Theorem, we assumed B to be convex. It is 376

THE DROP THEOREM interesting to know whether there exists a form of the Drop Theorem for the nonconvex case. An interesting result in this sense was recently obtained byJ.Guillerm[l]. Let (E,\\ I) be a normed vector space and A, B two subsets of E. We denote the distance defined by the norm of E by d, the distance between x and B by de(x), the distance between A and B by d(A,B) and the diameter of B by 8(5) As previously, D(x,B) is the drop from x to B. For a > 0, we denote the petal fromx to the function axdB by TPBa(x), that is, TP B a(x) = \u eE\dB(u) + a d(u,x)

We denote, by [u, v], the segment defined by u and v where u,v e E. Definition 5.1 We say that B is nearly convex from xeE if, for every e >0 and every y e B, it is possible to find two points u and v in B such that: Luis a projection ofx on B up to s, i.e. dB (x)< \\x - u\\ < dB (x) + s, 2. y belongs to [x, v] and [u, v] is contained in B. We say that B is nearly convex from A, if it is nearly convex from every point of A. Examples 5.1 1. A convex set is nearly convex from any point. 2. A star shaped set at x is nearly convex from x. 3. The set D = \(x,y) eR2 |jc|')#(l,-U v 2' ' ' ' V 2J j is not convex but it is nearly convex from any point of {0}x R. Theorem 5.12 Let E be a normed vector space and A c E a complete subset. Suppose that B^Eisa is a closed bounded subset, nearly convex from 311 'ill

VARIATIONAL PRINCIPLES A, and such that d(A,B)>0. Then, for every point a eE A, it is possible to find s^n£>(a,5) su a point xx,t sAr\D(a,B) eAf)D(a,B) such that ^n/>(fl,5)ni>(*.,fi)={*.}. Af)D(a,B)f)D{x.,B) = {x.}. Prooftf First, we prove that for every x e A and every u e D(x, B), we have dB\x) > (x)-d (u) > Mx)\\x (KX^JT^) > 0(5-10) dB (x) -dBB{u)>X {x)\\x -- 4, 4, with with X{x) X (x) == dd-j 1^—gJJ) ° ■ (5.10) x)+wg) > dB(x) + S(B)>l Indeed, let es >0 be arbitrary and u e D{x, D(x, B) such that u *x . Since B is nearly convex from A, we can choose y e B such that u E [X, y] and consider the points h, z corresponding by Definition 5.1 to s, e, x and v. y. Certainly, u e [x, z] and hence -- a)z, u= a x + + (l(l (l-a)z, a)z, for for some some aa ee [0,l]. [0,l].

(5.11) (5.H)

elementp=a + (l-a)z. Consider the element p=a h + (1-ar) z. Then, we have u-p u -p = a{x-h). a (xa(x-h). - h ).

(5.12) (5-12)

Sincep E B, we have dB{u)< (u)<\\u-p\\, \u - p\, and, using (5.11) and (5.12), we deduce (j\ B{u) J <..\ I\\x-h\\-s-\\u~p\\ IL. xJi daJ B(x)-d \x-h\-s-\u~{\ lljc-ftll-g-glls-ftH \\x-h\\-£-\\u~p\\ {x)-d \\x-h\\-e-g\\x-h\\ |LJ _C -til/ i | | --g I- |L| « - p J| | _ ||x ll-x -- h\\ /i|| -- _ge -- „iLaa ||x \\x-7--H A| h\\ | B Byx)-aB\u) s iiv-_„n IL_ „n ~ (i rvm JI \\x-u\ \\x-u\\ fl-flMr-dl \\x-u\\ \\x-4 (\-a)\\x-4 ( l a ) | | x ^ Ix-«|| '

(l-a)d (l-a)<sf U)-£ {l-a)d 5V-*/ B(x)-e \A u* B J**B — » ~(l-a)(rf ~(l-a)(d + S{B)) ff+ 0 such that, for every xe A and every ue D[x,B),we have

378

GENERALIZATION OF EKELAND'S PRINCIPLE dB(x)-d (x) -B(u)>a\\x-u\\. dB(u) > a \\x - u\\.

(5.13)

1 is increasing on. [d{A,B),+o\, Indeed, since the mapping?rt->-—TTivr [d(A,B),+x[, we

^7ri(5T '^TrkBj '

d B a:- mi\X{x)\x mf\X{pc)\x infU{x)\xeA}= mfU{x)\xeA}= ^ ), N . F: can takee or.= a:e eA)A\ ., {A'-BQ^EL From v(5.13), we obtain \.. 1 W l k.4; == ," X Wl ;j d(A,B) + S(B) ' d{A,B) + 5(B) d(A,B) + S(B) x that, for every xe A, the drop D(x, B) is contained in the petal *• TP,Ba\ PBa{x). ) ■

From this remark, if we choose some point a e A and apply Theorem 4.3 to the set Af)D(a,B), to the point a and to the function f{x)f{x)-a~ = a ldB{x) {x) (restricted to Af)D(a,B)), we obtain immediately the theorem. ■

6. Strong Forms and Generalizations of Ekeland's Principle In this section, we obtain two strong variants of Ekeland's principle as a consequence of Strong Phelps Lemma [Theorem 5.11]. In this sense, we will use the notations of Section 5. These results were proved for the first time in (Georgiev, P. [1]). Let [JB,| I] be a normed vector space, and consider the norm max |K-M)|L = max{|;c|,|f|}, {IMI>H}> f° |(;e,f)|| forr every (x,t) e EQQ, on the vector space EE0Q = ExR.

We denote : dd

0o ((*i. ((*1. h )> (*2 >h »'2 )) )) == II (*1. (*i >h 'l) )~~(*2 (*2. .'2 '2 4)IL» .

flrr = {(x,0)e£0 ||x| 0.

f^ *r>, == {(*,;) {(*,*) e£<>WMA)^4 e£oK((*.').$-)*4

Proposition 6.1 Given r, s,s, e> e> 0, 0, #zere #zere exz'sto exists £> S> 00 such SMC/Zthat //zatf 379

VARIATIONAL PRINCIPLES K({0,s),B )czK{{0,s),Br+£ ,Br+£ K[{0,s),BrS JczK[(0,s),B ).).).). rr^)0 >0 are are given given ,, consider consider 8£such thatto<s<—^—, 0<S<< r + s+e , r + ss+s' + s' andlet(x,?)eA:((0,,s),5 eKU0,s),B f< y). and let (.(x,t) rA

We have §$*s,s-S>0 & s, s-S>0 §±s,s-S>0

, s(r + S) d) ^
Because (x,/) (x,t) eK((0,s),B €ffl(0,5),B, ),, we we have have r sJ (x,t) = (x,f) = (0,*) (0,s) +A +X ,[(«„?,) {(xvti) -(0,5)] -(0,5)] for for some some XXxx> > 00 and and (x„f,) (fo,f,) x ^ ) ee 5B^. ^ .. r(S s-t, L

L — a nidd x —■■ S -xf>s-8 , > S - £ >>0 0 and and ani x^J22C==- =~i T -. Since t\ ^ s-t ^ anc 1x^— ?i < 8, 5, we have s-t^s-S s-t^s-d 2 hence A 2> 0. We get

