Theory of equations of neutral type

THEORY

OF

EQUATIONS

OF

NEUTRAL

TYPE

UDC 519.929

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii

In the p a p e r we p r e s e n t a s u r v e y of the investigations on the theory of equations of neutral type, i.e., equations f o r which the value of the d e r i v a t i v e at the p r e s e n t m o m e n t depends explicitly on the " p r e h i s t o r y ~ of the b e h a v i o r of the d e r i v a t i v e . The p a p e r consists of e l e ven s e c t i o n s .

INTRODUCTION In the t h e o r y of differential equations with deviating a r g u m e n t it is convenient to divide the c o n s i d e r e d equations into t h r e e types: r e t a r d i n g , advanced and n e u t r a l , although t h e r e are no a c c u r a t e l y e s t a b l i s h e d c r i t e r i a f o r such a delineation. C h a r a c t e r i s t i c r e p r e s e n t a t i v e s of the indicated types have the f o r m s

x'(t)=f(t, x(t--h)); x'(t)=f(t, x(t+h)); x'(t)=I(t, x(t--h), x'(t--h)) r e s p e c t i v e l y . The equations of the r e t a r d e d type a p p e a r m o r e frequently in applications than the other types and have been i n v e s t i g a t e d m o r e . In spite of the fact that they d e s c r i b e objects with an infinite-dimensional s p a c e of s t a t e s , t h e i r p r o p e r t i e s r e v e a l a g r e a t e r r e s e m b l a n c e with the p r o p e r t i e s of ordinary differential equations of n o r m a l f o r m . The equations of advanced type a r e e x t r e m e l y exotic and do not occur (directly) at the d e s c r i p t i o n of n a t u r a l l y d e t e r m i n e d p r o c e s s e s ; these equations have been investigated v e r y little. The equations of n e u t r a l type, to which this s u r v e y is devoted, occupy, by t h e i r p r o p e r t i e s and by t h e i r domains of applications, an i n t e r m e d i a t e position between the o r d i n a r y equations which a r e not solved with r e s p e c t to the d e r i v a t i v e and the equations of the r e t a r d e d type. In the l a s t 20 y e a r s , t h e i r t h e o r y has been shaped into an independent direction and the l i t e r a t u r e is a l r e a d y so v a s t that within the volume of the p r e s e n t p a p e r we can only p e r f o r m its c l a s s i f i c a t i o n and, in an e n t i r e l y b r i e f m a n n e r , outline the formulation of the fundamental p r o b l e m s and of c e r t a i n r e s u l t s . The l i m i t e d volume h a s not allowed us to include in this p a p e r a c o m p l e t e bibliographical list which, according to our information, contains o v e r 650 e n t r i e s ; in our opinion, the r e f e r e n c e s listed can r e p r e s e n t only to a c e r t a i n extent the accumulating directions of the theory of equations of neutral type. We also e m p h a s i z e that we could not include in the s u r v e y the investigations on d i f f e r ence and integral equations which have m a n y contingency points with the t h e o r y of differential equations of n e u t r a l type. We mention the m o n o g r a p h s , s u r v e y p a p e r s and conference p r o c e e d i n g s which are p a r t i a l l y or totally d e voted to equations of neutral type: [2-4, 7, 8, 10, 11, 28, 40, 46, 49, 60, 81-83, 101, 112, 139-140, 152-154, 167, 168, 188, 206, 207, 223, 226, 230, 232, 250, 259, 260, 262, 265, 269, 300]. 1.

Local

Solvability

of the

Cauchy

Problem

1.1. F o r m u l a t i o n of the P r o b l e m . In m o s t c a s e s , the c o n s i d e r e d Cauchy p r o b l e m f o r a f u n c t i o n a l - d i f f e r ential equation of neutral type can be w r i t t e n in the following f o r m :

x ' ( t ) = / ( t , x . x/, ~), t > o ; x(t)=(p(t),

t~0.

T r a n s l a t e d f r o m Itogi Nauki i Tekhniki, S e r i y a M a t e m a t i c h e s k i i Analiz, Vol. 19, pp. 55-126, 1982.

674

0090-4104/84/2406-0674507.50

9 1984 Plenum Publishing C o r p o r a t i o n

(1) (2)

Here the right-hand side of Eq. (1) depends on the time t, on the p r e h i s t o r y x t and x~ of the unknown function x and of its derivative x' (i.e., on the r e s t r i c t i o n s of these functions to [t - h, t] for some h E [0, co) or to ( - ~ , t]) and on the p a r a m e t e r p; ~ is the initial function, defined on [-h, 0] (or ( - ~ , 0]). Equation (1) can be an equation with a d i s c r e t e constant and variable lag or with a lag distributed over finite and infinite intervals. If by a solution of the p r o b l e m (1)-(2) we mean a continuously differentiable function, then a n e c e s s a r y condition f o r the solvability of this problem is the s o - c a l l e d ~pasting" or ~eonsistency ~ condition: ~'(0) =f(0, % q~').

(3)

Of course, this condition r e s t r i c t s sharply the class of possible initial functions ~0. However, in c e r t a i n c a s e s this r e s t r i c t i o n is not r e a l l y essential. F o r example, if Eq. (1) is an equation with a d i s c r e t e deviation of the argument

x'(t)=f(t, x(g(t)), x'(h(t))) and g(0), h(O) < 0, then one can easily redefine ~o in the neighborhood of z e r o , without disturbing the continuous differentiability, so that condition (3) should be satisfied. If h(0) = 0, then the question whether condition (3) is satisfied reduces e s s e n t i a l l y to the solvability of the equation qr

=f(O, ,p(g(O)), ,p'(O))

relative to ~o' (0) (and, obviously, it is solved affirmatively, f o r example in the case w h e n f satisfies a Lipschitz condition with a constant k < 1 with r e s p e c t to the last argument). In many investigations, by a solution of the p r o b l e m (1)-(2), instead of a continuously differentiable function one means an absolutely continuous function, a function with pieeewise continuous derivative, etc. In this case, condition (3) is not r e a l l y n e c e s s a r y . 1.2. P a r t i c u l a r i t i e s of the Cauchy P r o b l e m for Equations of Neutral Type. In c o n t r a s t to the case of the Cauchy problem, at the proof of the solvability of the p r o b l e m (1)-(2) for equations of the r e t a r d e d type there a r i s e s an essential difficulty consisting in the fact that, in general, the integral o p e r a t o r c o n s t r u c t e d f r o m the p r o b l e m (1)-(2) is not completely continuous. F o r certain c l a s s e s of equations of neutral type, this difficulty can be o v e r c o m e . F i r s t l y , here we have to c o n s i d e r equations of neutral type with a Nstrict rejection with r e s p e c t to the derivative"; this is the case when the right-hand side of the equation (1) depends not on x~ but only on x t ~ (6 0 > 0). Essentially, on segments of length 5 0, such aa equation m an equation of r e t a r d e d type. The Cauchy p r o b l e m m these and s~mllar situations has been c o n s i d e r e d m [46, 88, 251, etc.]. Secondly, we have the problems which reduce to equations with a contraction o p e r a t o r (in particular, these are the p r o b l e m s in which the o p e r a t o r f satisfies a Lipschitz condition with r e s p e c t to x t and x~ where the constant k with r e s p e c t to x~ is less than one). An existence and uniqueness t h e o r e m f o r the solution of the p r o b l e m {1)-(2) in the class of functions with bounded derivative as well as a t h e o r e m on the continuous dependence of the solutions on the initial data have been proved by Zverkin [79] by the method of s u c c e s s i v e approximations. Similar p r o b l e m s have been studied also in [195]. Thirdly, as mentioned by A. M. Zverkin, the conditions of the previous two types can be g e n e r a l i z e d in the following manner: one can require that the right-hand side of the equation should depend sufficiently well (for example, as in the previous case) only on the n e a r p r e h i s t o r y of the function x t and of its derivative x~. This may occur, f o r example, when the Lipschitz condition for the o p e r a t o r f is s a t isfied only on those functions x~ and y~ (or x t and Yt) which coincide up to the point t - 60 f o r some fixed 6 o > 0. Then, by the method of steps, the p r o b l e m r e d u c e s to an equation with a contraction. Fourthly, f o r some p r o b lems in c e r t a i n functional spaces one can s e p a r a t e convex c o m p a c t a which are invariant relative to the integral o p e r a t o r (see [67, 169, 254, 255]). ?

.

--u

0

.

.

.

.

.

.

.

.

.

1.3. Use of Condensing O p e r a t o r s . If one does not impose any r e s t r i c t i o n s on the m e m o r y of the equation (1) (considering it distributed over [t - h, t], 0 < h _< ~) and one does not a s s u m e in any f o r m that the Lipschitz condition is satisfied with r e s p e c t to x t, then one can show that, under sufficiently g e n e r a l conditions, the integral o p e r a t o r f o r the p r o b l e m (1)-(2) is condensing. T h e r e f o r e , f o r the investigation of p r o b l e m (1)-(2) it is possible to use the theory of n o n c o m p a c t n e s s m e a s u r e s and condensing o p e r a t o r s (see, f o r example, [191, 192]). Such an approach has been used in [43, 178]. In the last p a p e r one has a s s u m e d that f : R 1 • C [-h, 0] x C [-h, 0] ~ Rn and the o p e r a t o r f is continuous with r e s p e c t to the totality of the v a r i a b l e s ;

(4)

relative to x~ the weak Lipschitz condition is satisfied (in A. M. Z v e r k i n ' s form): there exist k < 1 and 6 o (0 < 5 0 ~ ) such that f o r a l l t , p , u , v 1 , v 2, w h e r e v1(s) = v 2 ( s ) , f o r t - h < s - < t - 6 0 , we have

675

If(t, u, vl, vt)--f(t, u, v2, ~)I~kllv,--v~lh;

,~'(0) =f(O, ,~, ,~', ~).

(5)

Under these conditions it is p r o v e d that t h e r e e x i s t s T O > 0 such that the s e t of the solutions of the p r o b l e m (1)-(2), defined on [0, To], is not empty, it is c o m p a c t in C%0 and it is connected. P r o b l e m {1)-(2) has been inv e s t i g a t e d by the method of condensing o p e r a t o r s also in [291, 292]. The obtained r e s u l t s a r e close to those f o r m u l a t e d above. See also [289]. 1.4. O p e r a t o r Equations of V o l t e r r a T y p e . One can combine the different f o r m s of the equations of neutral type into one by r e p r e s e n t i n g t h e m in the f o r m of a V o l t e r r a type o p e r a t o r equation. To this end, in [70, 72] one investigates the equation x ~r

(6)

where O is a V o l t e r r a type o p e r a t o r , i.e., (Ox)It0.q = (Og)Iio.q, if x, gEEt0.r 1, x [[0,q = g it0,q, te[0, T], EE0r j being s o m e functional s p a c e . We a s s u m e that the o p e r a t o r O is condensing with r e s p e c t to a m e a s u r e of n o n c o m p a c t h e s s , p o s s e s s i n g a s e r i e s of natural p r o p e r t i e s , andon the s t r u c t u r e of the s p a c e E[0.r , we impose n o n e s s e n tial r e s t r i c t i o n s , so that for E[n.r; one can take the space of smooth functions C m(0 ~< m < c~) , the Lebesgue s p a c e Lp(1..
. - ~ D (t, x t ) ~ f (t, xt)

(7)

has been c a r r i e d out by J. K. Hale with his coauthors (see [245, 258-261, 264, 265, 272] and a l s o [277]). This f o r m of Eq. (7) allows us to understand by a solution of the Cauchy p r o b l e m a function x, continuous on [-h, T] (T > 0), s u c h t h a t x 0 = ~ and f o r t E [0, T] we have t

D (t, xt) = D (0, ~) + I f (s, x~) ds. 0

Depending on the f o r m of the o p e r a t o r D, Eq. (7) may turn into an equation of r e t a r d e d , n e u t r a l , advanced or mixed type. Some applied p r o b l e m s (for e x a m p l e , f r o m e l e c t r i c a l engineering [236]) lead to p r o b l e m s of the f o r m (7). 1.6. Solving an Equation of Neutral Type Relative to the D e r i v a t i v e . S o m e t i m e s it is convenient to solve an equation of neutral type r e l a t i v e to the d e r i v a t i v e , i.e., to reduce p r o b l e m (1)-(2) to a Cauchy p r o b l e m f o r a c e r t a i n equation of r e t a r d e d type, since the m a j o r i t y of questions a r e solved much s i m p l e r f o r equations of r e t a r d e d type than f o r equations of neutral type. Many r e s u l t s in this direction have been obtained by N. V. Azbelev and his coauthors (see, f o r e x a m p l e , [2, 4, 5, 6, 8]) and a l s o [172]). In these p a p e r s one c o n s i d e r s p r o b l e m s of the type

x' (t) =f(t, x(h(t) ), x' (g(t) ) ), te[a, O], x(~)=q~(~), x'(~)=,(~), ~e[a, b].

(8)

The solution is sought in the s p a c e of functions absolutely continuous on [a, b] axed the values of the r i g h t - h a n d side of Eq. (8) lie in the s p a c e of functions which a r e s u m m a b l e on [a, b]. The conditions f o r a continuous junction need not be satisfied: x(a) =q~(a), x(b) =~{b), x'(a) =qD'(a), x'(b) = ~ ' ( b ) . Such a f o r m u l a t i o n of the p r o b l e m has r e s u l t e d in the n e c e s s i t y of a detailed investigation of the s u p e r p o s i t i o n o p e r a t o r (see, for e x a m p l e , [5, 683

(Sv)(t)={g(g(t)), O,

676

g(t)~la, b]; g (t)~[a, bi.

In [2] it is p r o v e d that the most important c h a r a c t e r i s t i c of Eq. (8) is its reducibility t o t h e equivalent equation ~ = Fix with a completely continuous o p e r a t o r F v In p a r t i c u l a r , this t u r n s out to be possible if Eq. (8) can be t r a n s f o r m e d into a quasilinear equation of the f o r m (Q.~) ( t ) + AC t) x (a) = f Ct, Tx), where

(Tx)(t)._{x(h(ot)),

h(t)~[a, b];

h (t)~[a, b];

Q:Lon-+Lp n , the columns of the m a t r i x A(t) belong to L~ (here I ~ is the space of v e c t o r - f u n c t i o n s x(t) = {xl(t) ..... xn(t)}, whose components are summable on [a, b] with power p (p > 1), and [[x ]]p= [ i ][x(t)l[Pdt[ l/p, and Q is a F r e d h o l m o p e r a t o r . In [132], with the aid of the implicit function t h e o r e m one shows that p r o b l e m (1)-(2) is equivalent to the Cauchy p r o b l e m f o r a certain equation of r e t a r d e d type. A fundamental condition f o r this is the unique invertibility of the o p e r a t o r I - f ~ ( . , 0, 0) (here f~ is the F r e c h e t derivative of the o p e r a t o r f with r e s p e c t t o t h e third argument, ~_= 0). Such an approach to solve the general nonlinear p r o b l e m (1)-(2) allows us, in p a r t i c ular, to use various results on the r e t a r d e d s p e c t r u m of linear o p e r a t o r s (see, for example, [126]). In this case, as a consequence, one obtains not only the majority of the known tests for the local solvability o f the problem (1)-(2), but also some new r e s u l t s . Some r e s u l t s in this direction can be found also in [23, 24]. 1.7. P r o b l e m of G. A. Kamenskii. If one imposes certain r e s t r i c t i o n s on the delay of the argument of the derivative x T of the unknown function, then condition (5) can be weakened in a significant manner. This was f i r s t o b s e r v e d by Kamenstdi in [89]. He has investigated an equation of the type

x' (t)= f (t,x (t), x (g, (t)) . . . . . x (gin (t)), x ' (h~ (t)) . . . . .

x' (t~,. (t))),

where O~ 0 and h ~ ( 0 ) = 0 ( i = ] . . . . . m). An existence and uniqueness t h e o r e m has been proved under the condition that the function f satisfies Lipschitz conditions with r e s p e c t to all the v a r i ables and

•

sup D h ~ ( t ) . s t < l

t ~ l O<.t4~

f o r some ~ > 0, where s i are the Lipschitz constants of the function f with r e s p e c t to the last m arguments and Dhi(t) are the upper derived n u m b e r s of the functions hi at the point t. Various refinements of the results of [89] can be found in [75, 76, 175, 177]. In [175, 177] one c o n s t r u c t s a special m e t r i c , relative to which the integral o p e r a t o r , c o n s t r u c t e d f r o m the p r o b l e m (1)-(2), is a contraction. This method together with s o m e other considerations allows us to generalize the r e s u l t s of [75, 76, 89] to the case when Eq. (1) is an equation with distributed deviations of the argument in the Banach space of p i e c e w i s e continuous functions having a Lipschitz constant with r e s p e c t to x'. Similar results can be found also in [70, 166, 167]. 1.8. Equations in a Banach Space. In the p a p e r s of Kwapisz and other Polish mathematicians [282,287], one investigates equations of neutral type in a Banach space: one c o n s t r u c t s s u c c e s s i v e approximations of the solutions, one proves t h e o r e m s on their convergence, on existence, uniqueness, andon the continuous dependence of the solutions on the right-hand side of the equations. In [77] one p r o v e s a t h e o r e m on the existence and uniqueness of solutions in the class of functions with bounded s t r o n g variation. In [63] one investigates a semiexplicit differential equation of r e t a r d e d type and also an equation of neutral type with a constant delay. One p r e s e n t s the method of the E u l e r polygonal lines and the method of s u c c e s s i v e approximations and one p r o v e s t h e i r convergence to the unique solution of the Cauchy problem. In [247-249] one investigates a l i n e a r equation in a Banach space and one gives f o r m u l a s for the r e p r e s e n t a t i o n of the solutions.

677

1.9. L i n e a r Equations. A detailed investigation of the boundary value p r o b l e m and also of the Cauchy p r o b l e m (which in c e r t a i n c a s e s is a s p e c i a l c a s e of a boundary value problem) for a linear f u n c t i o n a l - d i f f e r ential equation (see also Sec. 5)

s x=f, w h e r e ~ is a l i n e a r o p e r a t o r acting f r o m the s p a c e of absolutely continuous functions into the s p a c e of s u m mable functions, can be found in m a n y w o r k s of N. V. Azbelev, L. F. Rakhmatullina, etc. (see, f o r example, [1, 4, 7, 20, 137, 172, etc.]. In these works one m a k e s use widely of the techniques of functional analysis a n d t h e o r y of equations in a Banach space, in p a r t i c u l a r , S. M. N i k o l ' s k i i ' s t h e o r e m on the r e p r e s e n t a t i o n of a F r e d h o l m o p e r a t o r , the t h e o r e m on the branching of solutions, etc. In [69, 242, 254] one solves v a r i o u s p r o b l e m s f o r l i n e a r equations and s y s t e m s with constant delays and, in p a r t i c u l a r , one d i s c u s s e s the question of r e p r e s e n t a t i o n of solutions in an explicit f o r m . Regarding the r e p r e s e n t a t i o n of the solutions, see also [114, 171, 246]. In [135] one p o s e s and one i n v e s t i g a t e s a homogeneous p r o b l e m f o r an equation with v a r i a b l e delays. In [15] one c o n s t r u c t s an a p p r o x i m a t e solution of an i n t e g r o differential equation by the method of functional c o r r e c t i o n s and m i n o r a n t - m a j o r a n t functions. See also [213]. 1.10. Equations of Type P(x(t), x(t - 1))x'(t) +Q(x(t), x(t - 1))x'(t - 1) = o. A s e r i e s of p a p e r s by Vainberg, K a m e n s k i i , Myshkis (see [54, 55]) is devoted to the detailed investigation of equations of type

P(x(t), x(t--l))x'(l)+Q(x(t), x(t--1))x'(t--1)=O.

(9)

H e r e P and Q are defined in the entire plane and in any of its finite p a r t s t h e y s a t i s f y t h e Lipschitz condition; by a solution we m e a n a function which is continuously differentiable on (~, fi) ( - - o o < a < ~ - - l < o o ) . To Eq. (9) one a s s o c i a t e s the o r d i n a r y differential equation

P(x, y)dx+Q(x, g)dg=O. If we know the integral lines L of the last equation, then we can draw v a r i o u s conclusions about Eq. (9), and c o n v e r s e l y , in a s e r i e s of c a s e s , f r o m the a s y m p t o t i c b e h a v i o r of the solution x of the Eq. (9) one can make i n f e r e n c e s about the line L. One p r o v e s t h e o r e m s o n t h e existence, uniqueness and continUability of solutions of the Cauchy p r o b l e m f o r the equation (9). One obtains n e c e s s a r y and sufficient conditions f o r the existence of p e r i o d i c solutions: one shows that the solutions of equation (9) can be periodic only w i t h t h e p e r i o d s 1 / k or 2 / k , where k is a n a t u r a l n u m b e r . Many r e s u l t s r e m a i n valid also for g e n e r a l i z e d solutions which belong to the c l a s s of absolutely continuous functions with d e r i v a t i v e s having at m o s t a finite n u m b e r of discontinuities. 1.11. Equations of Superneutral Type. A significant n u m b e r of p a p e r s a r e devoted to the investigation of equations of s u p e r a e u t r a l type (the s i m p l e s t equations of t h e s e type a p p e a r in the automatic r e g u l a r i z a t i o n a t the a n a l y s i s of the s o - c a l l e d s y s t e m s with saturation) and also to e x t r e m a l - d i f f e r e n t i a l equations of s u p e r n e u t r a l type. As the s i m p l e s t r e p r e s e n t a t i v e of the f i r s t c l a s s , one can consider the following equation: x ' (t)-- f (t, x (t), x ' (t), x (Ao), x ' (Ao)), t > O;

(10)

where

~ = a ~ (t, x (t), x' (t), x (A~0, x' (Ak+~)), x(t), x'(t)), r e > l ;

(11)

k ~ 0 , 1. . . . . m - - l , Am~A,~(t, and of the second c l a s s

x'(t)=minmax f (p, q, t, X(~o), x'(t), x'(Ao)), t > O , q

(12)

P

w h e r e r 0 and A 0 a r e defined r e c u r r e n t l y as in (11). In [118, 158] one p r o v e s (basically, by the method of steps o r by the aid of the principle of contraction mappings) local and global t h e o r e m s f o r the e x i s t e n c e and uniquea e s s of solutions of Cauchy p r o b l e m s f o r Eqs. (10) and (12). See also [195]. 1.12. On Other Equations. In [122], etc., o n e p r o v e s t h e o r e m s on the existence (and s o m e t i m e s also on the uniqueness and on the continuous dependence on p a r a m e t e r s ) of solutions of integrodffferential equations; in [38] one i n v e s t i g a t e s a s y s t e m with an infinite n u m b e r of d i s c r e t e delays. In [252] one solves the p r o b l e m of the two bodies of c l a s s i c a l e l e c t r o d y n a m i c s which is d e s c r i b e d by a s y s t e m of functional-differential equations and one d i s c u s s e s the possibility of using v a r i o u s topologies for solving this p r o b l e m . In [22] one p r o v e s the existence of solutions of the Cauchy p r o b l e m f o r c e r t a i n equations of advanced type. See also [147, 167].

678

2.

C o n t i n u a b i l i t y , U n i q u e n e s s and D i f f e r e n t i a l

Inequalities

2.1. Continuability and Uniqueness. In many of the papers mentioned in Sec. 1, parallel with local existence theorems one has also proved theorems on the uniqueness and continuability of solutions (or on the existence of a unique global solution) (see, for example, [75, 76, 79, 89, 118, 175, 177,195, etc.]). In the majority of these papers, the theorems on the uniqueness and the continuability of solutions are obtained initially on a small segment and then the method of steps can be used. In [174, 176] one proves theorems by which the question on the eontinuability (oruniqueness) of the solutions of functional-differential equations of neutral type reduces to the investigation of the behavior of the solutions of scalar differential inequalities. For continuability, the fundamental condition has the form

If(t, xt, x/)l-.<~(t, IIx[I. jlx'l/,, llx II,-mo, ilx'!l,-~,>, and for uniqueness

I f ( t , xt, x / ) - - f ( t ,

Yt, V/)l'-
(11)

where x(s) =y(s) when s E(-oo, t - 6 0]. These conditions g e n e r a l i z e directly the results of the theory of o r dinary differential equations. Here, the estimating functions ~ are, roughly spealdag, Osgood functions. The functions ~ are such that essential r e s t r i c t i o n s a r e imposed only on the dependence of the r i g h t - h a n d side of Eq. (1.1) on the n e a r p r e h i s t o r y of x t and x[. The continuability of the solutions at a point has been investigated in [174, 289], but in c o n t r a s t to [174], in [289] one considers solutions which are not continuously differentiable. In [179] one gives an estimate of the solution of a differential inequality of neutral type which is useful in many p r o b l e m s . In [111] etc. one p r o v e s global existence t h e o r e m s and various t h e o r e m s on the estimate of the growth of solutions. 2.2. Successive Approximations. In addition to the above mentioned p a p e r s , we mention other ones in which one applies the method of s u c c e s s i v e approximations. In [155] one p r o v e s t h e o r e m s f o r the existence and uniqueness of solutions, f o r the convergence of the s u c c e s s i v e approximations, one gives an estimate for the rate of convergence of the method without imposing any r e s t r i c t i o n s on the delay of the arguments x and x', and the Lipschitz condition relative to x is replaced by a less r e s t r i c t i v e condition of type (1). In [14, 67], s i m i l a r p r o b l e m s are solved f o r linear equations, equations with a d i s c r e t e delay and f o r integrodifferential equations. 2.3. Differential Inequalities. In [41] one investigates the s t r u c t u r e of the integral vortex of the p r o b l e m (1.1)-(1.2) and its c o r r e s p o n d i n g differential inequalities. One c o n s t r u c t s an example which shows that, even in the case of a monotone function f , one cannot obtain, as a rule, c o r r e s p o n d i n g inequalities without imposing initial r e s t r i c t i o n s on the derivatives of the solution. One p r o v e s t h e o r e m s on s t r i c t and n o n s t r i c t differential inequalities, on the existence of upper and lower solutions with r e s p e c t to the cone of positive n o n d e c r e a s i n g functions in the space C~0 ' T], t h e o r e m s of existence of the upper A-solution and the lower A . - s o l u t i o n under the condition that the o p e r a t o r f is off-diagonal monotone: f~ (t, u~, ~a)--
u~
where a i is the i-th coordinate of the v e c t o r ~ = (a 1. . . . . an). 2.4. Other P r o p e r t i e s of the Solutions. The p a p e r s [189, 264, 286] are devoted to various p r o p e r t i e s of the solutions of the Cauchy problem; in [264] and [286] one gives conditions for the smoothness of the solutions; in [227] one gives examples of the branching and the cut-off of solutions. In [166, 208] one p r o v e s existence t h e o r e m s for a holomorphic solution. In [205], for equations with singularities one establishes e s t i m a t e s of the growth of solutions in the neighborhood of the singular point. In [66], f r o m the approximate solution of an equation of neutral (or retarded) type, one r e e s t a b l i s h e s the c o r r e s p o n d i n g initial function (see also [16]). In [160] one considers the question of the prediction of the behavior of s y s t e m s of equations of neutral type.

