288. A. Manitius, "Optimal control of h e r e d i t a r y s y s t e m s , " in: Contr. T h e o r y and Top. Funct. Anal., Vol. 3, Vienna (1976), pp. 43-178. 289. W. R. Melvin, "A c l a s s of neutral functional differential equations," J. Diff. Equations, 1_~2,No. 3, 524-534 (1972). 290. W. R. Melvin, "Toplogies f o r n e u t r a l functional differential equations," J . D i f f . Equations, 13.3, No. 1, 24-31 (1973). 291. R . D . Nussbaum, "A g e n e r a l i z a t i o n of the Ascoli t h e o r e m and an application to functional differential e q u a tions," J. Math. Anal. Appl., 3._~5,No. 3, 600-610 (1971). 292. R. D. N u s s b a u m , " E x i s t e n c e and uniqueness t h e o r e m s f o r s o m e functional differential equations of neutral type," J. Diff. Equations, 1_~.1,No. 3, 607-623 (1972). 293. R . D . N u s s b a u m , " P e r i o d i c solutions of autonomous functional differential equations," Bull. Am. Math. Soc., 7__~9,No. 4, 811-814 (1973). 294. W. J. P a l m and W. F. Schmitendorf, "Conjugate-point conditions f o r v a r i a t i o n a l p r o b l e m s with delayed a r g u m e n t , " J . Optim. Theory Appl., 1_~4,No. 6, 599-612 (1974). 295. L. Pandolfi, "Stabilization of n e u t r a l functional differential equations," J. Optim. Theory Appl., 20, No. 2, 191-204 (1976). 296. R. D. R a m a k r i s h n a , "Stability c r i t e r i a of n e u t r a l functional differential equations," P r e c . Nat. Inst. Sci. India, A3_~5,No. 1, 63-69 (1969). 297. M. Sendaula, "On the optimal control t h e o r y f o r neutral s y s t e m s , " in: P r o c . IEEE Internat. Syrup. C i r c . and Syst., Munich, Apr. 27-29, 1976, New York, N. Y., etc., pp. 408-411, 298. G. A. Shanholt, "Slowly varying l i n e a r n e u t r a l functional differential equations," Int. J. Control, 1_~8,No. 3, 559-562 (1973). 299. W. Snow, E x i s t e n c e , Uniqueness and Stability f o r Nonlinear Differential Equations in the Neutral C a s e , New Y o r k Univ., New York (1965). 300. W. A. H a r r i s , J r . and Y. Sibuya (eds.), P r o c . United S t a t e s - J a p a n S e m i n a r on Differential and Functional Equations, University of Minnesota, June 26-30, 1967, Benjamin, New York (1967).
MULTIVALUED
MAPPINGS
Yu. G. Borisovich, A. D. Myshkis, and
B. V.
D. V.
Gel'man, Obukhovs
UDC 517.988.52 kii
The p r e s e n t p a p e r is a s u r v e y of the c o n t e m p o r a r y state of a r t in the theory of multivalued m a p p i n g s . In it one c o n s i d e r s different f o r m s of continuity of multivalued mappings, one inv e s t i g a t e s differentiable and m e a s u r a b l e multivalued mappings, one c o n s i d e r s s i n g l e - v a l u e d continuous a p p r o x i m a t i o n s and sections of multivalued mappings, one studies fixed points of m u l t i v a l u e d mappings and other questions of this theory. One gives r e f e r e n c e s to the l i t e r ature r e g a r d i n g applications to the t h e o r y of g a m e s , m a t h e m a t i c a l e c o n o m i c s , the t h e o r y of differential inclusions and g e n e r a l i z e d dynamical s y s t e m s . The p a p e r contains an extensive bibliography.
INTRODUCTION The theory of multivalued mappings is a branch of mathematics which has been developed intensively in the last years and lies at the junction of topology, theory of functions of a real variable and nonlinear functional analysis. The concept of a multivalued mapping, assigning to the points of some set X a subset of another set Y, has arisen in a natural manner by refining the classical concept of a multivalued function. However, for a long time, in the above mentioned chapters of mathematics, one has studied systematically only the single-valued
Translated
from
Itogi Nauki i Tekhnild, Seriya Matematicheskii
0090-4104/84/2406-0719507.50
Analiz, Vol. 19, pp. 127-230,
9 1984 Plenum Publishing C o r p o r a t i o n
1982.
719
mappings and functions; m o r e o v e r , as it is known, p r e s e n t l y , without a special stipulation, the concepts of "mapping" and "function" m e a n s i n g l e - v a l u e d n e s s . Of c o u r s e , this gap can be c o m p e n s a t e d by that s i m p l e cons i d e r a t i o n that a multivalued mapping can be viewed as a s i n g l e - v a l u e d mapping into the family (supplied with some s t r u c t u r e ) of subsets of Y, so that there is hardly a need f o r an independent theory of multivalued m a p pings. N e v e r t h e l e s s , the mentioned a r g u m e n t helps v e r y little in the investigation of the specific p r o b l e m s which a r i s e naturally in this theory. The gradual r e a l i z a t i o n of these p r o b l e m s and the extension of t h e i r collection, the f o r m a t i o n of the fundamental concepts, the development of the n e c e s s a r y m a c h i n e r y , all these have led to the f o r m a t i o n of the theory of multivalued mappings, distinctly noticeable in the last decades. A large role in attracting attention to this theory has played the significant extension of the s p h e r e of its application and methods, which in c e r t a i n a r e a s (especially in the theory of g a m e s , m a t h e m a t i c a l economics and in the t h e o r y of optimal control) have become g e n e r a l l y accepted. In the p r e s e n t s u r v e y we have undertaken an a t t e m p t to sketch the fundamental ideas of the theory of multivalued mappings (possibly with a complete bibliography). In the f i r s t chapter we c o n s i d e r the e l e m e n t s of the analysis of multivalued m a p p i n g s . H e r e we d e s c r i b e the various topologies in the space of closed s u b s e t s and we d e t e r m i n e the f o r m s of the continuity of m u l t i valued mappings. We consider operations with multivalued mappings and we investigate the p r e s e r v a t i o n of the continuity p r o p e r t i e s . Then, we touch the p r o b l e m of the existence of continuous sections and of the a p p r o x i mations of multivalued mappings. We d e s c r i b e the concept of a differentiable multivalued mapping, we indicate the p r o p e r t i e s of m e a s u r a b l e multivalued mappings and we give the definition of a multivalued integral. The second chapter is devoted to t ~ theory of fixed points of multivalued mappings. We consider fixed point p r i n c i p l e s of multivalued mappings, defined on continua and p a r t i a l l y o r d e r e d s e t s , andwe give some g e n e r a l i z a t i o n s of the Banach principle of contraction mappings. A significant p a r t of the c h a p t e r is devoted to the p r e s e n t a t i o n of a p p r o x i m a t i v e and homological methods of the construction of the topological c h a r a c t e r i s t i c s of multivalued mappings and to the investigation, on this b a s i s , of v a r i o u s t h e o r e m s on fixed points. Unfortunately, the increasing volume of the m a n u s c r i p t has f o r c e d us to exclude f r o m the s u r v e y the a l r e a d y written c h a p t e r on the applications of multivalued mappings. Instead, we p r e s e n t a s u p p l e m e n t in which we give only r e f e r e n c e s to the l i t e r a t u r e devoted to s o m e of the fundamental directions of these applications. The authors p r o p o s e to extend l a t e r this s u r v e y by elucidating in detail the p r o b l e m s of the applications. At p r e s e n t , the v a r i o u s a s p e c t s of the theory of multivalued mappings have been v e r y extensively developed and, t h e r e f o r e , the authors do not p r e t e n d in any m e a s u r e the c o m p l e t e n e s s of the s u r v e y . We note that many of these a s p e c t s (this r e f e r s e s p e c i a l l y to sections, m e a s u r a b l e multivalued mappings, etc.) d e s e r v e undoubtedly a s e p a r a t e s u r v e y . CHAPTER THE 1.1.
