366.
L. Tzafriri, "Reflexibility in Banach lattices and their subspaces," J. Funct. Anal., I_~0, No. 2, 1-18 (1972 ). 367. B. Walsh, "Positive approximate identities and lattice-ordered dual spaces," Manuscr. Math., 14, No. I, 57-63 (1974). 368. B. Walsh, "An approximation property characterizes ordered vector spaces with lattice-ordered duals; Bull. Am. Math. Soc., 80, No. 6, 1165-1168 (1974). 369. H. yon Weizs~[cker, "Sublineare Abbildungen und ein Konvergenzsatz yon Banach," Math. Ann., 21__2, No. 2, 165-171 (1974). 370. A. W. Wickstead, "A characterization of weakly sequentially complete Banach lattices," Ann. Inst. Fourier, 2_6_6,No. 2, 25-28 (1976). 371. A. W. Wiekstead, "The s t r u c t u r e s p a c e of a Banach lattice," J. Math. P u r e s Appl., 5__66, No. 1, 39-54 (1977). 372. A. W. Wiekstead, "Weak and unbounded c o n v e r g e n c e in Banach l a t t i c e s , " J. A u s t r a l . Math. Soc., 2A4 , No. 3, 312-319 (1977). 373. A. Wirth, " O r d e r and n o r m c o n v e r g e n c e in Banach l a t t i c e s , " Glasgow Math. J . , 1_5_5, No. 1, 13 (1974). 374. A. Wirth, "Carriers and Archimedean classes in Banach lattices," J. London Math. Soc., i0, No. I, 121-124 (1975). 375. A. Wirth, "Relatively uniform Banach lattices," Proc. Am. Math. Soc., 5_~2, No. I, 178-180 (1975). 376. G. Wittstock, "Ordered normed tensor products," Leet. Notes Phys., 2_99, 67-84 (1974). 377. G. Wittstock, "Eine Bemerkung (iber Tensorprodukte von Banaehverb~nden," Arch. Math., 25, No. 6, 627-634 (1974). 378. P. Wojtaszczyk, "On Banach spaces properties of uniform algebras with separable annihilator," Bull. Acad. Polon. Sci., Set. Sci. Math., Astr. Phys., 25, No. 1, 23-26 (1977). 379. Yan-Chaen Wong and Kung-fu Ng, Partially Ordered Topological Vector Spaces, Clarendon Pres, Oxford (1973). 380. W. A. Woyczy~ski, "Geometry and martingales in Banaeh spaces," Leet. Notes Math., 472, 229-275 (1975). 381. W. A. Woyczynski, "A central limit theorem for martingales in Banach spaces," Bull. Acad. Polon. Sci., Set. Sci. Math., Astron. Phys., 23, No. 8, 917-920 (1975). 382. A. C. Zaanen, Integration, North-Holland, Amsterdam (1967).
CONDENSING R. A.
OPERATORS R. S.
Akhmerov, Potapov,
M. and
B.
I. Kamenskii, N. Sadovskii
UDC
517~988.52
This paper is devoted to a s u r v e y of the c u r r e n t s t a t e of the theory of m e a s u r e s of n o n c o m p a c t h e s s and condensing o p e r a t o r s .
IN TR ODU
C TI ON
Although the first concrete measure of noncompactness (in current terminology the Kuratowski measure of noncompactness) was introduced by K. Kuratowski in 1930 already, basically the theory of measures of noncompactness and condensing operators was constructured in the last 15 years. The definition of a condensing operator appeared as a natural generalization of the definitions of compact and contracting operators, connected with measures of noncompactness. It turned out that condensing operators are in many ways similar to compact ones: for mappings of the form I-f with condensing f one can properly define the topological degree, many theorems about fixed points of completely continuous operators generalize to condensing operators, for linear equations with condensing operators there is a generalization of the famous theorem of FredholmRiesz-Schauder. The stimulus for the development of the theory was the fact that condensing operators arose in various problems connected with differential equations in infinite-dimensional spaces, functional-differential equations of nontrivial type, integral equations, certain types of partial differential equations. As the bibliography given at the end of the paper shows, the literature on these questions is very extensive; there are surveys (ef., e.g., [116,174, 309]). 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 l d i Analiz, Vol. 18, pp. 185-250, 1980.
0090-4104/82/1804-0551507.50
9 1982 Plenum Publishing Corporation
551
The present paper is intended to s e r v e two ends: firstly, to give a possibly s i m p l e r and s h o r t e r d e s c r i p tion of the ideological side of the theory, not pretending at all to maximal generality or a complete elucidation of the facts, and secondly, to give an as complete as possible, but of necessity nouprofound s u r v e y of the l i t e r a t u r e , somewhat like a guide, appended to the list of literature. In a c c o r d with this, the whole text is divided into two chapters: nSurvey of the Theory" and "Survey of the L i t e r a t u r e . " In the f i r s t chapter we have tried to d e s c r i b e each concept and each fact in a maximally simple and concrete situation for which it does not lose meaning. All generalizations and also l i t e r a t u r e citations, we have tended to c a r r y out in the second chapter. In the second chapter the formulations a r e often c o a r s e n e d or given with incomplete list of conditions, so that as a r e f e r e n c e this part can only be used with cuation. The o r d e r of r e f e r e n c e s throughout the paper is alphabetical and in no way connected with priority. CHAPTER SURVEY I.I.
Measures
of
OF
THE
I THEORY
Noncompactness
1.1.1. Definition. Let (M, P) be a metric space. By the Kuratowski measure of noncompaetness ~(A) of the set A__~N/ is meant the greatest lower bound of those d > 0, for which A admits a finite subdivision into sets, whose diameters are less than d.* Obviously,
the set A is completely
bounded
if and only if ~ (A) = 0.
1.1.2. THEOREM. The intersection of a centered system of closed subsets of a complete metric space is nonempty, if in this system there are sets of arbitrarily small Kuratowski measure of noucompactness. Proof. We denote the given centered system by ~. We note that if in ~ there is a set A 0 with zero measure of noncompaetness, which by virtue of Hausdorff's test is compact, then the assertion follows trivially from the definition of compactness: it suffices to pass to consideration of the system ~'-- {A N A0: AE~} . In the general case we choose a sequence of sets AnE~, for which ~ (A n) ~ 0 as n ~ ~, and we show that the set
Ao=~
An
is compact
and the adjunction
of it to the system
~l leaves the system
centered.
Whence,
as above,
u=l
the assertion
of the theorem
will follow.
The compactness of A 0 follows from the fact that it is closed and the obvious equation ~ (A 0) = 0. Now we n ~A] NA0 is nonempty. We choose a sequence shall show that for any finite s u b s y s t e m ~ 0 ~ the set B~ .(AC_~ l
{xn} such that
n
n A)n (n A,).
sequence is relatively compact, since
for any N. Consequently, the s e t of its limit points is nonempty; but any of its limit points, as is easy to see, belongs to B. The t h e o r e m is proved. 1.1.3. Definition. The Hausdorff m e a s u r e of noucompactness ~((A) of the set A in the m e t r i c space (M, p) is the g r e a t e s t lower bound of those e > 0 for which the set A has in the space M a finite e-net. 1.1.4. R e m a r k s . a) It is easy to see that the Hausdorff m e a s u r e of noncompactness ~({A) depends on the choice of ambient space M, so m o r e c o r r e c t notation would be XM(A). Now the Kuratowski m e a s u r e of nonc o m p a c t n e s s is an intrinsic c h a r a c t e r i s t i c of the space (A, D). b) The Kuratowski and Hausdorff m e a s u r e s of noncompactness are connected by the obvious inequalities ~({A)..< a (A) 4 2~((A). Both of these inequalities a r e not improvable. It follows f r o m them, in p a r t i c u l a r , that the t h e o r e m proved above on the nonemptiness of the intersection of a centered s y s t e m is also valid for the Hausdorff m e a s u r e of noncompactness. c) In the definition of the Hausdorff m e a s u r e of noncompactness one can, in place of a finite e-net, speak of a completely bounded a-net. d) In the definition of the Kuratowski m e a s u r e of noneompaetness, instead of "whose d i a m e t e r s a r e less than d" one can say "whose d i a m e t e r s a r e not g r e a t e r than d"; in the s a m e way in the definition of the Hausdorff *infO = + ~ .
552
measure
of noncompaetness
it makes
no difference
how one defines ~-net, by open or closed bails of radius e.
1.1.5. Properties of the Kuratowski and Hausdorff Measures of Noneompactness rectly from the definitions, follow the following obvious properties of the measures X. By ,2 we denote ~ or X. a) Regularity:
:~(A) = 0 if and only if A is completely
b) Monotonically: c) Semiadditivity: d) Invariance e) Lipschitz
from
A~B
bounded.
follows that 9 (A) <_~ (B).
~(A [J B)=max{~(A),
with respect
in Metric Spaces. Diof noncompactness ~ and
,(B)}.
to closure:
~(A) = ~(A).
property:
[)~(A)--x(B)I..
w h e r e r is the H a u s d o r f f m e t r i c :
r (A, B) = inf {e > 0:
A ~ B , B~2A}
(A e and B e a r e ~ - n e i g h b o r h o o d s of the s e t s A and 13). 1.1.6. K u r a t o w s k i and H a u s d o r f f M e a s u r e s of N o n c o m p a c t n e s s in L i n e a r N o r m e d S p a c e s , B e s i d e s g e n eral properties of the functions ~ and )~ in metric spaces, we note several of their simplest properties for the case of linear normed spaces. f) Semihomogeneity: g) Algebraic
~(t.A) = I tl. ~(A) (t is a number).
semiadditivity:
~(A + B)~<~?(A) ~-~(B). h) I n v a r i a n c e w i t h r e s p e c t to t r a n s l a t i o n : ~(A + x 0) = ~J(A). In the n e x t two p a r a g r a p h s we e s t a b l i s h l e s s o b v i o u s p r o p e r t i e s of the f u n c t i o n s c~ and • in n o r m e d spaces. 1.1.7. T H E O R E M . L e t 13 be t h e u n i t b a l l w i t h c e n t e r a t z e r o in the l i n e a r n o r m e d s p a c e E. Then ~(13) = X(13) = 0, i f E is f i n i t e - d i m e n s i o n a l , and ~(13) = 2, ~(B) = 1 o t h e r w i s e . Proo~
The f i r s t a s s e r t i o n f o l l o w s f r o m the r e g u l a r i t y of the m e a s u r e s of n o n c o m p a c t n e s s c~ and X.
We s h a l l p r o v e the s e c o n d a s s e r t i o n f o r • X (13) = q < 1. T h e n
Bc__~J(xt@(q%-OB); 8>0
O b v i o u s l y , the s e t { 0} i s a 1 - n e t f o r 13, s o • (13) _< 1. L e t
i s c h o s e n s o t h a t q + ~ w i l l be l e s s t h a n 1~ Usirig p r o p e r t i e s of
the f u n c t i o n ~ w e g e t
q=)~ (B)~
diamS. Thus, let E be infinite-dimensional. Then, obviously, n
~(13) _< 2. If one assumes
that ~(13) < 2, then B~[J
A~, where
diamA
i < 2 for all i, while Ai~ without loss of
I~I
generality, can be assumed to be closed. Constructing a section of ]3 by a finite-dimensional ficiently large dimension, we get a contradiction with the theorem on antipodes. The theorem 1.1.8. THEOREM. In a linear normed space, the Kuratowski are invariant with respect to taking the convex hull: , (co A ) = ,
and Hausdorff
measures
subspaee of sufis proved. of noncompactness
(A).
P r o o f . F o r the H a u s d o r f f m e a s u r e of n o n c o m p a c t n e s s t h i s a s s e r t i o n c a n be p r o v e d v e r y s i m p l y : ff S is a f i n i t e e - n e t of the s e t A , t h e n c o S i s a c o m p l e t e l y b o u n d e d e - n e t of the s e t c o A .
553
We s h a l l p r o v e i t f o r the K u r a t o w s M m e a s u r e of n o n c o m p a c t n e s s .
L e t A = ~ A~ and d i a m A k -< d f o r
a n y k. It is e a s y to s e e t h a t c o A is the union of a l l s u m s of the f o r m
2 k ~ co A~, k~l. m
w h e r e the v e c t o r X = (X~, X2, . . . ,
Xm) r u n s t h r o u g h the s t a n d a r d s i m p l e x ~, i . e . , ~ > ~ 0 , ~ k ~ = l .
L e t e > 0.
The union of aI1 s u c h s u m s c a n be a p p r o x i m a t e d w i t h a n y a c c u r a c y 5 (~) (in the s e n s e of the H a u s d o r f f m e t r i c ) [5 (e) a s e ~ 0] by f i n i t e u n i o n s of the s a m e t y p e , w h e r e X r u n s t h r o u g h a f i n i t e a - n e t ~ of the s i m p l e x ~. Then f r o m p r o p e r t i e s of the K u r a t o w s k i m e a s u r e of n o n e o m p a c t n e s s , w e get:
~(cod)=~ =maxc,
~kcoAk
)/
)
~ c o A ~ ~<~ U ~ c o A ~ +2~(~)= +8(~)4max
"~
(coAk)+6(e)
d+25(0,
s i n c e d i a m (coA k) = d i a m A k _< d. W h e n c e i t f o l l o w s , by v i r t u e of the a r b i t r a r i n e s s The o p p o s i t e i n e q u a l i t y is o b v i o u s . The t h e o r e m is p r o v e d .
of e, t h a t c~ (eoA) _< ~ (A).
The p r o p e r t i e s of the n o n c o m p a c t n e s s m e a s u r e s e x p r e s s e d by t h e l a s t t h e o r e m and t h e i r i n v a r i a n c e w i t h r e s p e c t to c l o s u r e p l a y an e s p e c i a l l y l a r g e r o l e in a p p l i c a t i o n s . 1.1.9. M o r e G e n e r a l D e f i n i t i o n of M e a s u r e of N o n c o m p a c t n e s s . By a m e a s u r e of n o n c o m p a c t n e s s in a l i n e a r n o r m e d s p a c e E is m e a n t a f u n c t i o n ,~, s a t i s f y i n g t h e c o n d i t i o n s : 1) the d o m a i n of d e f i n i t i o n D('~) c o n s i s t s of the s e t of a l l s u b s e t s of the s p a c e E , 2) r = ~(A) f o r any s e t A. The K u r a t o w s k i and H a u s d o r f f m e a s u r e s general definition.
of n o n c o m p a c t n e s s c o n s i d e r e d a b o v e , a s w a s p r o v e d , s a t i s f y the
O b v i o u s l y the f o l l o w i n g two f u n c t i o n s w i l l a l s o be m e a s u r e s of n o n c o m p a e t n e s s in the s e n s e of t h i s d e f i n i tion: ~1 ( A ) ~
101 if A is completely bounded, if A is not completely bounded, ~2 (A) = diam A.
We note t h a t the m e a s u r e of n o n e o m p a e t n e s s ,/J2 is not r e g u l a r .
1.2.