T.et A .2 = Let /I ,= = st A,= A,= X

kl^^T^S)< e S)
+£

and (x,t) = (0 (0,s) +Aj[(x Ail[(x [(xll,t )-(0,s)] ==(0,a) (0,s)+ A {(A 221xx[(A sA2 )-(0,*)] )-(0,s)\ ]])-(0,s)] 22,s2x2,s-sZ 22)-(0,s)] (jc,f) = + (0,s)+A A,[(A ,*-5A f*) + lf,tf,)-(0,*)] = (0,J) + A 1 [ ( A 2 X * 22 , 0 ) - ( 00,,**A A222))]| = ( 0 ,,JJ ) + A 1 A 2 [ ( X 22 ,,00))- ( 0 , 5 ) ] ,

which (x,r) e K{{0,s),B ^(0,i),fir+sr+l )..).■■ ch evidently implies that (x,t) K[{0,s),B r+l.j. Theorem 6.2 [Amended Strong Ekeland's Principle] Let (D. d)bea f:D—>R\J{+oo} complete metric space and «rf f:D—>R\J[+ooj /:£>-> if U{+<»} -a lower semicontinuous Junction. Iff is bounded from below on D and not improper, then, for any function. from 8>0,y > 0 and any x00 e D, there exists a point x, e D such that : d~>0,y V f(x,) j[x*)< f(x, )\{x.}, D\{x.j, x x +(J 2) f(x f{x*) x x..t whenever whenever {*„ {x {*„} and*/(*„)+r f( f(x f(x„) d(x {Xn)+rd(x„X„)^f(x,). nn&N n}j neN n) + y 4**d(x,,*„)->/(*,). t,*„)->/(*,). , ,
380

GENERALIZATION OF EKELAND'S PRINCIPLE Proof. In the sequel, we are using the proof proposed by P. Georgiev [1]. First, we remark that, if /(x 0 ) = inf/(£>), then, taking x, =x0, we have conclusions (1) , (2) and (3) of the theorem satisfied. Assume f(x0) > inf/(DJ. We can suppose that d is bounded. Indeed, if this where not the case, we would consider the distance 1

d'= minis,d), where s = , ,—r ,—> r^T11 "' /[/{*>)f{D)\ ' /[/{*>)inf /(D)] r[f{x0)-mf inf f(D)] One can show that, if (1) , (2) and (3) are satisfied for d', then they are also satisfied for d, and evidently d'is bounded on D. The metric space (D, d) can be isometrically embedded into the Banach space of the bounded functions on D with the supremum norm from the mapping {x) = dx, where dx[y) = d(x,y) for x,y e D. Hence, we may suppose that D is a bounded closed subset of a Banach space [E,\ ||). Because of the fact that D is a closed set and/is lower semicontinuous, we have that epif is a closed subset of E0 = E x R ( endowed with the "max" norm as in Proposition 6.1). Consider 0 < 8 < y diam(D). Let s > 0,

5d

5d 5d ,_£0
i-L

n *

',' 5^ 11 ={(*»-*) =={(x,-£)e |W<J + ++<5 ^ S}},1, }, B {(x-e)eE 0||x|
B 2^ {(*,-*) e£ 0\\\x\\
K {^{Ab\ABR ^ ,x=K(b,B^) = ^ ( 0 , 5 , ) (={A6|AeJ^,&e£,}), {^{Ab\ABR++,b&B ,b&B11}), }), K K =K(0,B (={Ab\AeR (={Ab\AeR }). 2) ++,beB 2) +,beB K222=K(0,B =K(0,B (=Ub\AeR ,beB222}). }\ (={Ab\AeR 2}). 2) +,beB Because of Proposition 6.1, there exists 8? 82 >> 00 such such that that K33=\J^{{x,t)eE K =\J^{{x,t)eE \d0{{x,t),B )
381

VARIATIONAL PRINCIPLES We shall show now that there exists t0 e R such that ((x0,tQ) + K2K )f)epif *. 2)C\epif*f. f{x00)j < < +oo, +00, take take tt00 = = +00. +00. Since/is Since/is not Indeed, if /Jx f[x = j\x f(x0).0). suppose Suppose /\x f{x00)) = = +oo. since/ is not nol improper, there there exists exists aa point point Xi Xi ee D D such such that that /(*i) /"(JC.) < < +00. +oo. Take Take 3

F ] _ _ ^ o | „ _ *1 ~ *Q , // 1.

/ l 1i1^>_ f e — ^ , z = 3—*L and f00o = /J\n( * ,vu1; ) +V ^., 1i d , rf Then we have J (*o»'0) +*i (*i fo---444 IKM IKIN Mrf<*, (*i > /(*i)) ===(*o»'o) (* +^1 IK (*i./(*i)) .» 0. 'o) +^i(»i

which shows that (^^/(x,)) e(x00,t^o) (^./(x,)) e(x +K2. The set {(x0,t0) + K^r\epif is 0) a bounded subset of £E0 .• Indeed, if (x2,t2) e((x ^((x0,t0) + Ki)(\epif, K^Ciepif, then {x2,t2) ={x0Q,t,t0Q)) + + \[z \[Z22,-e), -E),

where where f(x /(x2)0, 2 >0, and we

have inf/(D) < - -£^<4 *<
+ St) +SS,) Sj) + x)

,

(r (t(f00-Mf{D)){d -inf/(D)Xtf +++<8?g,) 1x)) 0-inf/(D))(J

-— "■ - ro ro HM

£

-••

If we denote _,, || (f (r00-inf(£>))(rf -inf( £>))(<* ++
we obtain |x 2 ,f 2 )|£max{/ 0 ,f 3 }. Thus, we can apply the strong Phelps lemma [Theorem 5.11] and obtain a point 382

GENERALIZATION OF EKELAND'S PRINCIPLE (x.,a) e((x00,t,/00)+K )+/C ,f (x.,a) e((x )f)epif ( ( 10 ,^)+A",)Hep/ ^ 0))+K,)nep7/ +Auriga/ 3)f]epif 3 )nepz/ c ((x that: such that i,) ((x,, ((A:, , a) a) + +K K22)) fl D fl epif epz/ epif == {(x,, {(x,,a)), a)}, a)), and and whenever c W {w„ }„ejV (x.,ar) W KKLL# *"^"^(*•' (*•'°0'a)'whenever {"*}„ (**,a) ++KK22 and and ejV c (x.,a)

do{un,epif)^>Q. {xQ00,t,t00)) ++x{y-e), X[y,-e), where where XXI X>> 00 Obviously, / ( x * . ) = aa.. We have (x.,/(x.)j (x.,/(x,)) = \x {x X[y,-e), where and \y\ ||>|
/A t=—

~

and

x

*** ~xX00xX00 II II II II

llvll

flvfl Ml

X. X . ——x Xn nn\\ > J! ^1 ^ ,J7 i £f —

d + S,

« + O, ■

f{ -Xe
,/(,„)-,! *-4+!*gfSL
f{xo)-r\x*-x4JrS + S, Q) )~H|**-*o|| + *o) + £> which is a condition of (2) in Theorem 6.2. If v = 0, then x» = x0, and (2) is also fulfilled. Let x. lfy x» ^* x e D, and denote . ||x» |x» -- x|| x|| X x - x» X, J„/ \ ., L A , zz== and = /f\x*)( x . ) - / lXe. £. X ==--, and/ t= d |=d Then, ||z /; = /(x.) - A £■ = / ( x . ) - y | | x . - x | | = / ( x » ) - 7 d ( x t , x ) , f(x)>t = f{x,)-Xe= / ( x . ) - y ||x. -x\ = f(xt)-y d(xt,x), which is condition (1) of Theorem 6.2. To complete the proof, let which is condition (1) of Theorem 6.2. To complete the proof, let 383

VARIATIONAL PRINCIPLES WHEAT

P»

3Ild ^' n* ' *„ * X* 3Xld AXn) + 7 V* /(*»)••• !*♦ "~*n\\^> *XA^> J - > /(*») \X* ~ /(*»)