679

3.

Dependence

of t h e

Solutions

on P a r a m e t e r s

3.1. Continuity Relative to the P a r a m e t e r s and the Initial Data. In many of the above-mentioned p a p e r s , simultaneously with existence t h e o r e m s one has proved t h e o r e m s on the continuous dependence of the solution on the p a r a m e t e r . In this connection, in some of the papers one has imposed additional r e s t r i c t i o n s on the right-hand side of the equation; for example, in [251] and [255], for the formulation of the t h e o r e m on the continuous dependence of the solution on the p a r a m e t e r one has added, in particular, a Lipsehitz condition r e lative to xt. In [176], for the proof of such t h e o r e m s one r e q u i r e s only conditions (1.3)-(1.4). Under s i m i l a r assumptions, a somewhat different t h e o r e m on the continuous dependence on the p a r a m e t e r has been proved in [1041 . Various t h e o r e m s on the continuous dependence on the p a r a m e t e r , on the initial function, etc. can be found also in [71, 169, 266, 290]. In [71], the mentioned questions are studied for equations with an a b s t r a c t V o l t e r r a o p e r a t o r and the obtained r e s u l t s are applied to an equation of neutral type and to multidimensional integral equations. In [290] one considers an equation with d i s c r e t e delays whose solution lies in W~ but need not be continuously dependent in this space on the initial data. In the paper one c o n s t r u c t s a special topology, relative to which one has a continuous dependence on the initial data. 3.2. Differentiability with Respect to the P a r a m e t e r . In [179] one gives conditions under which the s o l u tions of p r o b l e m (1.1)-(1.2) are differentiable with r e s p e c t to the p a r a m e t e r . 3.3. Shift Operator along the T r a j e c t o r i e s of an Equation of Neutral Type. The shift o p e r a t o r is i n v e s tigated in [105, 109, 190]. In these papers one gives conditions under which the shift o p e r a t o r is condensing in a special noncompactness m e a s u r e , with a constant less than unity; the equation may depend on a p a r a m e t e r , in p a r t i c u l a r , on a small delay. One indicates applications of the obtained results to the Floquet theory for equations of neutral type, stability theory and the averaging principle. In [258] one investigates the conditions under which the shift o p e r a t o r along the t r a j e c t o r i e s of Eq. (1.7) can be r e p r e s e n t e d as a sum of a contractive and of a completely continuous operator. 3.4. Asymptotic Behavior with Respect to a Small Delay. In [58, 182] one investigates the behavior of the solutions of the equation

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

(1)

with a s m a l l constant delay At, one c o n s t r u c t s the expansion of the solution in powers of Bt, one introduces asymptotic e s t i m a t e s of the type O(Atn). In the case w h e n f a n d f ~ are continuous, and [ f4' I-.
x, ( t ) = a ( t ) x' (t--h) + b (t) x (t) + c (t)x ( t - h ) + a( t), where a, b, c, d are m a t r i c e s with smooth elements and h is a s m a l l delay. Linear problems are investigated also in [187]. In [59] one c o n s i d e r s the Cauchy p r o b l e m f o r s y s t e m s of type

v' (t)= Ag (t--h)+ F (t, v(t), y' (t-- T)), where A is a m a t r i x whose eigenvalues in absolute value do not exceed unity, and among the eigenvalues which a r e equal in absolute value to unity there are no multiple ones and none exactly equal to one. One c o n s t r u c t s an asymptotic r e p r e s e n t a t i o n of the solution f r o m where it follows that x(t, 7) has for T ~ 0 a limit x(t), where x(t) is a solution of the s y s t e m

x ( t ) = ( I - - A ) - ' F ( t , x, x), while ~(t, 7) does not tend to x(t) but oscillates around it with frequency 1/~-. In [59] one o b s e r v e s a relationship between the s y s t e m under consideration and a c e r t a i n equation with a small p a r a m e t e r at the derivative, 680

not containing a delay. An equation with a s m a l l p a r a m e t e r at the derivative is investigated in [217]. See a l s o [218]. 4.

Generalized

Solutions

4.1. Definition of G e n e r a l i z e d Solutions. One of the p a r t i c u l a r i t i e s of the Cauchy p r o b l e m f o r equations of neutral type is the following c i r c u m s t a n c e : f o r the existence of a solution of p r o b l e m (1.1)-(1.2) in the c l a s s i cal s e n s e (i.e., continuously differentiable), it is n e c e s s a r y that we have the gluing condition

~'(0) =f(0, % ~'). Obviously, the l a s t condition is not stable r e l a t i v e to s m a l l p e r t u r b a t i o n s of f and g0. Attempts to e l i m i n a t e this i m p r o p r i e t y in the formulation of the p r o b l e m have led to v a r i o u s definitions of g e n e r a l i z e d solutions (see, f o r e x a m p l e , [80, 99, 293]), reducing b a s i c a l l y to the fact that the solution m u s t s a t i s f y the equation a l m o s t e v e r y w h e r e . In [99], Myshkis and K a m e n s k i i have s u g g e s t e d that by a solution of the boundary value p r o b l e m one should m e a n a c e r t a i n g e n e r a l i z e d function (for details see Sec. 9 below). In [80] Zverkin defines a solution as a Lebesgue integrable function x' (t) such that the equation is s a t i s f i e d by the functions x' (t) and x(t) = t

(p(0)+$ x'(~)d~ at all the points of the s e g m e n t under consideration. 0 In [21], one introduces the definition of a g e n e r a l i z e d solution, consisting in the following. P a r a l l e l with Eq. (1.1), one c o n s i d e r s the p e r t u r b e d equation

x'(t)=f(t,

x o x/)+%(t)c,

t>O;

(1')

w h e r e c = , ' ( O ) - - f ( O , % q~'), v n ( t ) = { 1--t/R, 0 4 t ~ < ~ ; 0, t>~. The function xEC[o, rl ( T > 0 ) is called a g e n e r a l i z e d solution of the p r o b l e m (1.1)-(1.2) on [0, T], if a sequence of n u m b e r s 0In}, ~n-+0. and solutions of the p r o b l e m (1')-(1.2) f o r ~ = ~ such that Obviously, any c l a s s i c a l solution is a g e n e r a l i z e d one. It is p r o v e d that if one can talk of a solution theodory s e n s e (i.e., a solution satisfying the equation a l m o s t e v e r y w h e r e ) and if it is unique, then e r a l i z e d solution (see [74]).

there e x i s t s ci0. r l x,--~x. in the C a r a it is a g e n -

4.2. Investigation of G e n e r a l i z e d Solutions. A detailed investigation of g e n e r a l i z e d solutions is c a r r i e d out in [21, 73, 74, 176, 180]. One p r o v e s a local t h e o r e m on the existence of g e n e r a l i z e d solutions under conditions (1.4)-(1.5), t h e o r e m s on continuability, on uniqueness, differential inequalities, on the continuous dependence on a p a r a m e t e r , etc. The e s s e n t i a l s i n g u l a r i t i e s of the given definition of the concept of g e n e r a l i z e d solution a r e r e v e a l e d at the investigation of uniqueness (see [180]). One c o n s t r u c t s e x a m p l e s in which under natural a s s u m p t i o n s r e l a t i v e to the right-hand side of the equation (a Lipschitz condition with r e s p e c t to x t and x~ such that the Lipschitz constant r e l a t i v e to x~ is l e s s than one) one can have m a n y g e n e r a l i z e d solutions. This anomaly d i s a p p e a r s if the m e m o r y of the equation (relative to the derivative) is not delayed " f o r a long t i m e " n e a r the s i n g u l a r point t = 0 at which the gluing condition is violated. More exactly, a s s u m e that the Liptschitz condition with r e s p e c t to x~ is w r i t t e n in the f o r m

I f It, u, v , ) - f

(t, u, v~)l
w h e r e ~ ( t ) c ( - - o0, t] is the "true" m e m o r y of the equation at t i m e t, t[ vl -- v2 IIX(')= max1731 ( S ) - '0 2 (S)I, sEa(t) rues {tE[O, T]:r (t) N (0, ~l)= 0} (~" (t)= ~ (~,-z (t))) tends to z e r o for any n when V ~ O. It is p r o v e d that the addition of this condition to the condition of type (2.1) e n s u r e s the uniqueness of the g e n e r a l i z e d solution. 5.

Boundary-Value

Problems

5.1. F o r m u l a t i o n of the P r o b l e m . B o u n d a r y - v a l u e p r o b l e m s f o r equations of neutral type have been studied s t a r t i n g f r o m the end of the fifties, in connection with the v a r i a t i o n a l p r o b l e m s f o r functionals of n e u t r a l type with deviating a r g u m e n t . In [91] K a m e n s k i i has i n v e s t i g a t e d the equation

681

v"(x)=~(x, v(x), y'(x), V(x--A~(x)) . . . . . y(x--A~(x)), y'(x--A~(x)) . . . . . y'(x--A,,(x)), vr'(x--~(x)) . . . . 9 9.,V ~'~X t -- A

(1)

,,(x))),

A~(x) i>0, which can be c o n s i d e r e d as the E u l e r equation for the fundamental variational p r o b l e m . The b o u n d a r y value p r o b l e m is posed, f o r e x a m p l e , in the following manner: on the s e g m e n t [a, b] one has to find a solution y(x) of Eq. (1) which coincides on the initial s e t E a with the function go(x) and has a p r e s c r i b e d boundary value y(b) = Yb" Some v a r i a n t s of these conditions a r e also p o s s i b l e . F o r s i m i l a r p r o b l e m s one has obtained theor e m s on the existence arid the uniqueness of solutions. /

In [222] EltsgolTts has c o n s i d e r e d b o u n d a r y - v a l u e p r o b l e m s f o r s e c o n d - o r d e r differential equations of n e u t r a l type of the f o r m F (t, x (t - - . j (t)), x ' ( t - - ~j (t)), x " (t -- ~j (t))) = 0, x [e,. = % x (a) = b. Among the f i r s t investigations devoted to b o u n d a r y - v a l u e p r o b l e m s f o r equations of n e u t r a l type, we cite a l s o the p a p e r s of N e r s e s y a a [156] and Rozhkov [181]. In [156] one has c o n s i d e r e d a boundary-value p r o b l e m f o r a f i r s t - o r d e r l i n e a r equation and its conjugates. In [181] one has investigated a b o u n d a r y - v a l u e p r o b l e m for a f i r s t - o r d e r nonlinear equation of neutral type with a s m a l l delay. One has obtained solvability conditions f o r this p r o b l e m and o n e has c o n s t r u c t e d a s y m p t o t i c f o r m u l a s f o r its solutions. A boundary-value p r o b l e m f o r a f i r s t - o r d e r nonlinear differential equation of neutral type has been investigated also in [90]. 5.2. Boundary-Value P r o b l e m s with Infinite Defect. In [97] K a m e n s l d i and Myshlds have c o n s i d e r e d the s e c o n d - o r d e r l i n e a r differential equation of n e u t r a l type with s e v e r a l leading t e r m s ~ [ A ~ (t) z"(~l~ (t)) + Bz (t) z' (~ it)) + C t (t) z (~ (t))] -- d (t) = 0. i--0

(2)

As mentioned in this p a p e r , the i m p o r t a n c e of b o u n d a r y - v a l u e p r o b l e m s f o r differential equations with s e v e r a l leading t e r m s consists in the fact that exactly t h e s e p r o b l e m s a r e g e n e r a t e d by v a r i a t i o n a l p r o b l e m s with deviating a r g u m e n t . We d e s c r i b e the f o r m u l a t i o n f r o m [97] of the b o u n d a r y - v a l u e p r o b l e m f o r Eq. (2). Let a=minmine~(t), ~ = m a x m a x ~ ( t ) . On the s e g m e n t s In, a] and [b,~] (we a s s u m e t h a t a < a and fJ>b ; i

t

i

t

such a p r o b l e m is called a p r o b l e m with infinite defect) we define the fanctions go(t) and ~ (t), r e s p e c t i v e l y . On the s e g m e n t [a, b] we s e e k the solution z(t) of the Eq. (2), absolutely continuous t o g e t h e r with its derivative, satisfying (2) a l m o s t e v e r y w h e r e on [a, b] and such that the conditions

zC.~(t))=~C~,(t)),

if

o,(t)~
(3)

are satisfied. One obtains sufficient conditions f o r the validity of the F r e d h o l m a l t e r n a t i v e and one p r o v e s an existence and uniqueness t h e o r e m f o r the solution of p r o b l e m (2)-(3). In [98], the s a m e authors investigate a bounduryvalue p r o b l e m with infinite defect f o r the s e c o n d - o r d e r nonlinear differential equation of neutral type z" ( 0 = P (t, z (% (0) . . . . .

z" (o, (t)) . . . . .

z (am (t)), z' (~0 (t)) . . . . . dot

z" (ore ( t ) ) ) = F ( t ,

z' ( ~ it)),

lz (t)], Iz' (t)l, lz" (t)]),

(4)

a..
z(i)(~(t))=cp(J~(~(t)), z(i)(cri(t))=~J)(e~(t)), j=0,

l, 2,

if

~(t)-.
if ~l(t)>/b, i ~ 1 , 2 . . . . . m.

(5)

In dependence on the space to which the solution z(t) has to belong, one c o n s i d e r s s e v e r a l p r o b l e m s of f o r m (4)-(5) and one obtains sufficient conditions f o r t h e i r solvability. We f o r m u l a t e one of the r e s u l t s . We denote by E [ a , b] the collection of (vector) functions z(t), defined on [a, b], having on this s e g m e n t an absolutely continuous d e r i v a t i v e z'(t) such that z"(t) e x i s t s a l m o s t e v e r y w h e r e and is s u m m a b l e on [a, b] and such that v r a i s u p t z " ( t ) l < co . The n o r m in E [ a , b] is defined in the following manner: a ~ t.~b

682

It z II=max

[ 8

lr J,

2

!

i z It, ,r z,,jl,}

w h e r e IIz ]]1=vrairnax I z (t)f . Functions ~(t) and r (t) a r e a s s u m e d to belong to the s p a c e s E In, a] and E lb, 13], r e s p e c t i v e l y . The function F is a s s u m e d to be m e a s u r a b l e and e s s e n t i a l l y bounded with r e s p e c t to t f o r a r b i t r a r y fixed values of the r e m a i n i n g a r g u m e n t s and continuous with r e s p e c t to the totality of a r g u m e n t s , except t, f o r a l m o s t each tE [a, bI. Functions r p o s s e s s the p r o p e r t y that f o r any m e a s u r a b l e set E~[a, b[, the s e t s cri(E) (i = 1, 2 . . . . . m) a r e also m e a s u r a b l e . Then, we have the following t h e o r e m . A s s u m e that t h e r e exists a n u m b e r N such that f o r

rE[a, b] and f o r II z II~
vralmax IF (t, [z (t)], [z' (t) l,

[z" (t)J)[-.
and the function F s a t i s f i e s the Lipsehitz conditions r e l a t i v e to the v a r i a b l e s z" (a ~). . . . . z" (~m) with the constants r i . . . . . r m , r e s p e c t i v e l y , and rn

r=

rill.

Then, t h e r e e x i s t s at l e a s t one solution of the p r o b l e m (4)-(5) in the ball [[ z I < 1. F o r the proof of this t h e o r e m the authors m a k e use of the fixed point principle f o r condensing mappings [191]. 5.3. G e n e r a l i z e d Solutions in the Sense of G. A. K a m e n s l d i and A. D, Myshtds. Questions r e g a r d i n g f o r m u l a t i o n s of b o u n d a r y - v a l u e p r o b l e m s f o r the differential equation (2) a r e d i s c u s s e d also in [99]. One cons i d e r s t h e r e the boundary conditions

z(t)-----O, t
(6)

[the nonhomogeneous condition (3) can be r e d u c e d to these conditions with the aid of a s t a n d a r d substitution] and one p o s e s the question whether the solution of the p r o b l e m (2), (6) can be c o n s i d e r e d , in the s e n s e in which it was m e a n t (absolutely continuous t o g e t h e r with its derivative), as the continuation of a function defined by conditions (6). One gives e x a m p l e s when this is not so. In connection with this, the authors give a new definition f o r the solution of the b o u n d a r y - v a l u e p r o b l e m (2), (6), using the concept of g e n e r a l i z e d functions. Howe v e r , in this c a s e the s m o o t h n e s s of the solution is lowered. By a solution of p r o b l e m (2), (6) one m e a n s a g e n e r a l i z e d function, c o r r e s p o n d i n g to a usual absolutely continuous function z(t) (a
z(a) =z(b) =0,

(7)

which, being extended to the entire t axis by virtue of (6) by z e r o , a f t e r the i n s e r t i o n in the Eq. (2), yields a function whose support has no points on the interval (a, b). This definition be a l s o f o r m u l a t e d in an equivalent f o r m only with the aid of the concept of usual functions. solutely continuous function z(t) is called a solution of the boundary-value p r o b l e m (2), (7), (t), satisfying the conditions (7), one has the equality m

b

,_~

~,' (t)

/ +B'*(t)~l(t)j] + z*(a,(t))C,*(t)~l(t)ldt=jd*(t)q(t)dt.~

left-hand side of of a solution can Namely, an a b if f o r any function

b

;

(8)

where the a s t e r i s k denotes t r a n s p o s i t i o n . As mentioned in [100], an i m m e d i a t e extension of this f o r m u l a t i o n of b o u n d a r y - v a l u e p r o b l e m s to nonl i n e a r equations is, in g e n e r a l , not e a s i l y f e a s i b l e b e c a u s e of the n e c e s s i t y of n o n l i n e a r actions on g e n e r a l i z e d functions and, in the g e n e r a l q u a s i l i n e a r c a s e , b e c a u s e of the n e c e s s i t y of multiplying g e n e r a l i z e d functions by discontinuous ones. In [100], the f o r m u l a t i o n of the b o u n d a r y - v a l u e p r o b l e m f r o m [99] is extended to s p e c i a l q u a s i l i n e a r equations of the d i v e r g e n c e f o r m r(t,

z (%(t)) . . . . .

§ (I) (t, z (% (t)) . . . . .

z ( ~ (t)), z ' (% (t)) . . . . .

z ' (a~ ( t ) ) ) +

z ( ~ (t)), z' (o0 (t)) . . . . .

z'(Om(t))))=O

(9)

and one p r o v e s existence and uniqueness t h e o r e m s f o r such p r o b l e m s .

68 "~

5.4. Boundary-Value P r o b l e m s f o r L i n e a r Functional-Differential Equations. Various boundary-value p r o b l e m s have been studied by Azbelev, Rakhmatullina and t h e i r c o l l a b o r a t o r s [1-3, 5-9, 68, 137, etc.]. In these p a p e r s one investigates l i n e a r and nonlinear equations of f i r s t and higher o r d e r and one obtains r e s u l t s r e garding the s i n g l e - v a l u e d solvability of the b o u n d a r y - v a l u e p r o b l e m s , the continuous dependence of the solution of the boundary-value p r o b l e m on the p a r a m e t e r , the existence and the dimensionality of the fundamental s y s t e m , the conjugate boundary-value p r o b l e m , etc. We d e s c r i b e in m o r e detail the f o r m u l a t i o n f o r the linear boundaryvalue p r o b l e m and the a p p r o a c h used in its investigation [1, 5, 7-9, 137]. We consider the equation x ' (t)-- q (t) x ' [g (t)] + p (t) x [h (t)] = r (t),

t~[a, hi, (lO)

x(U=~(D,

x'CD=,CD,

~[a, b].

This equation can be r e p r e s e n t e d in the f o r m (11}

Lx = f , where the linear o p e r a t o r L is defined in the following m a n n e r . We set

hi; 0 , [g Ct)]

if g(t)fi[a, b], if g (t)~[a, bl;

xh(t)=

x [h (t)], 0

if h (Off[a, b], ~f h(t)-~[a, b];

~n(t)=

0 r

~f h(t)E[a, b], if h(t~[a, b].

,~ ( t ) =

Let

f (t)=r (t)--q( t) ~g (t)-- p(t)q~n (t}. Then, Eq. (10) can be written as

(Lx) (t) =x'(t)+q(t)x'a (t) +p(t)xh(t) =f(t).

(10')

Usually one i m p o s e s conditions on the functions in Eq. (10) so that the o p e r a t o r L turns out to be a continuous o p e r a t o r acting f r o m the space Dp of functions which a r e absolutely continuous on the s e g m e n t [a, b] and have d e r i v a t i v e s belonging to the space Lp of p - p o w e r s u m m a b l e functions into the s p a c e Lp. Since, for any fianction

xeDp , x(t) =x(a)+$x'(s)ds, t h e n l i n e a r o p e r a t o r L : Dp~Lp

has the r e p r e s e n t a t i o n

a

(Lx) (t) = Ax (a) + (Qx') (t),

(12)

where the linear o p e r a t o r Q:Lp-+L a is called the principal p a r t of the o p e r a t o r L arid in the case of Eq. (10) it has the f o r m b

(Qy)(t)=y(t)+q(t)vg(t)+ S p(t)x(t, s)y(s)ds,

(13)

a

X(t, s) being the c h a r a c t e r i s t i c function of the s e t {(t, s)C[a, bt • in, hi: a-.< s -,< h (t) ~
lx=O is called a boundary-value p r o b l e m for this equation.

684

(14)

F o r the boundary-value p r o b l e m (10'), (14) one can write out the conjugate b o u n d a r y - v a l u e p r o b l e m [with the o p e r a t o r which is conjugate to the o p e r a t o r (L, l) : Dp~LvXR '~ ]. In this c a s e it turns out that the F r e d h o l m p r o p e r t y of the o p e r a t o r Q is equivalent to the F r e d h o l m p r o p e r t y of the boundary-value p r o b l e m (109, (14). F r o m h e r e one obtains r e s u l t s which connect the s i n g l e - v a l u e d solvability of the boundary-value p r o b l e m (10'), (14) with the solvability of the homogeneous b o u n d a r y - v a l u e p r o b l e m Lx=0,

lx=O,

(15)

with the dimensionality of the kernel of the o p e r a t o r L, etc. 5.5. Boundary-Value P r o b l e m s f o r Nonlinear F a n c t i o n a l - D i f f e r e n t i a l Equations. n e u t r a l type

A nonlinear equation of

x' (t)= f (t, xlh(t)l, x'[g(t)]), te[a, b] (16) x (~) = q~(U,

x ' (~) = , (~), ~ [ a , b]

can be r e p r e s e n t e d also in the f o r m

x'=Fx

(17)

with an o p e r a t o r F acting f r o m the s p a c e Dp into the s p a c e Lp. In this case it is possible [2] that Eq. (17) can be reduced to the equivalent equation x' = q)x

(18)

with a c o m p l e t e l y continuous o p e r a t o r q) : D~-~Lp 9 Then the use of the "good" p r o p e r t i e s of c o m p l e t e l y continuous o p e r a t o r s allows us to obtain a s e r i e s of t h e o r e m s on the solvability of boundary-value p r o b l e m s . At the investigation of boundary-value p r o b l e m s , it is frequently convenient to use the s o - c a l l e d " w method" (see, f o r e x a m p l e , [8, 9]), which allows us to e s t a b l i s h a o n e - t o - o n e c o r r e s p o n d e n c e between the s o l u tions of the boundary-value p r o b l e m (17), (14) and the solutions of a c e r t a i n o p e r a t o r equation g = Gg

(19)

with the o p e r a t o r G : Lp-+Lp. The ideas r e l a t e d to the application of this method have been developed by a s e r i e s of authors and have been used for the investigation of boundary-value p r o b l e m s both f o r f i r s t - o r d e r equations of neutral type (16) and f o r higher o r d e r equations [4, 6, 8, 68, 120]. 5.6. V a r i o u s Questions. V a r i o u s b o u n d a r y - v a l u e p r o b l e m s f o r differential equations of neutral type have been investigated by m a n y authors [50, 87, 96, 122, 138, 141, 161, 281, etc.]. One has investigated existence and uniqueness questions f o r the solutions of boundary-value p r o b l e m s [122, 138], a s y m p t o t i c expansions of the solutions [181], the n u m e r i c a l determination of the solutions and the convergence of a p p r o x i m a t e methods [87, 96], the oscillation of the solutions [162], etc. Questions r e g a r d i n g the eigenfanction expansions of the solutions of a boundary-value p r o b l e m f o r equations of neutral type with a d i s c r e t e deviation of the a r g u m e n t have been c o n s i d e r e d in [141]. In [161] one has studied a boundary-value p r o b l e m of the S t u r m - L i o u v i l l e type with an infinite defect. In [87] one has studied the question of the c o n v e r g e n c e of the f i n i t e - d i f f e r e n c e method for the n u m e r i c a l solution of boundary-value p r o b l e m s for the l i n e a r equation

[k=!mak(t)x' ( t - - T k ) ] ' + + ~

[b~(t)x'(t--~k)+ck(t)x(t--~k)]=f(t),

0
(20)

k~tn

x(t)=O,

t~b.

(21)

In [96] one has obtained sufficient conditions for the convergence and e s t i m a t e s f o r the r a t e of convergence of the collocation method of the a p p r o x i m a t e solution of a b o u n d a r y - v a l u e p r o b l e m f o r equations of f o r m (2) with constant coefficients f o r the highest d e r i v a t i v e s and f o r special functions ~i and with homogeneous boundary conditions.

685

The question of the a p p r o x i m a t e solution by the method of oscillating functions of a b o u n d a r y - v a l u e p r o b l e m f o r an equation of neutral type has been studied in [216]. A boundary-value p r o b l e m f o r equations of neutral type in a Banach space has been c o n s i d e r e d in [281]. Finally, a s e r i e s of p a p e r s (V. 1% Skripnik, D. D. Baiaov, etc.) [119, 157, 159, etc.] a r e devoted to the investigation of b o u n d a r y - v a l u e p r o b l e m s f o r differential equations of s u p e r n e u t r a l type. 6.

