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.. 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-., and for uniqueness

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

Yt, V/)l'-~. 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~)l0,

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))~ 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] 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 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] (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 ~