Space
of Closed
ANALYSIS
OF
i
MULTIVALUED
MAPPINGS
Subsets
Let X be a topological s p a c e . We denote by P(X) the collection of all nonempty s u b s e t s of the s e t X, by C(X) the s e t of all of its nonempty closed subsets and by K(X) the s e t of all of its nonempty compact s u b s e t s . We note that C(X) \ C ( X \ A ) --- {F~C(X) IFNA ~ } . The set C(X) can be c o n v e r t e d into a topological s p a c e by v a r i o u s methods. The m o s t p r e v a l e n t are the following constructions (see, for example, [140]). 1.1.1. Exponential Topology. Space 2 X. The concept of the exponential topology h a s been introduced and investigated in [140, 141,938, 1211, etc.]. Let A0, A, . . . . . A , ~ X . We set B(Ao, A~. . . . . A.) = C ( A o ) N [ C ( X ) \ C ( X \ A ~ ) ] N . . . N [ C ( X ) \ C ( X \ A . ] , i.e., the s e t F lies in B(A0, A~. . . . . A,) if and only if FEC(X), F~A0, FNA~=/=:;2, i=1 . . . . . n. It is e a s y to v e r i f y that the s e t s B (Go. . . . . G~) , w h e r e 60 . . . . . G~ a r e open, f o r m the b a s i s of s o m e topology on the set C(X). The s e t C(X), p r o v i d e d with this topology, is denoted by 2X and is called the space of the closed s u b s e t s of X with the exponential topology.
720
1.1.2. THEOREM. The sets C(A) and C(X)\C(X\A) is open (closed) in X.
are open (closed) in the space 2X if and only if A
We note that when the set G runs through all the open sets of the space X, the sets C(G) and C ( X ) \ C ( X \ 6 ) f o r m aa open subbasis in the space 2X. 1.1.3. Upper Semifinite Topology. Space z X . investigated in [140, 240, 241, 938, etc.].
The concept of the upper semifinite topology has been
An open basis of this topology is f o r m e d by the sets C(G) when G runs through the collection of all nonempty open sets of the space X. The set C(X), provided with this topology, is denoted by z X . Thus, the topology z X is w e a k e r than 2X. We also note that z X is not a T l - s p a c e with the exception of the trivial case when X consists of one point. 1.1.4. Lower Semifinite Topology. Space XX. The concept of the lower semifinite topology has been introduced by Michael [938]. An open subbasis of this topology is f o r m e d by the sets C(X)\C(X\G), when G runs through the collection of all nonempty open sets of the space X. This is the weakest topology in which the sets C(K), where K is a closed set in X, are closed. The set C(X), provided with this topology, is denoted by kX. The space X X h a s a w e a k e r topology than 2X and it is not a T l - s p a c e , also with the exception of the trivial case when X consists of one point. 1.1.5. Hausdorff Metric. Space (2X)m. Let (X, p) be a m e t r i c space. We denote by C0(X) the set of nonempty closed bounded subsets in X. As the distance H(A l, A2) between A~, A2EC0(X) we take the l a r g e s t of the n u m b e r s sup p(x, A2) and supp(x, A1),
xEA,
xGA.
or, which is equivalent, /=I (A1, A2)= tnI {s l e > 0, A1~ U~ (As), A2cU~ (A1)}, where Ue(A) is an e-neighborhood of the set A. The function H satisfies all the axioms of a m e t r i c on the set C0(X) and turns this set into a m e t r i c space which is denoted by (2X)m. The m e t r i c H is called the Hausdorff m e t r i c (or s o m e t i m e s P o m p e i u - H a u s d o r f f ) , after Pompeiu and Hausdorff who have introduced and investigated it in [725, 1020]. T h e r e exists a t h e o r e m which establishes the connection between the topology of the space 2X and the topology of the space (23:)m. 1.1.6.
THEOREM [141]. Let X be a compact m e t r i c space. 2x
i.e., the identity mapping
Then,
.2x.
2x~(2x)~ is a h o m e o m o r p h i s m .
A more g e n e r a l fact is true. 1.1.7. THEOREM [141]. If K(X)c2 x, while (K(X)),~ is the set K(X) with the m e t r i c H, then K ( X ) ~ (K(X))~ for any m e t r i c space X. 1.1.8. Linear Operations on Sets. K X is a topological vector space, then in the set P(X) one can introduce in a natural m a n n e r the operations of addition and multiplication by a real n u m b e r (the s o - c a l l e d Minkowski operations). These operations p o s s e s s the following p r o p e r t i e s :
i)
AI+A2=A2+A1;
2) A~+ (A2+A3) = (A~+A2) +A3; 3) A~+O=A1 , where 0 is the z e r o of the space X; 4) 1,(AI+A2)=kAI+kA2; 5)
~(I~A)= (Mr)A;
6) I.A=A. F o r convex sets we also have the following property: 7) (~+p~)A=~,A+p,A if
~, ~t~>O. 72I
If Y is a subset of a v e c t o r space X, then Pv(Y) denotes the collection of all nonempty convex subsets of Y. If X is a topological v e c t o r space, then Cv(X) is the collection of all nonempty closed convex subsets of X, while Kv(X) is the collection of all nonempty compact convex subsets of X. 1.1.9. Conductivity of Linear Operations. We consider the mapping a : K(X) • generated by the addition of sets, andthe mapping ~ : RxK(X)-+K(X) , generated by the multiplication of a set by a number. 1.1.10. THEOREM. If the set K(X) is endowed with the topology of the space ~X, then the mappings and fl are continuous. 1.1.11. THEOREM. If the set K(X) is endowed with the topology of the space XX, then the mappings a and fi are continuous. 1.1.12. THEOREM. If set K(X) is considered in the exponential topology, then the mappings ~ and fi are continuous. The validity of this t h e o r e m follows f r o m T h e o r e m s 1.1.10 and 1.1.11 since the sets K(G) and K(X) \ K ( X \ G) f o r m an open subbasis of this topology when G runs through all the open sets of the set X. The space of subsets has been investigated f r o m different points of view also in [190, 240, 241,328, 342, 413, 417, 420, 439, 537,648, 666, 729, 732, 931, 970, 1121, 1125, 1151, 1168, etc.]. 1.2.