Linear

Equations

6.1. Operational Methods. F o r the investigation of l i n e a r equations of neutral type with constant coefficients, the application of the Laplace t r a n s f o r m turns out to be useful. This is r e l a t e d to the fact that the r e a l p a r t s of the roots of the c h a r a c t e r i s t i c quasipolynomial, c o r r e s p o n d i n g to the differential e x p r e s s i o n

(Lx) ( t) = aox' ( t) +alx' (t--o) + box ( t) + bix (t--co) f o r a 0 ~ O, are bounded f r o m above and, t h e r e f o r e , the exponential growth of the right-hand side of the equation

(Ix) (t) =/~(t)

(1)

implies the exponential growth of t h e solutions of this equation. Because of this, one can apply the Laplace t r a n s f o r m to Eq. (1). In this c a s e , if x s a t i s f i e s the initial condition

x(t)=g(O,

0..< t < o ,

while f o r f we have the e s t i m a t e

l f ( t ) l < c l e c,t, t > 0 ,

c1>0,

c2>0,

then f o r any sufficiently large real n u m b e r c we have c+IT

etsh -' (s) [p (s) + q (s)] as,

l

t>O,

(2)

C--

where

h (s) = aos + alse - ~ + bo + b~e-~, p( s)=aog(o~)e-~ + a~g(O) e - ~ + (aos + bo) ~ g( t)e-~tdt, 0

q (s) = ~ f (t) e-~tclt. o)

The d e s c r i b e d r e s u l t can be found in [46, 212]. The r e p r e s e n t a t i o n of s e v e r a l solutions of Eq. (1) in the f o r m of a finite s u m of e l e m e n t a r y functions can be found in [254]. In [69] one obtains f o r m u l a s s i m i l a r to (2), f o r s y s t e m s of equations of neutral type with s e v e r a l deviations of the a r g u m e n t s . The conditions which e n s u r e the possibility of the application of the Laplace t r a n s f o r m to equations of the f o r m (3)

Jy (x( t)-- Ax,)-~ Bxt + f ( t)

(A and B are l i n e a r bounded o p e r a t o r s , acting f r o m C h = C[_h, 0] into Rn) a r e indicated in [284]. The c a s e of the equations in a Banach s p a c e is c o n s i d e r e d in [247, 248]. If the deviation of the a r g u m e n t is v a r i a b l e , then the application of the Mikusinski o p e r a t o r turns out to be convenient. F o r e x a m p l e , the solution of the equation

x'(t)+a~x(t)+~ox'(t--h(t))+f~,x(t--h(t))=f(t),

t>O,

satisfying the initial conditions

x(t)=v(t)(inf(,--h(x))~O

X(m)(-~-O)=Xm

under c e r t a i n r e s t r i c t i o n s carl be r e p r e s e n t e d in the f o r m oo

(--1) k

~=0

w h e r e _J is the v a r i a b l e shift o p e r a t o r , defined by the f o r m u l a h

686

Bop+B, ,~kg,,

(re:O,

1)

.2r _(0, t < tt tl}' h J*--~f(t--h(t)), t > g is c o n s t r u c t e d f r o m f , y, x 0, and x 1, while a g . denotes the result of the application of the o p e r a t o r a to the function g (see [182]; in the same p a p e r one c o n s i d e r s also equations of a m o r e general form). 6.2. Representation of the Solutions. In the theory of l i n e a r equations of neutral type one establishes f o r m u l a s which are s i m i l a r to the f o r m u l a s of the variation of the a r b i t r a r y constant in the theory of o r d i n a r y differential equations. The derivation of these f o r m u l a s is complicated by the difficulties which are specific for the theory of equations of neutral type, related to the " p o o r n e s s " of the set of initial functions f r o m which one has to turn out the smooth solutions. T h e r e f o r e , in this case, by a solution we mean a function which satisfies, instead of the differential equation itself, some of its integral analogues (or one a s s u m e s that the derivatives of the solutions lie in a certain space of summable functions and the equation is c o n s i d e r e d in the sense of "almost e v e r y w h e r e " ) . We d e s c r i b e only some r e s u l t s . Thus, f o r the integral p r o b l e m t

t

x(t)=e~(O)--Ag+Ax,+ S Bx~ds+S h(s)ds, x,~ =

t >a,

(4) (5)

9

(A and B a r e linear o p e r a t o r s acting f r o m Ch into Rn), induced by the differential equation (3) with the initial condition (5), one has the following r e p r e s e n t a t i o n of the solution: t

x(~, 9, h ) ( t ) = x ( ~ , 9, o)(t)+j"

Here x(a, 9, h) is a solution of p r o b l e m (4)-(5), x(a, 9, 0) while

V(t)=

v(t-s)h(s)ds.

(6)

is a solution of the same p r o b l e m f o r h ( t ) ~ 0 ,

aW , where W is a solution of the following integral p r o b l e m at t

W(t)=Aw~+SBW~ds--tI,

t>0,

0

Wo=O. The fundamental condition imposed on the o p e r a t o r A is the condition of its nonatomieity at z e r o (see [272]). The d e s c r i b e d r e s u l t can be found in [269, 270, 272]. In the same works one has obtained r e p r e s e n t a t i o n s , s i m ilar to (5), in the case of a nonautonomous equation (4). In [114], a f o r m u l a of this type is proved f o r equations of the f o r m

x" (l) =Ax, + Bx,' + [ (t). The case of equations in a Banach space is c o n s i d e r e d in [249]. P a p e r s [20, 171, 246] are devoted to various questions r e g a r d i n g the r e p r e s e n t a t i o n of the solutions of linear equations of neutral type. Frequently, the initial p r o b l e m for an equation of neutral type can be reduced to the f o r m

(qx') (t)+a(t)x(a)=/(/),

t~[a, b],

(7)

where the o p e r a t o r Q acts in a certain space of functions, defined on the s e g m e n t in, b], with values in R n. The solvability of the initial p r o b l e m f o r Eq. (7) and r e p r e s e n t a t i o n f o r m u l a s for its solution can be investigated in t e r m s of the invertibility of the o p e r a t o r Q. F o r example, it is p r o v e d that the equation (7) with the initial condition x(a) =a is uniquely solvable for any p a i r ([, a) ~Lp ([a, b], R ~,) • - if and only if the o p e r a t o r Q, acting f r o m Lp (in, b], R~) into Lp ( in, b], R ~) , has a bounded inverse. In this case we have the following int e g r a l r e p r e s e n t a t i o n of the solution: x(t)=

I - - (Q-'a)(s)ds x(a)+ (Q-lf)(s)ds. a

t

Moreover,

(Cf ) ( t ) ~ f (Q-'f)(s)ds

is a V o l t e r r a o p e r a t o r if and only if Q and Q-1 are V o l t e r r a o p e r a t o r s .

In

a

the case of equations of the f o r m

( I - - S ) x ' + Kx' + A x = f ,

687

b

where (Kg)(t)=j'lr

s)x(s)ds

, the o p e r a t o r C is a V o l t e r r a o p e r a t o r if and only if the o p e r a t o r (I

S)-~

a

exists and I - S and (I - S)-i are Volterra operators. These results are described in [4, 8], etc. 6.3. Differential Operators of Neutral Type. As mentioned above, the fundamental properties of the solutions of equations of neutral type depend mostly on the properties of the operator containing the derivative of the desired function. The same situation is observed in many other problems. We consider, for example, the question of the validity of the Fredholm alternative in the problem regarding the periodic solutions. In general, for equations of neutral type the Fredholm alternative need not hold. Indeed, the simplest equation

x'(t)--x'(t--T) =0 has infinitely many l i n e a r l y independent T~periodie solutions. It is p r o v e d in [1631 that f o r the equation r~

co

~ B~( t) x' (t-- hk) + ~ x( t + s)d~p( t, s)= f ( t) x.,

(8)

the F r e d h o l m a l t e r n a t i v e in the s p a c e L 2 ( T - p e r i o d i c functions with values in R n with the n o r m

[[ x []~=

T

-~, f ' x(s) i2ds

) holds if and only if it holds f o r the equations

0

Ax = f andA*x ~ g , Fa

w h e r e (Ax)(t)=~.~ Bk(t)x(t--h~), k~I

m

while

(A*x)(t)=~x*(t+hk)Bk(t).

In p a r t i c u l a r , if o p e r a t o r A is invertible,

4=I

then, obviously, the Fred_h_olm a l t e r n a t i v e holds. (For equations of a m o r e g e n e r a l f o r m than (8), s i m i l a r condltions f o r the validity of the F r e d h o l m a l t e r n a t i v e are e s t a b l i s h e d in [164].) A s i m i l a r situation takes place also at the investigation of the s o - c a l l e d evolution solvability (see [125]) of the equation

Dx' + Bx=f,

(9)

w h e r e D, 13: C ~ C. As p r o v e d in [125, 129, 130], the evolution solvability of Eq. (9) is equivalent to the r e tarding invertibility of the o p e r a t o r D and does not depend on the o p e r a t o r B. Thus, the question on the s o l v ability of Eq. (9) r e d u c e s to the question on the invertibility of the o p e r a t o r D in the a l g e b r a of r e t a r d e d (i.e., Volterra) o p e r a t o r s , which, in turn, leads to the p r o b l e m of the computation of the r e t a r d e d s p e c t r u m (i.e., the s p e c t r u m in the a l g e b r a of r e t a r d e d o p e r a t o r s ) of the o p e r a t o r D or to the e s t i m a t i o n of the s p e c t r a l radius of the o p e r a t o r in this algebra. In [130] one c o n s t r u c t s a special s e m i n o r m in the a l g e b r a of r e t a r d e d o p e r a t o r s , which e s t i m a t e s f r o m above the s p e c t r a l radius of the o p e r a t o r in this algebra. One obtains effective f o r m u l a s f o r its computation. It is p r o v e d that the kernel of this s e m i n o r m contains the radical of the a l g e b r a of r e t a r d e d o p e r a t o r s . It is also shown that the radical of the a l g e b r a of r e t a r d e d o p e r a t o r s contains all the completely continuous o p e r a t o r s . In p a r t i c u l a r , this m e a n s that the p e r t u r b a t i o n s of the o p e r a t o r s D and B by completely continuous o p e r a t o r s do not affect the p r o p e r t y of the evolution solvability of Eq. (8). In m a n y c a s e s the s p e c t r u m of the o p e r a t o r D can be computed exactly. Thus, the s p e c t r u m of the o p e r a t o r of the f o r m

(Dx) ( t)-~ ~ A~x (h' (t)) k=O

(here h a ( t ) = h (h ~-I (t)) , while A k are constant n • n m a t r i c e s ) has the f o r m

@ ~ (f(X)) , where z~e(71

m

f (~')= Z k'm,, k~0

while a (T) is the s p e c t r u m of the o p e r a t o r T, defined by the f o r m u l a

(Tx) (t) = x (h (t)) (see [126] and also [133]). In [134, 196], for v a r i o u s r e s t r i c t i o n s on the m a t r i x Aij one w r i t e s out the s p e c t r u m of the o p e r a t o r 688

( D x ) ( I ) = ~ ~.~ A,i a'x (t +h,]) dti

i~0 j=0

In [128] one gives an example of an o p e r a t o r of the f o r m (Dx)(t) = x(t) -x(h(t)) having a one-dimensional kernel. Various questions, connected with the invertibility of the o p e r a t o r s D and construction of the o p e r a t o r s conjugate to D in various p r o b l e m s , have been investigated also in [4, 8, 20]. At the conclusion of this subsection we describe a generalization of the well-known t h e o r e m of F a v a r d . Let C be the space of functions, continuous and bounded on the entire axis, w i t h values in Rn and a s s u m e that S. : C ~ C ( ~ R ) is defined by the f o r m u l a (S,x) (t) =x(t+v) (t~R) . A linear bounded o p e r a t o r A: C ~ C is s a i d t o be a l m o s t periodic if the set {S,AS_. : ~R} is relatively compact in the space L(C) of linear bounded o p e r a t o r s in C. We c o n s i d e r an equation of neutral type of the f o r m

(~x) (t) = (Dx') (t)+(Bx) (t) =[(t)

(10)

with o p e r a t o r s D and B, almost periodic and continuous in the topology of uniform convergence on compacta. By H(D, B) we denote the c l o s u r e in L(C) • L(C) of the set {(S,DS_,, S,BS_J : ~R} . Assume, finally, that the s p e c t r a l radius of the o p e r a t o r D is less than one. It turns out (see [1501) that in this case the o p e r a t o r s is invertible (i.e., Eq. (10) has a unique bounded solution f o r any bounded function f ) if and only if the equation (~x') (t) + (Bx) (t) = 0 has no n o n z e r o bounded solutions f o r any (D, B)~H(D, B) 9 Moreover, i f f function ~ - ' [ .

is almost periodic, then so is the

6.4. Behavior of the Solutions at Infinity. F o r equations of neutral type, just as for o r d i n a r y differential equations, under certain conditions one has an exponential dichotomy of the solutions, i.e., the entire space of solutions splits into a direct sum of two subspaces such that the solutions f r o m one of the subspaces behave for t--.oo as O(e~,'), while the solutions f r o m the other subspace behave f o r t - - . - o o as O(e,,0 (a~ < 0 < a 2 ) . In [2741 one has obtained conditions for the exponential dichotomy of the solutions of the linear autonomous equation

dt Dx t = L x t

(11)

in the case when the zeros of the characteristic quasipolynomial, corresponding line Re k = c~ into two disjoint sets separated from this line, and

to Eq.

(II),

are divided by the

0

DqD=cp(0)-- I ~(s)dt~ (s), --h

w it h 0 f [d~t(s)]-->0 -E

for

and 0

0

,[

--h co

where

0 < hk-.< h

, while ~ [ A~I+ k=l

a,,)+ .[ / ~ = .,-"

--h

0

[ [ A(s)l ds < co. A n e c e s s a r y and sufficient condition for the exponential --h

dichotomy f o r autonomous equations of neutral type has been obtained in [48]. The case of a nonautonomous equation

(Lx) (t) = x' (t)-- A (t) x t -- B (t) x / = 0

(12)

is considered in [127], where it is proved that the exponential diehotomywith a , < 0 < a 2 under the condition of c o r r e c t solvability of Eq. (12) ( c o r r e c t solvability takes place, f o r example, if liB(t)l]
Here we mention a l s o the p a p e r s in which one has obtained conditions f o r the boundedness of the s o l u t i o n s of equations of n e u t r a l type on the s e m i a x i s . In [69], such conditions a r e i n d i c a t e d f o r equations of n e u t r a l type of the n - t h o r d e r with d i s c r e t e d e v i a t i o n s of the a r g u m e n t . In [234] one has e s t a b l i s h e d conditions under which f r o m the boundedness of the s o l u t i o n s of the equation m--1

m

x' (t + h.,)+ ~.~ A~(t) x' (t + h~)-[-~ B~(t) x (t-}-tq)=O I=0

i=0

(0=h0 < h~ < . . . < h~) t h e r e follows the boundedness of the s o l u t i o n s of the p e r t u r b e d s y s t e m trl--I

m

x' (t+ h,~)+ ~] IA, (t)+ C~ (t)l x' (t + k,) + ~ [B, (t)+D, (t)l x(t+ h,)=O. t=0

1~0

A l s o h e r e one has to mention the p a p e r [278]. In [261], f o r the l i n e a r equation of the f o r m

a-~-A(t, xt)~B(t, xt) with bounded c o e f f i c i e n t s , it is shown that none of its n o n z e r o s o l u t i o n s can tend to z e r o f a s t e r than any e x ponential as t---oo. The a s y m p t o t i c b e h a v i o r f o r t --* oo of the s o l u t i o n s of equations of n e u t r a l type has been s t u d i e d a l s o in [245]. A s y m p t o t i c a l l y autonomous l i n e a r equations have been s t u d i e d in [276]. 6.5. L i n e a r Equations with a P a r a m e t e r . We c o n s i d e r the s y s t e m

In this s e c t i o n we d e s c r i b e , b a s i c a l l y , the r e s u l t s of [217-219].

at A(~, OxU, e)+B(~, e) x(~--h, e)+ q-C(~, e) x'(~--h, e)+ f(z, e)em(l,~), A(v, e), B(~, e)

where

0(., ~)

and C(x, e) a r e r e a l n x n m a t r i c e s ,

is a given s c a l a r function such that d0 (t, ~)

tit

k (z)

f(.,e)

(13)

is a given f u n e t i o n w i t h v a l u e s m R n,

(here ~- = at and a is a s m a l l r e a l p a r a m e t e r ) ,

In addition, we a s s u m e that the c o e f f i c i e n t s of the s y s t e m (13) can be expanded in s e r i e s of p o w e r s of h>O. the s m a l l p a r a m e t e r a: c~

A(z, e)=%~s]A(])(~), B (~, e)=~ #BU) (T), j=o j=o c (~, ~)=]~ ~Jc(" (~1, f(~, ~1=~] ~Jf(" (~). j=0

j=0

F o r s y s t e m (13) one i n v e s t i g a t e s the i n i t i a l value p r o b l e m on the s e g m e n t [ - h , T]:

x

(~, ~)= q~(~, ~) (~G [ -

tz, Ol).

(14)

One assumes that

(,, e ) ~

eJ~u) (~). j=O

Since s y s t e m (13) contains in the r i g h t - h a n d side the t e r m A (~, e)x(T, e) , the method of s t e p s f o r the d e t e r m i n a t i o n of the solution in a c l o s e d f o r m is e i t h e r i n a p p l i c a b l e or l e a d s to an i m m e n s e amount of c a l c u l a t i o n s . T h e r e f o r e , one p r e s e n t s a m e t h o d which allows us t o w r i t e down an a s y m p t o t i c f o r m u l a f o r the solution at any s t e p . One i n v e s t i g a t e s both the r e s o n a n c e c a s e [when i k ( r ) c o i n c i d e s with one of the e i g e n v a l u e s of the m a t r i x A(T)] and the n o n r e s o a a n c e c a s e . In both c a s e s one obtains the r e p r e s e n t a t i o n of the solution x at the k - t h s t e p in the f o r m

x ( ~ , ~)--vm,~(~,

~)+~m-~(~, ~),

w h e r e V m , k ( r , a) can be computed in a c l o s e d f o r m , while ~ k ( T , a) a r e u n i f o r m l y bounded in the neighborhood of the point r = 0. S i m i l a r r e p r e s e n t a t i o n s a r e o b t a i n e d a l s o f o r equations of the f o r m

690

ex' ('v, e) = A ('~, e) x (~, e) + B (~, e) x (~-- h, e) + C ('~, e) x ' (~, e), with initial condition (14). In [219] one investigates the case of a variable deviation of the a r g u m e n t s . Here we mention also [170] in which one investigates the dependence on the p a r a m e t e r s a and b = b(0) of the solutions of the equation

x' (at)+ b (t)x'(t) + c (t) x (at) + d (t) x (t)=O, where b, c, and d are analytic functions while a is a real number. 6.6. Various Questions of the Linear Theory. The investigation of the behavior of the solutions of l i n e a r autonomous equations is simplified in an essential m a n n e r if some information is known about the s t r u c t u r e of the quasipolynomial and the set of its z e r o s . F o r example, f o r the equation t~

(a~x ~ (t)-

a~x (~ (t - h)) = o

k~O t~

the c h a r a c t e r i s t i c quasipolynomial can be split into two f a c t o r s : (1--e -h~)

and

~akk

k . As a consequence

of

this (see [228]), the solution x, satisfying the initial condition xt0 = ~p, can be r e p r e s e n t e d in the f o r m n

x ( t ) = , (t) + ~ c~x~ (t), i=O t~

where x i is the solution corresponding to the z e r o s of the polynomial

X ak~k ' ci a r e some constants, while k=0

n

r is the periodic continuation of the function e (t)-- ~ c~% (t).

The s t r u c t u r e of the set of the z e r o s of the quasi-

i~l

polynomials, c o r r e s p o n d i n g to various equations of neutral type, has been studied in [45] and in other p a p e r s . F o r equations of neutral type, as well as in the case of equations of r e t a r d e d type, the s e m i g r o u p s , g e n e r a t e d by the shift o p e r a t o r along the t r a j e c t o r i e s of the equation, turn out to be strongly continuous. In [42], for the equation

x' (t) = AlX (t) + A~x (t -- h) + A3x' (t -- h)

(15)

one has written out the generating o p e r a t o r of such a s e m i g r o u p . Unlike the case of ordinary differential equations, all of whose solutions, emitted at time zero, fill out the entire phase space Rn at any moment t, the solutions of equations of neutral type may, at some time td, fill out, instead of the entire space Rn, only some subspace of it. In these c a s e s one says that the equation is pointwise degenerate at the point t d. In the opposite case, one says that we have pointwise completeness. In [242] one indicates n e c e s s a r y and sufficient conditions for the pointwise d e g e n e r a c y of the s y s t e m (15) at some point, while in [241] one gives conditions which e n s u r e the pointwise completeness of s y s t e m s of the f o r m h

x' (t)= A~x( t ) + A2x ( t - - h ) + A3x' ( t-- h) + ~ k.x (t-- s) ds. 0

In [282] one has c o n s i d e r e d singular equations of the f o r m

(16)

A (z) g' (z)+ x [Bi (z) g(azz) + Ci (z) g' (~z)] = 0 l~l

with nonnegative shown that if

n • n matrix-functions A, B i, Ci, holomorphic in some c i r c l e Gr and continuous in Gr. It is

~liB~[[< ~,

~ ] ] C ~ { I < oo

i=I

i=1

and

while detA(z) has s z e r o s (eountingmultiplieities)in Gr, then the s y s t e m (16) has at least n - s dependent solutions, holomorphic in Gr.

linearly in-

691

7.

Problems

on the

Axis

7.1. Periodic Solutions. The p r o b l e m regarding the periodic solutions f o r equations of neutral type has a s e r i e s of p a r t i c u l a r i t i e s in c o m p a r i s o n with the analogous problem for ordinary differential equations and equations of r e t a r d e d type. F o r example, if the equation

x{"~ ( t ) = f ( t , x (t), x (t--Tt), x (t--~i), . . . , xVO (t--zi), xr

(1)

(here [ : R~+4~R ~ is a continuous function, T - p e r i o d i c with r e s p e c t to the f i r s t argument, z~=k,T (ki are natural numbers, T > 0), while 6 i > 0) is an equation of r e t a r d e d type (i.e., the f u n c t i o n f does not depend on the last two arguments), then it has at most an n - p a r a m e t e r family of T - p e r i o d i c solutions; however, in the case of an equation of neutral type, it is possible that any n times piecewise differentiable T - p e r i o d i c function is a solution of the equation (1) (see [229]). In the same paper and also in [220], one indicates another s e r i e s of p a r t i c u l a r i t i e s , inherent in sets of periodic solutions of equations of neutral type. Therefore, one of the i m portant steps in the investigation of the p r o b l e m on the periodic solutions of equations of neutral type is the isolation of the c l a s s e s of equations which do not p r e s e n t the pathological p a r t i c u l a r i t i e s of the above described type. Such c l a s s e s are usually isolated with the aid of conditions which e n s u r e the solvability of the equation in some sense, relative to the derivative. The simplest of these conditions is the condition of the stability of the contraction type with r e s p e c t to the variable x' o c c u r r i n g in the right-hand side of the equation. F o r example, one considers equations of the f o r m

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

(2)

where the o p e r a t o r f : R~X C~X Ch--*Rn , T - p e r i o d i c with r e s p e c t to the f i r s t variable, satisfies a Lipschitz condition with a constant k < 1 with r e s p e c t to the third variable"

I f ( t , tZ, vt)-- f (t, u,

z,~-)/-_<#l!v'-v=l G.