Condensing
Operators
1.2.1. Definition. Suppose in the Banach spaces E l and E 2 there are given measures of noncompactness and ~ with values in some partially ordered set (Q, _<). A continuous operator [:D(fi~EI-+E~ is called (~, r condensing, if from the inequality ~J[f(A)] .>_ ~(A) it follows that A is compact (A~D(f)). We one has
shall call f @, ~)-condensing
in the proper sense if for any set A_~D(f),
whose
closure is not compact,
I f (A)I < w (A). (In a p a r t i a l l y o r d e r e d s e t a < b m e a n s t h a t a _< b and a ~ b.) If the s e t Q is l i n e a r l y o r d e r e d , concepts obviously coincide. S u p p o s e in Q t h e r e is d e f i n e d m u l t i p l i c a t i o n by n o n n e g a t i v e s c a l a r s . c a l l e d (q, ~0, g~)-bounded, if f o r any s e t A~D(f),
then t h e s e two
The c o n t i n u o u s o p e r a t o r f w i l l be
, if (A)]-.< q~ (A). I r E 1 = E 2 and ~ = ~, then we s h a l l s a y s i m p l y " f - c o n d e n s i n g " and "(q, ~b)-bounded." I f q < 1, t h e n a (q, ~)b o u n d e d o p e r a t o r in t h i s c a s e is s o m e t i m e s c a l l e d ~ - e o n d e n s i n g w i t h c o n s t a n t q. The d e f i n i t i o n s of (q0, ~ ) - c o n d e n s i n g and {q, ~o, $ ) - b o u n d e d o p e r a t o r s e x t e n d n a t u r a l l y to f a m i l i e s of o p e r a t o r s f = { f x : X e A}; in t h i s c a s e , by f(A) is m e a n t U f x ( A ) :
554
Often, a f a m i l y of o p e r a t o r s f = { f}, : 2~ ~ A} is c o n s i d e r e d as an o p e r a t o r with two v a r i a b l e s f : A • E 1 ~ E z. And then, i n s t e a d of the w o r d s "{~a, $ ) - c o n d e n s i n g f a m i l y " o r "(q, ~, 0)-bounded f a m i l y " of o p e r a t o r s one u s e s the w o r d s " o p e r a t o r s w h i c h a r e "(9~, ~ ) - c o n d e n s i n g in the c o l l e c t i o n of v a r i a b l e s ~ o r ~ (q, 9), r in the c o l l e c t i o n of v a r i a b l e s . " 1.2.2. S i m p l e s t E x a m p l e s . A c o m p l e t e l y continuous o p e r a t o r , defined on a bounded s u b s e t of a B a n a c h s p a c e , is o b v i o u s l y tJl-condensing, w h e r e '1~1is a n o n c o m p a c t n e s s m e a s u r e f r o m 1.1.9; in e x a c t l y the s a m e way a c o n t r a c t i n g o p e r a t o r on a bounded s e t is r H o w e v e r , the m e a s u r e s of n o n c o m p a c t n e s s ~1 and ~z a r e not v e r y c o n v e n i e n t , s i n c e they do not have s u f f i c i e n t l y good p r o p e r t i e s . F o r e x a m p l e , as noted a b o v e , g~z is not r e g u l a r . It t u r n s out to be m o r e m e a n i n g f u l that the o p e r a t o r s of both indicated types a r e c o n d e n s i n g with r e s p e c t to the K u r a t o w s k i n o n e o m p a c t n e s s m e a s u r e . A c o m p l e t e l y continuous o p e r a t o r on a bounded s e t is a l s o X - c o n d e n s i n g . One should note, h o w e v e r , that a c o n t r a c t i o n of a bounded s e t m a y not be a X - c o n d e n s i n g o p e r a t o r (ef. [120]). We give an e x a m p l e of a c o n d e n s i n g f a m i l y of o p e r a t o r s . Let the o p e r a t o r s f0, ~:M~E~-->E2 be (q2, ~)c o n d e n s i n g in the p r o p e r s e n s e , w h e r e the d o m a i n of v a l u e s of the n o n c o m p a e t n e s s m e a s u r e s ~ and J~ a r e line a r l y o r d e r e d and @ is s e m i a d d i t i v e . Then the f a m i l y of o p e r a t o r s f = { f t : t E [0, 1]}, w h e r e ft(x) = (1 - t)f0(x) + ill(x), is (~, JJ)-condensing in the p r o p e r s e n s e . In fact, f o r any s e t A~_M, w h o s e c l o s u r e is not c o m p a c t , we have r
(A)I ~ ~ [ ~ (f0 (A) U f~ (A))I ~ r If0 (A) U f~ (A)] = max {~ If0 (A)I, ~ If~ (A)I} < ~ (A).
(We use the fact that if a n o n c o m p a c t n e s s m e a s u r e is s e m i a d d i t i v e then it is monotone.) The following t h e o r e m g i v e s , p r o b a b l y , the m o s t useful t e s t f o r a c o n d e n s i n g o p e r a t o r ; we f o r m u l a t e and p r o v e it f o r a f a m i l y of o p e r a t o r s . 1.2.3. T H E O R E M . Let the o p e r a t o r s of the f a m i l y f = {fk : ~ ~ A} be continuous and a d m i t a diagonal r e p r e s e n t a t i o n f~(x) : ,~(~, x, x) in t e r m s of an o p e r a t o r @ :A x M x E~ ~ E 2 ( E l , E 2 a r e B a n a c h s p a c e s , I?/GE~, A is an a r b i t r a r y set). F o r any y E E l , let the s e t r x M x {y}) be c o m p l e t e l y bounded and f o r any ~ ~ A, x ~ M the o p e r a t o r @()~, x, 9 ) s a t i s f y a L i p s c h i t z condition with c o n s t a n t q. Then the f a m i l y f(q, X) is bounded. P r o o f . One m u s t e s t a b l i s h that for any s e t AGN one has X [ f (A)I~
(i)
w h e r e the l e t t e r X on the left and the r i g h t d e n o t e s the H a u s d o r f f n o n e o m p a e t n e s s m e a s u r e in the s p a c e s E 1 and E 2 , r e s p e c t i v e l y . If the s e t A is not bounded, then (1) is obvious, s i n c e x(A) = + ~ . L e t A be bounded and S be a finite (x(A) + e ) - n e t of A in E~. We c o n s i d e r the s e t S 1 = ~(A x A x S). It is c o m p l e t e l y bounded, s i n c e it is the union of a finite c o l l e c t i o n of c o m p l e t e l y bounded s e t s r x A x {y}). We s h a l l show that SI is a q . (X (A) + e ) - n e t f o r f ( A ) . L e t z e f ( A ) , i.e., z = @ ( X , x , x ) , w h e r e X e A , x e A , a n d l e t y e S be s u c h t h a t I I x - y l l _ < x ( A ) + e. Then the e l e m e n t z~ = @(X, x, y) belongs to S 1 and
Hzl-zH--]lo(~, x, y)-r Thus, proved.
x, x)it-.
x[f(A)] <_ q(x(A) + ~), whence, by virtue of the arbitrariness
of e, (i) follows.
The theorem
is
1.2.4. Consequences. (a) If under the hypotheses of the theorem the set M is bounded and q < 1, then the family f is X-condensing. (b) The sum of a completely continuous and a contracting operator g, h : E l ~ E 2 on any bounded set M~E~ is a X-condensing operator. To conclude
this section we give a useful theorem
1.2.5. THEOREM. operator. Proof.
Let
The Fr~chet
f:D (f)GE~-+Ee,
derivative
on the derivative
of a (q, x)-bounded
at a point of a (q, x)-bounded
operator
operator.
is a (q, )/)-bounded
x be an interior point of D(f), A = f' (x). For sufficiently small h e El,
Ah~- f (x + h ) - - f (X)--o~(h), w h e r e w (h)/1[ hi[ ~ 0 as h ~ O. Hence for any bounded s e t A4~E~ and s u f f i c i e n t l y s m a l l e > O,
U s i n g p r o p e r t i e s of the n o n c o m p a c t n e s s m e a s u r e X and the (q, x ) - b o u n d e d n e s s of f, we get Z [A (N)I ~ qT, (M) ~- ~ [o) (eN)/e]. 555
Passing
to the limit as ~ --* 0, obviously gives
X [A (?d)] ~ qz (IV/). The t h e o r e m is p r o v e d . 1.3.
Index
of
Fixed
Points
of
a Condensing
Operator
1.3.1. Formulation of the Problem. In the theory of fixed points, important roles are played by the mutually close concepts of degree of a mapping, the rotation of a vector field, index of fixed points of an operator. Here we shall use the terminology connected with the last of these concepts, while in cases where this cannot cause misunderstanding, we shall simply nindex of an operator," instead of "index of fixed points of an operator."
Let E be a Banach space, R be a convex closed subset of E, V be a subset of R, open in the indueed topology of the space R. W e shall denote by V and ~z the closure and the boundary of this set in R. Let the o p e r a t o r f : V ~ R be X - c o n d e n s i n g and the s e t FIX (fl V) of fixed points of its r e s t r i c t i o n to V be c o m p a c t . A s u f f i c i e n t (but not n e c e s s a r y ) condition f o r this is the a b s e n c e of fixed points for the o p e r a t o r f on the b o u n d a r y V of the s e t V. In the r e s t of this s e c t i o n , i n s t e a d of " • we s h a l l w r i t e s i m p l y " c o n d e n s i n g . " F o r s u c h an o p e r a t o r we define an i n t e g e r ind R (f, V), c a l l e d the index of fixed points (for s h o r t , index) of the o p e r a t o r f on V with r e s p e c t to R. Since R in what follows will not c h a n g e , one can use the s i m p l e r notation ind (f, V). The index of a c o n d e n s i n g o p e r a t o r will have the following p r o p e r t i e s . 1 ~ The indices of h o m o t o p i c c o n d e n s i n g o p e r a t o r s coincide. Condensing o p e r a t o r s f0, fl : V ~ R a r e said to be h o m o t o p i c , if t h e r e exists a c o n d e n s i n g f a m i l y f = {fk : ;~ e [0, 1]}, w h e r e the map (X, x) ~ fk(x) is continuous and the s e t F i x ( f I v ) ~ U Fix(f~lv ~ is c o m p a c t . xEIo,l] 2 ~ Let Vi (i = 1, 2 , . . .
) be p a i r w i s e d i s j o i n t s u b s e t s of V w h i c h a r e open in R and the o p e r a t o r f have no
fixed points on V \ ~ V~ . Then the i n d i c e s ind if, Vi) a r e defined, only a finite s e t of t h e m a r e d i f f e r ent f r o m z e r o , and
ind(f, V)~
i n d ( f , V~). i~l
3 ~. If f(x) - x 0 E V, then ind (f, V) = 1. 4 ~ If f(x) ~ x 0 ~- V, then ind (f, V) = 0. 5 ~ If ind (f, V) ~ 0, then f has at l e a s t one fixed point. U n d e r the a s s u m p t i o n that the quantity d e s c r i b e d , ind R (f, V), e x i s t s , we can define an analogous c h a r a c t e r i s t i c u n d e r s o m e w h a t m o r e g e n e r a l conditions: it s u f f i c e s to a s s u m e that f is defined and X-condensing not on ~ , but only on V. In fact, one can, by definition, s e t ind R (f, V) = indR (f, W), w h e r e the r e l a t i v e l y open s e t W is s u c h that F i x ( / ) ~ W , ~ / c V . The i n d e p e n d e n c e f r o m W follows f r o m 2 ~ p r o p e r t i e s 1~ ~ a r e o b v i o u s l y p r e s e r v e d , w h e r e in 1 ~ it s u f f i c e s to a s s u m e that f0, fl, and f a r e defined on V. The definition of the index of a c o n d e n s i n g o p e r a t o r is b a s e d on the following c l a s s i c a l r e s u l t f r o m the t h e o r y of c o m p l e t e l y continuous o p e r a t o r s . 1.3.2. T H E O R E M . If in the p r e c e d i n g p a r a g r a p h we a s s u m e all the o p e r a t o r s a r e c o m p l e t e l y continuous and acting from an open subset U of a Banach space E into E, then a characteristic satisfying conditions 1~ ~ exists, is unique, and has properties 4 ~, 5 ~. This theorem, based on the Brouwer theory of various versions was proved by Leray and Schauder, proof can, for example, be obtained, using Theorems sel'skii and Zabreiko [74]: it suffices to set ind (f, U) tor field I-f on the boundary of a bounded domain W, An important role in the following constructions sel'skii, Zabreiko, and Strygin [50]. 1.3.3. Definition of Fundamental satisfies three requirements:
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the degree of mappings in a finite-dimensional space, in M. A. Krasnosel'skii, Rothe, Nagumo, Browder. Its 20.1, 20.2, 20.3, and 20.4 of the monograph of Krasno= T(I - f, W), where T(I -- f, W) is the rotation of the vecsatisfying the conditions Fix (f)~W, WcU. is played by the following concept,
Set. The set S is called a fundamental
introduced by Krasno-
set of the operator f:M
--E,
if it
) Fig. 1 1) S is nonempty, convex, compact;
2) f (M NS)ZS; 3) if
XoEM\S, then xoEco [{f (Xo)}US].
The last requirement says that upon applying the operator f, points of the set M "are not repulsed;' from S: if X o ~ M \ S , then f(x 0) does not lie in the shaded cone (el. Fig. 1). We note at once that a fundamental set S contains all fixed points of the operator. The concept of fundamental set will also be used in the obvious sense in relation to families of operators. 1.3.4. THEOREM. Let f~{f~:kEA} be a condensing family of operators (:M~E-~E) and K be a compact subset of E. Let A be a compact topological space and the map (h, x) ~ fh(x) be continuous. Then flM has a fundamental set containing K. Proof. We denote by ~ the collection of closed sets containing K and satisfying all the requirements fro from the definition of a fundamental set of the family f]M, except possibly for compactness. ~ is nonempty, since, e.g., the set T0=co [K Uf(M)] satisfies the conditions cited (without loss of generality one can assume that K~ 0, so that TO;~ Calso). We note, further, that if TE~l, then the set T,-=co[t(Uf(MNT)] also belongs to ~. In the verification one needs only r e q u i r e m e n t s 2) and 3) f r o m the definition of a fundamental set. T,c_T, since t
f (M NT 3 Z f (M N r ) Z c o [K U f (M rl r)l = r~. We verify 3). If x 0 e T, then the a s s e r t i o n follows f r o m the c o r r e s p o n d i n g property of T and the inclusion Now if x 0 E T, then f(x 0) ~ T~ and the a s s e r t i o n is obvious. Now we define the set S = belongs to ~'l and coincides with S. ease when MNS f r o m the equation
T~c_T.
N T and we shall show that it is fundamental for flM. It is easy to see that S r~-~
is in ~ a minimal set. Hence, by virtue of what was said above, the set S ~ = c b [K U f ( M N S ) ] But then ~ (M N S)~< X(S) = ~ [ f (M ~1S)], which for a condensing family f is possible only in the is compact. Since A is also compact and the map (X, x) ~ fx(x) is continuous, f r o m this and S = ~ [K U f (M N S)I follows the c o m p a c t n e s s of S. The t h e o r e m is proved.
In determining the index of a condensing o p e r a t o r we aIso use the following theorem of Dugundji [28, 188] on extensions of a continuous o p e r a t o r . 1.3.5. THEOREM. A continuous o p e r a t o r , defined on a nonempty closed subset of a metric space, and a s s u m i n g values in a locally convex space, can be extended continuously to the whole space while p r e s e r v i n g the convex hull of the set of values. 1.3.6. Definition of the Index of a Condensing Operator. We consider the situation described in 1o3.1. Let S be a fundamental set of the operator f l y , which exists by T h e o r e m 1.3.4, U be an open set in E such that U N R = V . Using Dugundji's t h e o r e m , we extend the o p e r a t o r f f r o m the closed subset UNS of the space U to all of U, with values in S. If UNS is empty, then the existence of such an extension is obvious. The o p e r a t o r f obtained acts f r o m U to S, and consequently, is completely continuous, while Fix (f) = Fix ( f I v)_c4.S- We set tnd (f, V) = Ind (f, U); the right side is defined here by virtue of T h e o r e m 1.3.2; on the left we use the notation ind (f, V) instead of the m o r e complete ind R (f, V).
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Our i m m e d i a t e p r o b l e m is to show that the index is independent of the c h o i c e s of U, S, and f. 1.3.7. W e l l - D e f i n e d n e s s of the Index. Let U1, $I, {1 and U2, $2, {2 s a t i s f y all the r e q u i r e m e n t s f o r m u l a t e d in the definition of the index. We shall show that ind (71, U , ) = i n d (f~, U2).
(1)
Without loss of g e n e r a l i t y one can a s s u m e that S~cS2, s i n c e one can c o n s t r u c t a fundamental set $3, c o n taining the c o m p a c t u m K=S1 uS2, and c o r r e s p o n d i n g U3, f3, and e s t a b l i s h that both s i d e s of (1) a r e equal to ind (fa, U3). F u r t h e r , we s e t U = U I NU2. Obviously, ind (fl, U1) = ind (fl, U) and ind (f2, U2) = ind (f2, U), as follows f r o m p r o p e r t y 2 ~ of the index. Hence to prove (1) it s u f f i c e s to e s t a b l i s h the equation ind (fl, U ) = i n d (f2, U).