Continuity
of M u l t i v a l u e d
Mappings
A multivalued mapping F of a set X into a set Y is a c o r r e s p o n d e n c e with a s s o c i a t e s to each point x~X a nonempty subset F(x)~_Y , called the image of the point x, i.e., it is a single-valued mapping F : F-~P(Y) F r o m now on, we shall call e v e r y mapping F : X-+P(F) an m-mapping (from X into Y). Since in the space of the subsets t h e r e exist various topologies, the c l a s s i c a l concept of the continuity of a single-valued function splits into various concepts when applied to multivalued mappings. In 1931-1932, Kuratowski [848] and Bouligand [424] have introduced the concept of semicontinuous multivalued mappings. Let X, Y be topological spaces. 1.2.1. Definition. An m-mapping F : X ~ P ( Y ) is said to be upper semicontinuous at the p6int x~X, if f o r any open set V~Y such that F(x)~V there exists a neighborhood U(x) of the point x such that
F(U(x))~V. An m - m a p p i n g F : X~P(Y) is said to be upper semicontinuous if it is upper semicontinuous at each point
x~X. 1.2.2. Definition. Let D~Y; the set F+~(D) (Fz ~(D)) :
by a small (full) p r e i m a g e of D under the m-mapping F : X.-,-P(Y) we mean
F+~(D)={xixEX, F ( x ) c D } (F-__'(D)= { x f xEX, F (x) N D @ ~5}). 1.2.3. THEOREM. The following statements are equivalent: (a) the m-mapping F is upper semicontinuous; (b) for any open set V ~ Y (e) for any closed set F/~I/
the set F_~~ (V) is open in X; the set ~-1 (W)
is closed in X;
(d) if D~Y, then F_-I(/3)~Fz~(D). 1.2.4. THEOREM, An m - m a p p i n g F : X~+C(Y) is upper semicontinuous ff and only if F is continuous as a mapping f r o m X into ~Y. 1.2.5. Definition. An m - m a p p i n g F :X-*P(Y) is said to be lower semicontinuous at the point xEX, if f o r any open set V ~ Y such that F(x)nV=/=;~, there exists a neighborhood U(x) of the point x such that F ( x ' ) N V # ~ for any x'eU(x) An m - m a p p i n g F : X--*P(Y) is said to be lower semieontinuous if it is lower semieontinuous at each point
xoX. 1.2.6. THEOREM. The following statements a r e equivalent:
722
(a) the m - m a p p i n g F i s l o w e r s e m i c o n t i n u o u s ; (b) f o r any open s e t V ~ Y
t h e s e t F-_~ (V) i s o p e n in X;
(c) f o r a n y c l o s e d s e t W ~ Y t h e s e t F_T~ (W) is c l o s e d in X; (d) if D ~ Y , t h e n F~(b)~F+~(D); (e) if A ~ X , t h e n
F(~)~F(-A).
1.2.7. T H E O R E M . A n m - m a p p i n g m a p p i n g f r o m X into X Y. 1.2.8. D e f i n i t i o n . t o be c o n t i n u o u s . 1.2.9. X into 2 Y.
THEOREM.
F : X-+C(Y) i s l o w e r s e m i c o n t i n u o u s if and only if it is c o n t i n u o u s a s a
If an m - m a p p i n g F : X~P(Y) is both u p p e r a n d l o w e r s e m i e o n t i n u o u s , t h e n it is s a i d An m - m a p p i n g
F : X-+C(Y) i s c o n t i n u o u s if and only if F i s a c o n t i n u o u s m a p p i n g f r o m
A n o t h e r i m p o ~ a n t c l a s s is f o r m e d by the c l o s e d m - m a p p i n g s . 1.2.10.
D e f i n i t i o n . An m - m a p p i n g
F : X-~C(Y) i s s a i d t o be c l o s e d if i t s g r a p h F F = {(x, b') [ (x, g) ~XX Y, b,6F(x))
i s a c l o s e d s e t in X x Y. 1.2.11.
THEOREM.
The f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :
(a) t h e m - m a p p i n g F is c l o s e d ; (b) f o r any p a i r xeX, y~y s u c h t h a t gCF(x) o t h e r e e x i s t s a n e i g h b o r h o o d U(x) of t h e p o i n t x a n d a n e i g h b o r h o o d V(y) of t h e p o i n t y s u c h t h a t F(U(x)) f~V(g) =~; (c) f o r any n e t s {x~}cX, {gJ~Y s u c h t h a t x~-~x, g~eF(x~), g~-+g,we h a v e g~F(x) . In the c a s e of m e t r i c s p a c e s it is s u f f i c i e n t to c o n s i d e r s e q u e n c e s . B e t w e e n c l o s e d and u p p e r s e m i c o n t i n u o u s m - m a p p i n g s t h e r e e x i s t s a c l o s e c o n n e c t i o n . 1.2.12. T H E O R E M . If the m - m a p p i n g F : X-+C(Y) is u p p e r s e m i c o n t i n u o u s and the s p a c e Y i s r e g u l a r , t h e n F is c l o s e d . If the m - m a p p i n g F h a s c o m p a c t i m a g e s : F : X-+K(Y) , t h e n in the g i v e n s t a t e m e n t t h e c o n d i t i o n of the r e g u l a r i t y of Y c a n be r e l a x e d : it i s s u f f i c i e n t to r e q u i r e t h a t it be a H a u s d o r f f s p a c e . 1.2.13. tinuous.
THEOREM.
If F : X-~K(Y) is a c l o s e d , l o c a l l y c o m p a c t m - m a p p i n g , t h e n i t i s u p p e r s e m i c o n -
W e m e n t i o n the f o l l o w i n g p r o p e r t i e s of c l o s e d and u p p e r s e m i c o n t i n u o u s m - m a p p i n g s . 1.2.14.