C
X

n\\

\ = f„ ==f{x.) \s. We denote 4, == ^ ' - , ^"" ,, zn = *" ^ - =- - and t„ f{x.) -- \s. fa. d ' A n rf and =[x.,f\x (x.,f(xt)) +A„(z„,-s), We have ||zj |||zznj|| == c/ cJ and (x„,t„) \x (xnn,t„) ,t„) = = [x*,f(x )) +\{z„,-s), + ^l(z„,-e), which which shows shows that that tt)) (xnn,t„)e(x.,f{x {x ,t„)e[x. K2. = (*.,/(*.)) tf{x.)) , /t)) ( * . ) ) +++*2-

/(*J-'„=/(*J-/W+^V^->°

We deduce f(x„)-t = f(x„)f(xn)-f(x.)+ f(x.)+ g|1**^ " * ,——->0, which implies implies that that /fix.)-t= k ) -n h=/(xj - /(*.)+ ^n|1 ->0, -> o, which di({xnn,tnn),epif)^0. di((x ),epif)^>0. ->0. Us Using (12), (i2), we we have (*.,/(*.)), and and the the Using have' (*„,/„)->• (*„.'„)->(*•./(*•))> theorem is proved. ad. ■I The amended strong Ekeland's principle implies naturally, a strong Flower Petal theorem. Given a metric space (D,d) and a function f:D—> f:D->R,R, we say that the set PrtS + f(x) 0, y > 0, XQ x0 e€ D, and / / Certainly, if ls d f(x) = d{x,b) d(x,b) for ib e D, D, then i^o'^o,/) Py,0(jc0j/) exactly the petal i^(x0)6)defined in Section 3.

W)

Let a:a:R ^>R R+ +-» R+ be an increasing real continuous function such that a(0) = 0. Let X a D be a non-empty subset. We say that f:X-+R / : X-> R is is a-H6lder continuous if \f{x) -- /(v)| < a(rf(x, |/(x) o(d(x,yj) v)) for all x, x, v e l . Theorem 6.3 [Strong Flower Petal Theorem] Let X be a closed subset of a complete metric space (D, (D, d), d), f : D -> -» R bt be a a-Holder continuous

384

GENERALIZATIONS OF EKELAND'S PRINCIPLE function bounded from below on X and let 5 > 0, y > 0 and x0 e X. X Then P x suc there exists a point x. x, eXC\PriS ,f) such rA(x0o>f) > that : (0

Pr0 (x ,f)C\X ,f)(\X = {x>},and {xt},and P.Jx 7,0(x Q n0.nr\X=lxA.and

(")

{x ^-x. whenever [x ^x. {x„} {xnn}\neN Py0(xt,f)andd(x d(xnn,X)^ 0. n}n£N neN neN c Pyy 00(x„f)and n,X)^0. i*»)„ {xt,f)and d[x ,X)^> ejv -> *'

Proof Applying Theorem 6.2, we obtain a point x* x. ee A"such A"such that that /f(x,) (x.) < < /j\xj ( x ) ++ yd{x yd(x , x), for _/br ee X, tt,x), j\x.) ^2^JC*,XJ, for every every xx eX, A,xx ** x*, x*, /(x») f(x )
(6.1) (6.2) (6-2)

„ - » x* .,-¥x whenever whenever \x„\ {x„\
„->xt t whenever n&N

and CU1U /(x„) + y d(xt,xn) -»

(6.3) (6-3) f(xt).

Condition (6.1) shows that, for each xxeX\{x e. e X\\x*}, X\\x*), t),

xx£P x£g PPyr00{x {x ,f), one has has 0,f), Q0{x 0,f), one

Py 0(x0,f)r\X={x,}. Condition (6.2) shows xt e ^ ( xPrjS \x/0,f). Condition(6.2) )ro,s* czP ^, X ) -—>0. » 0 . There exists y> |xnnn)\}neNczP |jc c Prfi/{x ) : „ / ) such that ^d\x _y„ >n„ eeX, X, rSJ Let \x ,f) 4(*xn„,X) > 0 (t\x*,f) «eN AT with withrf(x d\x ->0.0. Using Using(6.1), (6.1),we weobtain obtain dlxn,y„) ,y )-> d x ,y ) +n)-f(x,)< 0o <„) f{yn )-/(** )^yrd(x 4**>*«W n>y

f(y + f(x f{yn-n))-f{x + /U ) "- /(*„) + f(*„) -f{x.)
x + \f{y \f{ynnn)d(xnn,y ,ynn)) ++ a(d{y a(d(yn,x ,x„)) -> 0. 0. d{x )~/(x„)|< «*(*«»?») o-(c/(^,x„)) ^* rYr d( n>y / ( * J ^ ry d(x -> nn,ynn)n) + n)\ x n Now, condition (6.3) implies that {yn}n N~> —>x * t aand ^ hence, {xnn}neN x* neN -> x»

d(xn,x.) ^d(x 0. ■ since *») Another interesting generalization of Ekeland's principle can be obtained

385

VARIATIONAL PRINCIPLES following an idea that has been recently developed by W. Oettli and M. Thera [1]. Let (E, d ) be a complete metric space and F:E x E -» (-OO,-H»] be a function which is lower semicontinuous in the second argument and satisfies the following properties: ij) F(x,x) = 0,0 ,for f o rall a lxl xee ££,, i,) F(x,x F\x,y) < F[x,z) F{x,z) + + F(z,y) i2) F[x,y) F{z,y) for for all all x,y,z x,y,z ee E. E. To the mapping F, we associate the set-valued mapping Y:E -» E defined by, T(x) = {v eE\F(x,y) + d(x,y) < o) for all x e E. We then have the following generalization of Ekeland's principle. Theorem 6.4 [Ekeland] Let (E,d) be a complete metric space and F:Ex E —>(-oo,+oo] be a lower semicontinuous Junction in the second argument satisfying the assumptions (z'i) and (12). If there exists XQ e E such that inf F\x0,xj > -00, then there exists an element x,€ Esuch that xeE

F(x,,X) + d(x,,x)>0 F(x,,x) d(x,,x) > 0for for all all xx eE, <=E,x* x*x. x..

(6.4) (6.4)

Moreover, for every x e E, there exists x, e T{x) satisfying (6.4). Proof To prove this theorem, it is sufficient to remark that T is a dynamical system satisfying the assumptions (1) (4) of Theorem 3.1. We have that x eT(x), for every x eE, which implies that T is a dynamical system satisfying assumption (2) of Theorem 3.1. For every x e E,x eT(x) is a closed set since F is lower semicontinuous in the second argument and d is continuous. So, we have that assumption (1) of Theorem 3.1 is satisfied. To verify assumption (3), we must show that, for every y e r ( x ) , we have

r(v)crOc). Indeed, if v e F{x), we have F{x,y) + d{x,y)<0. F{x,y) d{x,y)<0. 386

(6.5) (6.5)

GENERALIZATIONS OF EKELAND'S PRINCIPLE Let z e T[y). Y[y). We then obtain F{y,z) + d{y,z)<0. F(y,z) d(y,z)<0.

(6.6) (6.6)

From (6.5) and (6.6), we deduce F{X,Z) + d(x,z)
+[F{y,z) + d{y,z)]< 0 + 0 = 0 , +\F(y,z) which implies that z e eY(x). Y(x). To verify assumption (4), we consider a sequence \xn}

a E such thai c= that

xn+x € r[xn), for alln all n e JV N and and with with xi *i arbitrary arbitrary in in E. E. We We have have F(jc„,x F(JC„,X„ +1)) n+1 +d(xn,xn+l+1) ) < +(i(x„,x„ < 0, 0, which which implies implies d(x„,xn+l )<-F(xn,xn+i ). n+1)<-F(x„,x n+1).