(2)
We c o n s i d e r on U the l i n e a r h o m o t o p y ft(x) = (1 - t)fl(x) + tf2(x) , effecting a t r a n s i t i o n f r o m fl to f2. The values of the o p e r a t o r s ft lie in S2, so that the f a m i l y F = { f t : t E [0, 1]} is c o m p l e t e l y continuous. We shall show that Fix (F) = Fix (flv); w h e n c e will follow the c o m p a c t n e s s of this set. I f x 0 E Fix (F), i.e., x 0 = (1 to)f-1(x o) + tof2 (Xo), then x 0 E S 2. Then f2 (Xo) = f(x0) and x0~co [{f (x0)} U $11, w h e n c e it follows that x 0 E $1, so that fl(xo) = f(xo). F i n a l l y , we get t h a t x 0 = f(xo), i . e . , x 0 ~ Fix (flv). Thus, Fix (F) ZFix (f Iy); the opposite inclusion is obvious. That the index is w e l l - d e f i n e d is proved. 1.3.8. R e m a r k . As a l r e a d y noted, along with the t e r m i n o l o g y "index of fixed points of an o p e r a t o r " one often u s e s t e r m i n o l o g y c o n n e c t e d with the c o n c e p t of r o t a t i o n of a v e c t o r field. If f is a c o n d e n s i n g o p e r a t o r , then the v e c t o r field I - f is called a c o n d e n s i n g v e c t o r field (I is the identity o p e r a t o r ) . By the r o t a t i o n of the v e c t o r field I - f on the s e t V r e l a t i v e to R in the c a s e when the o p e r a t o r f has no fixed points on V, is m e a n t the quantity 7 R ( I - f , V), defined by the equation Y~ (I - - f , V) = ind~ (f, V). 1.3.9. T h e o r e m on P r o p e r t i e s of the Index. The index of a c o n d e n s i n g o p e r a t o r , defined above, has p r o p e r t i e s 1~ ~ of P a r a g r a p h 1.3.1. All t h e s e p r o p e r t i e s follow e a s i l y f r o m the c o r r e s p o n d i n g p r o p e r t i e s of the index of a c o m p l e t e l y c o n tinuous o p e r a t o r . As an e x a m p l e we s h a l l prove the f i r s t of them. L e t the o p e r a t o r s f0 and fl be h o m o t o p i c and the h o m o t o p y be effected by a c o n d e n s i n g f a m i l y f = {fk : E [0, 1]}. By T h e o r e m 1.3.4 the f a m i l y fl V has g e n e r a l f u n d a m e n t a l s e t S. In addition, by v i r t u e of the d e f i n i tion of h o m o t o p y , the s e t Fix (fl V) is c o m p a c t . L e t U be an open s u b s e t of E s u c h that U f l R = V . We extend the o p e r a t o r s f~ of the f a m i l y f f r o m the s e t UAS to all of U, p r e s e r v i n g the convex hull of the d o m a i n of v a l ues of the family and the c o n t i n u i ~ of the m a p p i n g (;~, x) ~ f~(x). Then we g e t a c o m p l e t e l y continuous h o m o t o p y = { f~ : ~ E [0, 1]}, for w h i c h Fix (f) = Fix (fl V) is c o m p a c t . Hence ind (re, V ) = ind (f0, U ) = ind (fl, U ) = i n d (f~, V). 1.3.10. Fixed Point T h e o r e m . The t h e o r y of the index of c o n d e n s i n g o p e r a t o r s allows one to prove v a r i ous t e s t s f o r the e x i s t e n c e of fixed points. As an e x a m p l e we shall prove h e r e the g e n e r a l i z e d S c h a u d e r p r i n ciple. THEOREM. L e t T be a n o n e m p t y convex c l o s e d s u b s e t of the B a n a c h s p a c e E. Let the c o n d e n s i n g o p e r a t o r f c a r r y T into itself. Then ind T (f, T) = 1, i.e., f has in T at l e a s t one fixed point. P r o o f . We set R -- V = T. We fix an a r b i t r a r y point x 0 6 T and we c o n s i d e r the c o n d e n s i n g h o m o t o p y
F~ (x) = (1 -- t) f (x) + txo, effeeting a t r a n s i t i o n f r o m the o p e r a t o r f to the o p e r a t o r Fl~x) -- x 0. We shall show that the s e t of fixed points P = Fix (F) of the f a m i l y F = { F t : t E [0, 1]} is c o m p a c t . F i r s t l y , it is obviously c l o s e d . Secondly, P c c o i f ( P ) U {x0}], so ~ (P) _< • w h e n c e , by v i r t u e of the definition of c o n d e n s i n g o p e r a t o r , we get that P = ~ is c o m pact. T h u s , the quantity ind (Ft, T) is defined and independent of t. In p a r t i c u l a r , ind (f, T) = ind (F0, T) ~ ind (F1, T) = 1. Now using p r o p e r t y 5 ~ of the index, we get that f has a fixed point in T. The t h e o r e m is p r o v e d . 1.3.11. T h e o r e m on L o c a l C o n s t a n c y of the Index. L e t f = {fk : ~ E [0, 1]} be a c o n d e n s i n g f a m i l y of o p e r a t o r s , a c t i n g f r o m the c l o s u r e at of the r e l a t i v e l y open s u b s e t 12 of the convex c l o s e d s e t R , lying in the B a n a e h s p a c e E , in E, and let the m a p (~, x) ~-* fh(x) be continuous. Let us a s s u m e that the o p e r a t o r f0 has no fixed
558
p o i n t s on ~ . Then one c a n find a n u m b e r >'0 E (0, 1], s u c h t h a t f o r a I l ~ , s a t i s f y i n g the c o n d i t i o n 0 -< k _< k0, the i n d i c e s of the o p e r a t o r s fk on ~a r e l a t i v e to R a r e d e f i n e d and a s s u m e the s a m e v a l u e . P r o o f . It s u f f i c e s to e s t a b l i s h that t h e r e e x i s t s a k 0 > 0 s u c h t h a t f o r k E [0, k0] the o p e r a t o r s fa h a v e no f i x e d p o i n t s on a~. W h e n c e w i l l f o l l o w the c o m p a c t n e s s of the s e t of f i x e d p o i n t s of the f a m i l y F----{f~I~:XE[0, ~0]} a n d , c o n s e q u e n t l y , the f a c t t h a t any o p e r a t o r f r o m F is h o m o t o p i c w i t h the o p e r a t o r f0i~. L e t us a s s u m e the c o n t r a r y . Xn ~ i2 s u c h that
Then one c a n find a s e q u e n c e of n u m b e r s )~-+0 and a s e q u e n c e of p o i n t s ~-~
x~=f~(x~).
(3)
S i n c e f i s a c o n d e n s i n g f a m i l y of o p e r a t o r s , the s e q u e n c e {x n} is r e l a t i v e l y c o m p a c t . W i t h o u t l o s s of g e n e r a l i t y one c a n a s s u m e t h a t x~-+xo[{2. P a s s i n g in (3) to the l i m i t a s n - - ~ , and u s i n g the c o n t i n u i t y of the m a p (~, x) fh(x), we g e t
xo=fo(Xo), which contradicts 1.4.
Linear
the hypothesis Condensing
of the theorem.
The theorem
is proved.
Operators
In this section we study questions connected with the concept of Fredholm operator. It turns out that in studying the question of whether a bounded linear operator is Fredholm, measures of noneompactness play approximately the same role as the norm in considering the question of invertibility of an operator. We recall that a bounded linear operator A, acting from a Banach space E I into a Banach space E2, is called Fredholm, if: i) its kernel Ker (A) is finite-dimensional; 2) its image Im (A) has in E 2 finite codimension: dim (E 2/Ira (A)) < ~o. The image of a Fredholm operator is closed [96]. We
shall prove a test for Fredholmness
and study the Fredholm
cr.(A)={~:~EC, s
spectrum
is not Fredholm },
in p a r t i c u l a r , w e g e t a f o r m u l a f o r the " F r e d h o l m r a d i u s " : r| (A) = sup {I ~ I:kEa| (k)}. In c o n s i d e r i n g s p e c t r a the s p a c e is a s s u m e d to be c o m p l e x . 1.4.1. The O p e r a t i o n "+". L e t E be a B a n a c h s p a c e , BE be the l i n e a r s p a c e of a l l bounded s e q u e n c e s in E , KE be the s u b s p a c e of B E , c o n s i s t i n g of r e l a t i v e l y c o m p a c t s e q u e n c e s . We s e t E + = BE / K E . F u r t h e r , if A E L(E~, E2) , t h e n , by d e f i n i t i o n , A + is the l i n e a r o p e r a t o r a c t i n g f r o m E ~ to E l a c c o r d h l g to the r u l e A+X = A x + KE2, w h e r e x E X , x = (xl, . . . . x n, . . . ), A x = (Ax~ . . . . , Ax n .... ). That t h i s is w e l l - d e f i n e d f o l l o w s f r o m the f a c t t h a t A a c t s f r o m BE 1 to BE 2 and f r o m KE 1 to KE 2. ator s
1.4.2. T h e o r e m ( T e s t f o r F r e d h o l m n e s s ) . The o p e r a t o r A E L(E~, E 2) is F r e d h o l m if and only if the o p e r b i j e e t i v e a s a m a p f r o m E~ to E l , i . e . , K e r (A +) = { 0} and I m (A+) = E~§
P r o o f . a) L e t A be F r e d h o l m . We c o n s i d e r an e l e m e n t X e K e r (A+). L e t x E X. Then x t BE~, and y = A x E KE 2. S i n c e K e r (A) is f i n i t e - d i m e n s i o n a l , i t has a d i r e c t c o m p l e m e n t : E 1 = K e r (A) + E i. Then x = x " + x ' , w h e r e x " E B K e r (A) = K K e r (A), and x ' ~ BE~. We d e n o t e by A ' the r e s t r i c t i o n of the o p e r a t o r A to E~. Then x ' = (A')-~y E KE~, s i n c e A ' is a o n e - t o - o n e m a p p i n g of the B a n a c h s p a c e E 1 onto the B a n a c h s p a c e I m (A), and c o n s e q u e n t l y , the o p e r a t o r (A') -1 is bounded. T h u s , x = x " + x ' E K E j , i . e . , X = 0. Thus we have p r o v e d t h a t Ker(A +) ={0}. We shall show that Im (A+) = E~. Let Y E E2, + Y E Y. Since eodimlm(A) is finite and Im (A) is closed, there exists a finite-dimensionalsubspace E2 such that E2 = E2 + I m (A). Then y = y' + y", where y~ E BE~ = KE2, and y"E BIm(A). We s e t x = (A')-ly'', X = x + KEI, ThenA+X = y " + KE2 = y + KE2 = y . Thus, Im(A +) =E~. b) Now let is be known that Ker (A+) = { 0} and Im (A+) = E~. It is required to prove that the operator A i s is Fredholm. If Ker (A) is infinite-dimensional,then in it there exists a bounded but not completely bounded sequence x. Then X = x + KE1 is a nonzero element of Ker (A+), which contradicts the hypothesis. Before proving that the codimension of the linear manifold Im (A) in E2 is finite, we shall show that it s is closed. Let Yn = Axn converge to Yo as n ~ ~ and x n { El. The sequence {Xn} is bounded. In fact, if this is
559
n o t s o , t h e n w e c o n s t r u c t the n o r m a l i z e d s e q u e n c e z n = x n / l l x n l l , f o r w h i c h A z , , ~ O, i . e . , A+Z = A z
+ KE 2 =
KE 2, s o Z = K E I , and c o n s e q u e n t l y , z = {z n} i s c o m p l e t e l y bounded. P a s s i n g to a c o n v e r g e n t s e q u e n c e , we g e t f o r the e l e m e n t zo=limz,k,, on t h e one h a n d , zofiKer(A)flE;, i . e . , z 0 = 0, and on t h e o t h e r , tlz011 = 1. The c o n t r a d i c t i o n o b t a i n e d p r o v e s that A E 1, Now l e t us a s s u m e t h a t the (Banach) s p a c e E 2 / A E 1 is i n f i n i t e - d i m e n s i o n a l , and l e t us c h o o s e in it a b o u n d e d , but not c o m p l e t e l y bounded s e q u e n c e {Yn + AE1}. The e l e m e n t s Yn c a n be a s s u m e d to be c h o s e n s o t h a t the s e q u e n c e y is a l s o bounded and n o t c o m p l e t e l y bounded. U s i n g the s u r j e c t i v i t y of the o p e r a t o r A +, we find x ~ BE~ s u c h t h a t y - A x = z 6 KE 2. Then Yn + AE~ = Zn + AE1, w h e n c e it f o l l o w s t h a t the s e q u e n c e Yn + AE1 is c o m p l e t e l y b o u n d e d . The t h e o r e m is p r o v e d . 1.4.3. C O R O L L A R Y . If E is a c o m p l e x B a n a c h s p a c e and A is a bounded l i n e a r o p e r a t o r in E , t h e n
~r, (A) = ~ (A+); p o i n t s of the s p e c t r u m of the o p e r a t o r A , l y i n g o u t s i d e a c l o s e d c i r c l e of r a d i u s rff(A), can be only i s o l a t e d c i g e n v a l u e s of f i n i t e m u l t i p l i c i t y . P r o o f . The f i r s t a s s e r t i o n f o l l o w s e a s i l y o b t a i n e d f r o m the f o l l o w i n g known i n d e x of t h e o p e r a t o r (not to be c o n f u s e d if a t a t l e a s t one point X E G the o p e r a t o r s e t [45].
f r o m the t h e o r e m p r o v e d in an o b v i o u s w a y . The s e c o n d a s s e r t i o n is f a c t s : in a n y c o n n e c t e d c o m p o n e n t G of the o p e n s e t C \ ~ r ( A ) , the w i t h t h e index of f i x e d p o i n t s of the o p e r a t o r ! ) X I - A is c o n s t a n t , and XI - A is i n v e r t i b l e , then the i n t e r s e c t i o n of G w i t h r is a d i s c r e t e
1.4.4. D e f i n i t i o n . By a n o r m a l m e a s u r e of n o n c o m p a c t n e s s in a B a n a c h s p a c e E is m e a n t a n y s e m i n o r m q~ in B E , s a t i s f y i n g the c o n d i t i o n : ,# (x) = 0 i f and only if x ~ KE. It is o b v i o u s t h a t any n o r m a l m e a s u r e of n o n c o m p a c t n e s s ~, g e n e r a t e s a n o r m q,+ in E + a c c o r d i n g to the formula
~F+(X) = ~ (x), wt~erex~ X. By the ~ - n o r m o f the l i n e a r o p e r a t o r A : E ~ E is m e a n t the q u a n t i t y ( A ) = s u p { ~ (Ax):x6BE, vF ( x ) = 1}. It is e a s y to s e e that (A) = iF+ (A+), w h e r e on the r i g h t is the n o r m of the l i n e a r A + in the s p a c e (E +, q~+). 1.4.5. E x a m p l e s .
a) In BE w e i n t r o d u c e the n a t u r a l n o r m :
[[x][-----sup[]x~[} and f o r any x ~ BE we s e t
9 , ( x ) = M It x + y []. y~KE It is e a s y to s e e t h a t the function s o d e f i n e d in a n o r m a l m e a s u r e of n o n c o m p a c t n e s s , w h i c h is c l o s e l y c o n n e c t e d w i t h the H a u s d o r f f n o n c o m p a c t n e s s m e a s u r e :
r ( x ) = z ({x~:ni, a natural .umber}). b) The K u r a t o w s k i n o n c o m p a e t n e s s m e a s u r e a l s o g e n e r a t e s a n o r m a l n o n c o m p a c t n e s s m e a s u r e : r
= a ({Xn:tZ is a natural lamber }).
C) The n o r m ~ [ , g e n e r a t e d in E + by the n o r m a l n o n c o m p a c t n e s s m e a s u r e # l , o b v i o u s l y c o i n c i d e s w i t h the q u o t i e n t s p a c e BE / K E , and the n o r m 'I'+ is e q u i v a l e n t w i t h it:
(el. P a r a g r a p h 1.1.4). d) The r
of the l i n e a r o p e r a t o r A a d m i t s , a s is e a s i l y v e r i f i e d , the r e p r e s e n t a t i o n 9 ~( A ) = sup { ~ (Ax):ll x I i < 1}.