These equations are convenient for investigation also due to the fact that (see, for example, [190, 191]) one can apply to them the t h e o r y of the rotation of condensing v e c t o r fields (see [40, 191]). In this case, frequently, the p a r a m e t e r s which occur in the equation are the p a r a m e t e r s of the admissible homotopies of the c o r r e spending v e c t o r fields. We describe some o p e r a t o r s f o r which the existence of the fixed points is equivalent to the existence of T - p e r i o d i c solutions f o r the corresponding equations. The cited o p e r a t o r s will be condensing under natural r e s t r i c t i o n s , with a constant less than unity, relative to specially constructed sufficiently "good" m e a s u r e s of noncompactness (for terminology and facts regarding the theory of m e a s u r e s of noncompactness and condensing o p e r a t o r s , see [40, 191]). The f i r s t o p e r a t o r acts in the space of continuously differentiable T - p e r i o d i c functions and is defined by the f o r m u l a t

==

T

+ Si o

,<..-<. <'=t

t

S i <=, x.,

Its fixed points, and only they, are T-periodic solutions of the equation (2). The second o p e r a t o r acts in the space R n x PCT (PC T is the s p a c e o f T-periodic continuous functions with values in Rn) and has the f o r m F2 (~, Y)= (~-- M (y), ~ (X, y)), where T

(~ 0~, v)) (t) = f (t, (~ + Jr),, vD, 0

while #

(Yy) (t) = ; y (s) d s - - t M (y). o

The fixed points (k, y) of the operator F 2 and the T-periodic solution x of Eq. (2) are connected by the relations x' = y, x(0) = k. The shift operator turns out to be very useful in the problem of the periodic solutions of Eq. (2). At the same time, instead of the shift operator along the trajectories of Eq. (2) one considers usually the shifts along the trajectories of the equation

x'(t)=fG

692

x,, x,')+~,.(O (x'(O)--f(o, Xo, xo'))

(3)

(regarding the function ug, see Sec. 4.1) f o r which the c o n s i s t e n c y condition is s a t i s f i e d on the entire s p a c e of initial functions. The p e r i o d i c solutions of Eqs. (2) and (3) coincide. The p r o p e r t i e s of the o p e r a t o r s F 1 and F 2 a r e investigated in [190, 191] and those of the shift o p e r a t o r in [105, 190]. The conditions u n d e r which the shift o p e r a t o r along the t r a j e c t o r i e s of the equation d

- ~ D ( x 3 = f (t, x,) is condensing a r e given in [258]. P a r a l l e l with the t h e o r y of the rotation of condensing v e c t o r fields, a l s o other g e n e r a l i z a t i o n s of the c l a s s i c a l theory of the d e g r e e s of mappings of L e r a y - S c h a u d e r a r e useful for the i n v e s tigation of the p r o b l e m of the periodic solutions of equations of n e u t r a l type. In p a r t i c u l a r , we mention h e r e the application of tile t h e o r y of the coincidence d e g r e e of p a i r s to equations of n e u t r a l type (see [14, 271, 275]). Wittl the aid of topological methods, one p r o v e s easily existence t h e o r e m s f o r periodic solutions of q u a s i l i n e a r equations of neutral type in the ease when the l i n e a r p a r t of the equation does not have nonzero periodic s o l u tions. Results of this and s i m i l a r kind, obtained by both topological and a n a l ~ i c a l methods, can be found in [188, 191, 210, 220, 222, 223, 229]. A s e r i e s of r e s u l t s a r e connected with the p r e s e n c e of attracting points o r of an attracting s e t f o r the shift o p e r a t o r . Thus, it is p r o v e d in [220] that if f o r the equation

x'(t)=F(t, x(t), x(t--ht) . . . . . x'(t--h,~))

(4)

t h e r e exists a uniformly a s y m p t o t i c a l l y stable solution x * , which f o r sufficiently large t s a t i s f i e s the inequality Ix*(t) [ 0), then Eq. (4) has at l e a s t one periodic solution. An analogue of tale M a s s e r a - H a l a n a y t h e o r e m f o r l i n e a r equations of f o r m (2) is p r o v e d in [165]. The p r e s e n c e of repelling points can also lead to the existence of periodic solutions. equation

x'(t)= [ --ax(t--1)~

F o r example, the

a (xm+1(t__l))l(l__x2(t)) mk+ l dt

has a nontrivial solution of p e r i o d g r e a t e r than 2 if I

since for k = 0 the zero point is repelling for the shift operator tories of the equation

over a time larger than 2 along the trajec-

x' (t)= -- ax (t-- 1)(1 -- x2(t)) (see [293]). Various applications of the topological methods in the p r o b l e m of the periodic solutions of equations of neutral type a r e c o n s i d e r e d also in [49, 107, 191, 192, 275, 295]. Side by side with the cited p a p e r s , this p r o b l e m has been studied by analytical methods a l s o in [117, 203, 206]. The p r o b l e m of the p e r i o d i c solutions of l i n e a r equations with d i s c r e t e deviations of the a r g u m e n t (both c o m e n s u r a b l e mad n o n c o m e n s u r a b l e ) has been investigated in [18, 151]. 7.2. P e r i o d i c Solutions of Equations with a P a r a m e t e r . H e r e we d e s c r i b e two directions i n t h e i n v e s t i g a tion of the periodic solutions of equations with a p a r a m e t e r : the f i r s t one is connected with the application of v a r i o u s a s y m p t o t i c methods, methods of the s m a l l p a r a m e t e r ; the second one is connected with the application of topological methods. We d e s c r i b e briefly the r e s u l t s obtained in the f i r s t direction. In [210] one has i n v e s t i g a t e d periodic solutions of q u a s i l i n e a r equations of the f o r m x'(t) = A ( x . x , ' ) + f ( t , x. x,', e), w h e r e A : C~• I--h, 0], R")-~R" is a l i n e a r bounded o p e r a t o r , while f : R ~ • C~XL2( [--h, 0], R n) X[0, 1]'+R ~ is a nonlinear o p e r a t o r . In addition, in [210] one has obtained a f o r m u l a f o r the integral r e p r e s e n t a t i o n of the solution of the Cauchy p r o b l e m f o r the indicated equation. In [211] one investigates p e r i o d i c and a l m o s t periodic solutions of s i n g u l a r l y p e r t u r b e d equations of the neutral type of the f o r m

ex'(t) =[(t, x(t), x(t--eh), ex'(t--eh), e). E x i s t e n c e t h e o r e m s f o r p e r i o d i c solutions f o r s m a l l values of the p a r a m e t e r e f o r equations in a Hilbert s p a c e H of the f o r m 693

x'(t)+Ax~+Bx,'=f(t, x,, x,', e) (here A, B:C([--h, 0], H ) ~ H are l i n e a r bounded o p e r a t o r s , while f : R • (C([--h, 0], H))~• 1 ] ~ H is an o p e r ator periodic with r e s p e c t to the f i r s t a r g u m e n t and not containing linear t e r m s with r e s p e c t to the second and the t h i r d arguments) have been obtained in [85]. Also t h e r e one c o n s t r u c t s the asymptotics of the periodic solutions with r e s p e c t to the powers of a s m a l l p a r a m e t e r . S i m i l a r r e s u l t s f o r equations of the f o r m

x'(t)=[(t, x(t), x(t--h), x'(t--h), x'(t+h), ~), where /:R4"+Ix[O, 1]~R", h > 0 , are d e s c r i b e d in [17], while f o r the equation

x' (t) +axx' (t--h) +a2x(t) +a~x(t--h) = ex3 (t) -b~x3(t--h) +el (t) (a i and h are constants while ~ and /~ a r e small) they are d e s c r i b e d in [236]. The r e s u l t s obtained in the second direction of the investigations appear usually in the following manner. We consider, for example, an equation with a p a r a m e t e r of the f o r m

x'(t)=f(t,x, xt', ~)

(5)

(f : R~xChXChX [0, l i a r ~) and we a s s u m e that f o r ~ = 0 this equation has an isolated periodic solution x ~ with a n o n z e r o topological index with r e s p e c t to some equivalent o p e r a t o r equation (see previous subsection). Then, f o r s m a l l e, the equation (5) has a periodic solution xS, close to the solution x ~ As a rule, one also finds conditions in o r d e r that the topological index of the solution x ~ be different f r o m z e r o . Results of this type can be found ha [49, 190, 191, 220, 223, etc.]. See also the next two subsections and Subsection 9.2. 7.3. Bifurcation of the P e r i o d i c Solutions. We c o n s i d e r now the p r o b l e m of the birth of periodic solutions f r o m an equilibrium state. Assume that an equation of neutral type, depending on a p a r a m e t e r 5 E [0, 50], has f o r 5 = 0 a s t a t i o n a r y solution, andthe l i n e a r equation, obtained by the linearization of the initial equation along this solution, has a subspace of T0-periodic solutions. Then, at the variation of 5, just as in the t h e o r y of o r dinary differential equations, under certain conditions this subspace can be split into a finite n u m b e r of cycles of the initial nonlinear equation, of periods close to T O. We d e s c r i b e in m o r e detail only one result. We cons i d e r the equation

x' ( t ) - ~ x ' (t - h ) + ~x (t) + ~ v x (t - h) = ~ / ( x (t)),

(6)

in which f(0) = f ' ( 0 ) = 0. F o r the convenience of the formulation we p e r f o r m in it the change of variables 7 = Tt after which a statement on 2 v / T - p e r i o d i c solutions of equation (6) becomes a s t a t e m e n t on 2 ~ - p e r i o d i c solutions of the equation

r (x, ( t ) - ~ x ( t - 7t~))+ ~x (t)+ ~vx(t - r h ) = ~s" (x (t)).

(7)

It is p r o v e d in [235] that for 7 > fi > 0 there exist an infinite n u m b e r of p a i r s of real n u m b e r s (c~0, To) such that the c h a r a c t e r i s t i c equation kT ( 1-- ae -~rh) + ~ + czye-~rn ~- O, corresponding to the l i n e a r p a r t of Eq. (7), has f o r c~ = a 0 purely i m a g i n a r y roots XT = • 0. Let (o~0, T 0) be such a p a i r of n u m b e r s , T o ~ 0, and a s s u m e that there exists a r e a l solution a * ~ 0 of the equation 221

F(a*)=a*--'21To I f ( 2 a * c o s t ) c o s t d t : 0 0

suchthat

dF(a*)l>~>0 da

[

. Assume a l s o t h a t ~ = a 0 ( l + e )

and

[ f ' ( x ) l < l x [ for ixl<4a*. Then, for s m a l l e > 0 t h e r e exist functions e~. T (e} solution of Eq, (7) for T = T(~) and

and (t, e)~.x(t, e)

T(s)-+T0 and X(t, e)-~2a*cos t

for

such that x(-, 5) is a 2 v - p e r i o d i c

e-~0

(see details in [235]). In [237] one can find the expansions of T(5) and x(t, e) in powers of the small p a r a m e t e r e. In [86] one indicates a n e c e s s a r y and sufficient condition f o r the existence of a periodic solution of Eq. (6). More general equations have been c o n s i d e r e d in [267]. M o r e o v e r , in [267], the p r o b l e m on the apparition of the periodic solu694

tions for equations of neutral type r e d u c e s to the investigation of an analogous p r o b l e m f o r ordinary differential equations, c o n s t r u c t e d f r o m the eigenfunctions and eigenvalues of the l i n e a r i z e d equation. Nonautonomous equations have been considered in [84]; one has investigated questions of existence and uniqueness of a s m a l l n o n z e r o periodic solution of the equation rn

x' ( t ) - - ~ At(t, e)x' ( t - - h 3 = f (t, xt, x/, e) t=i

(here A~(t, e) : E-+E, [:RxC([--h, 0], E) x C ( [ - - h , 0], E) X [0, 1]-+E, E being a finite- o r an infinite-dimensional Banach space).

In [36] one presents a scheme for the investigation of the bifurcation of the periodic solutions of nonautonomous functional-differentialequations of neutral type with a small retardation and also of the bifurcation of the periodic solutions of equations of neutral type in the averaging principle. 7.4. Equations with Small Retardation. One of the most interesting cases, both in theory and in applications, is the case when the parameter in the equation is the deviation of the argument. In problems regarding periodic solutions, one has isolated conditions under which small variations of this quantity do not affect the qualitative picture of the existence and stability of the periodic solutions. In general, the investigation of equations with small retardation is complicated by the fact that equivalent operator equations contain operators which are only strongly continuous with respect to the parameter characterizing the magnitude of the deviation of the argument. In most of the investigations, by equations with a small deviation of the argument one means equations which can be written in the form

x' (t)-~/ (t, (V (e)x)t , (V(e)x'),), w h e r e f : R X C h X C h ~ R n, while (V(e)x)t

(V

(~) x)t (s) =

(8)

and (V(e)x'), are elements of the space Ch, defined by the f o r m u l a s x (t + ~ (~) s), (V (~) x'), (s) = x' (t + o~(~) s),

o~ is a function continuous at z e r o and defined on some s e g m e n t [0, ~0]. We note that if w(0) = 0, then to the z e r o value of the p a r a m e t e r ~ there c o r r e s p o n d s an equation which is an o r d i n a r y differential equation, not solved with r e s p e c t to lhe derivative. If o p e r a t o r f is continuous and satisfies the Lipschitz conditions with r e s p e c t to the t h i r d variable with a constant less than unity (the case when it is expanding with r e s p e c t to the third variable is c o n s i d e r e d in [106]), then a condition which e n s u r e s the p r e s e r v a t i o n of the periodic solutions for small e is (see [38]) that the topological index of the T - p e r i o d i c solution x ~ of the Eq. (8) f o r e = 0 be different f r o m z e r o . In the c a s e when the o p e r a t o r f is sufficiently smooth with r e s p e c t to the space v a r i a b l e s , then this condition is satisfied, for example, if Eq. (8), l i n e a r i z e d a l o n g x ~ f o r e = 0 does not have n o n z e r o T - p e r i o d i c solutions. Under these conditions, the indicated r e s u l t f o r equations of neutral type with a d i s c r e t e variable deviation of the a r g u m e n t is given in [184, 187]. In [184] one has investigated an equation with d i s c r e t e retardation, depending on the unknown function. In these c a s e s , under sufficiently smooth r i g h t - h a n d sides, one has c o n s t r u c t e d asymptotic expansions of the periodic solutions in the powers of the s m a l l p a r a m e t e r . In the case of deviations f r o m a nonzero retardation, s i m i l a r questions have been c o n s i d e r e d in [121]. The p r o p e r t i e s of the shift o p e r a t o r along the t r a j e c t o r i e s of Eq. (8) have been investigated in [109]. In p a r t i c u l a r , one has shown that the shift o p e r a t o r is condensing with r e s p e c t to the totality of the space variable and the variable c h a r a c t e r i z i n g the magnitude of the retardation; this fact allows us to investigate the p r e s e r v a tion of the stability p r o p e r t i e s f o r small variations of the deviations of the argument. In [39, 186] one has investigated the p r o b l e m of the periodic solutions of an autonomous equation of neutral type with a s m a l l deviation of the argument. The investigation of the ~autonomous" case is complicated by the following three c i r c u m s t a n c e s : f i r s t l y , the p e r i o d of the solution is not known a p r i o r i ; secondly, as a rule, autonomous equations do not have isolated periodic solutions; thirdly, as a rule, the topological index of the sets of periodic solutions of autonomous equations is equal to z e r o . These difficulties can be b y - p a s s e d by extending the method of the functionalization of the p a r a m e t e r , due to M. A. K r a s n o s e l ' s k i i , to equations of neutral type (this is done in [39]). If the equation

x'(t)=f((V(*)x)t, (V(e)x')t)

(9)

has a nonstationary T0-period~c solution x ~ then a fundamemal condition for the p r e s e n c e , for s m a l l e, of ~ periodic solutions of period close to T o is the absence, f o r the obtained linearization of Eq. (9) f o r e = 0 along the solution x ~ of the equation, of T0-periodic solutions linearly independent f r o m (x~ ' and also of Floquet s o l u tions a s s o c i a t e d to (x~ ' (see [39]).

695

An existence and uniqueness t h e o r e m for an almost periodic solution of functional-differential equations with a small lag has been p r o v e d in [27], while an existence and uniqueness t h e o r e m for a bounded solution of equations with a small discrete lag has been p r o v e d in [182]. 7.5. Floquet Theory. F o r linear ordinary differential equations with T - p e r i o d i c coefficients, any solution can be r e p r e s e n t e d as a linear combination of a finite n u m b e r of Floquet solutions, i.e., solutions of the form

xk~(t)=e

t -- ln~b T

i

~_~p,j(t)y,(t),

(10)

]=0

where X k (k = 1, 2 . . . . . /) are the eigenvalues of the shift o p e r a t o r along the t r a j e c t o r i e s of the initial equation l

/

over the time T, having multiplicities m k ( X r n ~ = n , n

is the dimension of the phase space), i = 1 , 2

m k ,

the polynomial Pij (t) has degree at most j, while the functions y] are T - p e r i o d i c . Since in the case of equations of neutral type the phase space is infinite-dimensional, it is natural to formulate the question on the c o m p l e t e ness of the s y s t e m of Floquet solutions, i.e., on the r e p r e s e n t a t i o n of the solutions in the f o r m of a convergent s e r i e s in functions of the f o r m (10). We describe results of this type. In [123] one considers an equation of the form

x' (t) = q (t) x (t --nT) + ax' (t-- roT),

(11)

where q is a T - p e r i o d i c function while m and n are natural n u m b e r s . It is p r o v e d that if a ~ 0 and q does not vanish ca intervals, then for the completeness of the s y s t e m of Floquet solutions in the space C [ - p T , 0] (p = max {m, n}) it is n e c e s s a r y and sufficient that the function q be of constant sign. If, in addition, the function q does not vanish at any point, then the s y s t e m of the derivatives of the Floquet solutions is complete in C [ - p T , 0]. This is sufficient in o r d e r that any solution of Eq. (11) should be approximated together with its derivative, on a n y finite interval, with any degree of a c c u r a c y , by a s y s t e m of Floquet solutions. In [124] one obtains sufficient conditions for the completeness in the space C I [ - T , 0] of the s y s t e m of Floquet solutions of the s e c o n d - o r d e r equation

x" ( t ) = p ( t ) x ( t ) + q(t) x ( t - - T ) + ax" (t-- T). In [104] it is shown that for equations of the f o r m

x' (t) = A (t) x t + B (t) xt' for liB(t)II
Stability

Theory

8.1. Definition of the Concept of Stability. The width of the class of equations of neutral type stipulates a g r e a t e r freedom in the choice of the definition of the stability concept than in the theory of ordinary differential equations and of equations of r e t a r d e d type. The p r e s e n c e of the delay in the derivative of the untmown function makes us, generally speaking, c o n s i d e r the concept of the stability of the solutions of equations of neutral type in the sense of the m e t r i c of the space C l (there a r e examples (see [145]) showing that the concepts of stability in the m e t r i c of C 1 and in the m e t r i c of C do not coincide). On the other hand, a sufficiently large class of equations of neutral type is described, for example the class of equations of the f o r m

Adt ( x ( t ) - - a ( t ,

x ( t - - 1 ) ) ) = f ( t , x(t), x ( t

1)),

in which one can construct a meaningful stability theory in the sense of the m e t r i c of the space C (see, for example, [244, 262]). By virtue of these c o n s i d e r a t i o n s , an entire s e r i e s of p a p e r s is entirely or partially de696

voted to the discussion of the various stability concepts and to the investigation of the interrelations between them [10, 11, 30-35, 113, 145, 221, 296]; here we also mention the p a p e r s [199, 200] in which one investigates the p r a c t i c a l stability (i.e., stability on a finite interval) of the solutions of equations of neutral type. Another specific p a r t i c u l a r i t y of equations of neutral type, generating a s e r i e s of difficulties in the t h e o r y of stability, is the impossibility to emit a smooth solution f r o m an a r b i t r a r y , initial, continuously differentiable function. One of the possible ways to bypass these difficulties consists in considering the stability of the solutions of the equation

x'(t)=f(t,

x,, xt' ) ( f : R X C h X C h - + R n)

as the uniform stability, with r e s p e c t to small values of the p a r a m e t e r p , of the solution of the following equation with p a r a m e t e r :

x'(t)=f(t,

x t, x / ) + v u ( t ) ( x _ ' ( O ) - - f ( O , Xo, Xo')),

where v~(t) is equal to z e r o f o r t>~t and to 1--W't f o r t~[O, >] (see details in [33-35]). One can also cons i d e r (see [30]) stability only on the set of initial functions f r o m which smooth solutions are emitted. Finally, one can consider the stability concept in the sense of the m e t r i c of the space W~ when by a solution we mean some g e n e r a l i z e d solution. 8.2. Stability of Linear Equations. Unlike the equations of r e t a r d e d type, the investigation of the stability of linear equations of neutral type is complicated by the possible p r e s e n c e of chains of z e r o s of the c h a r a c t e r istic quasipolynomial, approaching unboundedly the imaginary axis (the s o - c a l l e d critical case; regarding this, see Subsection 5 of this section). In the noncritical cases the results are s i m i l a r to the corresponding results in the theory of equations of r e t a r d e d type. F o r example, if the c h a r a c t e r i s t i c roots of the equation

ant

A~x' (t--hi) + . ~ B i x ( t - - h i ) = O t =0

t=O

lie in the left half-plane and are s e p a r a t e d f r o m the i m a g i n a r y axis, then all the solutions of this equation s a t isfy the relation

Ix (t)l--< Me-~' IIXo IIc, where M and Y are positive constants which do not depend on the initial function. Results of this type can be found in [112, 245, 277]. The stability p r o p e r t i e s of the differential o p e r a t o r of the f o r m

(Lx) (t) = x ' (t) + A (t) x, + B (t) x / a r e closely r e l a t e d to the " V o l t e r r a p r o p e r t i e s " of the inverse o p e r a t o r L -l. More p r e c i s e l y , if A(t) and B(t) are linear bounded o p e r a t o r s f r o m Ch into Rn, strongly continuous with r e s p e c t to t, sup J!A(t)ff < oo , and sup IJB (t)I [ < 1, then the z e r o solution of the equation * t

Lx=O is exponentially stable if and only if the o p e r a t o r L : C~-+C is invertible and its inverse is of r e t a r d e d type (i.e., f r o m the equality x(s) =y(s) f o r s < t follows the equality (L-'x) (t) = (L-lg) (t)). Regarding this, see

[131]. An entire s e r i e s of papers is devoted to the determination of sufficient ~coefficient n conditions f o r the stability of the solutions of linear equations of neutral type. We describe in detail one result. We c o n s i d e r the equation

"~! (t)= i x ( t - - s)dI~l (S)-~- f xl (t--S)dI~~ o o

(1)

Here we a s s u m e that 3

Ko (s)= i z (~) d~ + ~ a~z (s-- h~), 0

where )v is a continuous function, a n and h, > 0

n=I

are constants, while X is the Heaviside function. Let

697

(2)

% = ~ s~IctKj(s)l < co (i, j~{o, 1}) 0

and a s s u m e in addition that ce00 < 1. We denote co

~t]-~-~ staKi(s) (i, jE{O,

1}).

0

A s s u m e that e i t h e r fl01 < 0 and a l l + a00 < 1 or /301 < 0 and

01< ~ (1 --(1 + e -1) c,00)

while K 0 and K 1 a r e

monotonically nonincreasing and constant f o r s>~h. Then, the z e r o solution of Eq. (1) is a s y m p t o t i c a l l y stable. The d e s c r i b e d r e s u l t can be found in [113]. Also t h e r e one obtains s i m i l a r sufficient conditions f o r the stability of the solutions of s e c e n d - o r d e r equations of neutral type. Various sufficient stability conditions, e x p r e s s e d in t e r m s of the coefficients, f o r f i r s t - o r d e r equations of n e u t r a l type with a d i s c r e t e deviation of the a r g u m e n t a r e given in [146, 240, 298]. Conditions f o r the a s y m p t o t i c stability of the solutions f o r equations of the f o r m

~ [akxr (t) + bkx(~ (t-- h)l=O k=O

can be found in [149]. F o r e x a m p l e , f o r n = 1 t h e s e conditions a p p e a r a s :

d~12>bl 2, a02>b02 and

alao>blbo.

8.3. Lyapunov's Second Method. In the direction of the development of Lyapunov's second method for equations of n e u t r a l type, one can single out two cycles: a cycle connected with the d i r e c t t r a n s f e r of Lyapunov's second method to the indicated equations and a cycle devoted to the development of N. N. K r a s o v s t d i ' s ideas r e g a r d i n g the r e p l a c e m e n t of Lyapunov's functions by s o m e functienals defined on the (infinite-dimensional) phase s p a c e of p r e h i s t o r y - f u n e t i e n s . F r o m the f i r s t cycle we mention the p a p e r s [145, 296] in which the method of Lyapunov functions, in the g e n e r a l i z e d v a r i a n t s u g g e s t e d by B. S. Razumikhin, is c a r r i e d o v e r to equations of n e u t r a l type with a d i s c r e t e deviation of the a r g u m e n t . B . S . R a z u m i k h i n ' s g e n e r a l i z a t i o n r e d u c e s to the weal~ning of the r e s t r i c t i o n s on the d e r i v a t i v e ~V(t, x), by virtue of the s y s t e m of Lyapunov functions V: it m a y be negative not f o r all values of x and only on those solutions on which, roughly speaking, V(t, x(t)) does not d e crease. The investigations devoted to the developments in the second direction a r e m o r e extensive. The extension of the r e s u l t s of N. N. K r a s o v s k i i and S. N. Shimanov r e g a r d i n g L y a p u n o v ' s second method (in the "functional" variant) f o r equations of r e t a r d e d type to equations of n e u t r a l type is d e s c r i b e d in [142, 240]. One p r o v e s t h e o r e m s on stability, a s y m p t o t i c stability and instability and also a t h e o r e m o n inversion; it should be noted t h a t in these p a p e r s one i m p o s e s sufficiently rigid r e q u i r e m e n t s on the s m o o t h n e s s of the L y a p a n o v - K r a s o v s l d i fanetienals. In [244, 262], the method of Lyapunov functions ( m o r e p r e c i s e l y , L y a p u n o v - K r a s n o v s k i i functions) is justified f o r equations of the f o r m

ff-~-(x (t)- e (t, x (t-h)))= / (t, x~), w h e r e [ :RxCh---~R s p a c e C.

~ .

Ill this

c a s e , the stability of the solutions is understood in the s e n s e of the m e t r i c of the

In a s e r i e s of p a p e r s , the investigation of the stability of the solutions of equations of neutral type, in the s e n s e of the m e t r i c of the s p a c e C i, with the aid of the L y a p u n o v - K r a s o v s k i i functionals, is divided into two stages: the investigation of stability in the s e n s e of the m e t r i c of the s p a c e C and the investigation of stability of c e r t a i n equations and inequalities of o r d e r z e r o . See, f o r example, [115, 116, 145]. The method of the L y a p u n o v - K r a s o v s k i i functionals for equations of the f o r m d

-ji-A(t, xt)=B(t, xt),

(3)

w h e r e A and B act f r o m R • Ch into Rn, is substantiated in [116]. It should be mentioned that in this p a p e r , as well as in [115], the stability conditions a r e e s t a b l i s h e d under the weakened a s s u m p t i o n of the existence of only s i g n - c o n s t a n t funetionals and not of fanetionals of fixed sign. In [199], the method of the L y a p u n o v - K r a s o v s k i i functionals is extended to equations of the f o r m (3) for p r a c t i c a l stability (i.e., stability on finite i n t e r v a l s ) . Finally, we mention s u r v e y s devoted to the description of L y a p u n o v ' s second method f o r equations of neutral type [10, 11, 262]. 698

8.4. Stability Tests f o r the Solutions of Nonlinear Equations. We consider the following nonlinear equation of neutral type:

x' (t)-~ f x ( t - - s ) d K ~ ( s ) + 0

x' (t--s)dKo(s)+ f (t, xt).

(4)

0

Here K 1 and K 0 satisfy the r e q u i r e m e n t s listed in Subsection 8.2 of the p r e s e n t paper, while f : R )< C ((-- oo, 0], R")-+R ~ is continuous and satisfies the following conditions: 1) f ( t , 0 ) ~ 0 ; 2) f o r any % , E C ( ( - - co, 0], R") we have the inequality I f (t, ~)-- f (t, ~) I ~
nonincreasing. A s s u m e that the n u m b e r s ~ij (i = 0, 1; j = 0, 1, 2) a r e defined by the equalities (2). Assume that the following inequalities hold: C*oo+a11
aol+ao2<~,

alo+a21+a12< ~

i dK1 ~s) ~ yCZo~ i-~-~a~--~--~o. 0

Then, the trivial solution of Eq. (4) is asymptotically stable. This r e s u l t and also other "coefficient" tests f o r the stability of the solutions of Eq. (4) can be found in [115]. F o r stability t e s t s of periodic solutions of q u a s i l i n e a r equations of neutral type, see [283], etc. Now we c o n s i d e r an equation of neutral type of o r d e r n:

x(") (t)+ f (x(t), x' (t) ..... x(") (t))= = ~ ( t , x(t), x' ( t) . . . . . x(,) (t), x (t-h(t)), x' (t--h(t)) . . . . . x(,~) (t--h(t))). In [178] one establishes sufficient c r i t e r i a f o r the stability of the trivial solution of this equation. The basic r e s t r i c t i o n s are: the s m o o t h n e s s o f f and ~, the Lipschitz condition f o r ~ - f with a constant less than one with r e s p e c t to the argument x (n), the positivity of xf(0 . . . . . 0, x, 0) for x ~ 0, the qualified s e p a r a t i o n f r o m z e r o of the e x p r e s s i o n f (Xo, xl ..... x~_x, xn)--f (0, 0..... x~_~, O) Xn

the s m a l l n e s s of q~ and the following r e s t r i c t i o n on the delay: O
~-T (x ( t ) - g (t, x,) - ~ (t)) = f (t, x,) - ~ (t) have been obtained in [270]. Also there one has obtained e s t i m a t e s of the solutions at infinity. 8.5. Critical Cases. We c o n s i d e r the linear equation of neutral type with constant coefficients m 'i~

a~jx(t) ( t - h j ) = O .