THEOREM.
If t h e m - m a p p i n g
F:X-~C(Y) is c l o s e d and AeK(X) , t h e n F(A)~C(Y).
1.2.15.
THEOREM.
If t h e m - m a p p i n g
F :X-+K(Y) is u p p e r s e m i c o n t i n u o u s and AeK{X) , thenF(A)eK(Y).
1.2.16.
M a p p i n g s into a M e t r i c S p a c e .
L e t (Y, p ) be a m e t r i c s p a c e .
1.2.17. T H E O R E M . F o r t h e u p p e r (lower) s e m i c o n t i n u i t y of the m - m a p p i n g F : X-+K(Y) i t is n e c e s s a r y and s u f f i c i e n t t h a t f o r any e > 0 t h e r e s h o u l d e x i s t a n e i g h b o r h o o d U(x) of the p o i n t x s u c h t h a t F(x')~U.(F(x)) (F(x)cU,(F(x'))) f o r all x'cU(x). 1.2.18. D e f i n i t i o n . An m - m a p p i n g F:X-~Co(Y) is s a i d t o be c o n t i n u o u s in the H a u s d o r f f m e t r i c ff F is c o n t i n u o u s a s a m a p p i n g into t h e m e t r i c s p a c e (2Y) m . 1.2.19. T H E O R E M . dorff metric.
An m - m a p p i n g
F:X->K(Y) i s c o n t i n u o u s if a n d only if i t i s c o n t i n u o u s in the H a u s -
A s s u m e now t h a t the s p a c e Y i s s e p a r a b l e . L e t {r~}7=1 be a c o u n t a b l e , e v e r y w h e r e d e n s e s u b s e t of Y. E v e r y m - m a p p i n g F : X ~ P ( Y ) d e f i n e s t h e f u n c t i o n s {q~}7=1, q%:X--> R (Pi ( x ) = p (ri, F(x)). 1.2.20. T H E O R E M . F o r t h e u p p e r s e m i e o n t i n u i t y of an m - m a p p i n g F : X-+K(Y) it is n e c e s s a r y , and in the c a s e when X is c o m p a c t it i s a l s o s u f f i c i e n t , t h a t a l l the f u n c t i o n s ~0i be l o w e r s e m i c o n t i n u o u s (in the s i n g l e valued sense).
723
1.2.21. THEOREM. F o r the l o w e r semicontinuity of an m - m a p p i n g /: : X~K(Y) it is n e c e s s a r y , and in the case when X is c o m p a c t it is also sufficient, that all the functions ~i be u p p e r semicontinuous (in the s i n g l e valued sense). Different types and t e s t s f o r the continuity of the multivalued mappings have been studied by m a n y authors (see [4, 7, 11, 14, 15, 19, 131, 132, 133, 153, 240, 241, 356, 377, 378, 381-389, 409,413, 496, 498, 528, 536, 541, 578, 590, 592, 593,605, 621,645, 663, 705, 757, 795, 796, 811, 814, 849, 873, 875, 876, 934, 943, 958, 972, 1002, 1021, 1023, 1024, 1026, 1066, 1069, 1092, 1125, 1140, 1142, 1144, 1151, 1153, 1157]). We mention the s u r v e y s and monographs [19, 140, 141,407, 408, 931, 1151]. The p a p e r [408] contains, in p a r t i c u l a r , the original definitions of Kuratowski and Bouligand but it includes an i n c o r r e c t s t a t e m e n t r e g a r d i n g the equivalence of the concepts of closed and upper semicontinuous multivalued mappings. 1.3.
Operations
on Multivalued
Mappings
In this section we consider v a r i o u s continuity p r o p e r t i e s of multivalued mappings which a r e the r e s u l t s of s e t - t h e o r e t i c , topological or a l g e b r a i c operations on multivalued mappings. Operations on multivalued mappings and t h e i r p r o p e r t i e s have been investigated by Kuratowski [140, 141, 849] and by B e r g e [19, 407]. Let X, Y be topological s p a c e s ; let {Fj}jU, Fj:X-+P(Y)
be a family of m - m a p p i n g s .
1.3.1. THEOREM. (a) A s s u m e that the m - m a p p i n g s Fj a r e upper semicontinuous. finite, then the union of the m - m a p p i n g s j~jFy:X~P(Y),
If the index set J is
J{U is u p p e r semicontinuous. (b) Assume that the m - m a p p i n g s Fj a r e l o w e r semicontinuous. tinuous, (c) A s s u m e that the m - m a p p i n g s Ft:X-+C(Y ) a r e closed.
Then the union U F i is lower s e m i c o n j~s
If the index set J is finite, then the union
U F;:X~C(Y) is closed.
IEJ
1.3.2. THEOREM. (a) Assume that the m - m a p p i n g s F~:X~C(Y) dex set J is finite, the s p a c e Y is n o r m a l and
a r e upper semieontinuous.
If the in-
N Fj(x)--p~, vxEX, JEJ then the i n t e r s e c t i o n of the m - m a p p i n g s
N FI:X~C(Y), ~Es
is upper semicontinuous. (b) A s s u m e that the m - m a p p i n g s F j : X - ~ C ( Y )
a r e closed and
fl Fj(x)=/=Q~, VjEJ. iEJ Then, the i n t e r s e c t i o n
(] F i JEJ
is closed.
The following s t a t e m e n t holds. 1.3.3. THEOREM. A s s u m e that the m - m a p p i n g Fo:X~C(Y ) is closed, the m - m a p p i n g FI:X~K(Y) is upper semicontinuous and
Fo(x) NF1 (x)@ 0,
vxEX.
Then, the i n t e r s e c t i o n
Fo NFI:X---~K (Y) is upper semicontinuous. 724
1.3.4. C O R O L L A R Y . u p p e r s e m i c o n t i n u o u s and
A s s u m e t h a t the s p a c e Y i s H a u s d o r f f , the m - m a p p i n g s {FI}/Ej, F / : X ~ K ( Y )
are
N Fj (x) #- gO, vxEX. JCa Then, the i n t e r s e c t i o n
N F/:X-~K(Y) iCs is u p p e r s e m i c o n t i n u o u s . If t h e m - m a p p i n g s a r e l o w e r s e m i c o n t i n u o u s , then the i n v e s t i g a t i o n of the c o n t i n u i t y of t h e i r i n t e r s e c t i o n b e c o m e s m o r e c o m p l i c a t e d . E l e m e n t a r y e x a m p l e s show t h a t in the g e n e r a l c a s e s u c h an i n t e r s e c t i o n w i l l not be l o w e r s e m i c o n t i n u o u s ( s e e [212]) (in t h i s c o n n e c t i o n w e m e n t i o n the i n a c c u r a c y of s t a t e m e n t s in [ 1 9 , 4 0 7 ] ) . In o r d e r t o e l u c i d a t e the c o n d i t i o n s u n d e r w h i c h one c a n g u a r a n t e e the l o w e r s e m i c o n t i n u i t y of the i n t e r s e c t i o n of m - m a p p i n g s , t h e f o l l o w i n g c o n c e p t w i l l t u r n out t o b e u s e f u l ( s e e [212, 958]). 1.3.5.