1.4.6. R e m a r k s on the F r e d h o l m R a d i u s . L e t A be a bounded l i n e a r o p e r a t o r in E , ko be a n o r m a l n o n c o m p a c t n e s s m e a s u r e s u c h t h a t ~(A) < ~ and the s p a c e (E +, ~,+) is c o m p l e t e ; [r be the s e t of a l l n o r m a l
560
noncompactness Then
measures
equivalent with ~I, (i.e., the norms
generated
by them
in E + are equivalent with ,I~+).
r . (A) ~< ~I~(A), rr ( A ) = Iirn
~/-~-g--(An)
n---~oo
and r , (A) = in[ {0 (A):OE [~]}. This follows f r o m the known r e l a t i o n s b e t w e e n the s p e c t r a l r a d i u s and n o r m s of a l i n e a r o p e r a t o . r , s i n c e r~(A) = r(A +) and ~(A n) = ,]~+((A+)n). 1.4.7. S p e c t r a l R a d i u s T h e o r e m . S u p p o s e g i v e n i n the B a n a c h s p a c e E a n o r m a l n o n c o m p a c t n e s s m e a s u r e ~I,. Let the s e q u e n c e of o p e r a t o r s A n E L(E) c o n v e r g e s t r o n g l y to a n o p e r a t o r B and s a t i s f y the c o n d i t i o n ~' (g) ~< q ~ (x) for a n y x E BE and a n y s e q u e n c e y , e a c h e l e m e n t of w h i c h b e l o n g s to a n y of the s e q u e n c e s Anx. T h e n tt--mr (An) -.< max {q, r (B)}. P r o o f . Let us a s s u m e the c o n t r a r y : o > q, o > r ( B ) , w h e r e p = l i m r ( A ~ ) .
T h e s e i n e q u a l i t i e s s h o w , by
v i r t u e of P a r a g r a p h s 1.4.3 and 1.4.6, that for s o m e s u b s e q u e n e e of o p e r a t o r s A n k t h e r e a r e e i g e n v a l u e s Xnk, s a t i s f y i n g the c o n d i t i o n ] ~ k I >1 P - - 8 > max {q, r (B)}. By the B a n a e h - S t e i n h a u s p r i n c i p l e the s e q u e n c e of o p e r a t o r s is b o u n d e d , so the s e q u e n c e { Xnk } is b o u n d e d a l s o . Without l o s s of g e n e r a l i t y one c a n a s s u m e that X n k - - X0 as k ~ ~o I X01 > q. F o r the c o r r e s p o n d i n g n o r m a l i z e d e i g e n v e c t o r s x k one w i l l have
w h e r e Zk ~ 0 as k ~ ~ . Hence for the s e q u e n c e x of e i g e n v e c t o r s we get: I
(x) -.< igi q~; (x) + ~" (z) = ~ t
q~" (x),
i . e . , ~(x) = 0. C o n s e q u e n t l y , x i s a r e l a t i v e l y c o m p a c t s e q u e n c e , and we c a n a s s u m e that X k ~ X 0 as k ~ ~ . tt is c l e a r that x 0 is a n e i g e n v e c t o r of the o p e r a t o r /3 w i t h e i g e n v a l u e X0" W h e n c e , we get the c o n t r a d i c t o r y i n equality
r (B) > I~ol> , o - 6 > r (B).
The theorem 1.5.
is proved.
Applications
1.5.1. Formulation of the Problem. In this section we demonstrate the application of the technique developed to the study of differential equations with the example of two theorems on periodic solutions of equations with deflected argument of neutral type. We consider the equation
x' ( t ) = g (~, t, x (t - ~ ) , x' (t - @ .
(1)
R e l a t i v e to g we a s s u m e the following:
g : R 4-+ R;
(2)
g is continuous simultaneously in the first two arguments and T is periodic (T > 0) in the second argument;
(3)
g(8, t, xl, y3--g(e, t, x~, y~)14telxt-xzl-t-qlq~-g~.t (q < 1). Let us assume
(4)
that for e = 0, (1) has an exponentially stable T-periodic solution w.
(5)
561
We shall be interested in the question of the existence, uniqueness, and stability of a T-periodic solution of (I) for small positive e. 1.5.2. Translation Operator. In studying periodic solutions it is convenient to use the translation operator along trajectories of the differential equation, which can be defined by solving a Cauchy problem. It is easy to see, however, that the Cauchy problem for (1), corresponding to the initial condition
x(t)=rp(t)(--h~
(6)
= g(a, 0, ~o(-a), ~ ' ( - e ) ) . Hence, we shall
x ' (t) ~ g (e, t, x (t -- e), x ' (t -- e)) + Ix' (0) -- g (~, 0, x ( - - e), x ' ( - - e))l ~ (t),
(7)
w h e r e v is a p i e c e w i s e - l i n e a r function, equal to 1 f o r t = 0 and to 0 for t _> V > 0 (we shall always a s s u m e that the p a r a m e t e r s V and h s a t i s f y the inequalities V < h and V + h _< T). F o r s u c h an equation, the "gluing c o n d i tion n is s a t i s f i e d f o r any initial function 9~ ~ C~[-h, 0]. The t r a n s l a t i o n o p e r a t o r V~ : C~[-h, 0] ~ C l [ - h , 0[ along t r a j e c t o r i e s of (7) (e _< h) f o r time f r o m 0 to T is defined by the equation Ve (4)(s) = x ( T + s ) ( - h _< s < 0), w h e r e x is a solution of the p r o b l e m (7), (6). F r o m known t h e o r e m s on the Cauchy p r o b l e m for an equation of n e u t r a l type [63, 109] it follows that u n d e r the c o n d i tions (2)- (4). the operator V~ for any ~[0, /t] is defined on all of C'[ --//, 0], the mapping (e, q))~--~Ve (q~) is continuous, and the set
U
(8)
Ve(A) is bounded for any bounded set A ~ C 1[--[z, 01.
1.5.3. Auxiliary Inequality. The right side of (7) can be considered as an operator Ge, acting from C1[a h, b] to C[a, b] for any a and b, satisfying the condition 0 _< a < b [we note that for a _> h the second summand is actually absent, since u(t) - O]. We shall assume that the norm in C1[a - h, b] is defined by the formula llXllcl = Ix(a) l + llx' IIC, so as is easily verified, the Hausdorff noncompactness measure ){I(A) of any set A~C l [a--h, b] will coincide with the noncompactness measure ~ (A') of the set A' of derivatives of functions from A in C[a - h,
b]. It is asserted that the family of operators G = {Ge : e E [0, hi} is (q,)0-bounded on bounded sets, i.e., for any bounded A ~ C ~l[a--h, b[ one has
z lq (a)l ~< qz' (a) ( = qz (a'))
(9)
[we recall that by definition, G (A)= U G~(A)]. ~lO,hl F o r the p r o o f , we r e p r e s e n t the o p e r a t o r G e in the f o r m of a s u m f~ + fe, w h o s e s u m m a n d s a r e defined by the c o r r e s p o n d i n g s u m m a n d s on the r i g h t side of (7). The f a m i l y { f e } , obviously, is c o m p l e t e l y c o n t i n u o u s , so it s u f f i c e s to prove that the f a m i l y f = { f e : a E [0, h]} is (q, x)-bounded. But the o p e r a t o r s fe a d m i t a diagonal r e p r e s e n t a t i o n in t e r m s of the o p e r a t o r
9 (~, x, v) ( t ) = g (~, t, x (t - ~ ) , v (t - 0 ) , which, obviously, satisfies all the conditions b[). Thus, (9) is proved.
of Theorem
1.2.3 (as M one can take any bounded
set in C1[a - h,
1.5.4. Lemma on the Translation Operator. For sufficiently small h, the family of operators e ~ [0,-h]} is (ql, X') -b~ (with ql < I) on bounded sets. Proof. We need that h should satisfy the inequality nh _< T, where the natural number below.- For any bounded set A ~C I l--h, 01 we denote by A the set of solutions of the problem e E [0, h], ~0 E A, defined on [Th, TI:AcCI[--h, T]. Itis easy to see that for 0 <_ k _< n- i,
V = {V a :
n will be defined (7), (6) for all
A' ]tkh,n~O (A II(k--~k,rl). Using this inclusion for k = n - I, n - 2, . . ., 0, and (9), we get:
x ~[V (A)I ~< x' (a h(n-,)~,n) ~< qnx' (A). Further,
562
it is easy to verify that
(10)
i.e., Z1(A) ~ I ~
(11)
zl (A).
F r o m (10) and (11) follows the inequality xI[V(A)] <_ [qn/(1 - q)]xI(A), which also shows that for sufficiently large n (i.e., sufficiently small h), the family V is (ql, X1)-bounded with ql < 1. 1.5.5. Theorem on the Existence of T - P e r i o d i c Solutions. There exist positive h and a such that for ~ [0, h] the set ~l (~) of T-periodic solutions x of (1), satisfying the condition II • w llc1 _< a, is nonempty and lim sup iI x - - ~ I[c,=O.
(12)
P r o o f . We c h o o s e h > 0 s o that the f a m i l y V of t r a n s l a t i o n o p e r a t o r s Ve f o r e 6 [0, h] is (ql, Xl)-b~ with ql < 1. We note that the fixed points of the o p e r a t o r Va and only they, g e n e r a t e T - p e r i o d i c solutions of (1). The o p e r a t o r V0, by h y p o t h e s i s , has fixed point w 0 = wl[_h,0]. F r o m the exponential stability condition on w follows the e x i s t e n c e of an a 1 > 0 s u c h that in the open ball B with c e n t e r at w 0 and r a d i u s a~ in C ~ [ - h , 0] the o p e r a t o r V 0 has only one fixed point w 0 and for all s u f f i c i e n t l y l a r g e n a t u r a l n u m b e r s N one has
Vo:v ( B ) c B .
(13)
We note that V0, of c o u r s e , differs f r o m the t r a n s l a t i o n o p e r a t o r Vo along t r a j e c t o r i e s of (1) f o r a = 0, but (13) is n e v e r t h e l e s s valid, s i n c e ~ + h ~ T, and e o n s e q u e n t I y , V0N = ~ N - 1 v 0. F u r t h e r , by v i r t u e of the s a m e inequality ~ + h _< T, (7) for ~ = 0 and T - h _< t _< T is an o r d i n d a r y diff e r e n t i a l equation (it can be s o l v e d with r e s p e c t to the d e r i v a t i v e ) , f o r w h i c h , o b v i o u s l y , any bounded s e t of solutions is r e l a t i v e l y c o m p a c t in C t. Hence, the o p e r a t o r V 0 is c o m p l e t e l y continuous. To it we apply the t h e o r e m of Z a b r e i k o - K r a s n o s e l ' s k i i [74] (without loss of g e n e r a l i t y one can a s s u m e that N is a p r i m e } : 1 -----tnd (V0A', B) ~ ind (V0, B) (rood N).
Since this r e l a t i o n holds f o r all sufficiently l a r g e p r i m e n u m b e r s N, we get that ind (V0, B) = 1. F u r t h e r , s i n c e on 13 the o p e r a t o r V 0 o b v i o u s l y has no fixed points, one can u s e T h e o r e m 1.3.10 on the local c o n s t a n c y of the index, and we get the e x i s t e n c e of an h > 0 s u c h that f o r e ~ [% h] the o p e r a t o r Va has in B at l e a s t one fixed point, which g e n e r a t e s a T - p e r i o d i c solution of (1). The n u m b e r a, f i g u r i n g in the c o n c l u s i o n of the t h e o r e m , can be d e t e r m i n e d as s a = max {a~, a2}, w h e r e a2 = sup {ll x ~ w i/c':x (I-a,0] CFix (V) N
B}. It r e m a i n s to p r o v e (12). L e t us a s s u m e that it does not hold: t h e r e e x i s t s e q u e n c e s
e.-+0,
x,,C~I(e.) s u c h
that II x n - w[[c, >~6 > O.
(14)
We s e t q~n =Xnl[-h,0]. Since ~0n = V~n(~ n) and the f a m i l y V is X L c o n d e n s i n g on bounded s e t s , we get that the s e q u e n c e {cpn } is r e l a t i v e l y c o m p a c t , o r in o t h e r w o r d s ,
z' ({%,}) = ~ ({v~, (%,)}) < x' ({%}). Without loss of g e n e r a l i t y , one can a s s u m e t h a t % ~ 2 2 % ~ B . P a s s i n g , in the equation ~n = V~n@n), to the l i m i t as n - - ~ , we get % = V0@0), w h e r e , by v i r t u e of (14), % # w0, w h i c h c o n t r a d i c t s the c h o i c e of a 1. r e i n is p r o v e d .
The t h e o -
1.5.6. Equation with S m o o t h Right Side. L e t us a s s u m e now in addition that g is continuously differentiable in the collection of third and fourth arguments, (15) ! (0, t, w(t), w'(t)), s ~ = hi4 ) (0, t, w(t), w'(t)), w h e r e g(i) T is the d e r i v a and we i n t r o d u c e the notation: r~ = g(a) tive of g with r e s p e c t to the i - t h a r g u m e n t . We l i n e a r i z e (1) f o r s = 0 along the solution w; we get
u' = rO (t) y § sO (t) g', or
y ' - p (t) y, w h e r e p(t) = r ~ - s~ [ f r o m (4) it follows that t s~ we r e p l a c e by the s i m p l e r s u f f i c i e n t condition
(16)
i _< q < 1]. The condition of exponential s t a b i l i t y of w
563
T
(17)
p (t) dt < O. 0
1.5.7. E x p o n e n t i a l S t a b i l i t y T h e o r e m . If the m o r e s t r i n g e n t c o n d i t i o n s d e s c r i b e d in the p r e v i o u s p a r a g r a p h a r e s a t i s f i e d , then t h e r e e x i s t h > 0 and a > 0 s u c h that f o r e E [0, h] the s e t ~(e) c o n s i s t s of e x a c t l y one s o l u t i o n w e , w h i c h is e x p o n e n t i a l l y s t a b l e . P r o o f . We c h o o s e h a n d a a s in the p r e c e d i n g t h e o r e m , and w e fix in e a c h s e t ~(e) one s o l u t i o n we(w ~ = w). We c a r r y out the l i n e a r i z a t i o n of (7) a l o n g the s o l u t i o n we: v' ~t) = r~ (t) v (t - ~) + s' (t) v' (t - ~) + [v' (0) - r' (0) v ( - e) - s~ (0) v' (- ~)l v (t), I
(18)
!
w h e r e r e ( t ) = g(3)(e, t, w e ( t - e), w e ' ( t - e)), se(t) = g(4)(e, t, w e ( t - e), w e ' ( t - e)). It is known (ef. [109]) t h a t the t r a n s l a t i o n o p e r a t o r W a l o n g t r a j e c t o r i e s of (18) is the d e r i v a t i v e of V e a t the p o i n t w~ = Well_h,0], s o to p r o v e the e x p o n e n t i a l s t a b i l i t y of w e i t s u f f i c e s to e s t a b l i s h t h a t the s p e c t r a l r a d i u s r ( W e) of the o p e r a t o r W e f o r s u f f i c i e n t l y s m a l l e is l e s s t h a n one. T h i s f a c t w e e s t a b l i s h w i t h the h e l p of T h e o r e m 1.4.7 on the s p e c t r a l r a d i u s .
L e t us a s s u m e t h a t f o r s o m e
sequence {en} , en -- 0 and r(Wen) >_ I. Applying to the family W = {Wen: n = i, 2 . . . . ) the lemma on the translation operator 1.5.4, we get that this family is (q1, ~()-bounded (ql < 1) for sufficiently small h. Then, obviously, for the operators An = Wen one has the hypotheses of Theorem 1.4.7 holding with B = W0, and we have 1 ~ lira r (W~) ~ max (q,, r (W0)). n-.-~ o o
S i n c e ql < 1, i t f o l l o w s t h a t r(W0) ~ 1, b u t t h i s , a s is e a s y to v e r i f y , c o n t r a d i c t s (17). T h u s , h and simple calculation of the s o l u t i o n w a, of T h e o r e m 1 . 5 . 5 , g e t the u n i q u e n e s s
a c a n be c h o s e n s o t h a t f o r e E [0, h] a n y s o l u t i o n w~E~(e) w i l l be e x p o n e n t i a l l y s t a b l e . A of i n d i c e s now s h o w s t h a t w e in ~ (e) is unique. In f a c t , in v i e w of the e x p o n e n t i a l s t a b i l i t y the f i x e d p o i n t w e0 of the o p e r a t o r Ve is i s o l a t e d , w h i l e i t s i n d e x , a s e s t a b l i s h e d in the p r o o f is e q u a l to one. But ind (Ve, B) = 1 a l s o , w h e n c e by the a d d i t i v i t y p r o p e r t y of the index w e of w~ and w e. The t h e o r e m is c o m p l e t e l y p r o v e d . CHAPTER SURVEY
OF
THE
II LITERATURE
The following books and survey papers are partially or completely devoted to the theory of condensing operators and its applications: [6, 23, 24, 31, 32, 74, 111, 116, 121, 151, 152, 174, 199, 219, 227, 238, 243, 261, 282, 357, 386, 411, 426]. 2.1.