(5)

i=0 ]~0

We a s s u m e that all the c h a r a c t e r i s t i c roots of Eq. (5) have nonpositive real p a r t s . Unlike the equations of r e t a r d e d type, here we may have the s o - c a l l e d asymptotically critical cases {all c h a r a c t e r i s t i c n u m b e r s have negative real p a r t but there exists a chain of roots approaching unboundedly the imaginary axis) and the s u p e r c r i t i c a l c a s e s (there exists a chain of purely imaginary roots). In the asymptotically critical case, if all the " a s y m p t o t i c a l l y i m a g i n a r y " roots are simple, then we do not have an exponential stability of the solutions: there exist solutions which on c e r t a i n sequences of values of the a r g u m e n t behave as t -o~ when t - - ~ (~ > 0). Regarding this, see [256]. If, however, the multiplicity of the " a s y m p t o t i c a l l y i m a g i n a r y " roots is sufficiently large, then (see [299]) the z e r o solution of Eq. (5) may be simply unstable. The asymptotically critical c a s e s are investigated also in [64, 65, 239]. In p a r t i c u l a r , in the men699

tioned p a p e r s one gives conditions which ensure the stability or the instability of solutions and one finds the asymptotic behavior of the solutions when t ~ ~. Conditions which guarantee the p r e s e n c e of chains of " a s y m p totically imaginary" roots f o r the quasipolynomial corresponding to Eq. (5) a r e given in [65]. In the s u p e r c r i t i e a l case, depending on the behavior of the purely imaginary roots, the z e r o solution of Eq. (5) can be Lyapunov stable, but not asymptotically stable, or unstable (where the instability is observed on a countable, everywhere dense set in the space of coefficients of F_q. (5)). It should be mentioned also that in this case the stability of the solutions depends strongly on the relations between the magnitudes of the deviations of the argument in the equation. The investigation of the s u p e r c r i t i c a l case can be found in [64, 239]. The investigation of the stability of the zero solution of nonlinear equations of neutral type in the c r i t i c a l case of the p r e s e n c e of a simple or a multiple z e r o c h a r a c t e r i s t i c root f o r the linear p a r t is given in [144, 263]; f o r s i m i l a r results for the c r i t i c a l case of a pair of purely i m a g i n a r y roots, see [112]. In [273], the question of the stability of the solutions of a nonlinear autonomous functional-differential equation of neutral type in the critical case of s e v e r a l pairs of purely imaginary roots is reduced to the q u e s tion of the stability of the solution of a c e r t a i n ordinary differential equation, c o n s t r u c t e d f r o m the solutions of the l i n e a r i z e d equation, corresponding to the purely imaginary roots of the c h a r a c t e r i s t i c polynomial. 8.6. Stability T h e o r e m s for Constantly Acting Perturbations and Stability in the F i r s t Approximation. Here we r e s t r i c t ourselves only to the enumeration of the corresponding investigations. Stability t h e o r e m s for constantly acting perturbations and t h e o r e m s which are close in spirit to these r e s u l t s can be found in [142, 200, 267, 268, 278]. In [215] one considers the case when the unperturbed equation does not contain the deviations of the a r g u r m n t . Stability t h e o r e m s in the f i r s t approximation f o r equations of neutral type with a discrete deviation of the argument and under additional smoothness r e q u i r e m e n t s , a r e proved in [142, 146]. In [262] one p r o v e s such a t h e o r e m f o r equations of the f o r m

d--~(x(t)--O(t, x(t--h)))= f ( t, xt)

dt

(here

G:RXR"-+R"

, while f : R X C n ~ R " ) ,

while in [371 for the equation

x'Ct)-----fCt, xt, Xt') ( f : R >( Ch X Ca-~R")

; in the last paper the stability is understood in a certain g e n e r a l i z e d sense (see the end of

Sec. 8.1). 8.7. Stability of the Solutions of Equations with a P a r a m e t e r . In a s e r i e s of papers one has investigated the question of the p r e s e r v a t i o n of the stability p r o p e r t i e s of the solutions of equations of neutral type when a small delay is introduced. We consider the equation

x ' ( t ) = / ( t , V(~)xt, V (~)xt');

(6)

here f : R x C h x C h ~ R " , while V(~) acts for each e~[0, 1] f r o m Ch into Ch according to the f o r m u l a (V(e)x) (s) = x(es) (sE[--h, 0]) . In [34,106] one indicates conditions under which the periodic solutions xr of Eq. (6), close to the periodic solution x ~ f o r r = 0, inherit the stability p r o p e r t i e s of x ~ S i m i l a r results for equations of a p a r t i c u l a r f o r m are obtained in [215]. In connection with this, one has to mention [146, 221], in which one investigates the specific c h a r a c t e r of the dependence of the stability of the solutions on the behavior of the deviations and on the s t r u c t u r e of the equation. In [212] one investigates the stability of the p e r i o d i c solutions (in p a r t i c u l a r , one indicates a method for the calculation of the c h a r a c t e r i s t i c exponents) of linear equations of the f o r m

x' (t) = Ax, + B x,' + eC (t, ~) x, + ~D ( t, e) x / f o r small values of the p a r a m e t e r e. Similar p r o b l e m s are solved in [2011 for equations of the f o r m

x"(t)+co~x(t)= f (t)+~r x(t), x'(t), x"(t), x(t--h~), x'(t--h~), x" (t--hO . . . . . x(t--hp), x' (t--hp), x" (t - h~)), where ~ : RXR~3P*~'~R '~, o2=diag(ol 2. . . . . ~ ) ,

while ~ is a small p a r a m e t e r .

Finally, we mention h e r e the papers [32, 33, 35, 51-53, 213], d e s c r i b e d in Subsections 9.2-9.4, in which one investigates the stability of the periodic a n d a l m o s t periodic solutions of equations of neutral type in the averaging principle. 700

8.8. V a r i o u s Questions of the T h e o r y of Stability. In [108] one has investigated the orbital stability of the periodic solutions of autonomous equations of n e u t r a l type. One has c o n s i d e r e d the equation

x'(0 =f(x,, xt')

(7)

(~ : Ch• It has been p r o v e d that if Eq. (7) has a T - p e r i o d i c solution x ~ the shift o p e r a t o r along the t r a j e c t o r i e s of Eq. (7), l i n e a r i z e d along x ~ o v e r the t i m e T, has unity as a simple eigenvalue of multiplicity one, and all the r e m a i n i n g s p e c t r u m lies inside the unit c i r c l e (one also a s s u m e s t h a t f s a t i s f i e s a Lipschitz condition with r e s p e c t to the second v a r i a b l e with a constant l e s s than one), then the solution x ~ is orbitally stable. The p a p e r s [30, 143] a r e devoted to the investigation of the absolute stability of the solutions of the equations of neutral type (in the t h e o r y of equations with deviating a r g u m e n t , absolute stability is understood with two s e m a n t i c meanings: in the s e n s e of the independence of the stability p r o p e r t i e s f r o m the magnitudes of the deviations of the a r g u m e n t and in the s e n s e , c l a s s i c a l f o r the theory of automatic control, of uniform stability r e l a t i v e to s o m e c l a s s of equations; h e r e we have in mind the second meaning). In p a r t i c u l a r , in [30] one p r e sents a method f o r the investigation of the absolute stability of the solutions of equations of n e u t r a l type, s i m i l a r to the p r i n c i p l e of the absence of bounded solutions in the t h e o r y of o r d i n a r y differential equations, developed by M. A. K r a s n o s e l ' s k i i and A. V. P o k r o v s k i i . We give one of the s i m p l e s t r e s u l t s of the application of this method. We c o n s i d e r the equation

(8)

x' (t) =Ax(O +f(t, x(t--h), x" (t--h) ).

We denote by ~ s o m e s e t of mappings, satisfying the Lipschitz condition with s o m e fixed constants, acting f r o m R • 2n into R n. Let A be a Hurwitz n • m a t r i x and a s s u m e that f o r any f~t and any t~R, a, heR ~ we have the inequality

I f ( t , a, b)12-.~0, 1~< I). In this case, if

-=<,<=supII(itI--A)-lj].(~+

B(IIAII+=))I_8< 1 ,

then tile z e r o solution of Eq. (8) is absolutely stable r e l a t i v e to the family of the nonlinearities of ~. The investigation of the absolute exponential stability of equations of neutral type, in the s e n s e of the independence of the stability f r o m the magnitude of the delay, has been c a r r i e d out in [197]. In [224], one shows that the apparition of a s m a l l delay in the right-hand side of the equation

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

(9)

may lead to the loss of the stability p r o p e r t i e s of the solutions. If, however, f s a t i s f i e s a Lipschitz condition with r e s p e c t to the l a s t a r g u m e n t with a constant l e s s than one, then the stability p r o p e r t y is p r e s e r v e d . In p a r t i c u l a r , in [182] one shows that if the Eq. (9) has a bounded exponentially stable solution, then the equation

x' (t) =f(t, x(t-~), x' (t-~) ) h a s , f o r s m a l l ~, a close bounded, exponentially stable solution. Also t h e r e , one can find the expansion of this solution in the p o w e r s of the s m a l l p a r a m e t e r . In [262] one investigates the s t r u c t u r e of the w-limit s e t s of an autonomous equation of neutral type and one studies also the a s y m p t o t i c p r o p e r t i e s of the w-limit s e t s invariant relative to the shift o p e r a t o r . T e s t s f o r the uniform instability of solutions of equations of the f o r m (3) a r e p r e s e n t e d in [285]. The a s y m p t o t i c p r o p e r t i e s f o r t --* co of the solutions of equations of neutral type have been investigated in [111]; in p a r t i c u l a r , one indicates conditions f o r the existence of solutions with nonvertical a s y m p t o t e s . The stability domain in the s p a c e of the p a r a m e t e r s f o r equations 0s the f o r m

Ax' (t) +Bx(t) +Cx' (t--h) + Dx(l--h) --0 is indicated in [136]. Also there one has given conditions under which the stability is preserved when h varies. The stability of the equations of the motion of a pendulum in a viscous fluid (its motion is described by an equation of neutral type) has been investigated in [202]. The stability of the solutions of the equations of neutral type in connection with the problems of the theory of electric circuits has been investigated in [238]. 9.

Averaging 9.1. Averaging

Principle in a Finite Interval.

We

consider a Cauchy

problem

of the following form:

701

x'(t) =el(t, xt, x/),

(1)

Xo=g;

(2)

h e r e f : R•215 e is a s m a l l positive p a r a m e t e r and gECh is a given function. Let Ill be the natural imbedding o p e r a t o r of R n into Ch. We a s s u m e that f o r any a~R ~ t h e r e e x i s t s the finite limit

f o ( a ) = lira t ~ - 7 - 1 i f ( s , lha, O)ds o

and p a r a l l e l with p r o b l e m (1)-(2) we c o n s i d e r the s o - c a l l e d a v e r a g e d Cauchy p r o b l e m

x' (t) = e~o(x(t)),

(3)

x(O) =g(0).

(4)

We mention h e r e that the a v e r a g e d equation (3) r e p r e s e n t s an autonomous ordinary differential equation. A typical r e s u l t r e g a r d i n g averaging on a finite i n t e r v a l f o r equations of neutral type a p p e a r s as: the solutions of the p r o b l e m s (1)-(2) and (3)-(4) a r e close to e a c h other f o r s m a l l e, on an interval whose length has o r d e r e - l , in the m e t r i c of the s p a c e C 1. The averaging principle on a finite interval f o r equations with a d i s c r e t e delay can be found in [206, 207], f o r integrodifferential equations in [206] mid f o r equations in a Bmiach s p a c e with a d i s c r e t e delay in [173]; f o r equations of f o r m (1) the averaging principle in the c l a s s of g e n e r a l i z e d solutions (see Subsection 4.1) has been p r o v e d in [21]. The f o r m u l a t i o n of the a v e r a g i n g p r o c e d u r e f o r b o u n d a r y - v a l u e p r o b l e m s f o r equations of neutral type can be found in [44]. 9.2. Averaging P r i n c i p l e on the Entire Axis. P e r i o d i c Solutions. We a s s u m e that the o p e r a t o r f in Eq. (1) is T - p e r i o d i c r e l a t i v e to the f i r s t a r g u m e n t . A s s u m e that the a v e r a g e d equation (3) has a s t a t i o n a r y solution x~ ~ x * . Finally, a s s u m e that x* is an i s o l a t e d z e r o point of the v e c t o r field f0 of nonzero index. Under these conditions, Eq. (1) has f o r s m a l l e at l e a s t one T - p e r i o d i c solution x e , close in the m e t r i c of C ~ to the solution x ~ of the a v e r a g e d equation. If, in addition, the o p e r a t e r f is continuously differentiable with r e s p e c t to the s p a c e v a r i a b l e and the o p e r a t o r f~ (x*) does not have eigenvalues on the i m a g i n a r y axis, then such a p e r i o d i c solution which is close to x ~ is unique. M o r e o v e r , the solution x e of Eq. (1) inherits the s t a bility p r o p e r t i e s of the solution x ~ of Eq. (3). More p r e c i s e l y , the solution x e is strongly exponentially stable if the s p e c t r u m of the o p e r a t o r f~(x *) lies in the left half-plane and it is strongly unstable if the s p e c t r u m of ! , the o p e r a t o r f 0 ( x ) i n t e r s e c t s the right open h a l f - p l a n e . The foundations of the a v e r a g i n g principle for Eq. (1) in the c l a s s of periodic solutions can be found in [26, 31] and the investigation of the stability of periodic solutions in [32, 33, 35, 192]. 9.3. Averaging P r i n c i p l e on the Entire Axis. Almost P e r i o d i c mid Bounded Solutions. We consider the l i n e a r equation (1) with a l m o s t periodic coefficients: (L (e)x) (t) = x ' (t)--cA (t, xt)--eB (t, xt') =0, where A and B act f r o m R • Ch into R n, a r e a l m o s t periodic with r e s p e c t to the f i r s t a r g u m e n t , uniform with r e s p e c t to the second one, strongly continuous with r e s p e c t to the f i r s t a r g u m e n t , l i n e a r and bounded with r e spect to the second one. A s s u m e that the o p e r a t o r

t Aoa= ,-+~lim~ _ f A ( s ,

Iha)ds

(aER n)

does not have eigenvalues on the i m a g i n a r y axis. Then, f o r s m a l l e o p e r a t o r s L(~) a r e r e g u l a r , i.e., invertible in the space of a l m o s t periodic functions. One has also a s t a t e m e n t on the r e g u l a r i t y in the s p a c e of functions bounded on the entire axis, if the coefficients of the o p e r a t o r L(e) a r e bounded. In the c l a s s of a l m o s t periodic (and also bounded) functions, the averaging principle f o r Eq. (1) also holds (its f o r m u l a t i o n is analogous to the f o r m u l a t i o n of the averaging principle in the c l a s s of periodic functions; however, in this c a s e one has to impose s t r i c t e r s m o o t h n e s s r e q u i r e m e n t s on the o p e r a t o r f ) . The f o r m u l a t i o n of the a v e r a g i n g principle f o r equations of neutral type in the c l a s s of a l m o s t - p e r i o d i c functions, bounded on the entire axis, can be found in [23-25, 29]. 9.4. Equations Containing F a s t and Slow T i m e . We c o n s i d e r an equation of the f o r m x'(t) =,el(t, et, xt, xt'),

702

(5)

where [ : R x R • is a l m o s t periodic with r e s p e c t to the f i r s t and the second a r g u m e n t s , uniform with r e s p e c t t o t h e r e m a i n i n g v a r i a b l e s , smooth r e l a t i v e to the s p a c e v a r i a b l e of the mapping, while e is a s m a l l positive p a r a m e t e r . In this case the a v e r a g e d equation willnot be an autonomous a n y m o r e b u t a n o r d i n a r y differential equation

x' (t) =~f0(~t, x(t) ),

(6)

where #

io(,, a)=l' --# I i (s, , . --t

One has statements on the existence and the uniqueness of almost-periodic solutions x~ of Eq. (5), close, for small ~, to the almost-periodic solutions {~ of the averaged equation (6), on the inheritance of the stability properties of the solutions ~ by the solutions x~, on the bifurcation of the almost periodic solutions of Eq. (5) f r o m the periodic solution of Eq. (6), and on the a s y m p t o t i c expansion of the solutions x~ in p o w e r s of the s m a l l p a r a m e t e r ~. The mentioned r e s u l t s can be found in [47, 51-53, 209]. 9.5. Averaging Method f o r Equations in Nonstationary F o r m . Asymptotic integration p r o c e d u r e s , s i m i l a r to the a v e r a g i n g p r o c e d u r e , a r e p r o v e d also f o r equations of the f o r m

x'(t)=Ax,+Bx/+e[(t,

xt, x,'),

where A and B are linear bounded o p e r a t o r s , acting f r o m Ch into R n, while [:R• ~ is a function periodic with r e s p e c t to the f i r s t a r g u m e n t , under the a s s u m p t i o n that this equation has f o r ~ = 0 a stable t w o p a r a m e t e r f a m i l y of periodic solutions. One has also investigated various a s y m p t o t i c methods applied to singularly p e r t u r b e d s y s t e m s of equations of n e u t r a l type. F o r e x a m p l e , one has found conditions f o r the existence of periodic solutions of the following s y s t e m s of equations:

sx' ( t ) = f (t, x(t), x ( t - - e h ) , sx' (t--eh), e) and

x , ( t ) = / ( t , x(t), v(t), O, ~v'(t)=g(t, x(t), v(t), v ( t - e h ) , e v ' ( t - , a ) , O. Finally, one has investigated the a v e r a g i n g p r o c e d u r e f o r s e c o n d - o r d e r equations of neutral type, in p a r t i c u l a r , f o r equations of the f o r m

x" (t) + plx' Ct) + qtx ( t - h,) + p2 (t) + qzx' (t -- h2) + rx" (t -- ha) = = s t (t, x(t), {x(t--ht)}, x ' ( t), {x' (t--h~)}, x"( t), {x" (t--h~)}). Results of this n a t u r e can be found in [47, 213]. The K r y l o v - B o g o t y u b o v a s y m p t o t i c methods have been applied also to the investigation of the i n t e r a c t i o n p r o c e s s e s of two nonlinear o s c i l l a t o r y s y s t e m s under the p r e s e n c e of delays in the c o n s t r a i n t f o r c e s . F o r e x a m p l e , one has investigated the o s c i l l a t o r y p r o c e s s e s in the following s y s t e m s of equations of n e u t r a l type:

x" (t)+ o~2Cst) x (t) = ~/~ @t, x (0, x' (t), x" Ct), vCt-h), v'Ct-h), v"Ct-h)), y" (t)+o22 (et) y ( t ) = e f 2(et, x ( t - - h ) , x' ( t - - h ) ,

x " ( t - h ) , v(t), v' (O, v" (t)). These investigations can be found in [1881. 10.

Extremal

Problems

Variational p r o b l e m s f o r differential equations with deviating a r g u m e n t of neutral type and, in p a r t i c u l a r , p r o b l e m s of optimal control p r o c e s s e s d e s c r i b e d by such equations have been investigated intensively s t a r t i n g f r o m the end of the sixties and the beginning of the s e v e n t i e s . Up to now, a p p r o x i m a t e l y 500 p a p e r s have been published, devoted to a c e r t a i n extent to this topic. 10.1. C l a s s i c a l V a r i a t i o n a l P r o b l e m s . P r o p e r v a r i a t i o n a l p r o b l e m s , i.e., the p r o b l e m s of the d e t e r m i n a tion of the e x t r e m u m of a functional with deviating a r g u m e n t , have been i n v e s t i g a t e d in [92-95, 103,225, 233,

703

257, 294]. The fundamental variational p r o b l e m with deviating argument is posed in the following maimer: one has to find the e x t r e m u m of the functional b

[xl -- ~ F (t, x (t), x' (t), x (t--,),

x' (t -- ~)) dt

(1)

~t

with well-defined boundary conditions. At the same time, such a problem is the s o u r c e of a s e r i e s of boundaryvalue p r o b l e m s for differential equations of neutral type, which have been already discussed in the c o r r e sponding section. The initiator of the development of variational p r o b l e m s with deviating argument (and, in p a r t i c u l a r , of neutral type) has been L. ~. E l ' s g o l ' t s . In one of the f i r s t p a p e r s devoted to these questions [225], he has introduced the t r a n s v e r s a l i t y conditions in connection with variational p r o b l e m s for the functional (1). Subsequently, variational problems for functioaals of the f o r m (1) have been investigated in [103, 257, 294]. One has obtained n e c e s s a r y conditions for an e x t r e m u m , corresponding to the c l a s s i c a l conditions of Euler, W e i e r s t r a s s , Legendre, and Jacobi, and also sufficient minimality conditions for the functionals under consideration, f o r m ulated in t e r m s of the f i r s t and the second variations. One has also c o n s i d e r e d variational problems f o r functionals with variable lag. F o r the functional b

:~lvl=f F(x, V(x), V(x--~(x)),

V'(x), V' (x--~(x)))dx

(2)

a

with definite boundary conditions one has obtained n e c e s s a r y conditions for an e x t r e m u m and sufficient conditions for a weak relative e x t r e m u m (the analogues of the Legendre and Jaeobi conditions for the c l a s s i c a l variational problem). One has investigated the corresponding second variation of a quadratic functional and the related linear differential operator. One has obtained existence conditions for a self-adjoint extension of this o p e r a t o r and also convergence conditions f o r an a r b i t r a r y minimizing sequence for the mentioned quadratic functional. One has also studied the boundary-value problem generated by this variational problem; one has obtained conditions f o r its solvability. These and also s i m i l a r questions have been c o n s i d e r e d by Kamenskii in [92-95] and also by other authors (see, for example, [233]). 10.2. Maximum Principle. P a r a l l e l with the p r o b l e m s of the classical calculus of variations for equations with deviating argument of neutral type, one has developed methods of solution for the p r o b l e m s of optimal control. The p r o b l e m of control optimality in connection with a s y s t e m of equations of neutral type is formulated in the general case in a m a n n e r s i m i l a r to the c l a s s i c a l formulation of the p r o b l e m for equations without deviations of the argument: one has a controllable object, whose behavior is d e s c r i b e d by the equation

x'(t)=F(t, x, x. x/, u),

(3)

where x, f , u are (vector) functions. The (vector) function u(t) is called the control. Usually one a s s u m e s that on s y s t e m (3) there are imposed conditions which e n s u r e , for each admissible control, the existence and the uniqueness of the solution of the s y s t e m on a certain interval [t 0, T]. One also has a c e r t a i n functional T

S I x , u ] = S F ( t , x, u)dt,

(4)

to

which has the purpose to estimate the quality of the w o r k of tile object; f o r example, the s m a l l e r the value of 2/, the higher is the quality of the work. Finally, one gives certain boundary conditions

x,. =% x~=,.

(5)

Then the optimal control p r o b l e m is posed in the following manner. F r o m the set of admissible controls u one has to select one f o r which its corresponding solution x* of s y s t e m (3) satisfies the boundary conditions (5) and m i n i m i z e s functional (4). In this case, the solution x* is called optimal solution and the control u is called optimal control. In general, the optimal control p r o b l e m is a variational problem but the application of the methods of the calculus of variations f o r its solution is very difficult, just as in the case of the classical p r o b l e m of optimal control f o r ordinary differential equations.

704

The n e c e s s a r y conditions f o r the e x t r e m u m of the quality functional in the classical p r o b l e m of optimal control c a r r y the name of m a x i m u m principle. A large n u m b e r of investigations, devoted to problems of optimal control for objects d e s c r i b e d by differential equations with deviating argument of neutral type, are devoted to a certain extent to the t r a n s f e r of the m a x i m u m principle to this case [19, 102], [231, 232, 243, 278, 280, 288, 297], etc. In a s e r i e s of p a p e r s (see, for example, [232, 243, 279, 280]) one investigates the controlled system

x'(t) =Alx(t) +A2x(t--h) + A3x"(t--h) +Bu(t),

(6)

where A1, A 2, A 3, B are constant or variable m a t r i c e s and h is a constant lag. The interval [t 0, T] of the v a r i a tion of the time t is fixed. Often one a s s u m e s that the control u(t) is a function f r o m the space Lz([t o, T], R m) and that the boundary functions ~ and r belong to the Sobolev space W2(1)([-h, 0], Rn). We denote by x(~, u) the solution of Eq. (6) corresponding to the control u and emanating from 9~, i.e., xt0( ~, u) = 6. By the a c c e s s i b i l i t y set f r o m the initial element q~ we mean the set

A ~ = { x ~ ( % u)l ~eL~(lto, rI, R~)}. If the set A~ is closed in Wx(1)([-h, 0], R n) (for example, coincides with it), then f r o m this fact one derives that the optimal control problem with the corresponding quality functional has an optimal solution. F o r such p r o b lems one has a s e r i e s of c r i t e r i a for the closedness of the accessibility set A~ and one has f o r m u l a t e d n e c e s s a r y optimality conditions in the f o r m of a m a x i m u m principle. A s i m i l a r equation but with a variable lag has been c o n s i d e r e d in [297]. 10.3. Optimality Conditions of the Type of the Conditions of V. F. Krotov. Questions related to obtaining sufficient optimality conditions in the f o r m of the conditions of the type of those of V. F. Krotov, for s y s t e m s of neutral equations, has been c o n s i d e r e d in [102]. We give the p r e c i s e formulation of the p r o b l e m f r o m [102]. One considers the functional-differential equation of neutral type

x' (t)= f (t, x(t), x,(t), x;(t), u(t))

(7)

(t~It0, t,I, x = ( x ', x ~. . . . . x~ where

x,(t)=x(t--,(t)),

x ~ ( t ) - - - - - ~ ,=,-,r

and the functions

x(t), x~(t), u(t)

x(t)=~(t), x'(t)=n'(t), to-~(to)~
satisfy the conditions:

(8) (9) (10)

The sets Bl(t), B2(t), U are given; A(t) = t - 7(t) is a continuously differentiable and s t r i c t l y i n c r e a s i n g function. One has to minimize the functional tl

J Ix, u]= S fo(t, x(t), x,(t), x'~(t), u(t))dt.