(6.7)

From (i2), we get F F vv hence ^(vvo>* o^«n+ +FF(x + i) ^ ( o>x n) + n,xn+i), and o>xn) (x«>*«+i)» andhence

-F(x„,^„ -F(x )) < < F(vF(v F(v 0 ,x n,xn+1+1 0 ,^„)0,xn)-F(v 0,xn+1 n+l).

(6.8)

From (6.7) and (6.8), we obtain d(xnn,xn+1 F(vQ,xnn)) - F(v ).). d{x F(v0,x n+i n+x) < F(vQ,x 0,x n+1

(6.9) (6.9)

Now, because of the fact that inf infF(v F(v00,x)>-co, ,x)>-oo, we obtain, from (6.9) jre£ jre£

CD CO

Y,d(xn,xn+i) < +oo, which implies that assumption (4) of Theorem 3.1 is also Y,d(xn,xn+i) < +oo, which implies that assumption (4) of Theorem 3.1 is also satisfied. Hence, from the theorem first mentioned above, we obtain that Y satisfied. Hence, from the theorem first mentioned above, we obtain that Y 387

VARIATIONAL PRINCIPLES has a stationary point x, ee E, that is we have Y\xt) = \xt} which is exactly condition (6.4). The last conclusion of Theorem 6.4 is established by setting x0 = x in the proof of Theorem 3.1. ■ Corollary 6.5 If all the assumptions of Theorem 6.4 are satisfied, there exists x, x, ee hE such such that that exists x, ee rix T\x00]) and and F[x,,x) F[x,,xj + + d[x,,x) d\x, ,x) > d\x,,x) >000,,,for for all eEand and x *■ x^x,x, x, jor all allxxx eE eEand x*x, Remarks 6.1 1. From Theorem 6.4, we obtain, as a particular case, Ekeland's principle if F(X, V) = f(y) f{y)-f{x), F[x,y) - f(x),where/'is f (x),where/'is lower semicontinuous on E and bounded from from below. = y/{Ty-Tx), where £ is a normed 2. An interesting example is F[x,y) F\x,y)-y/\Ty-Tx), /?U{-K»} vector space, Te -»R U{+°°} iis a subadditive Te L(E, L(E, E), E), and and y/:E y/:E -> such that u/lO) = 0. mapping on E such that y/(0) = 0. Theorem Theorem 6.4 6.4 implies implies aa generalization generalization of of Caristi-Kirk Caristi-Kirk theorem theorem (Caristi, (Caristi, J. J. [1]), (Caristi, J. and W. A. Kirk [1]) and of Takahashi's Theorem [1]), (Caristi, J. and W. A. Kirk [1]) and of Takahashi's Theorem (Taka(Taka->/*U{+°o}is hashi, ->/fU{+°o} is hashi, W. W. [1]). [1]). Suppose Suppose again again that that aa function function F:ExE F:ExE ->/fU{+<»}is lower semicontinuous in the second argument and satisfies assumptions (ii) lower semicontinuous in the second argument and satisfies assumptions (ii) and and O2). O2). Suppose Suppose also also that that there there exists exists xxQQ ee E E such such that that inf inf F(x F(x00,x)>-co. ,x)>-co. Let T be the dynamical system defined by F as in Theorem 6.4. Let T be the dynamical system defined by F as in Theorem 6.4. Theorem 6.6 IfD is a subset ofE which satisfies the condition Theorem 6.6 IfD is a subset ofE which satisfies the condition

I:

{for every x e T(x0)) {\x [x e E such that x * [x e E such that x *

\\ D, there exists , v 3-c and x € T(x). T(x), x and x € T(x),

(6.10) (6-10)

then there exists x, ee T{x r(x0)C\D. Proof We obtain this theorem as a consequence of Theorem 6.4. Indeed, supposing the assumptions of Theorem 6.6 satisfied and applying Theorem 388

GENERALIZATIONS OF EKELAND'S PRINCIPLE 6.4, we obtain an element x* e r(x 0 ) such that F{X*,X) + d(xt,x) > 0, for all xeE, x*xt. From (6.10), it follows that x* e D. Hence, we have XteT(x0)r\DM

Theorem 6.7 [Takahashi] If the following condition is satisfied for fi every x e T(x0) with iir {\ inf F(x,x) < 0, there exists

(6.11) (6.11)

xeE

X xeE such that x*x and x e T(jc), xeE such that x*x and x e T(jc), then there exists x* e r(x 0 ) such that F[x,,x) > O,for all x e E. then there exists x* e r(jc0) such that F[x,,x) > 0,for all x e E. Proof Proof This theorem is a consequence of Theorem 6.6. Indeed, suppose the assumptions of Theorem 6.7 satisfied. Choose D- \ x e E inf F{x,x) > 0 f.

From (6.11), we obtain that condition (6.10) of Theorem 6.6 is satisfied, and, hence, there exists x. e T(x0)C\D. From the definition of D, we have that inf xeE

F(xt,x)>0.m

Theorem 6.8 [Caristi-Kirk] Let T : E —> E be a set-valued mapping satisfying the following condition :

({{:

r(x Q0 ) there exists for every x e T[x T{x) such that x e T(x). x e T(x)

(6.12)

Then there exists x* € T(x0) such that x, e-T\xt). Proof This theorem is also a consequence of Theorem 6.6. Indeed, if the assumptions of Theorem 6.8 are satisfied, we choose D- {x e E\x B T(X)}. We can show that condition (6.10) follows from condition (6.12), and, 389

VARIATIONAL PRINCIPLES applying Theorem 6.6, we obtain an element x. e r(x0)C\D. From the definition of D, we get that x, e r(;c.),and the theorem is proved. .■ From the generalization of Ekeland's principle (Theorem 6.4), we also obtain, as a natural consequence, a generalization of the Drop Theorem. Let (E, d) be a complete metric space and F:ExE^>R\J{+*o} F:E x£—» /fUj+ooJ 1be a mapping satisfying the following conditions: 1. F is lower semicontinuous in the second argument, 2. conditions (ii) and (i2) are satisfied, 3. there exists x0 e E such that inf F[x0,x) > -oo. xeE

For a > 0 and a e R, let Ta be the dynamical system defined by: For a > 0 and # e J?, let r a be the dynamical system defined by: ra(x) = [yeE\F(x,y) + ad(x,y)<0},forallxzE. ra(x) = {y eE\F(x,y) + ad(x,y)ZO},for all x eE. For a>0,a eR, and ^ c £ being a non-empty subset, let r , ^ be the dynamical system on A defined Ta A(x) = Ta (x) f)A, for all xeA. We have the following result: Theorem 6.9 6.9 Let F be a mapping as considered before, B c E be a bounded subset, and A czE be a closed subset such that x0 e A. Given b e E, we denote A(r0) = ix e A\d{b,x) < r0\, where r0 > d(b,x0). If sup{F(x,y)\x e A, y e B) = p < 0, then, for every0
Proof

14J

Let — . We have that x0eA(r Let a e 0, — and, applying <=A{r 0),0) \ rr00+r_ +r_ Theorem 6.4 with E replaced by A(r0) and d replaced by a d, we obtain an

390

GENERALIZATIONS OF EKELAND'S PRINCIPLE element x* &ra(x0)C\A(r0) x e A[rQ) and x*xt.

such that F(xt,x) + a d(x,,x)>0,

for all

Then x e A[r0), with xtx,, cannot be an element of

Ta(xtj, that is, we have r a (xt) - {xt}. If x e B, we have F(xt,x) + a d(x,,x)< p + a [d(xt,b) + d(b,x)] < p+a [r0 + r]<0, which implies that x e ra(jc*). ■ Remark 6.2 Suppose that [E,\ |) is a Banach space, F{u,) is a convex function, and c/(jc,_y) = |JC — _y||. In this case, r a (x») is a convex set, and hence it also contains, together with x* and B, the drop D(xt, B). The conclusion of Theorem 6.9 implies that D[x,,B)f]A(r0)={xt}.