Measures
of
Noncompaetness
2.1.1. Various Measures of Noncompactness. Analogs of the Kuratowski and Hausdorff measures of noncompactness in a uniform space E can be found in [Iii, 116]; these measures assume values in the set of nonnegative functions, defined on some family of continuous pseudometrics on E • E. We note here too that Theorem 1.1.2 is proved in [280]. The following set function is a measure of noncompactness on the metric space (M, M: I(A) = inf{ e > 0: any subset of the set A, the distance between any two points of which is less than e, is finite} (A~_M). For details and also the investigation of some properties of the measure I, cf. [36, 174,175,260]. The set function defined by the following formula will also be a measure of noncompactness of the Banaeh space E: ~0(A) = inf{e > 0: there exists a weakly compact subset C of the space E, such that A~_C+eB} (Bisthe unit ball in E, A~E). This measure of noncompactness has the following interesting property (ef. [178,184]): on the unit ball of the space E it is equal to zero, if E is reflexive; otherwise it is equal to one. Moreover, a decreasing (with respect to inclusion) sequence of weakly closed subsets of the space E, whose measure of noncompactness w tend to zero, has a nonempty intersection (cf. [184]). A series of other concrete
noneompactness
measures
can be found in [103, iii, 116, 149,176,251,403].
2 . 1 . 2 . F o r m u l a s f o r C a l c u l a t i n g N o n c o m p a e t n e s s M e a s u r e s in V a r i o u s S p a c e s . In the s p a c e C[a, b] of r e a l f u n c t i o n s , c o n t i n u o u s on the i n t e r v a l [a, b] w i t h the u n i f o r m n o r m , the n o n c o m p a c t n e s s m e a s u r e X of the s e t A c a n be c a l c u l a t e d a c c o r d i n g to the f o r m u l a :
564
)~ ( A ) = 12 ~Lmosup max [] x - - x~ 1[, x~A O...<,'r~6
w h e r e x r is the t r a n s l a t e of the function x:
x~
{ ()
if b--~
(cf. in t h i s c o n n e c t i o n [40, 4 1 , 1 1 1 , 116] et a l . ) . In t h e s p a c e l 2 of s q u a r e - s u m m a b l e
sequences the formula
~ (A) = l i m sup li Rnx II, w h e r e Rn x = Rn{xl . . . . . x n, Xn+ ~. . . . m e a s u r e (cf. [40, 41, 111, 116]).
) = (0, . . . .
0, Xn+ 1. . . .
) d e t e r m i n e s the H a u s d o r f f n o n c o r n p a e t n e s s
F o r m u l a s a l l o w i n g one to c a l c u l a t e the H a u s d o r f f n o n c o m p a c t n e s s m e a s u r e of s p e c i a l s e t s in [p c a n be found in [100]. F o r m u l a s f o r c a l c u l a t i n g v a r i o u s n o n c o m p a c t n e s s m e a s u r e s (in p a r t i c u l a r , the K u r a t o w s k i n o n c o m p a c t n e s s m e a s u r e ) in the s p a c e of f u n c t i o n s c o n t i n u o u s on a s e g m e n t w i t h v a l u e s in a B a n a c h s p a c e a r e g i v e n in [116, 1 3 9 , 3 1 3 ] . E s t i m a t e s of t h e K u r a t o w s k i n o n c o m p a c t n e s s m e a s u r e of s u b s p a c e s of the s p a c e of f u n c t i o n s s u m m a b l e a l o n g w i t h t h e i r p - t h p o w e r s , w i t h v a l u e s in a B a n a c h s p a c e , a r e o b t a i n e d in [356]. 2 . 1 . 3 . P r o p e r t i e s of N o n c o m p a c t n e s s M e a s u r e s . V a r i a n t s a n d g e n e r a l i z a t i o n s of T h e o r e m s 1 . 1 . 5 , 1 . 1 . 6 , 1.1.7, and 1.1.8 c a n be found in [ 4 0 , 1 1 6 , 1 7 7 , 2 1 6 ] et al. In a m e t r i c s p a c e the H a u s d o r f f n o n c o r n p a c t n e s s m e a s u r e w i l l a l s o be i n v a r i a n t w i t h r e s p e c t to p a s s a g e to the c o n v e x h u l l , if the c o n v e x hull is u n d e r s t o o d in the s e n s e of a c e r t a i n c o n v e x s t r u c t u r e on the g i v e n s p a c e (for d e t a i l s , cf. [403]). The n o n c o m p a c t n e s s m e a s u r e s c~ and • in a B a n a c h s p a c e , a s a l r e a d y n o t e d , a r e c o n n e c t e d by- the f o l lowing inequalities: a (A) ~ X (A) ~ 2a (A). V a r i a n t s o f t h e s e i n e q u a l i t i e s , and a l s o o t h e r i n e q u a l i t i e s c o n n e c t i n g c~, X, and I, c a n b e found in [ 4 1 , 1 1 1 , 1 1 6 , 174, 1 7 5 , 2 8 2 ] and o t h e r p a p e r s . The q u e s t i o n of the e x i s t e n c e of a u n i v e r s a l m e a s u r e of n o n c o m p a c t n e s s , i . e . , a m e a s u r e o f n o n c o r n p a c t n e s s , in t e r m s of w h i c h one can e x p r e s s f u n c t i o n a l l y a l l o t h e r m e a s u r e s of n o n c o m p a c t n e s s on a g i v e n s p a c e , is s t u d i e d in [22]. One s p e c i a l m e t h o d of c a l c u l a t i n g the n o n c o m p a c t n e s s m e a s u r e a is d e s c r i b e d in [197]. 2 . 1 . 4 . M o d i f i c a t i o n s of the C o n c e p t of M e a s u r e of N o n c o m p a c t n e s s . A s e r i e s of a u t h o r s c o n s i d e r e d another list of axioms defining noncompactness measures. We shall describe in detail one construction (el. [27]). Let E be a Banach space, 2 E be the set of closed subsets of E. A subset U of the set 2 E is called a #-system, if, firstly, the intersection of any family of sets of ~I belongs to U , and secondly, for any A ~ 2 E one can find A'GU containing A. Let K be a cone in some real linear topological space. A function ~ : 2 E ~ K is c a l l e d a d i s t i n g u i s h i n g m a p if it s a t i s f i e s the f o l l o w i n g f o u r r e q u i r e m e n t s : 1) if A~B, then ~(A) <_ ~(B); 2) if~(R) =0, then~(AUR)~(A);3)~(A)=~( N A') f o r a n y A ~ 2 E ; 4 ) ~ ( { x } ) = 0 for anyx~E. If a s U one A'~U, A,~A
t a k e s the s y s t e m of a l l c l o s e d s u b s e t s of E , then r w i l l be a m o n o t o n e n o n s i n g u l a r m e a s u r e of n o n c o m p a c t n e s s , h a v i n g p r o p e r t y 2). This s a m e c o n s t r u c t i o n a l l o w s one (cf. the c i t e d p a p e r ) to i n t r o d u c e the s o - c a l l e d p r o j e c t i v e m e a s u r e of n o n c o m p a c t n e s s [a m e a s u r e of n o n c o r n p a c t n e s s ~o is c a l l e d p r o j e c t i v e , if r = ~ ( t . A) f o r a n y n o n z e r o s c a l a r t]; p r o j e c t i v e m e a s u r e s of n o n c o m p a c t n e s s w e r e a l s o c o n s i d e r e d in [56]. C o n s t r u c t i o n s c l o s e to the one d e s c r i b e d a b o v e , in the c a s e of a m e t r i c s p a c e and 11=2 E w e r e s t u d i e d in [ 1 3 5 , 1 7 6 ] ; a n o t h e r a n a l o g o u s c o n s t r u c t i o n is s t u d i e d in [ 1 8 1 , 2 7 3 ] . In [194] the c o n c e p t of m e a s u r e of n o n c o n v e x i t y is s t u d i e d , w h i c h i s c l o s e in s o m e s e n s e to the c o n c e p t of m e a s u r e of n o n e o m p a e t n e s s . 2.2.
Condensing
Operators
2.2.1. Modifications and Generalizations of the Concept of Condensing Operator. In this section we shall describe certain classes of operators, close in their properties to condensing operators. In part such properties will be indicated directly below; some other properties will be considered in the following paragraphs.
565
An operator f: E ~ E (E is a Banach space) is called compatible with the distinguishing map ~ (cf. Paragraph 2.1.4), if, firstly, from A@II and ~(A) = 0 it follows that ~[f(A)] = 0, and secondly, if ~(A) a 0, then ~[f(A)] _ ~(A) for all A @ 2 E. It turns out (cf. [26, 27]), for an operator compatible with a distinguishing map, there is, roughly speaking, an invariant set A such that q~(A) = 0. The presence of an invariant compact set for a condensing operator relative to a monotone measure of noncompactness (and also for a condensing operator relative to an arbitrary measure of noncompaetness, such that the operator carries a convex closed set into itself) is proved in [111,116]. This property (i.e., the existence of an invariant compaetum) plays a very important role in a series of situations. Let M be an arbitrary subset of a Banach (or locally convex) space E, and f : M -- E. We construct a transfinite sequence of sets { T~ } in the following way: (C-Of (M), if
a = O, if ~-- 1 exists, a - - 1 does not exist.
r . = ~ / (M N T~, ), | ~ Tfh
if
The sequence { Ta }, as is easy to see, always "stabilizes": starting with some ordinal number 5 one has T~ = T5 for all c~ >_ 5. The set f~~ = T5 is called the limit domain of values of the operator f on M. The operator f is called (el. [111], 116]) limit compact, if its restriction to F~ is a compact operator, i.e., if the set f ( M n f ? (24)) is relatively compact. As already remarked, a condensing operator relative to a monotone measure of noncompactness is limit compact. One of the tests for limit compactness of an operator, defined on a subset M of a quasicomplete locally convex space E, is this (cf. [102]): the operator f is limit compact if and only if from the inequality c--of (24,q A) = A (ACE) follows the compactness of A We consider another class, more accurately countable set of classes of operators, close to condensing. Let ~ be s o m e set of subsets of the space E. The o p e r a t o r f : M ~ E is called a ~ - o p e r a t o r if for any T@~ and any A ~ E , f r o m the inequality c--oiT n f (A ~ 24)]= A follows the c o m p a c t n e s s of the set A. In the case when ~ = { ~ } and E is quasicomplete, the class of ~ - o p e r a t o r s , by virtue of what was said above, coincides with the c l a s s of limit c o m p a c t o p e r a t o r s . We denote by Kn the set of all subsets of the space E, containing exactly n elements, by K~ the set of all finite, and by K the set of all c o m p a c t subsets of the space E. By : ~ (respectively, d ~ and ~), we denote the set of all K n - o p e r a t o r s (respectively, K~o- and K - o p e r a t o r s ) . It is obvious that
It is known (el. [105]), that all the inclusions, with the possible exception of the last, are strict. It turns out that any K - o p e r a t o r (respectively, K ~ - o p e r a t o r , K0-operator) is condensing relative to a semiadditive nonc o m p a c t n e s s m e a s u r e which is invartant with r e s p e c t to addition of a compact (respectively, a semiadditive invariant with r e s p e c t to the addition of a finite set, semiadditive) m e a s u r e of noncompactness (cf. [103, 105]). We d e s c r i b e a n o n c o m p a c t n e s s m e a s u r e with r e s p e c t to which an a r b i t r a r y K - o p e r a t o r f : M - (E is a locally convex space) is condensing. We define in 2 E an equivalence relation R in the following way: (A~, A2) 6 R if one can find compact sets T 1 and T 2 such that A2c_co(TIUA1) and A,_cc--o(T2U A2). We denote by [A] the equivalence class (with r e s p e c t to R) containing A ~ 2 E, and by ~t the set of all equiv! 1 alence c l a s s e s , partially o r d e r e d in the following way: [A1] _< [A2] if one can find A s ~ [A1] and A 2 E [A2] such that A j c A '2 . We define a noncompactness m e a s u r e r on 2E by $(A) = [A]. The noncompactness m e a s u r e ,J~ is semiadditive (and consequently also monotone) and invariant with r e s p e c t to adding a c o m p a c t set, and the o p e r a t o r f is ~-condensing. More detailed information on the c i r c l e of questions discussed can be found in [102, 103,105]. To conclude this p a r a g r a p h we d e s c r i b e another v e r y broad class of o p e r a t o r s , a s s o c i a t e d in t e r m s of their p r o p e r t i e s with condensing o p e r a t o r s , the class of c o m p a c t l y c a r r i e d o p e r a t o r s . A bounded nonempty closed convex set R ~ E (E is a Banach space) is called a c a r r i e r for the o p e r a t o r [:M-+E (M~_E) with r e s p e c t to the set M, if R contains at least one set which is fundamental for f with r e s p e c t to M (this fundamental set can be empty if f has no fixed points in M), if [(MNR)~_R and if f is completely continuous on MflR. The o p e r ator f is called (cf. [74]) compactly c a r r i e d if it has at least one c a r r i e r . The definition given in P a r a g r a p h 1.2.1 of condensing o p e r a t o r were considered in one or another degree of generality in [33, 40, 41, 110, 1 1 1 , 1 1 6 , 177,214] and in a whole s e r i e s of later papers. Various modifications and generalizations of the concept of condensing operator can be found in [23, 27, 56, 78, 106, iii, 116,
123, 161, 169, 170, 174,176, 181,231,252,260,264,284, 285,291, 306,309, 324, 332, 354, 359, 416,425]. Connections between various definitions of condensing operator are studied in [424]. Operators, decreasing the measure of nonconvexity (cf. Paragraph 2.1.4), were considered in [194]. 566
2.2.2. Tests for a Condensing Operator. The known tests for a condensing cally, about Theorem 1.2.3 (cf. [108, 116, 160, 174, 224,309,416]).
operator
are grouped,
basi-
To the tests for a condensing operator, apparently, one should also refer the following assertion (cf, [418] and a close result in [55]): let f:E --E (E is a Banach space) and AcE. Then ~[f(A)] _< k if and only if for any e > 0 there exists a finite-dimensional operator S e : A -- E such that !If(x) - Se ~x)]i -< k + e for all x ~ A. We give another test for a condensing operator [284,285]. Let ~-(f) = inf{q > 0: f is (q, a)-bounded}. The operator f is called topologically strictly a-condensing if in E one can find a norm II9 ]II, equivalent with the original norm in E, such that Tl(f) < 1 [here Tt(f) is the number T(f), calculated not in the space (E, I]. lJ), but in the space (E, I] 9 One has the following fact: in order that the operator f be topologically strictly a-con] densing, it is necessary, and in the case of a linear f also sufficient, that one have lim[v(f~)]~-< I. n--~oo
2.2.3. Theorem on the Derivative of a Condensing Operator. The reader can find various and generalizations of Theorem 1.2.4 in [80, 116, 172, 174, 179, 183,302,307,322].
modifications
2.2.4. Properties of Condensing Operators. Let El, E2, and E 3 be Banaeh spaces with noncompaetness measures $i, $2, and :]J3, respectively, MbeasubsetofEi, and f:M~E2, g:f(M)~E 3. Among
the simplest
a) the composition $3)-bounded operator;
properties
of condensing
operators
of a (ql, 'bl, ']~2)-bounded operator
we note the following:
f and a (q2, ~J2, $3 )-b~
operator
g is a (ql, q2, ~1,
b) if the noneompaetness measure 'JJ2is monotone and algebraically semiadditive, then the sum $I, J~2)-bounded and a (q2, $i, ~2) -bounded operator is a (ql + q2, ~1, ~'2)-b~ operator;
of a (ql,
c) if f is a ff~1, gJ2)-condensing operator, g is a ($2, $3)-e~ operator, carrying completely bounded sets into completely bounded ones, the common domain of values of the measures of noncompaetness $i, $2, and :]~3is linearly ordered, and the measures $I and ~2 are regular, then the composition of f and g is a (r '[~3)condensing operator. Less RocR
trivial are the following
two assertions,
concerning
d) let OVI, D) be a metric space, R be a closed bounded and /~,,+~=f(/~NR) (n = i, 2 .... ); then limx(R~)=0.
properties
subset of M,
of X-condensing
operators
f:R ~ M be a X-condensing
e) let E be a Banaeh space, M be a closed subset of E, f:M--Ebe a X-condensingoperator;~:hen (~0 is the first transfinite ordinal number) the set T~ of transfinite sequences defined in Paragraph compact.