(11)

to

F o r this p r o b l e m one has found a sufficient condition fc~ an absolute minimum, which is a generalization of the c o r r e s p o n d i n g conditions of V. F. Krotov. F o r the p r o b l e m (7)-(9), (11) with a fixed right endpoint x(t 1) = x 2, when the set U does not depend on t, x, xT, x17, one f o r m u l a t e s a maximum principle. However, f o r such p r o b lems the m a x i m u m principle does not hold in the g e n e r a l case. 10.4. Various P r o b l e m s of Optimal Control. L i n e a r p r o b l e m s of optimal control have been c o n s i d e r e d also in [231, 288, 295]. One has c o n s i d e r e d p r o b l e m s of finite control f o r s y s t e m s d e s c r i b e d by h i g h e r - o r d e r differential equations of neutral type [61], the problem of the control of a s y s t e m of neutral type with s e v e r a l inputs of the group of o r d i n a r y dynamic r e g u l a t o r s , the p r o b l e m of the optimal control f o r a hyperbolic s y s t e m of partial differential equations with a deviating a r g u m e n t of neutral type [19] and c e r t a i n other p r o b l e m s . One has investigated the relation between the controllability of a c e r t a i n linear s y s t e m of neutral type and the c o r responding p e r t u r b e d nonlinear s y s t e m [253]. P r o b l e m s of optimal control for equations with deviating a r g u ment of neutral type in connection with the theory of g a m e s have been considered in [57]. P r o b l e m s of the controllability of linear and nonlinear differential s y s t e m s of neutral type with the aid of the selection of the initial functions have been investigated in [120]. In [62] one solves the problem of the selection of the coefficients d 0, d 1. . . . . dn_ 1 in the equation

705

coyr ~ (t) + . . . + cn_~y' (t) + g (t) = doyen>(t -- ~) + . . . + d._,g' (t - - . ) + / (t)

(12)

in such a m a n n e r that the l a r g e s t real p a r t of the z e r o s of the c h a r a c t e r i s t i c quasipolynomial of this equation should be minimized. ll. Partial

Differential

Equations

The list of r e f e r e n c e s devoted to partial differential equations of neutral type is relatively small. Apparently, the f i r s t investigations have been made in the m i d - s i x t i e s and since then a s e r i e s of authors have published p a p e r s in which they have studied questions of existence, uniqueness and continuous dependence f o r the solutions of initial and boundary-value p r o b l e m s , questions regarding the approximate determination of the solutions, questions of existence and determination of periodic solutions, etc. 11.1. Existence T h e o r e m s and Approximate Methods of Solution. T h e o r e m s on the existence, uniqueness, continuous dependence and approximate methods f o r the determination of the solutions f o r various p a r t i a l differential equations of neutral type can be found in [193, 194]. One c o n s i d e r s the Cauchy p r o b l e m for differential equations, the p r o b l e m of the determination of solutions with additional G o u r s a t - t y p e conditions arid some other boundary-value p r o b l e m s . Here is an example of a p r o b l e m and the method of its solution, investigated in [194]. One seeks the solution of the integropartial differential equation of neutral type:

Ottt (x, t)

oxot

fx

= / (t' x , , (x, t)) + ~ ~ [ K (x, t, ~, ~)tt (~, ~)+ m (x, t, ~, ~ o,,, (~, 6 (~))] d~dT,

(1)

CLa

satisfying the Goursat conditions:

tt(a, t)=q~(t),

on file initial set E h .

u(x, a)=r q~(a)=,Ca), a~u (x. t) o~ot = g ( x , t)

(2}

P r o b l e m (1)-(2) with the aid of the substitution tx

u(x, t)=o(x, t)+ y ~ v(~, x)d~d~

(3)

tX a

reduces to an integral equation which is solved approximately by the method of additive funetionals. Along these lines one succeeds to investigate the convergence of the p r o c e s s to the solution of the initial problem, to estimate the admissible e r r o r and to calculate the n u m b e r of iterations n e c e s s a r y to obtain an a p p r o x i m a tion with a p r e s c r i b e d a c c u r a c y . A s i m i l a r method can be applied also to linear equations of neutral type [193]. The p r o b l e m of the convergence of the approximate iterations to the solution of a functional-partial differential equation of neutral type in a Banach space has been c o n s i d e r e d in [110]. In [12] one introduces the concept of g e n e r a l i z e d solution f o r the hyperbolic equation of neutral type

O'uOt,(t,x) + X 02u (t--Z,O2t x)

•

x)

•.02u (t--z,~ x) ~ f ( t ,

X)

in a c e r t a i n domain and one indicates conditions for its existence and uniqueness. 11.2. Asymptotic Methods. A s e r i e s of p a p e r s [56, 208, 209] are devoted to the development of asymptotic methods f o r the construction of solutions of partial differential equations of neutral type. In them one considers partial differential equations with delay only with r e s p e c t to time, whose right-hand side r e p r e s e n t s an e x p r e s sion of the f o r m ~F, where ~ is a small positive p a r a m e t e r while F is a function depending on the time t, the space variable x, the function u of the variables t and x and on its partial derivatives up to and including the second o r d e r , computed both in the p r e s e n t and the preceding moments. The r e s u l t of the investigation is the r e p r e s e n t a t i o n of the solutions in the f o r m of expansions in powers of the small p a r a m e t e r e. F o r example, in one of the f i r s t p a p e r s devoted to these questions [56], one considers the equation

O'.(t,x) Ott

b2 a'.(t,x) ]-ctt(t--sh, x) =~f[~t,x,u(t,x),u(t--~,X),l Ox2

Ott (t, x) Ott (t z, O'u(t, x) O'u(t--z, x) ] Ox ,. ~ x),. ~ ' Ox' j'

706

O.(t,x) au(t--z,x) 0--7--' Ot ' (4)

where e is a small positive p a r a m e t e r , b, c, A, z are nonnegative constants, 0 <_ x _ l, f is a function, periodic with r e s p e c t to ut with period 2~, differentiable a sufficient n u m b e r of times with r e s p e c t to x and entire rational with r e s p e c t to the other a r g u m e n t s , with homogeneous boundary conditions

a~+f~u(t,

x)=O

fo~ x = O ,

Ou (t, x) ~z2 O ~ ~-~2U(t, x ) = O

for x = l ,

(5)

~ , o~2, /31, t32 being constants. F o r the solution of p r o b l e m (4)-(5), corresponding to a s i n g l e - f r e q u e n c y oscillation r e g i m e , one c o n s t r u c t s asymptotic f o r m u l a s . F o r this, f i r s t one w r i t e s down the partial solutions of the ~unperturbed" equation c)'u(t, x)

(6)

b2 O's(t, x) kerr(t, x ) = 0

Ot =

Ox ~

under the boundary conditions (5):

u. (t, x) = a.X. (x) co s (o~.t + %).

(7)

Then, the solutions of Eq. (4) are sought in the f o r m u (t, x) = aX~ (x) cos 4 + sul (.~t, x, a, 4) + ~2u2(vt, x, a, 4) + ~3u~(vt, x, a, ~;) + . . . .

(8)

In o r d e r to determine the quantities a , 4, ui, o c c u r r i n g in s e r i e s (8), one writes down the c o r r e s p o n d i n g e q u a tions. Similar r e s u l t s for a somewhat different equation have been obtained in [208]. The p r o b l e m of the r e p r e s e n t a t i o n of the solutions in the f o r m of s e r i e s for various equations has been c o n s i d e r e d also in [13]. 11.3. Periodic Solutions. In a s e r i e s of p a p e r s , MitropolTskii, Tkach [147, 203, 204] investigate the p r o b lem of the existence and of the determination of solutions, periodic with r e s p e c t to the time t and of p e r i o d T, for equations of the f o r m O2u (t. OtoxX) -- f ( t , X, U(t, x), u ( t - - ~ , X), Ou(t,ot x) , Ou(t--~,x) Ou(t,x) Ou(t--~,x) O=u(t--~,x) \ Ot ' Ox ' " Ox ' OtOx ]' where u, f

a r e n - v e c t o r s , with the i n i t i a l - b o u n d a r y conditions

u(o, x)=uo(O)+v(x), u(t, O)=uo(t)+v(o). Here u0(t) is a given function and the function v(x) belongs to a c e r t a i n set of functions ensuring the existence of solutions u(t, x) which are periodic with r e s p e c t to t of p e r i o d T. In [204], an analogous p r o b l e m has been c o n s i d e r e d f o r the equation or~,~ (t, ~) =B~(t, x)u(t, x)+B2(t, OtOx ~

+ f { t , x , Ou(t,x) Ot

O'n (t, x) O t O x = 't

O=u(t,x)

Ou(t--~,x) '

'

Ot

O'u (t--x, x) OtOx e

x)u(t--x, x)+

'

OtOx

6mu (t, x) , 99

om+lu (t--z, d t d x ra

O t O x m _ l "~ X)

O'u(t--r,x) '

OtOx

'

Omu (t--~. x) OtOxra_ 1

,

.~ ]

where B 1, B 2 are m a t r i c e s .

LITERATURE 1. 2. 3. 4.

CITED

N . V . Azbelev, "On linear boundary-value problems for functional-differential equations,n Differents. Uravn., 10, No. 4, 579-584 (1974). N . V . Azbelev, "On nonlinear functional-differential equations," Differents. Uravn., 12, No. 11, 1923-1932 (1976). N . V . Azbelev, "Boundary-value p r o b l e m s f o r nonlinear functional-differential equations," in: D i f f e r ential Equations with Deviating Argument [in Russian], Naukova Dumka, Kiev (1977), pp. 5-11. N . V . Azbelev, L. M. Berezanskii, and L. F. Rakhmatullina, won a l i n e a r functional-differential equation of evolution type, n Differents. Uravn., 13, No. 11, 1915-1925 (1977). 707

5. 6.

7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22.

23. 24. 25. 26. 27. 28. 29.

30.

708

N . V . Azbelev and G. G. I s l a m o v , "On a c e r t a i n c l a s s of functional-differential equations," D i f f e r e n t s . Uravn., 1_.22,No. 3 , 4 1 7 - 4 2 7 (1976). N . V . Azbelev and V. P. Maksimov, "A p r i o r i e s t i m a t e s of the solutions of the Cauchy p r o b l e m and the solvability of boundary-value p r o b l e m s f o r equations with r e t a r d e d a r g u m e n t , " Differents. Uravn., 1._55, No. 10, 1731-1747 (1979). N . V . Azbelev and L. F. Rakhmatullina, " L i n e a r functional-differential equations," in: Differential Equations with Deviating A r g u m e n t [in Russian], Naukova Dumka, Kiev (1977), pp. 11-19. N . V . Azbelev and L. F. Rakhmatullina, " F u n c t i o n a l - d i f f e r e n t i a l equations," Differents. Uravn., 1._!4,No. 5, 771-797 (1978). N . V . Azbelev, L. F. Rakhmatullina, and A. G. T e r e n t ' e v , "The W - m e t h o d in the investigation of a d i f f e r ential equation with deviating a r g u m e n t , " Trudy T a m b o v s k . Inst. Khim. M a s h i n o s t r . , No. 4, 60-63 (1970). N . L . A l e k s e e v s k a y a and P. S. G r o m o v a , " L y a p u n o v ' s second method f o r differential equations with deviating a r g u m e n t , " T r . S e m . T e o r . Differents. Uravn. Otklon. A r g u m e n t o m , Univ. Druzhby Narodov P a t r i s a Lumumby, 1_.00,3-40 (1977). N . L . A l e k s e e v s k a y a and P. S. G r o m o v a , "LyapanovVs second method f o r differential equations," in: Differential Equations with Deviating Argument [in Russian], Naukova Dumka, Kiev (1977), pp. 19-34. A . G . A l e s k e r o v and R. M. Aliev, NOn a g e n e r a l i z e d solution of a m i x e d p r o b l e m f o r a hyperbolic equation of neutral type," Izv. Akad. Nauk AzSSR, S e t . F i z . - T e k h . Mat. Nauk, No. 2, 68-74 (1978). R . M . Alley and N. M. Alley, "The foundation of the averaging principle f o r q u a s i l i n e a r differential e q u a tions of neutral type," in: Functional Analysis and Its Applications [in Russian], Baku (1971), pp. 30-35. T. Amankulov, "On a method for solving integrodifferential equations of neutral type with deviating a r g u m e n t , " Trudy Kirgiz. Gos. Univ. Ser. Mat. Nauk, No. 7, 3-6 (1970). T. Amankulov and L. E. Krivoshein, "On the theory of linear integrodifferential equations of neutral type," Trudy Kirgiz. Gos. Univ. S e t . Mat. Nauk, No. 6, 6-15 (1967). K . I . Anastasova, rAn i n v e r s e p r o b l e m f o r a linear equation of neutral type," Godislmik Vyssh. Uchebn. Zaved. Prilozhen. Mat., 1~0, No. 2, 77-89 (1974(1975)). P . Z . Anachkov, " P e r i o d i c solutions of nonlinear s y s t e m s of differential equations of n e u t r a l type with deviating a r g u m e n t , " G o d i s h n i k V y s s h . Uchebn. Zaved. P r i l o z h e n . Mat., 1_~.1,No. 3, 19-26 (1975(1977)). A . B . Antonevich and V. B. Ryvldn, "On the n o r m a l solvability of the p r o b l e m of the periodic solutions of a linear differential equation with deviating a r g u m e n t , " Differents. Uravn., 10, No. 8, 1347 -1358 (1974). S . S . Akhiev, wOptimal p r o c e s s e s in distributed s y s t e m s with neutrally r e t a r d e d a r g u m e n t s , " Ueh. Zap. Azerb. Univ. Ser. F i z . - M a t . Nauk, No. 3, 17-23 (1971). S . S . Akhiev, "On a method of r e p r e s e n t a t i o n of the solution and of the d e r i v a t i v e of the solution of c e r t a i n s y s t e m s of f u n c t i o n a l - d i f f e r e n t i a / e q u a t i o n s , not solved with r e s p e c t to the d e r i v a t i v e , " Nauch. T r . Azerb. Univ. Vopr. P r i M . Mat. i Fdbernet., No. 1, 125-131 (1979). R . R . Aldlmerov, "On the a v e r a g i n g principle f o r functional-differential equations of neutral type," Ukr. Mat. Zh., 2__5,No. 5, 579-588 (1973). R . R . Akhmerov, "On the existence of solutions of the Cauchy p r o b l e m f o r a c l a s s of f u n c t i o n a l - d i f f e r ential equations of advanced type," in: M a t e r i a l s of the Ninth Scientific Conference of the D e p a r t m e n t s of P h y s i c s - M a t h e m a t i c s and Natural Sciences, 1973, Moscow (1974), pp. 5-7. R . R . Akhmerov, " A l m o s t - p e r i o d i c solutions in the a v e r a g i n g p r i n c i p l e f o r equations of neutral type," in: Trudy N a u c h n . - I s s l e d . Inst. Mat. Voronezh. Gos. Univ., No. 15 (1974), pp. 3-8. R . R . A k h m e r o v , "The averaging principle and a l m o s t periodic solutions of s y s t e m s of f u n c t i o n a l - d i f f e r ential equations of neutral type," D i f f e r e n t s . Uravn., l_!l, No. 2, 211-221 (1975). R . R . A k h m e r o v , "On the r e g u l a r i t y of a l m o s t periodic differential o p e r a t o r s of n e u t r a l type that a r i s e in the averaging principle," Ukr. Mat. Zh., 2_.~8,No. 5, 663-670 (1975). R . R . A k h m e r o v , "A r e m a r k on the a v e r a g i n g principle on the whole r e a l line f o r equations of neutral type," Differnts. Uravn., 1__33,No. 8, 1506 (1977). R . R . Akhmerov, " A l m o s t - p e r i o d i c solutions of equations of neutral type with a s m a l l delay," in: T e o r . i P r i M . Vopr. Differents. Uravn. i Algebra, Kiev (1978), pp. 3-6. R . R . Akhmerov, "On the theory of equations of n e u t r a l type," in: The T h e o r y of O p e r a t o r Equations [in Russian], V o r o n e z h (1979), pp. 3-16. R . R . A k h m e r o v and N. G. Kazakova, "The averaging principle on an unbounded interval f o r functionaldifferential equations of n e u t r a l type," Godishnik V y s s h . Uchebn. Zaved., P r i l o z h e n . Mat., 1_44, No. 3, 217228 (1978). R . R . A k h m e r o v , N. G. Kazakova, and A. V. P o k r o v s k i i , " P r i n c i p l e of the absence of bounded solutions and the absolute stability of equations of neutral type," Serdica, 4, No. 1, 61-69 (1978).

31. 32. 33.

34.

35.

36. 37.

38.

39.

40. 41. 42. 43. 44.

45. 46. 47.

48.

49. 50. 51.

52.

53. 54.

R . R . A k h m e r o v and M. I. Kamenskii, "On the second t h e o r m of N. N. Bogolyubov in the a v e r a g i n g p r i n ciple f o r functional-differential equations of neutral type," D i f f e r e n t s . Uravn., 1__0,No. 3, 537-540 (1974). R . R . Atahmerov and M. I. Kamenskii, ' The a v e r a g i n g principle and the stability of periodic solutions of equations of neutral type, n T r . N a u e h n . - I s s l e d . Inst. Mat. Voronezh. Gos. Univ., No. 15, 9-13 (1974). R . R . A k h m e r o v and M. I. Kamenskii, "On the stability of solutions of equations of n e u t r a l type," in: The Fourth All-Union Conference on the T h e o r y and Applications of Differential Equations with Deviating A r g u m e n t [in Russian], Naukova Dumka, Kiev (1975), pp. 20-21. R . R . A k h m e r o v and M. I. Kamenskii, "On the question of the stability of the e q u i l i b r i u m state of a s y s t e m of functional-differential equations of neutral type with s m a l l deviation of the a r g u m e n t , n Usp. Mat. Nauk, 30, No. 2, 205-206 (1975). R . R . A k h m e r o v and M. I. Kamenskii, nOn a c e r t a i n approach t o t h e study of the stability of periodic s o l u tions in the a v e r a g i n g principle f o r functional-differential equations of n e u t r a l type,n C o m m e n t . Math. Univ. Carolin., 1_~6,No. 2, 293-313 (1975). R . R . A k h m e r o v and M. I. Kamenskii, nOn the question of bifurcation in s t r o n g l y continuous f a m i l i e s of o p e r a t o r s ," in: Seventh All-Union Topology Conference, Minsk (1977), p. 13. R . R . A k h m e r o v and M. I. Kamenskii, "The stability in the f i r s t a p p r o x i m a t i o n of an e n s e m b l e of dyn a m i c a l s y s t e m s , n in: P r o c . Fifth All-Union Conf.-Sem. on the Control of L a r g e S y s t e m s , A l m a - A t a (1978), pp. 114-115. R . R . A k h m e r o v and M. I. Kamenskii, "On equations with a s m a l l deviation of the a r g u m e n t , " in: AllUnion Conference on Asymptotic Methods in the T h e o r y of Singularly P e r t u r b e d Equations [in Russian], P a r t 2, A l m a - A t a (1979), pp. 10-12. R . R . Akhmerov, M. I. Kamenskii, V. S. Kozyakin, and A. V. Sobolev, " P e r i o d i c solutions of s y s t e m s of autonomous functional-differential equations of n e u t r a l type with a s m a l l delay," Differents. Uravn., 10, No. 11, 1923-1931 (1974). R . R . A k h m e r o v , M. I. Kamensldi, A. S. Potapov, and B. N. Sadovsldi, "Condensing o p e r a t o r s , " Itogi Nauki i Tekh., S e t . Mat. Anal., 1_~8, 185-250 (1980). R . R . A k h m e r o v and A. E. Rodkina, ~Differential inequalities and the s t r u c t u r e of an integral v o r t e x f o r s y s t e m s of functional-differential equations of n e u t r a l type," Differents. Uravn., 9, No. 5, 797-806 (1973). A . L . Badoev, "On sernigroups g e n e r a t e d by linear s y s t e m s of neutral type, ~ Soobshch. Akad. Nauk Gruz. SSR, 5._~.1,No. 1, 15-18 (1968). A . L . Badoev and B. N. Sadovsldi, "An e x a m p l e of a condensing o p e r a t o r in the theory of differential e q u a tions os n e u t r a l type with deviating a r g u m e n t , " Dokl. Akad. Nauk SSSR, 186, No. 6, 1239-1242 (1969). D . D . Bainov and S. D. Milusheva, ~The application of the a v e r a g i n g method to a two-point boundary value p r o b l e m f o r s y s t e m s of integrodifferential equations of F r e d h o l m type, not solved with r e s p e c t to the d e r i v a t i v e , ~ Ann. Univ. Sci. Budapest. EStvSs Sect. Math., 20, 47-57 (1977). N . Z . Balukhov and E. M. Markushin, ~The R o u t h - H u r w i t z p r o b l e m f o r f i r s t - o r d e r equations of n e u t r a l type," in: A l g o r i t m i z . i Avtomatiz. P r o t s e s s o v i Ustanovok, Kuibyshev, No. 5 (1971), pp. 65-68. R. Bellman and K. L. Cooke, D i f f e r e n t i a l - D i f f e r e n c e Equations, Academic P r e s s , New Y o r k (1963). G . V . Belykh and V. I. Fodchuk, "The application of a s y m p t o t i c methods to differential equations of n e u t r a l type," Tr. Sere. po Mat. Fiz. i Nelinein. Kolebaniyam, Inst. Mat. Akad. Nauk Ukr. SSR, No. 3, 25-39 (1969 (1970)). S . E . Birkgan, nOn the exponential dichotomy of solutions of l i n e a r d i f f e r e n t i a l - d i f f e r e n c e equations of neutral type with constant coefficients," in: Studies in Stability and Theory of Oscillations, Y a r o s l a v i T (1978), pp. 3-18. Yu. G. Borisovich, "On c e r t a i n nonlinear p r o b l e m s in the t h e o r y of functional-differential equations, n in: Differential Equations with Deviating A r g u m e n t [in Russian], Naukova Dumka, Kiev (1977), pp. 42-51. V . G . Brodovskii, "On a boundary-value p r o b l e m for differential equations with deviating a r g u m e n t , " Izv. Akad. Nauk Kaz. SSR, S e t . F i z . - M a t . , No. 5, 25-30 (1970). V. Sh. Burd, "The bifurcation of the a l m o s t - p e r i o d i c oscillations of differential equations with a f t e r e f f e c t of n e u t r a l type with f a s t and slow t i m e , " in: Studies in Stability and Theory of Oscillations, Y a r o s l a v l ' (1976), pp. 143-153. V. Sh. Burd, "The a v e r a g i n g principle on an infinite interval f o r differential equations with a f t e r e f f e c t of neutral type with f a s t and slow t i m e , " in: Differential Equations with Deviating A r g u m e n t [in Russian], Naukova Dumlaa, Kiev (1977), pp. 62-72. V. Sh. Burd, "The method of a v e r a g i n g on an infinite interval f o r differential equations with a f t e r e f f e c t of neutral type," Nelinein. Kolebaniya i T e o r i i Upr., Izhevsk, No. 2, 57-77 (1978). N . G . M . V a i n b e r g and G. A. K a m e n s k i i , nThe qualitative study of the solutions bf a q u a s i l i n e a r d i f f e r ential equation with deviating a r g u m e n t of n e u t r a l type," T e m a t . Sb. Nauclm. Tr. Mosk. Aviats. Inst., No. 419, 33-37 (1977). 709

55. 56. 57. 58. 59.

60.

61.

62.

63. 64. 65.

66. 67.

68. 69. 70. 71. 72. 73.

74. 75. 76. 77. 78.

79.

710

N . G . M . Vainberg, G. A. KamensMi, and A. D. Myshkis, "On a differential equation of neutral type, p o s s e s s i n g an integrating f a c t o r , " Differents. Uravn., 9, No. 11, 1956-1965 (1973). S . A . Vasilishin and V. I. Fodchuk, 7On the construction of a s y m p t o t i c solutions for q u a s i l i n e a r p a r t i a l differential equations with time lag," Ukr. Mat. Zh., 19, No. 4, 108-113 (1967). F . P . V a s i l ' e v , "On existence conditions f o r a saddle point in d e t e r m i n i s t i c g a m e s for integrodifferential s y s t e m s with delay of neutral type,7 Avtomat. T e l e m e k h . , No. 2, 40-50 (1972). A . B . V a s i l ' e v a , ' B o u n d a r y - l a y e r p h e n o m e n o n and vibration p r o c e s s e s in equations of neutral type with a s m a l l d e l a y , ' Abh. Deutsch. Akad. W i s s . Berlin K1. Math. P h y s . Tech., No. 1, 201-205 (1965). A . B . V a s i l ' e v a , V. I. Rozhkov, and A. A. Plotnikov, "Boundary l a y e r and o s c i l l a t o r y p r o c e s s e s for e q u a tions of neutral type with a s m a l l lag," T r . Sere. T e o r . Differents. Uravn. Otklon. A r g u m . Univ. Druzhby Narodov P a t r i s a I ~ m u m b y , 5, 18-20 (1967). The Fourth All-Unien Conference on the Theory and Applications of Differential Equations with Deviating Argument, Kiev, Sept. 23-26, 1975, A b s t r a c t s of Reports (Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR), Naukova Dumka, Kiev (1975). M . K . Gadzhiev, 7Finite control f o r s y s t e m s with deviating a r g u m e n t , " in: P r o e . of the Young R e s e a r c h e r s of the Institute of C y b e r n e t i c s , Baku (1978), pp. 70-72. M a n u s c r i p t deposited at VINITI, Aug. 21, 1979, No. 3121-79 Dep. L . S . Gnoenskii and G. A. K a m e n s k i i , nOn the influence of the lag in computational devices on the quality index of control s y s t e m s , 7 T r . Sem. T e o r . D i f f e r e n t s . Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a I m m u m b y , 7, 28-42 (1969). I. G o m a , "The investigation of the solutions of the Cauchy p r o b l e m f o r s e m i e x p l i c i t differential equations with r e t a r d e d a r g u m e n t , " Differents. Uravn., 1__~2,No. 3, 552-554 (1976). P . S . G r o m o v a , "A study os the stability of the solutions of s y s t e m s of l i n e a r d i f f e r e n t i a l - d i f f e r e n c e equations of neutral type in special e a s e s , " Zag. Drgan. Nielin., No. 14, 149-158 (1973). P . S . G r o m o v a , "Conditions f o r the existence of infinite chains of z e r o s of quasipolynomials which a p p r o a c h unboundedly the i m a g i n a r y axis,n T r . Sere. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 1_~0,69-78 (1977). I . F . Dorofeev, nOn the r e c o n s t r u c t i o n of the initial function," T r . Univ. Druzhby Narod. P a t r i s a Lumumby, 2_!1, 209-212 (1967). M . E . Drakhlin and T. K. P l y s h e v s k a y a , "The e x i s t e n c e , uniqueness and convergence of s u c c e s s i v e a p p r o x i m a t i o n s f o r differential equations of neutral type with r e t a r d a t i o n , 7 Differents. Uravn., 9, No. 9, 1583-1592 (1973). M . E . Dral~hlin and T. K. P l y s h e v s k a y a , "A b o u n d a r y - v a l u e p r o b l e m f o r a nonlinear differential equation of neutral type," Differents. Uravn., 11, No. 6, 986-996 (1975). O . A . Dyshin, "On the operational method f o r solving l i n e a r d i f f e r e n t i a l - d i f f e r e n c e equations with constant coefficients," Differents. Uravn., 2, No. 9, 1193-1200 (1966). Yu. A. Dyadchenko, non the local solvability of o p e r a t o r equations," in: Qualitative and Approximative Methods of Investigation of O p e r a t o r Equations [in Russian], No. 3, Y a r o s l a v l ' (1978), pp. 48-60. Yu. A. Dyadchenko, "On the dependence of the solutions of an o p e r a t o r equation of V o l t e r r a type on the p a r a m e t e r , " P r i b l i z h . Metody Issled. Differents. Uravn. P r i l o z h e n . , No. 4, 40-47 (1978). Yu. A. Dyadchenko, nOn the solvability of a nonlinear o p e r a t o r equation of V o l t e r r a type," in: Theory of O p e r a t o r Equations [in Russian], Voronezh (1979), pp. 22-23. Yu. A. Dyadehenko and A. E. Rodldaa, "The continuability and the continuous dependence on a p a r a m e t e r of g e n e r a l i z e d solutions of equations of neutral type, W T r . N a u c h n o - I s s l e d . Inst. Mat. Voronezh. Univ., No. 12, 28-31 (1974). Yu. A. Dyadchenko and A. E. Rodldna, "On g e n e r a l i z e d solutions of equations of neutral type," T r . Nauc h n o - I s s l e d . Inst. Mat. Voronezh. Univ., No. 15, 24-32 (1974). L . A . Zhivotovskii, ' E x i s t e n c e t h e o r e m s and uniqueness c l a s s e s of solutions of functional equations with h e r e d i t y , 7 Differents. Uravn., 7, No. 8, 1377-1384 (1971). L . A . Zhivotovskii, "On the question of the existence of solutions for differential equations with deviating a r g u m e n t of neutral type," Differents. Uravn., 8, No. 11, 1936-1942 (1972). T . A . Zamanov, "On o p e r a t o r - d i f f e r e n t i a l equations with deviating a r g u m e n t , " Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 42-48 (1969). Z . D . Z a p r y a n o v and D. D. Bainov, 7Sufficient conditions f o r the stability of the solutions of equations of neutral type of an a r b i t r a r y o r d e r , ~ in: P r o c . Fifth Int. Conf. on Nonlinear Oscillations, Kiev, Vol. 2 (1970), pp. 166-169. A . M . Zverkin, "Existence and uniqueness t h e o r e m s f o r equations with deviating a r g u m e n t in the c r i t i c a l c a s e , " Tr. Sere. T e o r . Differents. Uravn. Otldon. A r g u m . Univ. Druzhby Narodov P a t r i s a Lumumby, 1, 37-46 (1962).