Definition.
An m - m a p p i n g
F : X-~P(Y) is s a i d to be q u a s i o p e n at t h e p o i n t
xO.X if
int F (x) =# a n d if f o r any yeint F(x) one c a n f i n d a n e i g h b o r h o o d V(y)cY and a n e i g h b o r h o o d
U(x)cX, s u c h t h a t
V (V) = F (x') f o r a l l x'~U(x) 1.3.6.
. If F i s q u a s i o p e n at e a c h p o i n t x~X , t h e n i t w i l l be c a l l e d q u a s i o p e n .
T H E O R E M [958]. The f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :
(a) t h e m - m a p p i n g F : X-*P(Y) i s q u a s i o p e n ; (b) i n t F ( x ) = # ~
forall
xeX
a n d the g r a p h of t h e m - m a p p i n g
intF:X~P(Y),
(int F) (x) =int F(x) i s open in X x Y. The c o n c e p t of a q u a s i o p e n m - m a p p i n g a l l o w s us to g i v e t h e f o l l o w i n g c r i t e r i o n f o r l o w e r s e m i c o n t i n u i t y . 1.3.7. T H E O R E M [212]. L e t Y be a m e t r i c s p a c e . An m - m a p p i n g F : X ~ P(Y) is l o w e r s e m i c o n t i n u o u s a t t h e p o i n t x E X if a n d only if f o r a n y ~ > 0 t h e m - m a p p i n g F, : X--,P(Y), F,(x) =U,(F(x)) is q u a s i o p e n at the p o i n t x. 1.3.8. T H E O R E M [212, 958]. L e t Y be a f i n i t e - d i m e n s i o n a l t o p o l o g i c a l v e c t o r s p a c e . An m - m a p p i n g F : X ~ Cv(Y) i s q u a s i o p e n a t the p o i n t x s X if a n d o n l y if int F(x) =/=Zf a n d F i s l o w e r s e m i c o n t i n u o u s a t t h e p o i n t x. Now we f o r m u l a t e a c o n d i t i o n f o r t h e l o w e r s e m i c o n t i n u i t y of the i n t e r s e c t i o n of m u l t i v a l u e d m a p p i n g s ([212, 958]): 1.3.9. T H E O R E M . A s s u m e t h a t the m - m a p p i n g Fo:X-+P(Y) is l o w e r s e m i c o n t i n u o u s at the p o i n t x 0 E X, t h e m - m a p p i n g F I : X ~ P ( Y ) is q u a s i o p e n at x 0 a n d a s s u m e t h a t
Fo(x) NF1(x) :/:f~. for all x E X and Fo (xo) NFL (xo) c F o (xo) Nint F, (xo). Then, the intersection
FoNF~ i s l o w e r s e m i e o n t i n u o u s a t t h e p o i n t x o.
1.3.10. C O R O L L A R Y [212]. L e t Y be a f i n i t e - d i m e n s i o n a l t o p o l o g i c a l v e c t o r s p a c e , a s s u m e t h a t the m m a p p i n g F0, F~ : X ~ C v ( Y ) i s l o w e r s e m i c o n t i n u o u s a n d a s s u m e t h a t
Fo(x) nF~ (x) ~ for all x E X and Fo (x0) Nint F 1(Xo) :7~
1.3.11. f o r s o m e x 0 E X.
Then, t h e i n t e r s e c t i o n
725
FonF, : X-~Cv(Y) is lower semicontinuous at the point x 0. We mention that the intersection of m - m a p p i n g s is investigated in certain p r o b l e m s of mathematical economics and the condition 1.3.11 e n s u r e s a strong condition of A r r o w - D e b r e u type (see, for example, [213]). 1.3.12. COROLLARY [212]. Let Y be a m e t r i c space, assume that the m - m a p p i n g s F0, FI : X ~ P ( F ) are lower semicontinuous and that Fo (x) n U, (F, (x)) ::~ for s o m e ~ > 0 at each point x 6 X. Then, the m - m a p p i n g Foil (F,), : X...~P(Y), (FoN(F,).) (x) =Fo(x)NU,(F, (x)) is lower semicontinuous. We c o n s i d e r the continuity p r o p e r t i e s of the compositions of multivalued mappings. Let X, Y, Z be topological spaces. 1.3.13. THEOREM. If the m - m a p p i n g s Fo : X-~P(Y), FI : Y-+P(Z) a r e upper (lower) semicontinuous, then t h e i r composition
F, Fo : X--~P(Z), o
F, ~ Fo(X) =F, (Fo(x)) is upper (lower) semicontinuous. 1.3.14. THEOREM. If the m - m a p p i n g Fo : X'-*K ( Y) is upper semicontinuous and the m-mapping FI:Y --* C(Z) is closed, then t h e i r composition
FI . Fo : X ~ C ( Z ) is closed. Now we consider the operation of taking the C a r t e s i a n product of multivalued mappings. 1.3.15. THEOREM. t h e i r C a r t e s i a n product
(a) If the m - m a p p i n g s
Fo :X.--+P(Y),F1 :X-+P(Z) a r e lower semicontinuous, then
FoXF, : X-,-P(YXZ), (Fo>(F,) (x) =Fo(x) XF, (x) is lower semicontinuous. (b) If the m - m a p p i n g s F0 : X-+C(Y), F, : X.-+C(Z) are closed, then the m-mapping
FoXF, : X--+C(YXZ) is closed; (c) if the m - m a p p i n g s F0 : X--)-K(Y), F,:X---)-K (Z) are upper semicontinuous, then the m-mapping
Fo•
: X-~-K(YXZ)
is upper semicontinuous. We investigate algebraic and some other operations. cal v e c t o r space. 1.3.16.
THEOREM.
Let X be a topological space and let Y be a topologi-
(a) If the m - m a p p i n g s F0, Fl : X-~P(Y)
are lower semicontinuous, then t h e i r sum
Fo+F, : X ~ P ( Y ) (Fo+F,) (x) =Fo(x) -}-F, (X) is lower semicontinuous. (b) If the m - m a p p i n g s F0, F, : X.-+K(Y) are upper semicontinuous, then t h e i r sum Fo+F1 : X-)-K(Y) is upper semicontinuous.