(cf. [113]): map,
for o~-~ r 2.2.1 is
The connection between contracting and X-condensing maps is illuminated by the following assertion [71]. Let f:M ~ M[(M, p) is a complete metric space] be a X-condensing operator, satisfying the follow[ngthree requiremerits: a)there exists a compact set K2]V/ such that from the equation f(A)=A follows the inc!usionA~K; b) slap p (f~ (x), x) < 0o for any x EM; c) there exists a neighborhood
U of the set K such that sup diam f~ (U) ~ ~ . Then one
n
n
can find a complete
metric space T, a point p in it, and a continuous
map g:M ~T
such that: a) g(x) = p forx E K; b)g is a
homeomorphism between M\/( and T\{p}; c) the map gfg-i is a contraction on T. In this same paper there are given conditions under which a X-condensing operator, acting in a metric space, is contracting in an equivalent metric. For results connected with the given circle of questions, cf. also [366]. Finally, we describe two properties of s-condensing maps. The first of them relates to the so-called property of minimal displacement. Let f:M --M be a continuous (q, a)-bounded map of the convex bounded subset M of the normed space E into itself and q > I. Then
lnf {[I x Results
conneeted
with the given assertion
The second assertion relates bounded retraction of the unit ball tence of a homotopy H : S x [0, !] ~ t])] _< tqa (A) for any A~_S and any
j" (x)l[: x e M } < (1 -- q-')" z (M).
can be found in [142,209,211,217,301,348].
to the geometry of Banach spaces. It turns out that the existence of a (q, a)B onto the unit sphere S of the Banach space E is equivalent with the exisS, connecting the identity and a constant mapping, such that a [H(A • [0, t E [0, I] (el. [211,212]).
567
Various p r o p e r t i e s of condensing o p e r a t o r s w e r e also studied in [ 1 1 1 , 1 1 6 , 1 7 4 , 2 5 9 , 3 0 9 , 3 2 3 , 3 2 4 , 419] et al. In [141] i n t e r m s o f " s m o o t h n e s s " of the boundary of the domain ~ R " , conditions a r e f o r m u l a t e d under which the o p e r a t o r of inclusion of W~(~) in L2(~) turns out to be {q, x)-bounded. Various p r o p e r t i e s of condensing d i s s i p a t i v e o p e r a t o r s w e r e studied in [36, 3 7 , 2 3 8 , 2 4 2 , 2 7 0 ] . 2.2.5. Multivalued Condensing Mappings. A whole s e r i e s of p a p e r s is devoted to m u l t i v a l u e d condensing m a p s . We shall not dwell on the d e s c r i p t i o n of these papers due to the r e c e n t publication of the detailed s u r vey [24], devoted to the theory of multivalued o p e r a t o r s . We r e s t r i c t o u r s e l v e s m e r e l y to r e f e r e n c e s to p a p e r s connected with the theory of multivalued condensing o p e r a t o r s (below, in the c o r r e s p o n d i n g places we shall r e p e a t these r e f e r e n c e s , without s t r e s s i n g that the given p a p e r s r e l a t e to multivahied o p e r a t o r s ) : [35, 53, 909 5 , 1 0 1 , 1 2 6 - 1 2 8 , 1 6 4 , 179, 180, 182, 1 8 3 , 2 0 3 - 2 0 5 , 2 3 0 , 2 3 1 , 2 3 4 , 2 3 5 , 2 4 5 , 2 5 1 , 2 5 2 , 2 6 9 , 2 7 7 , 2 7 9 , 2 8 4 , 2 8 9 , 293,295-299,320,321,331,334,347,363,375,396,403,405,420,421]. 2.3.
Theory
of
Rotation
2.3.1. Properties of Rotation. In Paragraph fields were described. Below we shall give some
1.3.1 some properties of the rotation of condensing more important properties of the rotation.
vector
The first of the properties described relates to the question of the homotopy classification of condensing vector fields. Suppose there is given on a set U of the Banach space E a homotopy f: U • [0, I] ~ E, which is condensing in the collection of variables, all of whose fixed points are contained in some compact set K. Then (cf. [124]) there exists a condensing homotopy F : U • [0, I] • [0, I] ~ E such that: i) all fixed points of the map F(., t, s) for all t, s E [0, i] are contained in the same set K; 2) F(x, t, s) = f(x, t) for all (x, t, s) E U •
[0, 1] X {0} UKX [0, 1]X[0, 1]; 3) the map F ( . , . ,
1) is c o m p l e t e l y continuous.
F r o m this p r o p e r t y of homotopies follows the following a s s e r t i o n : if the condensing o p e r a t o r f : VR ~ R (VR is the r e l a t i v e c l o s u r e of V • R in R) has no fixed points on VR, then t h e r e exists a homotopy F : VR x [0, 1] ~ R , condensing in the collection of v a r i a b l e s , joining the map f with a c o m p l e t e l y continuous map, and such that F{x, t) r x for x ~ VR; i.e., in the c l a s s of fields, homotopic to a condensing v e c t o r field I - f , there e x i s t s a c o m p l e t e l y continuous v e c t o r field. Whence and f r o m Hopf's t h e o r e m for c o m p l e t e l y continuous v e c t o r fields, it is e a s y to deduce the following r e s u l t : Let fl and f2 be condensing m a p s , acting f r o m V into E, where V is the c l o s u r e of the J o r d a n domain V in E (being J o r d a n m e a n s that E \ l ? is connected); let fl and f2 not have fixed points on V and 7 ( I - f i , V) = 7 ( I - f 2, V), then the fields I - f i and I - f 2 a r e homotopic. With the help of the construction d e s c r i b e d above one can c a r r y over to condensing v e c t o r fields other homotopy invariants of c o m p l e t e l y continuous v e c t o r fields. F o r details, cf. [124], and also [63]. As an example of this p r o c e s s , we give the following a s s e r t i o n (cf., e.g., [74]): The rotation 7 R ( I - f , V) of the condensing v e c t o r field I - f on V r e l a t i v e to R depends only on the behavior of the o p e r a t o r f on VR. We r e c a l l (cf. P a r a g r a p h 1.2.2) that if fl and f2 a r e condensing o p e r a t o r s , then the homotopy F(x, t) = t)fl(x) + tf2 (x) is condensing in the collection of v a r i a b l e s . This p r o p e r t y allows one in the m a j o r i t y of c a s e s to c a r r y o v e r to the c a s e of condensing vector fields a s s e r t i o n s for c o m p l e t e l y continuous vector fields in whose proof one u s e s only linear homotopies. (1 -
We d e s c r i b e s o m e m o r e p r o p e r t i e s of condensing v e c t o r fields, analogous to p r o p e r t i e s of completely continuous v e c t o r fields. Let U1 and U2 be open bounded subsets of the Banach s p a c e E. Let f : Ui ~ E and g : ~2 ~ E be condensing maps such that g ( I - f ) + f is condensing and (I--f)U~ ~U2 Finally, let {V~:~A} be the s e t of connected components of the s e t U2\(I--f)(~O. We denote by 7 ( I - f , U1, k) the rotation on Ui of the v e c t o r field I - f - z , where z is an a r b i t r a r y point f r o m VX. Then
y((I--g)(I-- f), U~)= ~ ?(I--g, Vr)7(I-- f , U1, ~,). This a s s e r t i o n , and also others close to it, can be found in [300,407]. Now we c o n s i d e r a linear v e c t o r field I - A , where A is condensing with constant k < 1. Then the index of z e r o as a fixed point of the o p e r a t o r A is equal to (--1)/~, w h e r e fl is the s u m of the multiplicities of the r e a l eigenvalues of the o p e r a t o r A which a r e l a r g e r than one. T h e o r e m s on the rotation of a linear condensing v e c tor field of the type given above can be found in [ 1 7 4 , 3 9 1 , 3 9 5 ] . The following a s s e r t i o n connects the rotation of the fields I - f and I - f p, where p is a p r i m e . Let f be a condensing o p e r a t o r with constant k < 1, defined on the c l o s u r e of the domain U, and Fix (fP) be the set of fixed 568
points of the o p e r a t o r i p, lying in U. Let fP have no fixed points on I~ and f(Fix(ff))cU. Then (cf~ [50,388]) ( [ - - f , U) ~ ? ( I - - fP, U) (rood p). To conclude this p a r a g r a p h we give a t h e o r e m on the finite-dimensional approximation of a X-condensing v e c t o r field (cf. [116,310]). Let the 1-condensing o p e r a t o r f : ~R - - R have no fixed points on VR. Let us a s sume that in E t h e r e are sequences of finite-dimensional subspaces {En: n = 1, 2, . . .} and o p e r a t o r s Pn: E ~ En (n = i, 2, . . . ), satisfying the following conditions: a) Pn satisfy Lipschitz conditions with constant one; b) lira Pnx = Jr for any x E E ; c) Pnf(X) ~ R for any natural number n and any x e ~r Then for sufficiently n--eve
large n,
~ (I - - P J , V)=yR (l--f, V). We also note here that the formula given can s e r v e as the definition of the d e g r e e of the map f. This a p p r o a c h in the g e n e r a l situation was studied in detail in [30-34, 1 6 5 , 2 0 0 , 2 0 1 , 3 1 8 , 3 9 1 , 4 1 1 , 4 2 1 , 4 2 2 ] and other papers. The theory of the index of fixed points of condensing maps, acting on Banach manifolds, is c o n s t r u c t e d in [38, 39, 150]. The following p a r a g r a p h s of the p r e s e n t section a r e devoted to the d e s c r i p t i o n of the construction of the theory of rotation for various c l a s s e s of o p e r a t o r s , connected in one way or another with condensing ones. 2.3.2. Compactly C a r r i e d Mappings. Let f : U -- E be a map, compactly c a r r i e d on U (cf. P a r a g r a p h 2.2.1), and R be some c a r r i e r of f relative to U. We set ] ~ f I g N , ~. The o p e r a t o r f, by virtue of Dugundji's t h e o r e m , can be extended to a completely continuous o p e r a t o r f defined on ~ , where if f has no fixed points on U, then f also has no fixed points on U. The rotation of the compactly c a r r i e d v e c t o r field I - f is defined by the f o r m u l a
y ( l - - f , U)=y(I--], d). This turns out to be well-defined (i.e., independent of the choice of carrier R and extension f). :['he rotation constructed has properties 1~ ~ from Paragraph 1.3.1. We dwell now on the homotopy property. Those homotopies I-f?,(k E [0, I]), for which the operators f?~ have no fixed points on U and there exist 0 = k 0 < ki < 9 . 9 < Xk+ I = 1 such that the operators f~ for k E [ki, Xi+l] (i = 0, i, .... k) have common carrier, willbe admissible homotopies in the present theory. For details, cf. [74]. 2.3.3. Limit Compact Operators. As already noted above, a limit compact operator, described in Paragraph 2.2.1, can serve as an example of a compactly carried operator. In the class of admissible homotopies there are the homotopies of the form I-f~, where f)t(x) r x (x E I~) and the operator(h, x) ~ fk(x) is continuous in the collection of variables and limit compact in the collection of variables in the natural sense. A systematic account of the theory of rotation for limit compact 116, 174, 331, 420] and other papers.
vector fields can be found in [95, iiI,
A close approach in the case when the operator is defined on an absolute neighborhood a Banach space is described in [303,304,306, 309, 310].
retract lying in
2.3.4. K-Operators. The theories of rotation described in Paragraphs 2.3.2 and 2.3.3, unfortunately, has at least two deficiencies: firstly, the rotation, in contrast with the case of condensing vector fields, depends, generally speaking, on the behavior of the operators inside the domain, and secondly, linear homotopies frequently turn out to be not admissible. The theory of rotation for Kn-operators (n >_ 2), K~o-operators ~ and Koperators (cf. 2.2.1) lacks the first of these deficiencies. The construction of these theories is based on the concept of fundamental set (for details, cf. [101, 102,106]). Moreover, a construction of the theory of rotation for limit compact vector fields not having the indicated deficiency is in some sense in possible (cf. [102]). The construction in the analogous way (under minimal for K-operators (cf. [102]) lacks the second deficiency.
additional assumptions)
of the theory of ro~ation
2.3.5. Remark. There are other versions of the theory of the degree of a mapping and the description of their connections with the theories described above in [23, 25, 26, 50, 58, 66, 122,1231 142, 151,153, 174, 198, 202,204, 206,219,234,246, 247,307,316,319,333,357,407,421,425]. 2.4.
Applications
to
the
Theory
of
Operator
Equations
2.4.1. Fixed Points. We begin the description of applications of the theory of condensing operators with the most extensive domain, the theory of fixed points. We note immediately, that any test for the difference
569
of the rotation of the vector field I-f from zero (in a "good" theory of rotation) is automatically existence of fixed points of the operator f.
a test for the
2.4.2. Analogs of 8chauder's Principle. Besides Theorem 1.3.9, various theorems on fixed points of operators carrying a convex closed bounded set (or its boundary) into itself and which are in some sense of other condensing, can be found in [26, 27, 74, 78, 102, iii, 116,120,154, 165, 167,168, 177,181,184, 193, 203. 208,217,251,252,256,263,265,287,301,303,304,308,312,331,337,351-353,359,368,370,375,379,388, 396,405,416]. In the following paragraphs of this section, if nothing is said to the contrary, f denotes an ~- or X-condensing operator, defined on a closed ball B of the Banach space E with center at zero. 2.4.3. THEOREM. Let f satisfy the following requirement f(x) =kx, then~_< i. ThenT(I-f, B) =I.
(the Leray-Sehauder
For various versions and generalizations of this theorem, cf. [173,174, 256,276-278,288,290,297,320,324, 327,335,343,346,347,350,416]. 2.4.4.
191,203,204,220,222,231,
T H E O R E M . S u p p o s e f o r a l l x E ]~ one h a s l l x - f(x)it 2 _< I] f(x)]] 2 --tlxII
F o r v a r i a n t s and g e n e r a l i z a t i o n s , 2.4.5. T H E O R E M . f, B) = 1 is odd.
condition): if x E B and
2. Then 7 ( I - f , B) = 1.
cf. [ 2 6 3 , 2 8 8 , 3 6 7 ] .
L e t E be a H i l b e r t s p a c e and s u p p o s e f o r a l l x E ]3 one has if(x), x) _< Ilxll 2. Then 7 ( I -
F o r v a r i a n t s and g e n e r a l i z a t i o n s , cf. [27, 263, 288]. 2.4.6. T H E O R E M .
S u p p o s e f o r a l l xE B one h a s f ( - x ) = - f ( x ) . T h e n 7 ( I - f , B) i s odd.
F o r v a r i o u s g e n e r a l i z a t i o n s of this t h e o r e m , cf. [59, 74, 111, 1 1 6 , 2 3 1 , 3 1 0 , 3 2 6 , 3 3 6 , 3 6 7 , o m i t the g e n e r a l i z a t i o n of the B o r s u k - U l a m t h e o r e m [192].
420]. H e r e we
2.4.7. T H E O R E M ( S c h a e f e r ' s P r i n c i p l e ) . L e t f be an ~ - o r X - c o n d e n s i n g m a p in a B a n a c h s p a c e E and l e t t h e r e e x i s t X0 E [0, 1] s u c h t h a t the e q u a t i o n x = ;~0f(x) h a s no s o l u t i o n s . Then the s e t {(x, ~) ~ E • R : x = Xf(x), 0 < ~ < ;~0} is unbounded. F o r v a r i a n t s and g e n e r a l i z a t i o n s ,
cf. [79, 9 7 , 2 2 0 , 2 2 8 , 3 6 7 ] .