80.

81. 82. 83.

84. 85. 86. 87.

88. 89.

90.

91.

92.

93. 94. 95. 96.

97. 98.

99.

100. 101.

102.

A . M . Zverkin, "On the definition of the concept of solution f o r equations with deviating argument of neut r a l type," T r . Sere. T e o r . Diiferents. Urava. OLklon. Argttm. Univ. Druzhby Narodov P a t r i s a Lumumby, 4, 278-283 (1967). A . M . Zverkin, "Differential equations with deviating argument," in: Fifth S u m m e r Math. School, JuneJuly 1967, Kiev (1968), pp. 307-399. A . M . Zverkin, G. A. Kamenskii, S. B. Norkin, and L. ~,. Eltsgoltts,t "Differential equations with deviating argument, ~ Usp. Mat. Nauk, 17, No. 2, 77-164 (1962). A . M . Zverkin, G. A. Kamensk--~i, S. B. Norkin, and L. ~ . E1Tsgollts, t "Differential equations with deviating argument. II," T r . Sere. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 2, 3-49 (1963). M. Ilolov and T. Sabirov, ~On the p r o b l e m of the branching of the periodic solutions in c e r t a i n c l a s s e s of differential-difference s y s t e m s of neutral t y p e , ' Dold. Akad. Nauk TSSR, 1_~8,No. 12, 3-6 (1975). M. Ilolov and T. Sabirov, "On the continuability with r e s p e c t to the p a r a m e t e r of the periodic solutions of differential equations of neutral type in a Hilber~ space, ~ Dokl. Akad. Nauk TSSR, 20, No. 3, 3-6 (1977). M. Imanaliev and M. Dzhuraev, "Branching theory for a periodic solution of a certain nonlinear differential equation of neutral type and its applications," Abh. Akad. Wissensch. DDR, No. 3, 359-363 (1977). A.G. Kamenskii, G. A. Kamenskii, and A. D. Myshkis, "On the convergence of the finite-difference method for the numerical solution of boundary-value problems for linear differential-difference equations," Dokl. Akad. Nauk SSSR, 233, No. 2, 280-282 (1977). G.A. Kamensldi, ~On the general theory of equations with deviating argument," Dokl. Akad. Nauk SSSR, 120, No. 4, 697-700 (1958). G.A. Kamenskii, "Existence, uniqueness, and continuous dependence on initial conditions of the solutions of systems of differential equations with deviating argument of neutral type," Mat. Sb., 5_.55,No. 4, 363-378 (1961). G.A. Kamenskii, "A boundary-value problem for a nonlinear first-order differential equation with deviating argument of neutral type, ~ Tr. Sere. Teor. Differents. Uravn. Otklon. Argumentom. Univ. Drazhby Narodov Patrisa Lumumby, 3, 39-46 (1965). G.A. Kamenskii, "On the uniqueness of the solution of a boundary-value problem for a nonlinear secondorder differential equation of neutral type with deviating argument," Tr. Sere. Teor. Differents. Uravn. Otklon. Argumentom, Univ. Druzhby Narodov Patrisa Lumumby, 4, 275-277 (1967). G.A. Kamenskii, "A sufficient condition for a weak extremum for functionals with a deviating argument," Tr. Sem. Teor. Differents. Uravn. Otldon. Argumentom, Univ. Druzhby Narodov Patrisa Lumumby, 9, 94-97 (1975). G.A. Kamenskii, "On extrema of functionals with deviating argument," Dokl. Akad. Nauk SSSR, 224, No. 6, 1252-1255 (1975). G.A. Kamensk~i, "On the conditional extremum of a functional with deviating argument," Dokl. Akad. Nauk SSSR, 23__55,No. 2, 266-268 (1977). G.A. Kamenskii, ~Variational problems for functionals with a deviating arguraent," in: Differential Equations with a Deviating Argument [in Russian], Naukova Dumka, Kiev (1977), pp. 139-148. G.A. Kamenskii and ~. D. Kononenko, non the application of the collocation method to a boundary-value problem for a linear equation with a deviating argument of neutral type, ~ Ukr. Mat. Zh., 2__99,No. 3, 306312 (1977). G.A. Kamenskii and A. D. Myshkis, "Boundary-value problems with infinite defect for differential equations with a deviating argument," Differents. Uravn., 7, No. 12, 2143-2150 (1971). G.A. Kamenskii and A. D. Myshlds, "Boundary-value problems for a nonlinear differential equation with deviating argument of neutral type," Differents. Uravn., 8, No. 12, 2171-2179 (1972). G . A . Kamenskii and A. D. Myshkis, 'On the formulation of boundary-value p r o b l e m s for differential equations with deviating a r g u m e n t and s e v e r a l highest t e r m s , " Differents. Uravn., 10, No. 3, 409-418 (1974). G . A . Kamenskii and A. D. Myshkis, "A boundary-value problem for s e c o n d - o r d e r quasilinear differential equations in divergence f o r m with deviating argument," Differents. Uravn., 10, No. 12, 2137-2146 (1974). G . A . Kamenskii, S. B. Nortdn, and L. E. ~l'sgolTts, "Certain directions of r e s e a r c h of the theory of differential equations with deviating argument (on the fifteenth a n n i v e r s a r y of the s e m i n a r ) , " T r . Sem. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 6, 3-36 (1968). G . A . Kamenskii and E. A. Khvilon, "Optimality conditions for s y s t e m s of neutral type with deviating a r gument," Tr. Sere. Teor. Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 6, 213-222 (1968).

711

103. G. A. K a m e n s k i i and E. A. Khvilon, "On the optimal p r o c e s s e s f o r s y s t e m s d e s c r i b e d by differential equations with deviating a r g u m e n t of neutral type," T r . Sere. Teor. Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 8, 83-88 (1972). 104. M. I. Kamensldi, "The evaluation of the index of an isolated solution of the Cauchy p r o b l e m f o r a functional-differential equation of n e u t r a l type," in: Functional Analysis and Its Applications, No. 1, Voronezh (1973), pp. 6-13. 105. M. I. Kamenskii, "On the shift operator along the trajectories of equations of neutral type," in: Collection of Papers of Postgraduates on the Theory of Functions and Differential Equations, Voronezh (1974), pp. 19-22. 106. M. I. Kamenskii, nOn the theory of perturbations of bounded operators in a Banach space," Sb. Rabot Aspirantov. Voronezh. Univ., No. 2, 18-23 (1974). 107. M. I. Kamenskii, nOn an approach to the theory of periodic solutions for functional-differential equations of neutral type," Tr. Nauchn.-Issled. Inst. Mat. Voronezh. Gos. Univ., No. 15, 44-52 (1974). 108. M. I. Kamenskii, nOn the Andronov-Witt theorem for functional-differential equations of neutral type," Tr. Nauchn.-Issled. Inst. Mat. Voronezh. Gos. Univ., No. 12, 31-36 (1974). 109. M. I. Kamenskii, "The shift operator along the trajectories of equations of neutral type, depending on a parameter," Uch. Zap. Tartus. Univ., No. 448/21, 107-117 (1978). ii0. M. Kvapish (M. Kwapisz) and Ya. Turo (J. Turo), "On approximate iterations for a partial differentialfunctional equation of hyperbolic type in a Banach space," Differents. Uravn., ii, No. 9, 1626-1640 (1975). 111. L . N . Kitaeva, "On the existence of solutions with linear a s y m p t o t i c s of integrodifferential equations with deviating a r g u m e n t , " Tr. F r a n z . Politekhn. Inst., No. 76, 70-75 (1974). 112. Yu. S. Kolesov and D. I. Shvitra, Self-Oscillation in S y s t e m s with Lags [in Russian], Mokslas, Vilnius (1979). 113. V. B. Kolmanovskii, nOn the stability of linear equations of neutral type," Differents. Uravn., 6, No. 7, 1235-1246 (1970). 114. V. B. Kolmanovskii, "The representation of solutions of equations of neutral type," Ukr. Mat. Zh., 24, No. 2, 171-178 (1972). 115. V. B. Kolmanovskii and V. R. Nosov, "On the stability of nonlinear oscillation systems, described by equations of neutral type," in: Proc. Fifth Int. Conf. on Nonlinear Oscillations, 1969, Vol. 2, Kiev (1970), pp. 228-232. 116. V. B . Kolmanovskii and V. R. Nosov, "The stability of systems with deviating argument of neutral type," Prikl. Mat. Mekh., 43, No. 2, 209-218 (1979). 117. M. M. Konstantinov and D. D. Bainov, "On a boundary value problem of second order for systems of differential equations of superneutral type," Soobshch. Akad. Nauk Gruz. SSR, 74, No. 2, 285-288 (1974). 118. M. M. Konstantinov and D. D. Bainov, nOn the existence of solutions of s y s t e m s of differential-functional equations of s u p e r n e u t r a l type with an i t e r a t e d delay," Dokl. Akad. Nauk SSSR, 217, No. 3, 519-521 (1974). 119. M . M . Konstantinov and D. D. Bainov, "On a boundary value p r o b l e m with a g e n e r a l i z e d boundary condition f o r s y s t e m s of differential equations of s u p e r n e u t r a l type," Dokl. Bolg. Akad. Nauk, 28, No. 7, 889892 (1975). 120. M . M . Konstantinov, S. P. P a t a r i n s k i , P. Khr. Petkov, and N. D. Khristov, " P r o b l e m s of the control of the motion with the aid of an initial s y s t e m of functions, d e s c r i b e d by differential-functional equations," T e o r . P r i m e n . Mech., 1, 33-39 (1975). 121. M. F. Korolev,~Periodic solutions of s y s t e m s of equations of n e u t r a l type in the critical c a s e . " Univ. Druzhby Narodov P a t r i s a Lumumby, Moscow {1977). Manuscript deposited at VINITI, Feb. 14, 1978, No. 503-78 Dep. 122. L . E . Krivoshein, "On a method of solving boundary-value p r o b l e m s for ordinary nonlinear i n t e g r o d i f f e r ential equations," T r . Kirg. Univ. S e r . Mat., No. 9, 17-22 (1974). 123. N. A. Kulesko, non the c o m p l e t e n e s s of the Floquet s y s t e m of solutions of equations of neutral type," Mat. Zametki, 3, No. 4, 297-306 (1968). 124. N. A. Kulesko and B. Ya. Levin, nOn the c o m p l e t e n e s s of the Floquet solutions for s e c o n d - o r d e r d i f f e r ential equations with deviating a r g u m e n t , " Sib. Mat. Zh., 1._~8,No. 2, 321-326 (1977). 125. V. G. Kurbatov, nOn the concept of r e t a r d i n g invertibility in the theory of l i n e a r equations of neutral type," in: Fourth All-Union Conference on the Theory and Applications of Differential Equations with Deviating A r g u m e n t [in Russian], Kiev (1975), pp. 134-135. 126. V. G. Kurbatov, nOn the s p e c t r u m of an o p e r a t o r with c o m m e n s u r a b l e deviations of the a r g u m e n t and with constant coefficients," D i f f e r e n t s . Uravn., 13, No. 10, 1770-1775 (1977). 127. V. G. Kurbatov, "On the dichotomy of solutions of equations of neutral type," in: Studies in Stability and Theory of Oscillations [in Russian], Y a r o s l a v l ' (1977), pp. 156-166.

712

128. V. G. Kurbatov, "On a conjecture in the theory of functional-differential equations," D i f f e r e n t s . Uravn., 1_~4, No. 11, 2074-2075 (1978). 129. V. G. Kurbatov, "On the invertibility of l i n e a r autonomous functional-differential o p e r a t o r s in s p a c e s of functions continuously differentiable on the axis," in: Methods of Solution of O p e r a t o r Equations [in Russian], Voronezh (1978), pp. 88-93. 130. V. G. Kurbatov, "On the invertibility of r e t a r d i n g o p e r a t o r s , " in: The T h e o r y of O p e r a t o r Equations [in Russian], Voronezh (1979), pp. 43-52. 131. V. G. Kurbatov, "A c r i t e r i o n f o r the stability of functional-differential equations," Voronezh. Univ., V o r o n e z h (1980). Manuscript deposited at V I N r r I , May 29, 1980, No. 2139-80 Dep. 132. V. G. Kurbatov and A. E. Rodkina, "On the solvability of nonlinear functional-differential equations," Voronezh. Univ., Voronezh (1980). Manuscript deposited at VINITI, May 19, 1980, No. 1911-80 Dep. 133. V. G. Kurbatov and I. S. F r o l o v , "On the s p e c t r a l mapping t h e o r e m in a t e n s o r product of Banaeh s p a c e s , " Voronezh. Univ., Voronezh (1977). Manuscript deposited at VINITI, June 6, 1977, No. 2215-77 Dep. 134. V. G. Kurbatov and I. S. F r o l o v , non the invertibility of d i f f e r e n t i a l - d i f f e r e n c e o p e r a t o r s in a Banach s p a c e , " Voronezh. Univ., Voronezh (1977). Manuscript deposited at VINITI, March, 13, 1978, No. 849-78 Dep. 135. Yu. V. Lomakin and S. B. Norkin, "On the multiple z e r o s of the solutions of a homogeneous initial value p r o b l e m f o r a s e c o n d - o r d e r differential equation with deviating a r g u m e n t of netural type," T r . Sere. T e o r . Differents. Uravn. Otklon. A r g u m e n t o m , Univ. Druzhby Narodov P a t r i s a L u m u m b y , 8, 115-128 (1972). 136. Yung-Ching Liu, "Stability of solutions of d i f f e r e n t i a l - d i f f e r e n c e equations," Adv. Math., 4, No. 2, 297-303 (1958). 137. V. P. M a k s i m o v and L. F. Rakhmatullina, "The conjugate equation f o r the g e n e r a l l i n e a r b o u n d a r y - v a l u e p r o b l e m , " D i f f e r e n t s . Uravn., 1._33,No. 11, 1966-1973 (1977). 138. F. G. Maksudov, I. D. Mardanov, and A. Kh. Shamilov, "A special boundary-value p r o b l e m f o r nonlinear differential equations with deviating a r g u m e n t of n e u t r a l type," Dokl. Akad. Nauk AzSSR, 3._44,No. 6, 13-17 (1978). 139. M a t e r i a l s of the Second All-Union Intercollegiate Conference on the Theory and Applications of D i f f e r ential Equations with Deviating Argument, Sept. 1968 (Academy of Sciences of the Ukrainian SSR, C h e r novtsy Univ.), Chernovtsy (1968). 140. M a t e r i a l s of the T h i r d All-Union Intercollegiate Conference on the Theory and Applications of Differential Equations with Deviating Argument, Sept. 1972 ( A c a d e m y o f Sciences of the Ukrainian SSR, C h e r n o v t s y Univ.), Chernovtsy (1972). 141. P. K. Matsenko, "Expansion in the eigenfunctions of a differential equation with deviating a r g u m e n t of neutral type," in: Studies in Differential Equations and Theory of Functions [in Russian], No. 4, Saratov. Univ. (1974), pp. 61-80. 142. A. F. Misnik, " L y a p u n o v ' s second method f o r equations of neutral type," T r . Sere. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby N a r o d o v P a t r i s a Lumumby, 6, 78-108 (1968). 143. A. F. Misnik, "The absolute stability of nonlinear a u t o m a t i c control s y s t e m s of n e u t r a l type," T r . Sem. Teor. Differents. Uravn. Otklon. A r g u m . Univ. Druzhby Narodov P a t r i s a Lumumby, 7, 92-106 (1969). 144. A. F. Misnik, "A study of the stability in the critical case of s e v e r a l z e r o roots f o r s y s t e m s of d i f f e r ential equations of neutral type," Ukr. Mat. Zh., 2._~4,No. 2, 260-268 (1972). 145. A. F. Misnik and V. R. Nosov, "On the question of the stability of d i f f e r e n t i a l - d i f f e r e n c e equations of neutral type," Sb. Nauchn. Rabot Aspirantov. Univ. Druzhby Narodov P a t r i s a Lumumby, Fak. F i z . - M a t . E s t e s t v . Nauk, No. 1, 43-55 (1968). 146. A. F. Misnik and V. R. Nosov, "On the stability of s y s t e m s of differential equations of n e u t r a l type," Sb. Nauchn. Rabot Aspirantov. Univ. Druzhby Narodov P a t r i s a Lumumby, Fak. F i z . - M a t . E s t e s t v . Nauk, No. 1, 73-97 (1968). 147. Yu. A. M i t r o p o l ' s l d i and B. P. Tkach, " P e r i o d i c solutions of nonlinear s y s t e m s of p a r t i a l differential equations of neutral type," Ukr. Mat. Zh., 2.~1, No. 4, 475-486 (1969). 148. Yu. A. M i t r o p o l ' s k i i and A. N. Sharkovskii, "The development of the theory of differential equations with deviating a r g u m e n t at the Institute of M a t h e m a t i c s of the Academy of Sciences of the Ukrainian SSR," in: Differential Equations with Deviating A r g u m e n t [in Russian], Kiev (1977), pp. 215-221. 149. I. I. Morozov, "Algebraic stability c r i t e r i a f o r s y s t e m s with a constant lag," Nauchn. Zap. D n e p r o p e t r . Univ., 5,5, 87-89 (1961). 150. E. Mukhamadiev, "On the invertibility of functional o p e r a t o r s in s p a c e s of functions bounded on the axis," Mat. Zametki, 1.~1, No. 3, 269-274 (1972).

713

g

151. E. Mukhamadiev and B. N. Sadovskii, "On the e s t i m a t e of the s p e c t r a l radius of an o p e r a t o r r e l a t e d to equations of neutral type," Mat. Zametki, 1._33,No. 1, 67-78 (1973). 152. A. D. Myshkis, L i n e a r Differential Equations with a R e t a r d e d A r g u m e n t [in Russian], Nauka, Moscow (1972). 153. A. D. Myshkis, nOn c e r t a i n p r o b l e m s of the theory of differential equations with deviating a r g u m e n t , " Usp. Mat. Nauk, 32, No. 2, 173-202 (1977). 154. A . D . Myshkis an'd-L. E. ~ l ' s g o l ' t s , "The status and the p r o b l e m s of the theory of differential equations with deviating a r g u m e n t , " Usp. Mat. Nauk, 2._~2,No. 2, 21-57 (1967). 155. E. M. N e p o m n y a s h c h a y a and B. N. Sadovskii, nOn the method of s u c c e s s i v e a p p r o x i m a t i o n s for functionaldifferential equations of neutral type," T r . Mat. Fak. V o r o n e z h s k . Univ., No. 6, 59-68 (1972). 156. A. B. N e r s e s y a n , NOn c e r t a i n boundary-value p r o b l e m s f o r equations of neutral type," T r . Sere. Teor. Differents. Uravn. Otklon. A r g u m . Univ. Druzhby Narodov P a t r i s a L u m u m b y , 2, 118-135 (1963). 157. T. St. Nikolova and D. D. Bainov, "On the existence and uniqueness of the solutions of b o u n d a r y - v a l u e p r o b l e m s f o r integrodifferential equations of s u p e r n e u t r a l type," Ukr. Mat. Zh., 28, No. 4, 540-546 (1976). 158. T. St. Nikolova and D. D. Bainov, " E x i s t e n c e and uniqueness t h e o r e m s f o r the solutions of c e r t a i n s y s t e m s of integrodifferential equations of s u p e r n e u t r a l type," Icy. Vyssh. Uchebn. Zaved., Mat., No. 1, 75-80 (1978). 159. T. St. Nikolova and D. D. Bainov, "On a p r o b l e m f o r e x t r e m a l - i n t e g r o d i f f e r e n t i a l equations of s u p e r neutral type," Bul. Inst. Politechn. Gh. Gheorghiu-De], Bucuresti, Ser. E l e c t r o t e c h n . , 41, No. 1, 17-22 (1979). 160. T. St. Nikolova, D. D. Bainov, and M. M. Konstantinov, "Some r e m a r k s on prediction f o r equations of neutral type," Publ. Electrotehn. Fak. Univ. Beograd. S e t . Mat. Fiz., No. 461-497, 143-146 (1974(1975)). 161. S. B. Norkin, " S t u r m - L i o u v i l l e type b o u n d a r y - v a l u e p r o b l e m with an infinite defect," T r . Sere. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 9, 120-128 (1975). 162. S. B. Norkin, "The oscillation of the solutions of differential equations with deviating a r g u m e n t , " in: Differential Equations with Deviating Argument [in Russian], Kiev (1977), pp. 247-255. 163. V . R . Nosov, " P e r i o d i c solutions of a s y s t e m of l i n e a r equations of g e n e r a l f o r m with deviating a r g u ment," Differents. Uravn., 7, No. 4, 639-650 {1971). 164. V. R. Nosov, "On the existence of periodic solutions f o r l i n e a r s y s t e m s of general f o r m with distributed deviation of the a r g u m e n t , " D i f f e r e n t s . Uravn., 7, No. 12, 2168-2175 (1971). 165. V. R. Nosov, "The M a s s e r a - H a l a n a y t h e o r e m f o r l i n e a r s y s t e m s of neutral type," in: M a t e r i a l y IX Nauclm. Konf. F - t a F i z . - M a t . E s t e s t v . Nauk Univ. Druzhby Narodov im. P a t r i s a Lumumby, Moscow (1974), pp. 42-43. 166. G. P. Pelyukh, "On holomorphic solutions of s y s t e m s of nonlinear differential-functional equations," in: The Method of Integral Manifolds in Nonlinear Differential Equations [in Russian], Kiev (1973), pp. 182194. 167. G. P. Pleyukh and A. N. Sharkovskii, Introduction to the Theory of Functional Equations [in Russian], Naukova Dumka, Kiev (1974). 168. E. Pirmey, O r d i n a r y D i f f e r e n c e - D i f f e r e n t i a l Equations, Univ. of California P r e s s , B e r k e l e y (1958). 169. T. K. P l y s h e v s k a y a , "On the question of the continuous dependence on the p a r a m e t e r s of the solution of a functional-differential equation, ~ Sb. Nanchn. T r . P e r m . Politekh. Inst., No. 170, 121-127 (1975). 170. V. M. Polishehuk, "The dependence on p a r a m e t e r s of the solutions of l i n e a r d i f f e r e n t i a l - d i f f e r e n c e e q u a tions," in: The Method of Integral Manifolds in Nonlinear Differential Equations [in Russian], Kiev (1973), pp. 204-212. 171. V. M. Polishchuk and A. N. Sharkovskii, "The r e p r e s e n t a t i o n of the solutions of l i n e a r d i f f e r e n t i a l - d i f f e r e n e e e q u a t i o n s of n e u t r a l type," D i f f e r e n t s . Uravn., 9, No. 9, 1627-1645 (1973). 172. L. F. Rakhmatullina, "On the canonical f o r m s of l i n e a r functiomal-differential equations," Differents. Uravn., 11, No. 12, 2143-2153 (1975). 173. P. I. Rashkov, S. I. Bilchev, D. T. Dochev, and D. Tr. Dochev, "Averaging in differential equations of neutral type with an a s y m p t o t i c a l l y l a r g e lag in a Banach s p a c e , " Godislm. Vyssh. Uehebn. Zaved. Prilozh. Mat., 1_~1,No. 3, 95-101 (1975(1977)). 174. A. E. Rodkina, "On the continuability of the solutions of a functional-differential equation of n e u t r a l type," Differents. Uravn., 1._00,No. 2, 361-363 (1974). 175. A. E. Rodkina, "On the Cauchy p r o b l e m f o r equations of neutral type," in: Fourth All-Union Conference on the Theory and Applications of Differential Equations with Deviating Argument [in Russian], Kiev (1975), p. 203. 176. A. E. Rodkina, non the continuability, uniqueness, and continuous dependence on a p a r a m e t e r of the s o l u tions of equations of neutral type," Differents. Uravn., 1..1.1,No. 2, 268-279 (1975).