726
1.3.17. THEOREM. (a) If the m - m a p p i n g F : X--.P (Y) is lower semicontinuous and the function t : X-+R is continuous, then the product f.F : X-.+P(Y), ([.F) (x)=/'(x).F(x) is lower semicontinuous. (b) If the m - m a p p i n g F : X---~K(Y) is upper semicontinuous and the function T : X--~R the product
is continuous, then
[.F : X-+K(Y) is upper semicontinuous. Let Y be a topological space. 1.3.18.
THEOREM. An m-mapping
F : X ~ P ( Y ) is lower semicontinuous if and only if the closure 1: : X..+C(Y), P(x) =F(x),
is lower semicontinuous. Let Y be a metric space. 1.3.19. THEOREM.
K the m-mapping
F : X-*P(Y) is lower semicontinuous, then the m-mapping
F, : X---~P(Y), Fe(x)=U~(F(x)) is also lower semicontinuous. Let Y be a n o r m e d space.
1.3.20. THEOREM.
Let F:X-+K(Y) be a compact, upper semicontinuous
m-mapping.
Then, for any
e > 0 the m - m a p p i n g F~:X-+C(Y),
~, (x) = u~ (F (x)), is closed. Let Y be a complete locally convex space. 1.3.21. THEOREM.
If the m-mapping
F:X-*K(Y)
is upper (lower) semicontinuous, then the convex
closure coF:X ~Kv(Y), co F (x) = co (F (x)),
is upper (lower) semicontinuous. We conclude this section by a m a x i m u m t h e o r e m , s o m e t i m e s called the continuity principle of optimal solutions, which plays an important role in the applications of the multivalued mappings to game t h e o r y and to mathematical economics. 1.3.22. THEOREM. Let X, Y be topological spaces, let ~ : X.-,-K(Y) be a continuous m - m a p p i n g and let f : X X Y - + R be a continuous function. Then, the function (p:X-~R,
(x)---- max f (x, y) ~Eo(x) is continuous, while the m - m a p p i n g
F : X-~K(Y),
F(x) = ~ l y ~ r
f(x,
y)--~(x)}
is upper semicontinuous. Among other works devoted to the investigation of the various p r o p e r t i e s of the operations on m - m a p p i n g s , we mention [17, 145, 146, 168, 259,302, 303, 379, 380, 383, 384, 386,387, 417, 590, 592, 614, 849, 876, 1033,
1078, I156]. Various topological properties of m-mappings have been investigated in [5, 6, 16, 17, 19, 42, 50, 84, 123, 136, 139, 144, 153, 154, 158, 178, 185, 186, 240, 241, 248, 260, 262, 282, 287, 288, 314, 316, 341, 348, 355, 374, 434, 469, 486, 514, 515, 520, 533, 535, 546, 587, 596,627,638,643, 644, 662, 665, 696, 719, 731, 732, 751, 752, 763, 768, 777, 785, 794, 797, 810, 812, 834, 854, 887, 898, 901, 912, 959, 960, 961, 973, 990, 997, 1022, 1033, 1034, 1068, 1073, 1075, 1076, 1098, Ii01, 1125, 1134, 1146, 1151, 1154, 1156, 1159, 1162, 1163, 1164, 1165, 1174, 727
1198, 1213, 1216, 1217, 1235, 1238, 1239, 1240, 1243]. The homotopic p r o p e r t i e s of m - m a p p i n g s have been investigated in [432,693, 1061, 1062, 1063, 1171]. We mention that many topological p r o p e r t i e s of spaces can be e x p r e s s e d in t e r m s of multivalued mappings (see, for example, [184, 187, 191, 242, 243, 284, 315, 418, 419, 525, 531, 583, 794, 886, 957, 971, 980, 1237, 1242]). 1.4.
Continuous
Sections
of Multivalued
Mappings
Let X, Y be topological spaces and let F : X--.~P(Y) be an m-mapping. 1.4.1. Definition. A continuous single-valued mapping ~ : X-+Y be said to be a continuous section (selector, sample) of an m-mapping F if f (x) ~F (x) for any x E X. A classical t h e o r e m on the existence of a continuous section of an m - m a p p i n g is Michael's t h e o r e m [940]. Since we have not found a Russian translation of this significant result, having numerous applications, we give it in its entirety. 1.4.2. THEOREM. The following p r o p e r t i e s of a T 1 space X are equivalent: a) X is p a r a c o m p a c t ; b) if Y is a Banach space, then every lower semieontinuous m-mapping F : X-+Cv(Y) has a continuous section. The proof of this t h e o r e m is based on the following statement. 1.4.3. LEMMA. Let X be a p a r a c o m p a c t space, let Y be a n o r m e d space and let F : X-~Pv(Y) be a lower semicontinuous m-mapping. F o r any ~ > 0 there exists a continuous single-valued mapping [, : X-+Y such that f , (x)fiU~ (F (x)) for any x 6 X. Proof. Assume that
u,={xl x~X, v~UdF (x))} for any y E Y. These sets are open by virtue of the lower semtcontinuity of the m-mapping F.
The s y s t e m
{Uy}uEr f o r m s an open covering of the p a r a c o m p a c t space X. Let {Uui}jEs be a locally finite subcovering of this covering, We c o n s i d e r {%J]JU' the partition of unity c o r r e s p o n d i n g to the covering IUvj}jU. We define a continuous mapping f ~ : X ~ Y in the following m a n n e r : A(x)= ~ %j (x).vj. J(~S
It is easy to v e r i f y that f e is the d e s i r e d mapping. 1.4.4. P r o o f of T h e o r e m 1.4.2. 1) F i r s t we prove that a)=~ b). We construct an inductive sequence of continuous mappings {f,~}~'=I, f
k:X-+ Y,
satisfying the conditions:
1.4.5. 11f~+t(x)--f~(x)ll < 1/2"% 1.4.6. f k(x)fiU1/~ (F(x)) f o r each x fi X. The existence of f~, satisfying the condition 1.4.6, follows f r o m L e m m a 1.4.3. If f l . . . . . f k have been already constructed, t h e n f k + l is c o n s t r u c t e d in the following manner. By the induction hypothesis, we have
F~+I(x) = F (x) ClU~-~(A (x)) § O, vxeX. F r o m 1.3.12 it follows that the m-mapping Fk.~:X-+Cv(Y) is lower semicontinuous. Then, f r o m l_emma 1.4.3 there follows the existence of a continuous mapping fk.1:X--+Y such that fk.~ (x)fiU2-k-~ (F,+~(x)) for each xEX But then, f o r e a c h xEX we have 728
1
1
I
IIft,+, (x)-- f k(x)ll < - 7 + - ~ - 7 <.2,_------7 i . e . , c o n d i t i o n t . 4 . 5 h o l d s , and fk+l (x)EU:-~-~ ( F ( x ) ) , i . e . , c o n d i t i o n 1.4.6 h o l d s . F r o m c o n d i t i o n 1.4.5 it f o l l o w s t h a t the s e q u e n c e {f~}~_l c o n v e r g e s u n i f o r m l y t o a c o n t i n u o u s f t m c t i o n f f o r w h i c h , by v i r t u e of t h e c o n d i t i o n 1.4.6 and of t h e e l o s e d n e s s of the i m a g e s of t h e m a p p i n g F , w e h a v e f(x)EF(x) f o r e a c h xEX, i . e . , f is t h e d e s i r e d c o n t i n u o u s s e c t i o n . 2) W e show t h a t b) =~ a). F o r t h i s it i s s u f f i c i e n t t o e s t a b l i s h t h a t e a c h open c o v e r i n g a d m i t s a s u b o r d i n a t e p a r t i t i o n of u n i t y . L e t
~ of t h e s p a c e X
Y=l~ (Gg)={ yI y:~'+ R' VE~l[Y(U)] < oo }. Taking
Ilvll= u(!'~ ,~lv(U)l
, we
convert
Y into a Banach
space.