If we

have the particular case where F{x, y) = d(b,y) - d(b,x), then, the condition suplF^yjpc e A,y e Bj = p < 0 implies that sup[6,y) <: mf(b,x)
X&A

with center b and radius r0, and therefore every point of D[xt,B)r\A must be in A(r0). In this case, the conclusion of Theorem 6.9 gives D(xt,B)C\A= {x,} , which is the Drop Theorem. We will now give a generalization of Ekeland's e-principle [Theorem 4.6] and of Borwein-Preiss' smooth s-variational principle, replacing the distance and the norm by a "gauge-type" lower semicontinuous function. This new unification and generalization was recently obtained by Y. X. Li and S. Z. Shi [1] using a refinement of the classical proof of Ekeland's evariational principle. In 1987, J. M. Borwein and D. Preiss [1], while studying some differentiability problems about convex functions, proved the following smooth evariational principle. 391

VARIATIONAL PRINCIPLES Theorem 6.10 [Borwein-Preiss] Let (E,d) (E,d) be a complete metric space, / : £ - > / ? /?U{+°o} U { + ° o } be< f:E—> be a lower semicontinuous function bounded from below, X > 0, andp > 1. Ther, X>0,andp> Then, for every x0 e E and e>0 such that ) < mff(x) f(x0)<mff(x) f{x) + + €, e£, there exists a sequence \xn}
which converges to

xeE

some xe e E and a function

m go

pp

where ju jun>0 for all n = - 1,2, \,2,....and ....and Z //„ = 1 such thai that: where ,) d{x d(xd(x < 1. ij) U) d(x ,xQ0)<X, )
n=i

Y xeE

h) + ~

f(x + — jjjcp (ppp{x (x£), p(x)> £) f(xe) is) /(*) /(,)+ h). f{x) f{x) + -~(Ppp{x)>f(x + e) pj-

0, then, for any 0 < X < 1, there exists x£ e E such that vl||a - xe\ < f(a) - f(xe) < e and/(jc) + A\x- xx\ > f(x£), whenever x *■ xe. To obtain this form, we consider in Theorem 4.6 the distance d defined by the norm | | and replace d by Ad. Considering this form of Ekeland's E-principle, we remark that, even if/is differentiable, the perturbed version

/ + A|(-)-JCJ

will generally not be

differentiable at xe. A reasonable smooth variant has long been sought. The problem has been solved by Borwein-Preiss' smooth e-variational principle. This principle has very interesting applications (Borwein, J. M. and D. Preiss [1]), (Phelps, R. R. [12]). Remark 6.3 Remark 6.3 Ekeland's e-variational principle is not an exact consequence

392

GENERALIZATIONS OF EKELAND'S PRINCIPLE of Borwein-Preiss' result. The next result is a generalization of both Ekeland's e-variational principle and Borwein-Preiss' smooth e-variational principle, together with some significant improvements. Let (E, d) be a complete metric space and pr.E p.E xxE^>R E -> +R+ U{+°°} U{+°°} be a function. Definition 6.1 We say that p is a gauge-type function if the following conditions are satisfied: 1) p(x,x) = 0 for all x eE, 2) for all T] > 0, there exists S> 0 such that for ally, z e Ewe have d{y,z) <JJ, p[y,z) <S => d(y,z) 3) —> p[y,z) isislower lowersemicontinuous. semicontinuous. 3) for for every every zz ee E, E, the the function function yy — » p(y,z) The next result was proved in (Li, Y. X. and S. Z. Shi [1]). The next result was proved in (Li, Y. X. and S. Z. Shi [1]). f:E^R\J{+oo} Theorem 6.11 Let (E,d) be a complete metric space and f:E —> /?U{+°o} be a lower semicontinuous function bounded from below. Suppose that a gauge-type function p on E and ia sequence sequence \S\Sn)n) ofofre* real numbers are given such that S0>0 and S„>0 >0 for all n=l,2,~. n=l,2,-. 1Then, for every x0 € E and e>0 such that

wr

£, f{x )<mff(x)
(6.13)

there exists a sequence \xn\\ c Ewhich converges to some xe e E such that p(x £,x0)<^ p(**>*o)^-f In general,

(6.14)

°0 X ,xX )<^r; M=0 p(x n=0,l,2,.-, p(*«»*«) P(*«»*J^«7-; "=0,1,2,.., £ nn)^^-> p( e> "=0,1,2,", (*«»*«) ^ ^ - ; "=0,1,2,.. >

(6.15)

393

VARIATIONAL PRINCIPLES 00

n=0

00

00

n=0

1=0 n=0

tS ZS f(x) + Y.S Y,5„p ZS„p x*x npn p (x,xn) > f(xE £) + ZS„ npp (x£c,x„) for all x*x £, e£,

(6.16) (6.17)

and, when Sk>0 and 8 • = 0 for all j > k>0, (6.17) may i e replaced by: for allx ?±xe, there exists m >ksuch that k-\ 4-1

)>f{x fix) Sip{x,x +dSkkkkp(x,x fix,) f{x)+J+JJJ8JIiiisp{x,x ) x,) p(x,x m tp iii(x, kp (x,x m) >e)f{x f(x)+j m eee) m)> iy +^S ;=o i. -=n0

If — 1

ZSS,p Z + Sip ,xt) ++5S6kkpp(x pS,,o(x..x_\ (*,, + 1Iy.S:o(x^x.) (x ip(x(x £,xeesi),x,) e,x ee,mx).mxmm).). 1=0 1-0

(6.18)

Proof We must consider two cases for \5 [Snn } . Case I: S„ > 0, for all n e A (^ a N and \A\ = <x>. In this case, we can assume, without loss of generality, that 8n > 0, for all n e N. We define ):={xeE\f{x)-^ T(x0):= [x [X eE\f(x) &E\f(x) SS00p(x,x )< 00)
(6.19)

Since xQ e T(x0), it follows T(x0) is non-empty, and, since / a n d p(-,x0) are lower semicontinuous, T[x0) is a closed subset of E. We also have s (y,xQ0)< ))

Take x\ e T(x0) such that

394

GENERALIZATIONS OF EKELAND'S PRINCIPLE f{xi) S0p(Xl,x0)< iinf [f( p(x,x )] + +-~ -^-, /{x^ SoP^x^ SSQ0p(x,x + / M + *oP(*i»*b)* n f l[f(x) / Wx) ++
and define again

(6.21) (6.21)

2S Zd00

r ( xXll):=^x ,):=ixBT{x ) : = \x { x eBT{X r ( 0x)f[x)+I p{x,x )zf(x ) l)/ ( * l ) ++ Sd00p( xxXl00)\. r(x,):= StP{x,x p( T{ sT{x f{x)+is S0Xlf p(x ,x00)\, )\.. (6.22) '6.22) i)^f(x Xu 000) ) f{x)+T itlp(x,x ii)
)and 7,(^:„_ (x„_ In general, suppose that we have defined x„_, JC„_, er(x„_ eT(xn_22)and (^„_1) such that T(x^):=\x e rn_(n_7xn_)f(x)+Y5,.p(x,xA _ 2 ) | /+(^xi+p(x,x )+ E ^i)iP ( x{,xx^) ,) l\x Tlx ,):=UeT(x nn^):=<xsT{x 2n)\J{x) T(x^):=\x ^T{x "id 2)\f{x) ) ;—n I

1=0 i=0

«-2 n-2

1

1=0 1=0 1=0

J JJ

^^/(v.)+ZMv..* / ( V , ) + Z M V . ^ ,( ) •• ^/(v,)+EMv^A

(6.23) (6.23)