714

177. A. E. Rodkina, "On the Cauchy p r o b l e m for an equation of neutral type," in: Collection of A r t i c l e s on Applications of Functional Analysis [in Russian], V o r o n e z h (1975), pp. 155-160. 178. A. E. Rodkina and B. N. Sadovskii, "On the K r a s n o s e l ' s k i i - P e r o v principle of connectedness," Tr. Mat. Fak. Voronezh. Gos. Univ., No. 4, 89-103 (1971). 179. A. E. Rodkina and B. N. Sadovskii, "On the differentiation of the o p e r a t o r of t r a n s i t i o n along the t r a j e c t o r i e s of an equation of neutral type, ~ T r . Mat. Fak. Voronezh. Gos. Univ., No. 12, 31-37 (1974). 180. A. E. Rodkina and B. N. Sadovskii, NOn the theory of g e n e r a l i z e d solutions of equations of n e u t r a l type," in'. Collection of A r t i c l e s on Applications of Functional Analysis [in Russian], Voronezh (1975), pp. 161166. 181. V. I. Rozhkov, "A boundary-value problem for equations of neutral type with small delay," Tr. Sere. Teor. Differents. Uravn. Otklon. Argumentom, Univ. Druzhby Narodov Patrisa Lumumby, 3, 47-56 (1965). 182. V. I. Rozhkov, "Equations of neutral type with small delay on an infinite interval," Tr. Sem. Teor. Differents. Uravn. Otldon. Argumentom. Univ. Druzhby Narodov Patrisa Lumumby, 3, 186-203 (1965). 183. V. I. Rozhkov, "The asymptotic expansion in the powers of the lag of the solution of an equation of neutral type with several small delays," Tr. Sere. Teor. Differents. Uravn. Otklon. Argumentom, Univ. Druzhby Narodov Patrisa Lumumby, 6, 45-54 (1968). 184. V. I. Rozhkov, "The asymptotic expansion in powers of the lag of the periodic solution of an equation of neutral type with small delay," Differents. Uravn., 4, No. 7, I, 50-1257 (1968). 185. V. I. Rozhkov, "The construction of an asymptotic expansion for the solution of a linear system of equations of neutral type with small lag, avoiding the boundary layer," Mat. Zametki, 6, No, i, 109-113 (1969). 186. V. I. Rozhkov, "On the periodic solutions of autonomous systems of equations of neutral type with small lag," Differents. Uravn., 7, No. 3, 446-452 (1971). 187. E. Yu. Romanenko and T. S. Feshchenko, "On the asymptotic behavior of the solutions of linear differential-functional equations of neutral type in the neighborhood of a singular point," in: Qualitative Methods in the Theory of Differential Equations with Deviating Argument [in Russian], Kiev (1977), pp. 101-123. 188. V. P. Rubanik, Oscillations of Quasilinear Systems with Lag [in Russian], Nauka, Moscow (1969). 189. I. I. Ryabtsev, "On a variable lag," Izv. Vyssh. Uchebn. Zaved., Mat., No, i, 164-173 (1959). 190. B. N. Sadovskii, "The application of topological methods in the theory of periodic solutions of nonlinear differential-operator equations of neutral type," Dekl. Akad. Nauk SSSR, 200, No. 5, 1037-1040 (1971). 191. B. l~I.Sadovskii, "Limit-compact and condensing operators," Usp. Mat. Nauk, 2_7.7,No. i, 81-146 (1972). 192. B.N. Sadovskii, 'Condensing operators and differential equations," Preprint, Inst. Mat. Sib. Otd. Akad. Nauk SSSR (School on Theory of Operators in Function Spaces), Novosibirsk (1979). 193. K. Sarygulov, "Solving a class of linear integropartial differential equations with the aid of a variant of Yu. D. Sokolov's method," Tr. Kirgiz. Univ. Ser. Mat. Nauk, No. 8, 93-99 (1973). 194. K. Sarygulov, "The approximate solution of a class of integropartial differential equations with deviating argument" Sb. Tr. Aspirantov i Soiskatelei. Kirgiz. Univ. Ser. Mat. Nauk, No. i0, 37-42 (1973). 195. V. P. Skripnik, "On equations with transformed argument of neutral and superneutral type," Ukr. Mat. Zh., 22, No. 5, 611-624 (1970). 196. V. E. Slyusarchuk, "On the invertibility of differential operators in the space of functions infinitely differentiable on the axis," Differents. Uravn., 15, No. i0, 1796-1803 (1979). 197. V. E. Slyusarchuk, "The absolute exponential stability of linear differential equations of neutral type in a Banach space," Differents. Uravn., 16, No. 3, 462-469 (1980). 198. M. A. Soldatov, "On the properties of the solutions of linear differential-difference equations," Sib. Mat. Zh., 8, No. 3, 669-679 (1967). 199. S. N. Sorokin, "Stability in finite differential systems of neutral type," Prikl. Mat. Mekh., 40, No. i, 44-54 (1976). 200. S. N. Sorokin, " P r a c t i c a l stability f o r constantly acting p e r t u r b a t i o n s of p r o c e s s e s of neutral type," T r . Mosk. A v t o m o b . - D o r . Inst., No. 143, 33-37 (1977). 201. N. Stoyanov and D. Bainov, ~The stability of the periodic solutions of q u a s i l i n e a r n o , autonomous s y s t e m s of neutral type," in" T h e o r e t i c a l and Applied Mechanics, I National Congress, Varna, 1969, Vol. 2, Sofia (1971), pp. 107-112. 202. T . G . Strizhak, "The dynamic stability of a pendulum filled with a viscous fluid," Ukr. Mat. Zh., 21, No. 1, 129-133 (1969). 203. B. P. Tkach, "On the periodic solutions of s y s t e m s of differential equations with deviating a r g u m e n t of n e u t r a l type," Ukr. Mat. Zh., 21, No. 1, 73-85 (1969). 204. B. P. Tkach, "On the periodic solutions of a c l a s s of h i g h e r - o r d e r s y s t e m s of p a r t i a l differential equations of neutral type," in: D i f f e r e n t i a l - D i f f e r e n c e Equations [in Russian], Kiev (1971), pp. 152-162.

715

205. T . S . Feshchenko, nOn the asymptotic behavior of the solutions of a class of differential-functional equations in the neighborhood of a singular point," in: Asymptotic Methods in the Theory of Nonlinear Oscillations. P r o c . o f All-Union Conf., Katsiveli, 1977, Kiev (1979), pp. 220-226. 206. A. N. Filatov, Asymptotic Methods in the Theory of Differential and Integrodifferential Equations [in Russian], Fan, Tashkent (1974). 207. V. I. Fodchuk, "The averaging method for differential-difference equations of neutral type," Ukr. Mat. Zh., 20, No. 2, 203-209 (1968). 208. V. I. Fodchuk, "Application of asymptotic methods to partial differential equations of neutral type," in: Differential-Difference Equations [in Russian], Kiev (1971), pp. 184-193. 209. V. I. Fodchuk, "Existence t h e o r e m s and methods of construction of bounded solutions and integral manifolds of p e r t u r b e d differential-functional equations," in: Differential Equations with Deviating Argument [in Russian], Kiev (1977), pp. 279-289. 210. V. I. F odchuk and A. Kholmatov, "Behavior of the solutions of differential-functional equations of neutral type in the neighborhood of e l e m e n t a r y stationary points," Ukr. Mat. Zh., 25, I~o. 5, 610-620 (1973). 211. V. I. Fodchuk and A. Kholmatov, " P e r i o d i c and a l m o s t - p e r i o d i c solutions of singularly p e r t u r b e d differential equations of neutral type," Differents. Uravm, 1__22,No. 6, 1029-1027 (1976). 212. V. I. Fodchuk and A. Kholmatov, "A study of the stability of s y s t e m s of differential-functional equations of neutral type with periodic coefficients," Mat. Fiz. Resp. Mezhved. Sb., No. 20, 71-76 (1976). 213. V. I. Fodchuk and A. Kholmatov, "The averaging method for singularly p e r t u r b e d equations of neutral type," in: Analytic Methods in the Theory of Differential Equations [in Russian], Kiev (1977), pp. 189-200. 214. S. G. Khristova and D. D. Bainov, "Globally bounded solutions of s y s t e m s of integrodifferential equations of superneutral type," Godishn. Vyssh. Ucheb. Zaved., Prilozhen. Mat., 1__33,No. 4, 77-86 (1977(1978)). 215. D. Ya. Khusainov, "On the stability of the solutions of s y s t e m s of differential equations with deviating argument of neutral type under a small deviation of the argument," in: Qualitative Methods of the Theory of Differential Equations with Deviating Argument [in Russian], Kiev (1977), pp. 156-167. 216. N . V . Sharkova, PAn approximate solution by the method of oscillating functions of the boundary-value problems for linear differential equations with deviating argument," Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 120-127 (1967). 217. M. I. Shkil' and Yu. P. Pidchenko, "On the asymptotic solution of s y s t e m s of linear differential equations of neutral type with slowly varying coefficients," in: Approximate Integration Methods of Differential and Integral Equations [in Russian], Kiev (1973), pp. 163-183. 218. M. I. Shkil' and u P. Pidchenko, "O11 the approximate solution of s y s t e m s of linear differential equations of neutral type with slowly varying coefficients," Vychisl. PriM. Mat. (Kiev), I~o. 22, 155-168 (1974). 219. M. I. Shkil' and Yu. P. Pidchenko, "On the question of the construction of f o r m a l solutions of nonhomogeneous s y s t e m s of equations of neutral type," in: Approximate Methods of Mathematical Analysis [in Russian], Kiev (1977), pp. 138-147. 220. L. E. E l ' s g o l ' t s , "Certain p r o p e r t i e s of the periodic solutions of linear and quasilinear equations with deviating argument," Vesta. Mosk. Univ. Set. Mat. Mekh. Astron. Fiz. Khim., No. 5, 229-234 (1959). 221. L. 4. ]~l'sgol'ts, "On the theory of the stability of differential equations with deviating arguments," Vesta. Mosk. Univ. Set. Mat. Mekh. Astron. Fiz. Khim., No. 5, 65-71 (1959). 222. L. ~,. E l ' s g o l ' t s , "On boundary-value problems for ordinary differential equations with deviating a r g u ments," Usp. Mat. Nauk, 15, No. 5, 222-224 (1960). 223. L. 4. ~ l ' s g o l ' t s , "Basic d'~rections of r e s e a r c h in the theory of differential equations with deviating a r gument," Tr. Sem. T e o r . Differeats. Uravn. Otklen. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 1, 3-19 (1962). 224. L. ~. ~ l ' s g o l ' t s , "On the question of the effect of a small deviation of the argument on stability, ' Tr. Sem. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby I~arodov P a t r i s a Lumumby, 1, 114-115 (1962). 225. L. t~. E l ' s g o l ' t s , "Variational p r o b l e m s with deviating argument and moving boundaries," Tr. Sem. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 3, 239-241 (1965). 226. L. E. ]~l'sgol'ts, "Basic directions of r e s e a r c h in the t h e o r y of differential equations with deviating a r gument," in: P r o c . 2rid All-Union ConI. on Theoretical and Applied Mechanics, 1964, Survey of Reports, No. 2, Nauka, Moscow (1965), pp, 158-164. 227. L. ~. ~ l ' s g o l ' t s , "A note on branching and cut-off of solutions of equations with deviating argument," T r . Sem. T e o r . Differents. Uravn. Otklen. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 5, 242-245 (1967). 228. L. E. ~ l ' s g o l ' t s , "Cases of the expansion of the c h a r a c t e r i s t i c quasipolynomial into polynomial and quasipolynomial f a c t o r s , " Tr. Sere. T e o r . Differents. Uravn. Otklon. Argum. Univ. Druzhby Narodov P a t r i s a Lumumby, 5, 246-249 (1967).

716

,,'

f

229. L . E . E l ' s g o l ' t s , "A note on c e r t a i n periodic solutions of equations with deviating a r g u m e n t , " T r . Sere. T e o r . D i f f e r e n t s . Uravn. Otldon. Argum. Univ. Druzhby Narodov P a t r i s a L u m u m b y , 7, 189-193 (1969). 230. L. ~. E l ' s g o l ' t s and S. B. Norkin, Introduction to t h e Theory and Application of Differential Equations with Deviating A r g u m e n t s , Academic P r e s s , New Y o r k (1973). 231. B. A. A s n e r , J r . and A. Halanay, " D e l a y - f e e d b a c k using d e r i v a t i v e s for m i n i m a l t i m e linear control p r o b l e m s , " J. Math. Anal. Appl., 4_~8,No. 1, 257-263 (1974). 232. H. T. Banks and M. Q. J a c o b s , "An attainable s e t s a p p r o a c h to optimal control of functional differential equations with function s p a c e t e r m i n a l conditions," J. Diff. Equations, 13, No. 1, 127-149 (1973). 233. H. T. Banks and G. A. Kent, "Control of functional differential equations of r e t a r d e d and neutral type to t a r g e t sets in function s p a c e , " SIAM J. Control, 1_.9.0,No. 4, 567-593 (1972). 234. R. Bellman and K. L. Cooke, "Stability theory and adjoint o p e r a t o r s f o r l i n e a r d i f f e r e n t i a l - d i f f e r e n c e equations," T r a n s . Am. Math. Soc., 9/2, No. 3, 470-500 (1959). 235. R. K. Brayton, "Bifurcation of periodic solutions in a nonlinear d i f f e r e n c e - d i f f e r e n t i a l equation of neutral type," Q. Appl. Math., 2__~4,No. 3, 215-224 (1966). 236. R. K. Brayton, "Nonlinear oscillations in a distributed network," Q. Appl. Math., 2..44,No. 4, 289-301 (1967). 237. R. K. Brayton, "On the bifurcation of periodic solutions in nonlinear d i f f e r e n c e - d i f f e r e n t i a l equations of neutral type," in: Differential Equations and Dynamical S y s t e m s , Academic P r e s s , New York (1967), pp. 215-220. 238. R. K. Brayton, " S m a l l - s i g n a l stability c r i t e r i o n f o r e l e c t r i c a l networks containing l o s s l e s s t r a n s m i s s i o n lines," IBM J . R e s . Develop., 1_22,No. 6 , 4 3 1 - 4 4 0 (1968). 239. W. E. B r u m l e y , "On the a s y m p t o t i c b e h a v i o r of solutions of d i f f e r e n t i a l - d i f f e r e n c e equations of neutral type," J. Diff. Equations, 7, No. 1, 175-188 {1970). 240. W. B. C a s t e l a n and E. F. Infante, "Liapunov functional f o r a m a t r i x neutrM d i f f e r e n c e - d i f f e r e n t i a l e q u a tion with one delay," J. Math. Anal. Appl., 7.1, No. 1, 105-130 (1979). 241. A. K. Choudhury, "Pointwise d e g e n e r a c y of l i n e a r t i m e - i n v a r i a n t d e l a y - d i f f e r e n t i a l s y s t e m s , " Int. Conf. Differ. Equat., 1974, New York etc. (1975), pp. 806-807. 242. A. K. Choudhury, "On the pointwise c o m p l e t e n e s s of d e l a y - d i f f e r e n t i a l s y s t e m s of neutral type," SIAM J. Math. Anal., 7, No. 6, 913-929 (1976). 243. M. A. Connor, " T i m e optimal control of n e u t r a l s y s t e m s with control integral c o n s t r a i n t , " J. Math. Anal. Appl., 4.~5, No. 1, 91-95 (1974). 244. M. A. Cruz and J. K. Hale, "Stability of functional differential equations of neutral type," J. Diff. E q u a tions, 7, No. 2, 334-355 (1970). 245. M. A. Cruz and J. K. Hale, "Exponential e s t i m a t e s and the saddle point p r o p e r t y f o r neutral functional diff e r e n t i a l equations," J. Math. Anal. Appl., 34, No. 2, 267-288 (1971). 246. P. C. Das and N. Parhi, " R e p r e s e n t a t i o n f o r solutions of l i n e a r neutral functional differential equations," Math. S y s t e m s Theory, 7, No. 3, 232-249 (1973). 247. R. Datko, " L i n e a r d i f f e r e n t i a l - d i f f e r e n c e equations in a Banaeh s p a c e , " in: P r o c . IEEE Conf. Deeis. and Contr. Incl. 15th Symp. Adapt P r b c e s s . , C l e a r w a t e r , Fla., 1976, New York, N. Y. (1976), pp. 735-742. 248. R. Datko, " L i n e a r autonomous n e u t r a l differential equations in a Banach s p a c e , " J. Diff. Equations, 2._~5, No. 2, 258-274 (1977). 249. R. Datko, " R e p r e s e n t a t i o n of solutions and stability of l i n e a r d i f f e r e n t i a l - d i f f e r e n c e equations in a Banach s p a c e , " J. Diff. Equations, 29, No. 1, 105-166 (1978). 250. K. Sehmidtt (ed.), Delay and Functional Differential Equations and T h e i r Application, Academic P r e s s , New York (1972). 251. R. D. D r i v e r , " E x i s t e n c e and continuous dependence of solutions of a neutral functional-differential e q u a tion," Arch. Rational Mech. Anal., 1__9,No. 2, 149-166 (1965). 252. R. D. D r i v e r , " T o p l o g i e s for equations of n e u t r a l type and c l a s s i c a l e l e c t r o d y n a m i c s , " Univ. R. I., Techn. Rep., No. 60 (1975). 253. R. D. Gahl, "Controllability of nonlinear s y s t e m s of n e u t r a l type, ~ J . Math. Anal. Appl., 6.__~No. 1, 33-42 (1978). 254. M . J . G o o v a e r t s , W. Meensen, and J. De Kerf, "Solving the nonhomogeneous f i r s t - o r d e r linear d i f f e r e n t i a l - d i f f e r e n c e equation with constant coefficients," Utilitas Math., 6, 75-86 (1974). 255. L. J. G r i m m , " E x i s t e n c e and continuous dependence f o r a c l a s s of nonlinear n e u t r a l - d i f f e r e n t i a l e q u a tions," P r o c . Am. Math. Soc., 2-9, No. 3, 467-473 (1971). 256. W. Hahn, " U b e r D i f f e r e n t i a l - D i f f e r e n c e GleichLmgen mit anomalen LSsungen, ~ Math. Ann., 133, No. 3, 251-255 (1957). 257. A. Halanay, " T h e o r e m e de Noether p o u r des p r o b l e m e s variationnels a a r g u m e n t s deplaces," Ann. Polon. Math., 2-9, No. 2, 189-198 (1974).

7t 7

258. J. K. Hale, "A c l a s s of neutral equations with the fixed point p r o p e r t y , " P r o c . Nat. Acad. Sci., U.S.A., 6..~7, No. 1, 136-137 (1970). 259. J. K. Hale, "Functional differential equations," in: Leer. Notes Math., 183, 9-22 (1971). 260. J. K. Hale,~Functional differential equations (Appl. Math. Sci., 3), New York etc., S p r i n g e r - V e r l a g (1971). 261. J. K. Hale, " F o r w a r d and backward continuation f o r neutral functional differential equations ," J . Diff. Equations, 9, No. 1, 168-181 (1971). 262. J . K. Hale, "Stability and oscillations in functional differential equations," in: P r o c . 5th Annual P r i n c e t o n Conf. Inform. Sci. and Syst. (1971), Princeton, N,J., etc., pp. 220-223. 263. J. K. Hale, " C r i t i c a l c a s e s f o r n e u t r a l functional differential equations," J. Diff. Equations, 10, No. 1, 59-82 (1971). 264. J. K. Hale, "Smoothing p r o p e r t i e s of n e u t r a l equations," An. Acad. B r a s i l . Cienc., 45, No. 1, 49-50 (1973). 265. J . K. Kale, "o~-contractions and differential equations," in: Actes Conf. Internat. Equa.-Diff. 73, Hermann, P a r i s (1973), pp. 15-41. 266. J. K. Hale, "Continuous dependence of fixed points of condensing m a p s , " 3. Math. Anal. Appl., 46, No. 2, 388-394 (1974). 267. J. K. Hale, " B e h a v i o r n e a r constant solutions of functional differential equations," J. Diff. Equations, 15, No. 2, 278-294 (1974). 268. J . K. Hale, " P a r a m e t r i c stability in difference equations," Boll. Un. Mat. Ital., 11, No. 3, 209-214 (1975). 269. J. K. Hale, " T h e o r y of functional differential equations," Appl. Math. Sci., 3 (1977). 270. J. K. Hale and M. A. Cruz, "Asymptotic b e h a v i o r of n e u t r a l functional differential equations," Arch. R a tional Mech. Anal., 3_~4,No. 5, 331-353 (1969). 271. J. K. Hale and J. Mawhin, "Coincidence degree and periodic solutions of neutral equations," J. Diff. Equations, 1-5, 295-307 (1974). 272. J. K. Hale and K. R. M e y e r , "A c l a s s of functional equations of neutral type," Mere. Am. Math. Soc., No. 7 6 (1967). 273. A. H a u s r a t h , "Stability in the c r i t i c a l case of purely i m a g i n a r y roots for neutral functional differential equations," J. Diff. Equations, 13, 329-357 (1973). 274. D. Henry, " L i n e a r autonomous neutral functional differential equations," J. Diff. Equations, 1-5, No. 1, 106128 (1974). 275. G. Hetzer, "Some applications of the coincidence degree f o r s e t - c o n t r a c t i o n s to functional differential equations of n e u t r a l type," C o m m e n t . Math. Univ. Carolin., 16, No. 1, 121-138 (1975). 276. A . F . Iz6 and N. A. de Molfetta, " A s y m p t o t i c a l l y autonomous neutral functional differential equations with t i m e dependent lag," in: Dyn. Syst. Int. Syrup. (1976), pp. 127-132. 277. A. F. Iz~ and O. F. Lopes, " F u n c t i o n a l - d i f f e r e n c e equations," in: Funct. Anal., New York (1976), pp. 191-214. 278. A. F. Iz6 and J. G. dos Reis, "Stability of p e r t u r b e d neutral functional differential equations," Nonlinear Anal., 2, No. 5, 563-571 (1978). 279. M . Q . J a c o b s , "Controllability and optimization of neutral s y s t e m s , " in: P r o c . IEEE Conf. Decis. and Contr. Incl. 15th Syrup. Adapt. P r o c e s s . , C l e a r w a t e r , Fla., 1976, New York, N. Y. (1976), pp. 743-745. 280. M . Q . J a c o b s and C. E. Langenhop, " C r i t e r i a f o r function s p a c e controllability of l i n e a r neutral s y s t e m s , " SIAM J. Control Optim., 1_~4,No. 6, 1009-1048 (1976). 281. T. Jankowski and M. Kwapisz, "On the existence and uniqueness of solutions of boundary-value p r o b l e m s f o r differential equations with p a r a m e t e r , " Math. Nachr., 7_~.1,237-247 (1976). 282. T. Jankowski and M. Kwapisz, " C o n v e r g e n c e of a p p r o x i m a t e iterations f o r functional equations in a Banach s p a c e , " Rev. Roumaine Math. P u r e s Appl., 2.1, No. 8, 1029-1048 (1976). 283. J. L. Kaplan and J. A. Yorke, "On the stability of a periodic solution of the differential delay equation," SIAM J. Math. Anal., 6, No. 2, 268-282 (1975). 284. F. Kappel, " L a p l a c e - t r a n s f o r m methods and l i n e a r autonomous functional-differential equations," B e t . Math.-Statist. Sel~. F o r s c h . G r a z . , No. 64, 1-62 (1976). 285. M. Kisilewicz, " C e r t a i n p r o b l e m s of the uniform instability of g e n e r a l i z e d s y s t e m s with aftereffect," D i s c u s s . Math., 2.1, No. 1, 43-48 (1975). 286. K. Kunisch, " A b s t r a c t Cauchy p r o b l e m s and a b s t r a c t integral equations f o r neutral functional differential equations," Arch. Math., 3.1, No. 6, 580-588 (1979). 287. M. Kwapisz, "On the existence and uniqueness of solutions of a c e r t a i n integral-functional equation," Arm. Polon. Math., 3_~_1,No. 1, 23-41 (1975). 9

718

1!

288. A. Manitius, "Optimal control of h e r e d i t a r y s y s t e m s , " in: Contr. T h e o r y and Top. Funct. Anal., Vol. 3, Vienna (1976), pp. 43-178. 289. W. R. Melvin, "A c l a s s of neutral functional differential equations," J. Diff. Equations, 1_~2,No. 3, 524-534 (1972). 290. W. R. Melvin, "Toplogies f o r n e u t r a l functional differential equations," J . D i f f . Equations, 13.3, No. 1, 24-31 (1973). 291. R . D . Nussbaum, "A g e n e r a l i z a t i o n of the Ascoli t h e o r e m and an application to functional differential e q u a tions," J. Math. Anal. Appl., 3._~5,No. 3, 600-610 (1971). 292. R. D. N u s s b a u m , " E x i s t e n c e and uniqueness t h e o r e m s f o r s o m e functional differential equations of neutral type," J. Diff. Equations, 1_~.1,No. 3, 607-623 (1972). 293. R . D . N u s s b a u m , " P e r i o d i c solutions of autonomous functional differential equations," Bull. Am. Math. Soc., 7__~9,No. 4, 811-814 (1973). 294. W. J. P a l m and W. F. Schmitendorf, "Conjugate-point conditions f o r v a r i a t i o n a l p r o b l e m s with delayed a r g u m e n t , " J . Optim. Theory Appl., 1_~4,No. 6, 599-612 (1974). 295. L. Pandolfi, "Stabilization of n e u t r a l functional differential equations," J. Optim. Theory Appl., 20, No. 2, 191-204 (1976). 296. R. D. R a m a k r i s h n a , "Stability c r i t e r i a of n e u t r a l functional differential equations," P r e c . Nat. Inst. Sci. India, A3_~5,No. 1, 63-69 (1969). 297. M. Sendaula, "On the optimal control t h e o r y f o r neutral s y s t e m s , " in: P r o c . IEEE Internat. Syrup. C i r c . and Syst., Munich, Apr. 27-29, 1976, New York, N. Y., etc., pp. 408-411, 298. G. A. Shanholt, "Slowly varying l i n e a r n e u t r a l functional differential equations," Int. J. Control, 1_~8,No. 3, 559-562 (1973). 299. W. Snow, E x i s t e n c e , Uniqueness and Stability f o r Nonlinear Differential Equations in the Neutral C a s e , New Y o r k Univ., New York (1965). 300. W. A. H a r r i s , J r . and Y. Sibuya (eds.), P r o c . United S t a t e s - J a p a n S e m i n a r on Differential and Functional Equations, University of Minnesota, June 26-30, 1967, Benjamin, New York (1967).

MULTIVALUED

MAPPINGS

Yu. G. Borisovich, A. D. Myshkis, and

B. V.

D. V.

Gel'man, Obukhovs

UDC 517.988.52 kii

The p r e s e n t p a p e r is a s u r v e y of the c o n t e m p o r a r y state of a r t in the theory of multivalued m a p p i n g s . In it one c o n s i d e r s different f o r m s of continuity of multivalued mappings, one inv e s t i g a t e s differentiable and m e a s u r a b l e multivalued mappings, one c o n s i d e r s s i n g l e - v a l u e d continuous a p p r o x i m a t i o n s and sections of multivalued mappings, one studies fixed points of m u l t i v a l u e d mappings and other questions of this theory. One gives r e f e r e n c e s to the l i t e r ature r e g a r d i n g applications to the t h e o r y of g a m e s , m a t h e m a t i c a l e c o n o m i c s , the t h e o r y of differential inclusions and g e n e r a l i z e d dynamical s y s t e m s . The p a p e r contains an extensive bibliography.

INTRODUCTION The theory of multivalued mappings is a branch of mathematics which has been developed intensively in the last years and lies at the junction of topology, theory of functions of a real variable and nonlinear functional analysis. The concept of a multivalued mapping, assigning to the points of some set X a subset of another set Y, has arisen in a natural manner by refining the classical concept of a multivalued function. However, for a long time, in the above mentioned chapters of mathematics, one has studied systematically only the single-valued

Translated

from

Itogi Nauki i Tekhnild, Seriya Matematicheskii

0090-4104/84/2406-0719507.50

Analiz, Vol. 19, pp. 127-230,

9 1984 Plenum Publishing C o r p o r a t i o n

1982.

719