We
C= {y[ yEY, y(U)>~O, vUE3s
consider
uE~XY(U)=I}
a c o n v e x c l o s e d s u b s p a c e of Y. F o r a l l x E X, l e t
F (x)=CN{YIy&, g(U)=O ;r Clearly,
xeU}.
F(x)ECv(Y) vxEX. We show t h a t the m - m a p p i n g F i s l o w e r s e m i c o n t i n u o u s .
F i r s t of a l l w e s h o w t h a t f o r any gEC, e > 0 t h e r e e x i s t s g'EC s u c h t h a t [ I g - - g ' J ] < e and f o r at m o s t a f i n i t e n u m b e r of U ~ . In o r d e r to f i n d s u c h a yT it is n e c e s s a r y to s e l e c t U, . . . . . g(U3>O, l~
l--e/2, a n d t h e n t o d e t e r m i n e g'EC f r o m t h e
U~Eeg, relations
u~{u~. . . . .
r
v'(u,)=v(u~)+
y,(U~)=y(u~), It is e a s y t o s e e t h a t
g'(U)>O
u~}; 1-~;
i = 2 . . . . . n.
Ily--y'i[<2(1--6)<e.
A s s u m e now t h a t xeX, y & ( x ) a n d t h e r e i s g i v e n ~ > 0. A s s u m e t h a t yV a n d U 1. . . . . Un s a t i s f y the above m e n t i o n e d c o n d i t i o n s . If U =U1N ... NU, , t h e n , due to v(U,) >0, l
by the r e l a -
1.4.7. T h e r e e x i s t e x a m p l e s (see [940]) w h i c h show t h a t t h e c o n d i t i o n s of the c o m p l e t e n e s s of the s p a c e Y, of t h e c l o s e d n e s s a n d c o n v e x i t y of the i m a g e s of t h e m - m a p p i n g F , the c o n d i t i o n of l o w e r s e m i e o n t i n u i t y of the m - m a p p i n g F a r e n e c e s s a r y f o r the e x i s t e n c e of a c o n t i n u o u s s e c t i o n . A t the s a m e t i m e it is e a s y t o g i v e e x a m p l e s of m u l t i v a l u e d m a p p i n g s w i t h c o n v e x i m a g e s , w h i c h a r e n o t l o w e r s e m i c o n t i n u o u s but a d m i t c o n t i n u o u s s e c t i o n s . W e g i v e a new r e s u l t in t h i s d i r e c t i o n , o b t a i n e d by V. D. G e l ' m a n . L e t X be a m e t r i c s p a c e and l e t Y be a c o n v e x , c o m p a c t s u b s p a e e of the B a n a c h s p a c e E. L e t F : X-+Kv(Y) be an m - m a p p i n g , F. (x) = U. (F (x)). F o r e a c h p o i n t x 0 E X we define the s e t L(F) (x 0) a c c o r d i n g t o t h e f o l l o w i n g rule:
L (F)(xo)= N (U ( e>0 ~>0
N F~(x))),
x~U6(xo)
The s e t L(F) (xO m a y be e m p t y f o r s o m e x 0 E X. 1.4.8.
LEMMA.
F o r any p o i n t x0E X t h e s e t L ( F ) ( x ~ is a c o n v e x , c l o s e d s u b s e t of F ( x ~ .
1.4.9. L E M M A . A s s u m e t h a t t h e m - m a p p i n g W : X-~Kv(Y) is l o w e r s e m i c o n t i n u o u s . i n c l u s i o n F(x)~-~(x) h o l d s , t h e n L(F)(x)~_W(x) f o r a n y x E X .
If f o r any x E X the
1.4.10. THE ORE M. In o r d e r t h a t an m - m a p p i n g F, : X--~ Pv (Y) should have a continuous section for any c > 0, it is n e c e s s a r y and s u f f i c i e n t t h a t L(F) (x)=/=~ f o r a n y x E X.
729
We mention that the nonemptiness of the sets L(F) (x) f o r any x E X does not guarantee the p r e s e n c e of a continuous section f o r the m-mapping F. 1.4.11. Example. ditions:
Let F be the m - m a p p i n g of the segment [-1, 1] into R 3, defined by the following conI(x, 1, 0); (x, - - 1, x)l.)l ' x>0, xeQ; [(x, 1, O);(x, - - 1 , - - x x>O, xe[--1, 1]\Q; F ( x ) = l(O, 1, 0); (0, -- 1, 0)I, x=O; /l(x, - 1 , 0 ) ; ( x , 1, x)], x < 0 , x~Q;
I
(l(x, --1, 0); (x, 1, - x ) l , x
Obviously, t{(x, 1, o)},
x > o;
L(F)(x)---~[(O, 1, 0); (0, -- 1, 0)], x = O ; [{(x, -- 1, 0)}, i.e., L (F)(x)4: •
x < O,
for any x E [-1, 1]. However, the m - m a p p i n g F does not have a continuous section.
We c o n s i d e r now the iterates of L:
LO(F) = F,
L" (F) = L (L"-' (F)), n > 1.