We choose xn e T{x r(^nn_j) _x) such that

+ f{xntt))) + + n"'ZS ,x inf j / ((xx). ))++ ^"j:s x( ,^x^i,)\.)) l K if( f(x tSV inf +I5^(^^)} S/ 7V(ip(x,x T ip{x i)< 4 I ( v ^ j p f i/W +++ -^--*^-, ^^ <,) ip{x tln,x l)< i=o ^K-iK '=o J 22
^K-iK

<=°

>

»o 0

(6.24) (6.24)

and we set n 7{x 7{x y.= y.=\x \x xE l£,.p(*,*,.) (x,x) << f{x /(*„) + +nis ±S ,x) ) |. \.. (6.25) 7(x Tlixj\f(x) ^ J I / U ) ++ts £<3>(x,x + l%5 kiP/{x >(x (*„,*,) B):= iP(x,x lP iP tt,nXl l{xnBs}.= eElixj\f(x) lix^lfix) is ,x)\. n )+ [P () ( )
1=0 i=0 i=0

1=0 1=0 i=0

>J >

For every n E N, T(x T(X„)n) is a closed and non-empty set. From (6.24) and (6.25), we get, for ally ah> E e T(xnn), Snnp(y,x p(y,xnn)< )<

//(*„)+I
L L

i=o i=o

JJ

L L

i=o i=o

395

VARIATIONAL PRINCIPLES

inf

r / ( x ) +/=0 Z\f{x)^5 ^ p ( x ,iPx{x,x , . ) {l)\<5 < ^ -nd^-^-, -)

er J **er{x K-i)L n-i)L

i=0 i=o

'=0

-IJ

*2 oo0

which implies p{y,x n)<^j~ /> U* J *^

" ; ;forforalla11v^€e Tfo). ^«)-

(6.26) (6-26)

From condition (2) of Definition 6.1, we deduce that d(y,xn) - » 0 , and hence the diameter of 71(x„) is convergent to zero. Since E is complete, using Cantor's

Theorem, it follows that there exists a unique xe e

f]T[xn)

n=0

which, from (6.20) and (6.26), satisfies (6.14) and (6.15). Obviously, we have {*„} -> xc. For any ^ i t , w e have that x £ f) T[x„), and hence there n=0

exists an m e N such that

f{x) + II.S /(x) < lyp(x,x , x , )m>)/ ( x J ++ l £ , .ZS p (lxp(x ip(x i)>f(x m ,mx,x( l)),, i=0 1=0

1=0 i=0

and, from (6.19), (6.23) and (6.24), we can deduce that, for any q>m,v/e have //(*o) K ) ^ */ (/(*«) S £/./>(x ^E*,/> ( x m (*-*/)* , x,x,) , ) >> / (/(*,) xf(x a )) / (x*J«+) + m

q

+ + ±8tp(xffffe,x,), ,x,), ( *£e)), ) + t p ( xqq,x,) q,x\ > + Xtt S8iP(x 8iP(x ,x,) > /f(x f(x + ff «?,p(x «?,p(x , Xi ), i=0 1=0 i=0

1=0 i=0

i=0

which implies that (6.16) and (6.17) hold. We now consider the second case. Casell: S„ > 0 for all n e B c N,\B\ 0 and £ . = 0 for all j > k > 0. Without loss of generality, we suppose that S! > 0, for all i k, we choose x„ e 7^x„_,) such that 396

GENERALIZATIONS OF EKELAND'S PRINCIPLE

//(*»)+ZAP ( x j / xl kk//>> (( *x (*«.*/)* „ , *x , ) < 1=0 (=0 1=0

if(x)+Z6 inf \f{x) +ip(x,x % %p{x,x)\ i)}x>Xi)\ Pl{x,x)\ iP( •.*»)}-++ ^ r , , ,=0 i=0

1 jx^T[x e rVn-l)^ ^n,_)x)^ -

JiJ

2I

o0 O °0

(6.27)

and define ana we we aenne T\xxn):Ax __lI)\f(x) ++&1,6, %6 71 . H * ee T\x 7tx 71 nBx„_, )|/(x) ^(x)+ . / 7 t(p{x,x p x Jx,x,. x (t) \. *.

i=0

k

+S + + ^^kpp p{x,x ( x(x,x„) , xn)
> J

(6.28) (6.28)

Now, by the same deductions as above, (6.14) - (6.16) also hold. If x * xe, it may follow that there exists an m > k such that k-\ ft —1 + ^ /d?k(p{x,x // (( ** )) ++ IY.S ^ . /ip{x,x 7 ( x , Xi), . ) + x , X mm)) i=0 1=0

£-1

>f{xm)+1S )+lSip(x ,xi)i) ip(xmm,x

>/(».)+I^(w) 1=0 k-\

** //(**) ( * , ) ++ k-\ II S,p ,xj) + p (x„ (xE,xxm),), S,p {x (xet,x,) +6 £Ak/> m * /(**) + i=0 I S,p (x£,xi) + Skp (xE,xm), 1=0 i=0

that is (6.18) holds, and therefore the theorem is proved. ■ Remarks 6.4 l.In Theorem 6.11, the sequence {5n} may be any positive numerical sequence, which may not even be bounded. 2. If, in Theorem 6.11, we consider p(y,z) - s d(y,z), 60 = 1 , and, for all n e N, 8n = 0, we obtain Ekeland's s-principle [Theorem 4.6]. 3. If in Theorem 6.11, we consider

397

VARIATIONAL PRINCIPLES

x P M = ^ M V ( * o ) = ^/M pM=^M''/W=iS/( ) + e'< + £,<Mf(x) inf /fix) ( x ) ++ s,s, SS00 == £,S p.p.0,= =Pi=l-y>—, ,= 11-- yy >>—, —, S£

xeE

with y > 0 and 8n = p n+]> 0, we obtain the theorem of Borwein-Preiss with some significant improvement, which replaces (i2) and (i3) by f{xe) + -^9 {xEJ) f(xXs) + ^i

f{ /(**) p ( *PP*{ )£eh > for f( )+-jjPp\ e) ^-^
When (E,d) is a Banach space [E,j |) from Theorem 6.11, we obtain the following result: Theorem 6.12 Ief(.E,J Z,e/(is,| |)||j be a Banach space, /:£->J?U{+«>} f:E —> RU{ +°°} be a lower semicontinuous function bounded from below and A > 0. Suppose that /?':£-> fl+U{+°o} p': E -> R+ U{ +°°} zs a lower semicontinuous Junction such that

p'(o) fO ^(o) == ooO wfe) P'(O)

U)

c /or for

[i2) / "' even* every {y k}] {'2) {^i}

c £, £E, p' ccz \\y I1-» k)^>0implies t) -> >p'(y P'(>(>"t) ->0°implies zwp/z'e^\\y ||vkk1I\\-» ->000

(629) (6.29)

and that 80>Q, Sn>0, n = 1,2,...is a sequence of non-negative real numbers. Then, for every x0 e E and e>0, such that /(*„)< inf/(*) = £, *, xeE

(6.30)

there exists a sequence \xn) c E which converges to some x£ e E, such that

398

GENERALIZATIONS OF EKELAND'S PRINCIPLE , s

)<jfL. p'(x -x000)<^. )<^-. p'(x£££-x Moreover, in general, p\x£e-x-xn)<-—; )<^-; PV n)
s

22 d d0

(6.31)

So

n = 0,1,2,...., 0,1,2,....,

(6.32)

0

)+ + T,S„p'(x f(x,)+ f(x ts T.8„p'(x )
(6.33)

fix) T.S p'(xn)>f(x - xn)e)+ts > f(x f(xn£pix ) e-x + n2ZS 2ZS p'{x££-x-£n.),for xn), for all all x*x, x±x£. f(x)+ / ( * ) ++ l8 T,S ),forallx±x £)+ np\x-x nnp'(x-x„)> nnp'{x