2.4.8. M a p p i n g s D i r e c t e d I n w a r d . L e t U be a n o n e m p t y c o n v e x c l o s e d s u b s e t of the B a n a c h s p a c e E and x E E . We d e n o t e the s e t {zoE:~ (g~U, a~O)[z--x+a(g--x)]} by J u ( x ) . In a s e r i e s of r e s u l t s the t r a d i t i o n a l c o n d i t i o n of i n v a r i a n e e (of the s e t U) w i t h r e s p e c t to the o p e r a t o r f : U ~ E is r e p l a c e d by the l e s s r e s t r i c t i v e c o n d i t i o n v (x~U) i f ( x ) o ] v ( x ) ] (such m a p p i n g s a r e s a i d to be d i r e c t e d i n w a r d ) . T H E O R E M . L e t f be an ~ - o r X - c o n d e n s i n g m a p p i n g d i r e c t e d i n w a r d and l e t f(U) be bounded. Then f h a s a f i x e d point in U'. Various theorems on fixed points of mappings which are condensing in one sense or another, directed inward (or in some sense or other weakly directed inward) can be found in [203,334,345,346,348]. 2.4.9. THEOREM. Let U be a nonempty closed convex subset of a Hilbert space H, and f:U - - H be a X-condensing operator. Let the following (Frum-Ketkov) condition hold: There exists a bounded subset A of the s e t U s u c h that p(f(x), A) _< p(x, A) f o r a l l x ~ U [ h e r e p(x, A) =inf IIx'yt} ]. Then f h a s in U a t l e a s t one f i x e d point. Y6A
M o d i f i c a t i o n s and g e n e r a l i z a t i o n s of this t h e o r e m , and a l s o r e s u l t s on f i x e d p o i n t s of c o n d e n s i n g o p e r a t o r s s a t i s f y i n g the F r u m - K e t k o v c o n d i t i o n o r its a n a l o g s , c a n be found in [46, 73, 1 5 4 , 3 1 2 , 3 3 5 ] . 2.4.10. T H E O R E M . L e t (M, p) be a c o m p l e t e m e t r i c s p a c e of f i n i t e d i a m e t e r , ~ : M • M - - R be a l o w e r s e m i c o n t i n u o u s f u n c t i o n , and f : M --* M be an ~ - c o n d e n s i n g m a p s a t i s f y i n g the c o n d i t i o n ~ if(x), f(y)) < ~ (x, y) f o r a l l x , y E M, x ~ y. Then f h a s a unique f i x e d point in M. F o r v a r i a n t s and g e n e r a l i z a t i o n s ,
cf. [170, 1 7 6 , 2 1 4 , 2 6 4 , 2 6 6 , 2 9 1 , 3 8 4 ] .
2 . 4 . 1 1 . R e m a r k . T h e r e is known a w h o l e s e r i e s of p a p e r s d e v o t e d to p r o v i n g f i x e d point t h e o r e m s f o r m a p p i n g s of the t y p e " c o n t r a c t i n g + c o m p a c t ~ : [75, 86, 89, 147,153, 155,164, 245, 248, 264, 2 6 7 , 2 7 9 , 3 2 7 , 350, 378, 379, 381]. 2 . 4 . 1 2 . T H E O R E M . L e t E be a B a n a c h s p a c e , R be a c o n v e x c l o s e d s e t in E , V be an open s e t in E, V and be the c l o s u r e and b o u n d a r y of t h e s e t V N R in the r e l a t i v e t o p o l o g y of the s p a c e R. L e t the o p e r a t o r f : V---R 570
be c o n t i n u o u s and l i m i t c o m p a c t (cf. P a r a g r a p h 2 . 2 . 1 ) , and the f a m i l y of o p e r a t o r s fn : V - - R be l o c a l l y e q u i c o n t i n u o u s . L e t the f o l l o w i n g c o n d i t i o n s hold: a)
f~ (x)-[2:f (x)
f o r any x ~ V;
b) if x* = f(x*) and x ~ V, t h e n fn(x) + x* - fn(X*) e R f o r n >_ N(x*); e) the o p e r a t o r ] : A X V - + 1~ (A ----{N (x*), N (x*) ~- 1. . . . .
f(a, ~'
""1 =
~ }), d e f i n e d by the f o r m u l a
[.f.(x)--]-x*--f~(x*), If (x),
if
if
a=n,
a = c~,
is l i m i t c o m p a c t in the c o l l e c t i o n of v a r i a b l e s f o r a n y f i x e d point x* of the o p e r a t o r f; d) x ~ f(x) f o r x E ~} and 7 R ( I - f ,
V) ~ 0;
e) any of the o p e r a t o r s f(n, 9 ) h a s no m o r e than one f i x e d point. Then the s e t of f i x e d p o i n t s of the o p e r a t o r f is n o t e m p t y , c o m p a c t , and c o n n e c t e d . V a r i a n t s and g e n e r a l i z a t i o n s , and a l s o o t h e r r e s u l t s on c o n n e c t i o n s of s e t s of f i x e d p o i n t s of o p e r a t o r s w h i c h a r e c o n d e n s i n g in s o m e s e n s e o r o t h e r c a n be found in [107, 1 0 8 , 1 1 6 , 2 0 1 , 3 2 3 , 3 2 5 , 361,362,364,371]. 2.4.13. O p e r a t o r s w i t h N o n c o n v e x D o m a i n of V a l u e s . L e t R be a c l o s e d s u b s e t of the B a n a c h s p a c e E , r e p r e s e n t a b l e in the f o r m of a l o c a l l y f i n i t e union of c o n v e x c l o s e d s e t s ; U~_R be an open s u b s e t of the s p a c e R w i t h m e t r i c , i n d u c e d f r o m E ; f : U - - U be a c o n t i n u o u s m a p . THEOREM.
Let: a) the s e t
U~= ~ fn(U)
be n o n e m p t y , c o m p a c t , a n d c o n t a i n e d in U; b) the m a p f be
l o c a l l y a - c o n d e n s i n g ( i . e . , f o r eac'h point x E U one c a n find a n e i g h b o r h o o d Vx s u c h t h a t the r e s t r i c t i o n flVx is an a - c o n d e n s i n g map); c) t h e r e e x i s t a c o m p a c t s e t K s u c h that U~KDU~, U~ is h o m o l o g i e a l l y t r i v i a l in K and U~o is a c o m p a c t s u b s e t of U. Then ind R (f, U) = 1. We note t h a t ind R (f, U) d i f f e r s f r o m the i n d e x d e f i n e d in P a r a g r a p h 1 3 . 6 , s i n c e in t h e p r e s e n t s i t u a t i o n the s e t R i s not c o n v e x . C l o s e r e s u l t s c a n be found in [ 2 4 7 , 2 4 9 ] .
2.4.14. Remark. A series of theorems on fixed points of condensing maps, and also theorems on the solvability of operator equations containing maps which are condensing in some sense or other, can be found in the following papers [43, 44, 58,148, 158,161, 170,172,176,189,202,214,215,217,221,224,225,227, 229,230,253,254,261,264-266,268,269,272,273,292,301,308,310,312,315,326,327,329, 332,335,339342,348,349,351-353,361,365,372,374,375,380,382-385,387,389,404,412,419]. 2.4.15. Equations with a Parameter. parameter of the solution of the equation
In this paragraph we consider the question of dependenee
on the
x = f ( x , x);
(1)
f o r s i m p l i c i t y we s h a l l a s s u m e that X e [0, 11. L e t f : l] x [0, 1] -~ E ( U i s an open s u b s e t of t h e B a n a e h s p a c e E) b e an o p e r a t o r w h i e h is c o n d e n s i n g in the c o l l e c t i o n of v a r i a b l e s . T H E O R E M . S u p p o s e f o r X = 0 (1) h a s a s o l u t i o n x 0 of n o n z e r o t o p o l o g i e a I index w i t h r e s p e c t to the v e c t o r f i e l d I - f ( - , 0). Then f o r s u f f i c i e n t l y s m a l l X, (1) h a s a s o l u t i o n x x , w h e r e I j x ~ - - x 0 [ ] ~ 0 . T h i s r e s u l t is a s i m p l e c o r o l l a r y of T h e o r e m 1.3.11. T h e o r e m s of t h i s t y p e , and a l s o o t h e r t h e o r e m s on t h e c o n t i n u o u s d e p e n d e n c e of s o l u t i o n s on a p a r a m e t e r c a n be found in [111, 116, 143, 144, 1 5 9 , 2 4 0 , 2 5 5 ] . 2.4.16. B i f u r c a t i o n . Now we d e s c r i b e a r e s u l t c o n n e c t e d w i t h the q u e s t i o n of the n u m b e r of s o l u t i o n s x x of (1), c l o s e to x 0. F o r the d e s c r i p t i o n w e c o n s i d e r a v e r y s p e c i a l e a s e of d e p e n d e n c e on the p a r a m e t e r : it w i l l be a s s u m e d t h a t f(x, X) = (1 + X)g(x), w h e r e g is a c o n d e n s i n g o p e r a t o r w i t h c o n s t a n t k < 1. M o r e o v e r , we s h a U a s s u m e that g(x) = A x + B(x), w h e r e A is a b o u n d e d l i n e a r o p e r a t o r , and B is a c o n t i n u o u s o p e r a t o r , s a t i s f y i n g the c o n d i t i o n II B x l [ / l l x l l ~ 0 a s Ilxll ~ 0. O b v i o u s l y , u n d e r t h e s e c o n d i t i o n s x 0 = 0 w i l l be a s o l u t i o n of (1) f o r any X.
THEOREM. Let 1 be an eigenvalue of the operator A of odd multiplicity. (i) has at least one nonzero solution x x such that I[x~ II_-~ 2.
Then for sufficiently small X
571
Modifications, generalizations, and also other theorems on bifurcation of solutions of equations containing operators which are condensing in some sense or other, can be found in [15, 26,199,213,330,393,394, 406,408,418]. 2.4.17. Theorems on Surjectiveness and Invariance of Domain. Of the many series of results on surjectiveness of mappings containing condensing ones, we give only one. THEOREM. Let f: E ~ E (E is a Banach space) be a condensing operator. Let there exist two numerical sequences {rn} and {mn} such that mn,,--'~_~ ~ and
II/(x)-~x
[I> m.
for any p > 1 and x E E, IIxII = rn. Then the map I - f is surjective. .Close to t h e o r e m s on s u r j e c t i v e n e s s in spirit are t h e o r e m s on invariance of domain. We give a typical result. THEOREM. Let the o p e r a t o r f : E ~ E be condensing, f(0) = 0 and the map I - f be injective. Then for any neighborhood of z e r o W, the set if-f)(W) is a neighborhood of z e r o . Various t h e o r e m s on s u r j e c t i v e n e s s and invariance of domain can be found in [173,174, 1 8 5 , 1 9 1 , 2 1 8 , 223,230,231,266,267,288,289,310,322,326,332,333,359,368,370,375,413,417,420]. 2.4.18. Operators in Spaces with a Cone. Now we give two t h e o r e m s f r o m the theory of positive condensing o p e r a t o r s . Let K be a cone in the Banach space E. We r e c a l l that the inequalities
Ax-.<x (!lxll=r, xEK)
(2)
and
Axe(1 +~)x (H.xt]=p, x@7(; e > o is a fixed number
(3)
define c o m p r e s s i o n (for r < 0) and dilation (for r > 0) o p e r a t o r s of the cone. THEOREM. Let the condensing o p e r a t o r f be a c o m p r e s s i o n or dilation of the cone. Then f has in K at least one fixed point. THEOREM. Let the condensing o p e r a t o r f satisfy (2). Then one can find ~ > 1 and x E K, x ~ 0, s u c h t h a t f(x) = ;~x. These and other r e s u l t s resulting to the theory of positive condensing o p e r a t o r s can be found in [26, 54, 80-82, 85, 87, 104, 129, 131, 136-138, 1 9 0 , 1 9 6 , 2 0 4 - 2 0 7 , 2 2 2 , 2 3 5 , 2 8 6 , 2 9 3 , 2 9 4 , 3 3 8 , 3 4 4 , 3 6 8 , 3 6 9 ] . 2.4.19. R e m a r k . We note finally that for condensing v e c t o r fields the general topological principle that fields having different rotations have opposite directions at at least one point of the boundary of the set r e m a i n s valid. Hence, the known a s s e r t i o n s about the existence of eigenvalues of nonlinear condensing o p e r a t o r s , following f r o m this principle, r e m a i n valid. Results of this type, and also a s s e r t i o n s about the existence of eigenvalues of nonlinear condensing o p e r a t o r s , proved on the basis of other c o n s i d e r a t i o n s , can be found in [ 2 0 5 , 2 0 6 , 2 3 0 , 2 3 5 , 2 8 6 , 2 8 8 , 2 9 4 , 3 4 4 , 3 5 4 , 369]. 2.4.20. Linear Theory. A s y s t e m a t i c account of the r e s u l t s described in Sec. 1.4 can be found in [116]. Besides the approach d e s c r i b e d in this section, based on passage to the space E +, an a p p r o a c h to the study of F r e d h o l m properties of linear o p e r a t o r s based on factorization of the space of bounded linear o p e r a t o r s by the subspace of completely continuous linear o p e r a t o r s has been developed (cf. [40, 4 1 , 2 8 2 , 305] et al.). Let E l and E 2 be Banach s p a c e s , S be the unit sphere in El, and A : E 1 ~ E 2 be a bounded linear operator. We set
I[Aii~ini{q>O:A is (q, x)-bounded}, [[A[l~-=inf{q>O:A is (q, ~)-bounded}, [IAI]t=lnf{l[B--A II:B:EI~E2 is a linear completely continuous o p e r a t o r } , IIA II~=inf{ll AlL ][:L. is a subspace of El of finite codimension}.
572
The f o l l o w i n g a s s e r t i o n s hold:
a) {I A [[~< [i A Ht; b) if E~ is a s p a c e w i t h a b a s i s , t h e n 11A lit ~< b I[ A I~, w h e r e b ~-~tim [1Q~ 11, and Qk :E2 ~ E2 a r e d e f i n e d by the f o r m u l a Q k x = Qk(Xl, . . . , x k, Xk+l, . . . ) = (0 . . . . 1
c) ~ II A I[x<[l A* [Iz<21[ A IIx, w h e r e the c o n s t a n t s 1 / 2 d) if E 1 -- E 2 = Lp, then
, 0 , Xk+ 1. . . .
);
and 2 in t h e s e i n e q u a l i t i e s a r e s h a r p ;
][All•
e) if E 1 and E 2 a r e r e f l e x i v e , then [1A lit ~-[[ A* I[t ; now i f a t l e a s t one of t h e s e s p a c e s i s n o n r e f l e x i v e , t h e n e q u a l i t y is i m p o s s i b l e ;
f) I[A*{[~
h) if E 1 = E~ is a
Hilbert s p a c e , then ]l A II~=11 A* Ilk=l] A*A ti~;
i) II A [ l z = Z (AS). These,
and also other facts close to those listed above,
can be found in [40, 41, Iii, 116,125,282,305,
390].
Modifications and generalizations of Theorem 1.4.6 can be found in [40,116,125, 140,205,371,373]. From Theorem 1.4.6, in particular, it follows that any linear condensing operator, acting in a (real or complex) Ba~aeh space, can be represented as a sum of a finite-dimensional and a compressing operator in an equivalent norm (cf. [116, 125]). The representation of a condensing linear operator as the sum of a completely continuous and a compressing operator (in the original norm) may be impossible (cf. [41]). In connection with this one should also note the following fact (cf. [418]): if A : E ~ H (E is a Banach and H is a Hilbert space) is a bounded linear operator, then k = inf{q > 0 :A is a (q, c~)-bounded operator} if and only if for any e > 0 there exists a representation of the operator A in the form Ke + Lr where Ke is a completely continuous linear operator, and the operator Le satisfies the condition k ~< IILall -< k + e. The description of the spectra of linear operators in terms of properties connected with the condensing properties of their iterates, can be found in [40, 77,116,125,163,257,258,284,285,305]. From Theorem 1.4.6 also follows the Fredholm alternative for equations of the form ?vx - Ax = f, where A is an operator which is condensing with constant q, and I kl > I/q (cf., e.g., [116], and also [282,332,419]). Now here we note that analogs of the Fredholm alternative for nonlinear condensing operators are given in [174, 295,296,328,329,332,371]. In a series of papers various questions connected with perturbations of Fredholm operators by condensing ones are studied. The basic result in this direction reduces, roughly speaking, to the following assertion: For any Fredholm operator A there exists a number 0 > 0 such that for any operator B, satisfying the inequali .t3r IDBll 4.