(6.34)

*x eG t£

n=0 n=0 n=0

n=Q n=0

«=0

f^Tiew, 8k > 0 awe? £ 7 = 0 ybr all j> k>0, (6.34) way £>e replaced by : for all x &xe, there exists m>k such that jJfc-l f c - 1l

//(*) ( * ) ++ T,8 )>f(x f(xe£ee))) 2ZS p'{x-x lip'(x-x i)i)i)i) + m)> mm m)>f(x + ZS HSip'(x-x + 8kp'(x-x SkSp'(x-x ip'(x-x kkp'(x-x n=0 ~° k i *-l , , , . (*, 0' (x (x, +1S + 1I 8t£,/?' [x 585kiykkp' 0p' +1 Sfi,tfi0 (x (x,e£ --xXixX (x£E£ -- *xxxmmm).).). i)t)() + £

(6.35)

n=0 n=0

Proof In the proof of Theorem 6.11,vie consider" p\x,y) x - y) for p{x,y)== p'{ p'{x-y) all x,yeE. jc.^eE. ■ The study of generic differentiability, that is on a dense Gg subset of the domain of non-convex functions, has been considered by many authors. An interesting result in this sense was proved in (Georgiev, P. [2]). His result is essentially based on the following variant of Borwein-Preiss' £-variational principle. Theorem 6.13 Let (E,\\ ||j |) be a Banach space, DczE a closed non-empty set, and /:£>-> f:D—»l?U{+°°} l?U{+°o} a lower semincontinuous function. Given ££>0, ■ >0, A>0,p>l, A>0,p>l, andxx0 0 eD eDsuch that >1, and 399

VARIATIONAL PRINCIPLES i n f | / ( x ) + —||x inf j^\\x --xxx0\\"| f |[ < f{x / ( x00) ) << inf/(x) inf / ( x ) + *, e, 0\\" jre£>

JTSD

there exists rr,x > 0 where for every r2>0, > 0, there f/zere exists exwta a point usD, a sequence {x„}" {x„}™==Q Q cD convergent to u, and a sequence {/"„}"=0 c[0,l] 00

with 2_, V H„ fin = 1 a«
r a// e D M //°^ "A * * e D , w/zere v> /f((u")+J) +7i4A(MA)*(") * //((**) )++44 AAM >where 7

A,

A

A(*) = I > J * - * T . n=0

2J 2) V || u | M- -xX00|| < | <2A,,
x x

n>l,

r

V) Iki-^ol^l\\ \-x o\\^ \x

4

)

\ \- o\hi

Proof We fix sx < s with the property that / ( x 0 ) < inf f(x) + £sx, and we proof We fix sx < s with the property that / ( x 0 ) < xeZ) inf/(*) + i > and we define define ju junn == (\-q (\-qxx)q" )q"

■m

Ml,!^\-\-> ..

xeZ)

for for all all nn = = 0,1,2,.., 0,1,2,.., where where qqxx ee 0,min-h, 0,min-h,

1

For fixed 8* 0 , / ( x 0 ) - i n f | / ( x )

+

^||x-x0l} •

we denote D, = | x e D | / ( x )

+

^|x-x0|r
+

^|x-x0|r+(5}j.

Since/is lower semicontinuous, the set Ds is closed, and, since we have

400

x0£Ds,

GENERALIZATIONS OF EKELAND'S PRINCIPLE r, := dist(x0,Dg) > 0.

r

/

j]r

0,min ssuch that, for q:= — Let r2 >0 be fixed and q2 e O.murw,,— Let rr22 >0 be fixed and qg22 e 0,min
■=-fer-^r<« "-*•<-»■ s:=[A±_.l

._L.lV]

P

< 1 and ^ 9 < r '■-{^) MT) \\-q ) \-q ^s) <—*.<*■

y^ ,,

(6.36) (6.36)

x

We define by induction the functions {/„}"=0 and the points {*„}" {x„}"=0 as follows: Jo'~ Jo'='■ ~J J'' ' Jo

/« = f(/ )M + J r f-jjpMnfct|*-*T fn +l(i(*) ) = n\ » f

(6.37) (6-37)

/ffnn„(x (x ( nxn)<mf j)<Mf dist(x which isis exactly exactly xx&D i\\>dist(x 0,D s0),Ds) ==rllr, i, which Since xx^eD \\x00-x-x x\\>dist\x 0,Dg) condition (4). (4). From (6.37) and 6.38), we deduce X X ft . J CKn+\ + l ~Xn\ n\\ n|| =~ - / fn+\\ nfn+\\ + l lxX^n+\)~ n+\) n+lj fin~7^V =

X

X

X l)-f $ n) X = /fn+ = « l{+ lnn+l)-fn (+^ + n+ l )+-\(/ n) «+l(^) n+

++

/nl^n+l fn\XXn+$ n+\)J fn\ X

XX

<£ <£

< \) n \+S f +\) / n ( J C « ) /*{*„)-fr,{ - /fn«{ (n^)-fn{ ^ + l +++\+£ «nn', + l )nn+ +

which implies

401

VARIATIONAL PRINCIPLES

"W

..-.f«-.+o"'-,fifa±flrw"' iu i ± L±£ 1 =*Ur(wJJ -Jfi&±a A k „+1i _ lJ
V

1"

"

V

\i/p

"" I e% Mn J

" l %

(l-?.)J U, (l-<7.)J (i-?.)J

J

Ui(i-«i)J

-M^T vVl-gJ l—^j \\-q ) y x

^e) ^

^

7f^/' V uVf/ \SJ

and, considering (6.36) for m m > n, we obtain \\x \\x | * mmm-x--x | | < ^ ( l -
(6.39)

The last inequality shows that {*„}"=0 is a Cauchy sequence, and therefore, there exists a point u sD such that w = lim xn. n->oo

Now, we remark that the assertions (2) and (3) follow by (6.39). To show (1), we suppose y> 0 to be given. Since/is lower semicontinuous and A is continuous, there exists S>0 such that f(u) ++ jFFA(u)
+ + jFFA(x) + + ^ /(«)+JrA(«)
(6.40) (6-4°)

whenever |JC - w|| < S. We choose n to be sufficiently large such that; ^enn<—, n - «|| < 8 and < j 5 \x Ik-"!<<*

(^)i>*k-**r<7r (^)|/*i*»-**r<jFor every x e D, using (6.40), (6.37) and (6.38), we obtain For every x e D, using (6.40), (6.37) and (6.38), we obtain

/( U ) + ^A( U )
402

GENERALIZATIONS OF EKELAND'S PRINCIPLE

++ yy, ,

and (1) is proved. ■ It is also important to remark the following variant of Borwein-Preiss' evariational principle, which is important in the study of the smooth drop property. This variant was considered by P. Georgiev, D. N. Kutzarova, and A. Maaden [1]. Theorem 6.14 Let (E,\ |) be a Banach space and DcE \E,\ ||j 9 c £ a closed nonempty set. Let Let /:£>-> f:D—> flU{+°o| R U{+°°} be a lower semincontinuous bounded from below function. Then, for every ss >0, >0, XX >> 0, 0, and and xXi D, there exist x ee D

W «,2 C j D a w J {>"«Li

such that

1) f>» 1) 1 ^ = = l.1n=\

eN, nsN, 2) /(*„) x, € D (in the norm topology), £

444

°°

xxx

M ££ °° °°

X

X

r al1 x e D

2 /fcO + iiI T rr ll2X -*«P /(*•) T Z forallxeD, A A-* n=l A

5

X* X&X,,X,,

n=l n=l

A

«=1 «=1

) /w+nxxni^/fa)£ °° °°

A A A

n=l n=\ B=I n=l

Proof The proof is similar to the proof of Borwein-Preiss' variational principle proposed in (Phelps, R. R. [2]. ■ Another interesting variational principle obtained recently, is the DevilleGodefroy-Zizler variational principle. This principle has also important applications. Before this principle can be presented, some notions need to be to introduced.

403

VARIATIONAL PRINCIPLES Let (is,I (is,I I) be a Banach space and and f:D-> RU{+<»} function.We Wesay saythat that t:D-+K\JRU{+<»} -H» a£afunction.

Eifif f(y) f{y) = = inf mff{x) f strongly strongly exposes exposes the point point yy ee E 'mff(x) f(x) and and | y - j d | - » 0 jre£

whenever //(«„) ( « „ ) -> f(y) ■ In this this case, case, we say thaty thaty is a strong strong minimum. minimum.InIn many optimization problems we are interested in finding a strong strong minimum. minimum. We say that a function
is open for alln alln e N. Theorem 6.15 [Deville-Godefroy-Zizler] [Deville-Godefroy-Zizler] Let \E,\ \j be a Banach space and \F,\ |L) be a Banach space of real valued continuous functions on E such that the following assumptions are satisfied: sup[\g(x)\ 1) for each geF, \\g\\F > \\g\\x = sup \\g[x)\

xeE], xeE\,

e &

2) if g F and x eE, then T rxxg:E g:E -> - » R, given by T txg[t) +1) is in Ig[t) = g[x +1) Fand\\Txg\\F=\\g\\ =\\g\\ F, & F and a eR, then h:E —> R given 3) if g s eF given by h{t) hit) = g(at) si at) is in F, F.

4) there exists a bump function q>

/?U{+°°} is a lower semicontinuous

function Junction

bounded from

below and where dom(f) is non-empty, then the set of all geF such that / + g attains a strong minimum on E contains a dense Gg-subset G ofF. f+ Proof We follow the ideas of the original proof proposed in (Deville, R., R, G. Godefroy and V. Zizler [2]). We define, for every n e N,

^Un == L LeF3 e FXn3eE,{f x „ € £ , ( / ++gg)(x ) ( ^n)<Mhf nM \\>^\. . Using assumption (1), we can show that U„ is open. We show that U„ is

404

GENERALIZATIONS OF EKELAND'S PRINCIPLE dense in m F. To show this, we consider an arbitrary g e F and an arbitrary 8 > 0, and prove that we can find h e F with \h\FF < s and xnn e E such that {f + g + (f + h)(x h)(x„)<mfl{f h)(x )<M\{f ++ g + g+h){x)\\\x-x\\^\. h){x)\\x-xn\\>±\. nn)<mfl{f Using assumptions (2) and (4), we can find a bump function

+$\y - 4 <- ((il -- 01k - c ll+P4 ll+M\xx - 4+Pi 4+P{11 - 01k - "II HI • \\x - y\\ +fty 625

MAXIMAL ELEMENT PRINCIPLES

From assumption (2) we have \\c - u\\ < diam(C) < a inf {h(x) \ x e A) < ah{x), and, combining the last relations, we obtain fx - y\\ + 0iy P\\y -- u\ u\ <<(1 (1 -- t)\x t)\x -- c\\ c\\ + +fit\\x fit\\x-- u\\ u\\ +pcc{\ +pa{\ -- t)h(x) t)h(x) <[[(1 < ( ! --/t)) ++/&pt++pa{\ pa{\ - - t)]h(x) t)]h(x)==ph(x) ph(x) (considering the definition of p). Because u e C was arbitrary, we obtain that Ibt - jv|| F a mapping. We also suppose given a function q> from R+ into

626

NEW MAXIMAL ELEMENT PRINCIPLES R+. For every x,y e E with x *y, the segment [x,y] is considered as a metric space where the metric is the restriction of the norm | 1 given on E. We say that T is: - directionally closed if and only if, for every x,ye D, the restriction 7*1 rx i has a closed graph; - directionally (p-lipschitzian if and only if, for every x,y e D, x^y there is an element u G Jxj'] such that \\T(x)-T{u)\\< R+ is super-additive if and only if / ( / , +t2)> /(*,) f(tx) + f(t22)) for for all all tthhtt22 €e R R++.. + f(t An example of a super-additive mapping is cp(t) = atk with a > 0, k > 1. Theorem 4.44 [Turinici]

Let (E,\\ |) and [F,\ |) be Banach spaces,

DczEa non-empty convex set and T : D -> F, cp : R+ -+ R+ mappings. If the following assumptions are satisfied:

627

MAXIMAL ELEMENT PRINCIPLES 1) T is directionally closed, 2) T is directionally q>-lipschitzian, 3)

tf+ by O(w) O(u) = [\\u-y |w —0\\>-0for || forallallu w eE eE and ¥(u)
((4-33) 4J3)

and for every iu/ e\x e [ xQ,yA, u** z, z, either 0j0],a < \\z - u\\ > O(z) - Q>(z)-<&(u)or W\T(z)-T(u)\\>V(z)-V(u). \\T(z)-T(u)\\>V(z)-V(u). («). Suppose z *y0. For every ue]z,y0], the first relation in (4.34) implies ||z - wfl > 4>(z) - 3>(w) = ||z - y0j - \\u - y0\\ = \\z - wfl

628

(4.34) (4.34)

NEW MAXIMAL ELEMENT PRINCIPLES which is impossible. Hence, considering assumption (3), we must have \\T(z)-T(u)\\>V(z)-V(u)T (»t \\T(z)-T(u)l>V(z)-V(u)

= 9J(llz<-^o|)-o|) 4z-yo\h^\u-yot) = pflk - H- IAb " - y-0fj>ol) =
>^(l| -4 >^(||z-zw |),

which contradicts assumption (2). Hence, z = y0, and, by (4.33), we have ^ x0b-- >^ b1O) | ^ o ) - ^ o ) h ^ o ) - ^ o ) = ¥(*6) -=* Q since cp(0) = 0. As xo,yo e D were arbitrary, the theorem is proved. ■ Theorem 4.44 has an interesting generalization for functions defined between two metric spaces. Before giving this generalization, we must introduce some preliminaries. Let (E,d) be a complete metric space. For every x,y e E let [x,y]m denote the metrical segment between x and y, i.e. [x,y]m = [z G E | d(x,z) d{x,z) + d(z,y) = d(x,y)}. Every metrical segment is a closed and bounded subset of E. The following property is true: Ifx, y, z € E are such that z e [x,y]m then necessarily we have, [x,z] [x,z\mm,, [z,y]m, <=[x,y]m. Indeed, by assumption d(x,z) + d(z,y) = d(x,y) d(x,y), , if «u e [x,z]m, then 629

MAXIMAL ELEMENT PRINCIPLES d(x,u) + d{u,z) = d(x,z), which implies d(x,u) + d(u,z) + d(z,y) d{z,y) = d{x,y). d(x,y).

(4.35)

On the other hand, we have d{x,y)
(4.36)

+ d(u,y).

From (4.35) and (4.36), we obtain d(u,z) + d(z,y)

+d(z,y).

Hence, (4.37)

d{u,z) + d(z,y) d{z,y) = d(u,y). From (4.37) and (4.35), we deduce d(x,u) + d(u,y) = d(x,y),

that is u e [x,y]m. We have a similar proof for the inclusion [z,y]m, a [x,y]m.

m We say that a subset D c E is metrically convex if and only if, for every x, v G D, we have [x,y]m c D. Let (F, S) be another complete metric space and T : D -> F a mapping. We consider [;c, v]m as a metric space taking as metric, the restriction of the metric d. We suppose given a mapping

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