573
2 . 5 . A p p l i c a t i o n s to D i f f e r e n t i a l
Equations
We begin with the description of applications of the theory of noneompactness measures and condensing operators to the theory of differential equations with deflected argument of neutral type. Without going into the "classificational" details, we shall mean by an equation of neutral type the following equation
x' (t)= f (t, xt, x't);
(1)
h e r e f : R•215 ~ (Ch=C[--h, 0] i s the s p a e e of functions, eontinuous on the s e g m e n t i - h , 0] with values in R n with u n i f o r m norm); xt and x~ a r e the e l e m e n t s of the s p a c e Ch defined by the equations xt(s) = x(t + s), x~(s) = x ' ( t + s) (s E i - h , 01). Below we shall a s s u m e that the o p e r a t o r f is continuous in the collection of v a r i a b l e s and s a t i s f i e s a Lipsehitz condition with constant k < 1 in the third v a r i a b l e . F i r s t we d e s c r i b e c e r t a i n o p e r a t o r s g e n e r a t e d by v a r i o u s p r o b l e m s for (1). 2.5.1. Cauchy P r o b l e m . One s e e k s a function x e C l [ - h , T], satisfying (1) for t E [0, T] and satisfying the initial condition
x(t)=~(t)
( t e l - h , 01L
(2)
w h e r e ~ is a given function. The p r o b l e m (1)-(2) ean be redueed in many ways to the p r o b l e m of fixed points of various condensing o p e r a t o r s . We d e s c r i b e one such o p e r a t o r . With this goal, for any y E C [ - h , T] we set t
(Jr) (t) = g) (0) @ ~ ~t (s) ds, 0
(3)
;~' (t) + f (0, (Yg)o, go)-- r (0) for t6[-- h, 0], (Fg) (t)= i f (t, (dr)t, Vt) for tQ[0, T I. Then the o p e r a t o r F acts in C [ - h , T] continuously and is condensing on any bounded s u b s e t of the s p a c e C [ - h , T]. F o r details, el. [108,116]. We note that for the existence of a continuously differentiable solution of (1)-(2) it is n e c e s s a r y that one have the "gluing" condition: q0'_(0) = f (0, % q~').
(4)
If this condition holds, then the solvability of (1)-(2) on the s e g m e n t i - h , T] is equivalent with the existence of a fixed point of the o p e r a t o r F in the s e t {y E C [ - h , T] :y(t) = q)T(t) for t ~ i - h , 0]}. Equation (4) often turns out to be r e s t r i c t i v e , so it is frequently useful to pass f r o m (1) to
x ' ( t ) = f ( t , xt, x;),
(5)
where
if(t, xt, x't)= f (t, xt, x'~)+v~(t)ix'_ ( 0 ) - - f ( 0 , x0, fro)], and v~ (t) = 1 - u - i t
for t E [0, p) and vp (t) = 0 for t E (p, ~).
This equation now s a t i s f i e s the gluing condition for
any initial function ~r The o p e r a t o r F, constructed f r o m f with the help of (3), differs f r o m the o p e r a t o r F by the c o m p l e t e l y continuous o p e r a t o r
(~ fv(O)--f(O, JV)o, Vo) for t E l - - h , 0 I, crv) (t)= lv~ (t) [v (O)-- f (O, (Jr)o, g0)] for tfi [0, T I. Hence F is a l s o •
in C [ - h , T] on any bounded s u b s e t .
Various condensing operators arising in studying the Cauehy described in [I, 4, 21, 47, 48, 116, 119, 133, 134,241,311].
problem
for equations of neutral type are
2.5.2. Problem of Periodic Solutions: Translation Operator. If the operator f is such that (5) for any initial function (p ff Cl[-h, 0] has on the segment i-h, T] a unique solution x(-, q~), then for (5) one can define the translation operator U T along trajectories from the point 0 to the point T according to the following rule
( u r , e ) ( s ) = x ( r + s, co) (se[-tt, oIL The o p e r a t o r U T acts f r o m C l [ - h , 0] to C l [ - h , 0] and is condensing with r e s p e c t to the m e a s u r e of n o n c o m p a e t ness r defined by 574
, , (A) = ~ @A'), where c~ is the Kuratowski noncompactness m e a s u r e in the space Oh, ~ is a linear function on the s e g m e n t i - h , 0], satisfying the conditions ~<(-h) = 1, 1 < ~(0) < k, and by x A ' we denote the s e t { y E Ch:y(t) = x(t)x'(t), x E A}. The problem of existence of a T - p e r i o d i c solution of (1) (in the case of a function f which is T - p e r i o d i c in the f i r s t argument) is equivalent with the p r o b l e m of existence of fixed points of the o p e r a t o r UT. The t r a n s lation o p e r a t o r also turns out to be useful in a s e r i e s of other problems. The a s s e r t i o n d e s c r i b e d can be found in [65]. Other r e s u l t s connected with conditions under which the translation o p e r a t o r along t r a j e c t o r i e s of one equation or another of neutral type is condensing are contained in [38, 69, 115, 119, 236, 244].
2.5.3. Problem of Periodic Solutions: Integral Operators. Now let the operator f be T-periodic in the first variable. By PC T we denote the space of T-periodic functions with values in Rn with uniform norm, and T
by P C T the space of continuously differentiable T - p e r i o d i c functions. Let y E PC T. We set and
'fo
M(g)=-~- g (s)ds
t
(J y) (t) ~- ~ tJ (S) d$ - - t ~ (~]). o We define the o p e r a t o r
(p:R"xPCT-+RnxPCr
by the f o r m u l a
q~(a, v) = ( a - M (v), f (a, V)), where F (a, y) (t) = f (t,
a-+-(Jy)t, b't).
Fixed points (a, y) of the o p e r a t o r 9 and T - p e r i o d i c solutions x of (1) are connected by the following relations: x(0) = a, x ' = y. It turns out that the o p e r a t o r ~ is condensing on any bounded set U~PC/ with r e s p e c t to the uoncompactuess m e a s u r e g)2, defined by
{Y6PCr:S(aER ~) [(a, y) 6A]}.
~2 (A) = Z e c r Now let x E PC~. We set t
,
(o)+I
t
(v -
I
0
T
i i (s, .<;) ds. 0
Fixed points of the o p e r a t o r | and only they a r e T - p e r i o d i c solutions of (1). The o p e r a t o r | is condensing on any bounded subset of P C ~ with r e s p e c t to the uoncompactness m e a s u r e ~0~, defined by
*a.(A) = )tpcr {vEPCr :~ (x~A) [v = x']}. These and other condensing o p e r a t o r s connected with the p r o b l e m of periodic solutions of (1) can be found in [39, 115, 116, 1 1 9 , 2 4 6 , 3 1 4 , 3 1 5 , 3 1 7 ] . 2.5.4. Boundary P r o b l e m s . We consider only one boundary problem for the equation of neutral type of the f o r m
x" ( t ) = g (t, x (G0(t)), x' (~1 (0), x" (~2 (t))),
(6)
~vhere g : [0, 1) x 1t3 - - R and cTi: [O, 1] - - i t (i = 0, 1, 2) are continuous functions. Let a=min
min~(t),
fi=max
i=0,1,2 tCl0,1l
max~i(t)
i = 0 , 1 , 2 tC[OAI
and c~ < O, fl > 1. We c o n s i d e r (6) t o g e t h e r with the following boundary condition: x(,) ( t ) = o
(toil, o] U I1, ~I; i = o , 1, 2).
(7)
We define on the space W2[0, 1] an o p e r a t o r R by the formula 1
(Ru) (t) = S O (t, s) g (s, ~ (% (s)), ~' (~ (s)), ~" (~2 (s))) ds; 0
here
s) t,
if
s>t, 575
and by u, u ' , u ' are denoted, r e s p e c t i v e l y , extensions to the interval [c~, fi] of the functions u, u', u" by z e r o . It turns out (cf. [57]) that the o p e r a t o r R is condensing with r e s p e c t to the noncompactness m e a s u r e X. Now fixed points of the o p e r a t o r R a r e solutions of (6)-(7). Close results can be found in [29, 57,237]. Now applying to o p e r a t o r equations one or another fixed point t h e o r e m , one can get various existence t h e o r e m s for solutions of various problems for equations of neutral type. There are results of this s o r t in [1, 6, 7, 19-21, 29, 39, 47, 48, 57, 9 1 , 1 1 5 , 1 3 3 , 1 6 2 , 2 3 6 - 2 3 9 , 2 4 1 , 2 4 3 , 2 4 6 , 2 7 1 , 3 1 1 , 3 1 4 , 3 1 5 , 3 1 7 ] . We give here one existence t h e o r e m for each of the problems listed above. The proofs of these t h e o r e m s a r e based on applying T h e o r e m 1.3.10 to the c o r r e s p o n d i n g o p e r a t o r equations. 2.5.5. THEOREM. Let the o p e r a t o r f be continuous in the collection of v a r i a b l e s , satisfy a Lipschitz condition with constant k < 1 in the third variable, and the initial function 9) satisfy (4). Then for some T > 0 the p r o b l e m (1)-(2) has on the segment i - h , T] at least one solution. 2.5.6. THEOREM. Suppose irl addition to the hypotheses of the preceding t h e o r e m the o p e r a t o r f is bounded and T - p e r i o d i c in the f i r s t variable. Let A be a Hurwitz n • n matrix. Then the equation
x' ( t ) = A x ( t ) + f (t, xt, x't) has at least one T - p e r i o d i c solution. The following t h e o r e m r e l a t e s to the solvability of the boundary problem (6)-(7). 2.5.7. THEOREM. Let the functions g and ~i satisfy the conditions listed in P a r a g r a p h 2.5.4 and in addition let the function g be bounded. Then (6)-(7) has at least one solution. 2.5.8. Equations of Neutral Type with a P a r a m e t e r . F o r equations containing a p a r a m e t e r , applications based on T h e o r e m 2.4.15 or its analogs turn out to be very interesting. T h e o r e m 1.5.5 can s e r v e as an example of an a s s e r t i o n of this sort. Below we formulate a s e r i e s of typical r e s u l t s , which are obtained on this path. F o r simplicity we shall always a s s u m e that the p a r a m e t e r figuring in the equations is ~ e [0, 1]. 2.5.9. Cauchy P r o b l e m . We c o n s i d e r the equation
x' (t)= f (t, xt, x't, ~), where f: RXChXChX [0, I]--~R n is continuous and satisfies a Lipschitz condition with constant k < 1 in the third variable. Suppose for ~ = 0 that Eq. (8)* has on the segment i-h, T] a unique solution x ~ satisfying the initial condition (2). THEOREM. F o r sufficiently small ~ and all g, the equation
x' (t)• f (t, xt, x't, ~) + v~ (t) ix'__(O)-- f (O, Xo, x'o, ~)] with the initial condition (2), has on the s e g m e n t i - h , T], at least one solution x~, where ]i x~--x~
~0
We note that in the p r e s e n t situation the index of the solution x ~ with r e s p e c t , e.g., to the o p e r a t o r F (cf. p a r a g r a p h 2.5.1) is equal to 1. F o r details, cf. [63]; close r e s u l t s a r e in [17,238]. 2.5.10. Equation with Small Deflected Argument. We consider the autonomous equation of neutral type with small deflected a r g u m e n t x ' (t) = f (V (~) xt, V (~) x',, ~),
(9)
where f : C , , X C h X [0, II-+R n is continuous and satisfies a Lipschitz condition with constant k < 1 in the second variable, and the o p e r a t o r V(}) for each ~ ~ [0, 1] acts in Ch by the formula
iv (~) ul (s)--- u (~s) (sGI- h, 0l). Suppose for ~ = 0 the (ordinary differential) equation (9) has a T0-periodic solution x ~ and the o p e r a t o r f ( . , 9 0) is uniformly continuously differentiable in a neighborhood of the set {(V(0)x~ V(O)x~ Tol}. We denote by a(t) [respectively, b(t)] the derivative of the o p e r a t o r f ( . , 9 0) with r e s p e c t to the f i r s t (respectively, second) variable at the point (V(0)x~,V(0)x~'). Let the equation
v' (t) = a (t) v (t) + b (t) v' (t) not have solutions linearly independent f r o m x ~ with period T o and not have solutions of the f o r m y(t) = ( t / T 0) • x~ + v(t), where v is a T0-periodic function. 9No Eq. (8) is indicated in Russian original - P u b l i s h e r .
576
THEOREM.
For sufficiently small }, (9) has a T~-periodic solution x~, where
(o)i +max l* o'
T~ ~ ItT~ \ l
' kT;-. ) l + l r -To
Modifications and generalizations of this assertion, and also other results on periodic solutions of equations of neutral type with parameter can be found in [6, 17, 18, 60, 66,115-117,243,249]. 2.5.11. Averaging Principle. In this paragraph we give a theorem, usually called the averaging principle in the problem of periodic solutions. We consider the equation with parameter
x' (t)= ~f (t, xt, x',).
(lO) T
We
assume
that f satisfies the hypotheses
of Theorem
2.5.6. For any a ~ I~n we set f0(a)=,-~I f(s, a, O)ds 0
(here a is the constant function from Ch identically equal to a). We consider, along with (i0), the ordinary differential equation x ' (t) = fo (x (t)).
(11)
THEOREM. Let (11) have the s t a t i o n a r y solution x~ =- x 0 and the index of the z e r o x 0 of the f i n i t e - d i m e n sional v e c t o r field f0 be different f r o m z e r o . Then (10) for sufficiently s m a l l ~ has a T - p e r i o d i c solution x~, w h e r e It x~ - x0 ilc,-+ 0. ~0
Modifications, g e n e r a l i z a t i o n s , and close r e s u l t s can be found in [2, 3, 5, 9, 11, 70, 121]. 2.5.12. Stability Theory. Without dwelling on details, we only indicate papers in which applications of 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 to the study of the stability of solutions of equations of n e u t r a l type a r e d e s c r i b e d : [6, 8, 10, 12-14, 16, 17, 64, 67, 69, 120]. We d e s c r i b e two m o r e e x a m p l e s of applications of 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 , now to the t h e o r y of differential equations in infinite-dimensional s p a c e s . 2.5.13. Cauchy Problem for Equations in Banach Spaces. Let E be an arbitrary Banaeh space. We consider a Cauchy problem of the form
x' (t)= f (t, x(t)), x (0) = x0,
(12) (13)
where the operator f acts from [0, T] • R to E and for any bounded set A it is uniformly continuous in the collection of variables (t, x) 6 [0, T] • M. Let us assume that for any bounded set AcE one has
x(f(t, A))..
z (0) = 0
has on the s e g m e n t [0, T] only the z e r o solution. THEOREM. F o r any x 0 6 E one can find r/(x 0) > 0 such that the p r o b l e m (12)-(13) has on the s e g m e n t [0, rT(x0)] at l e a s t one solution. Close r e s u l t s can be found in [49, 51, 61, 90, 112,114, 116, 126, 128, 1 3 9 , 1 4 5 , 1 5 6 , 1 6 6 , 180, 182,186, 193,195,197,336,358,397-402,415]. 2.5.14. Boundary P r o b l e m . We consider a boundary p r o b l e m of the form_
x"(t)= f (t, x(t), x' (t)) (tG[0, 1]), a~x ( i ) + ( - - ly +t bzx t (i)~d~ (i=O, I),
(14) (15)
w h e r e the o p e r a t o r f: [0, 1] • E • E - - E is continuous and bounded, ai, b i > 0, d i ~ E (i = 0, 1). Suppose for any bounded s e t s At, A2~E a ( f ([0, 1] X AI X A2))-_<~-max {a (A~), a (A2)}. Let, in addition, 2~. max ]G(t, s ) ] < I; h e r e , by G(t, s) we denote the G r e e n ' s function g e n e r a t e d by the o p e r a t o r t,s(~IO,ll
b2/~t 2 and the boundary conditions (15). 577
THE OREM. The problem (14)-(15) has at least one solution. Various results connected with the applicability of the theory of condensing operators to the theory of boundary problems for differential equations in a Banach space can be found in [146, 157, 187, 409]. 2.5.15. Other Applications. For results relating to the solvability of various problems for differential equations in infinite-dimensional spaces, obtained with the help of the apparatus of the theory of measures of noncompactness and condensing operators, cf. [36, 37, 42, 51, 52, 86, 90, 126-128,132, 138,147, 180, 182,187, 197,224,226,229,242,270,283,338,360,392,401]. The description of applications of measures of noncompactness and condensing operators to the theory of integral equations can be found in [55, 56, 84, 88,136, 1 7 1 , 2 7 4 , 2 7 5 , 3 0 2 , 4 1 7 ] , to the theory of boundary problems for ordinary differential equations in [ 5 2 , 2 0 2 , 2 0 5 , 2 0 6 , 2 8 6 , 3 9 3 , 3 9 4 ] , to the theory of partial differential equations in [ 1 3 6 , 2 0 2 , 2 0 5 , 2 9 6 , 2 9 9 , 330,417]. LITERATURE i. 2.
3. 4.
5. 6. 7.
8. 9.
I0. 11. 12.
13. 14.
15. 16.
17.
578
CITED
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