We continue this p r o c e s s for each transfinite n u m b e r of the f i r s t type, while f o r a transfinite n u m b e r of the second type we s e t L~ (F) (x) = fl L~ (F) (x). ~,;cr
We shall say that the sequence {L~ (F)} is stabilized at the step ~0 if
L~"(F) (x)-----L~+I (F) (x) f o r any x E X. 1.4.12. THEOREM. In o r d e r that an m - m a p p i n g F : X-~Kv(Y) should have a continuous section, it is n e c e s s a r y and sufficient that the sequence {L~(F)} be stabilized at some (finite or countable) step a0 and that L~o (Y) (x) =/=Z f o r any x E X. 1.4.13. R e m a r k . It p r e s e n t s i n t e r e s t to determine under which additional assumptions on Y is the s e quence {L~(F)} stabilized in a finite n u m b e r of steps. We mention also the following interesting p r o p e r t y of quasiopen m - m a p p i n g s (see [212]). 1.4.14. Assume that X is a p a r a c o m p a c t space, Y is a m e t r i c v e c t o r space, F : X.--~P(Y) is quasiopen and int F(x) EPv(Y), vxeX. Then, F has a continuous section. An extensive l i t e r a t u r e is devoted to the development of Michael's t h e o r e m as well as to other t h e o r e m s on continuous sections (see, for example, [70, 155-158,198-203, 289, 293, 2 9 9 , 3 0 2 , 3 0 3 , 3 2 7 , 3 2 9 , 3 3 0 , 331, 334, 335, 336, 346, 354, 356, 470, 4 7 1 , 4 8 0 , 4 8 7 , 4 9 3 , 5 2 8 , 530, 562, 574, 585, 622, 625,639, 649, 656, 677, 680, 724, 728, 739, 802, 851, 853, 888, 896, 897, 898, 899, 932, 9 3 9 , 9 4 1 , 9 4 2 , 944, 945-947, 955, 969, 970, 979, 981, 992, 1009, 1057, 1104, 1105, 1110, 1125, 1161, 1168, 1169, 1194, 1227, 1245]). We mention the s u r v e y [992] where one also gives some applications of the concept of section. 1.5.
Continuous
Approximations
of Multivalued
Mappings
It is e a s y to see that, in general, upper semicontinuous and closed m - m a p p i n g s do not admit continuous sections. The path to their investigation by means of single-valued mappings is opened by single-valued approximations. Let (X, Pz), (Y, 97) be m e t r i c spaces.
The m e t r i c p i n t h e product space X•
p((x, v), (x', v'))=max{~x(x, x'), o~-(v, v')}-
730
is defined by the equality
1.5.1. Definition. Let F : X ~ P ( Y ) be an m - m a p p i n g . A continuous mapping f ~ : X ~ Y is said to be an s - a p p r o x i m a t i o n of the m - m a p p i n g F if
, where s > 0,
p. ( r j , r e ) = sup p (V, re) <~, v~r~s where
F / , FF are the graphs of the mappings
/~, F.
The p r o b l e m of the existence of s - a p p r o x i m a t i o n s , r a i s e d by yon Neumann [977] and important f o r applications, has been investigated by many authors (see, for example, [83, 488-492, 867, 983, 1112, 1113]). One of the m o s t general r e s u l t s is the following statement. 1.5.2. THEOREM [867]. Let X be a m e t r i c space and let Y be a m e t r i c LCS (locally convex space). Then, for any s > 0, e v e r y upper semicontinuous m - m a p p i n g F : X-~Cv(Y) has an s - a p p r o x i m a t i o n ~, : X-~Y such that f , ( x ) ~ coF (X). We mention the following important p r o p e r t y of e - a p p r o x i m a t i o n s . 1.5.3. THEOREM. Assume that the m - m a p p i n g F:X-+C(Y) is upper semicontinuous and that f ~ / X - , - Y is a sequence of a t - a p p r o x i m a t i o n s f o r F, e i ~ 0 . If {x~}T=I~X, l i m x t = x 0 , and lirnf~t(x~)=V0 , then
VoEF(Xo).
i-~
i~
T h e o r e m s of existence of s - a p p r o x i m a t i o n s , for c e r t a i n c l a s s e s of upper semieontinuous m - m a p p i n g s wlth nonconvex images, a r e contained in [39, 73, 82, 196]. The s o - c a l l e d acute-angled approximations are another suitable class of continuous s i n g l e - v a l u e d approximations. Let X be a p a r a c o m p a e t topological space, let E be a Hausdorff LCS and let E* be its topological conjugate. By (/, y), where I~E*, y~E, we denote the v a l u e / ( y ) . 1.5.4. THEOREM [212, 869, 870]. Assume that the m - m a p p i n g F : X - * C v ( E \ 0 ) is upper s e m i c o n t i n u ous. Then, there exists an acute-angled approximation f o r it, i.e., a continuous mapping ~ : X-.E* such that at each point x E X, f o r ali y E F(x) we have (f(x), y) > 0. The proof of T h e o r e m s 1.5.2 and 1.5.4 can be found in the s u r v e y [37]. 1.6.
Differentiability
of Multivaiued
Mappings
One of the most natural definitions of the differentiability of m - m a p p i n g s is the definition of 7r-differentiability which o c c u r s in [376]. If E is a topological v e c t o r space, then the s e t Kv(E) is a s e m i l i n e a r space (see 1.1.8). We c o n s i d e r l(v~(E) =Kv(E) • Kv(E) and we introduce in this set an equivalence relation according to the following rule:
A+D=B+C. The equivalence class of the element (A, 13) will be denoted by (A, B). Let ~ = K v 2 ( E ) / ~ ; we introduce (A, B ) ~ ( C , O),
if
in t~ linear operations according to the following rule:
(A,B) + (C,D) = (A+C,B+D); a(A,B)~
{(~A,c~B>
<[atB,[a[A }
if if
r a
It is easy to verify that the set E, with the operations defined in this m a n n e r , is a v e c t o r space. Let n : Kv(E)-*E
be the mapping defined by:
n(A),=(A, 0), A~Kv(E), where 0 is the z e r o of the space E. Obviously, this mapping is s e m i l i n e a r , i.e.,
(aA+f3B) =a~(A) +[3~(B), a, 1~>0. If E is a n o r m e d space, then in Kv(E) t h e r e is defined the tIausdorff m e t r i c H, which g e n e r a t e s in/~ a n o r m according to the following rule:
731
II(A, B>II = H ( A , B). Then II--(C, D>II = H ( A + D , B + C ) . It i s e a s y to s e e t h a t t h e m a p p i n g v is an i s o m e t r y , i . e . ,
H(A; B) = [l~ (A)--~(B)11, f o r any A, B6Kv(E). 1.6.1. isometry
THEOREM.
If E is a linear n o r m e d space, then there exist
a linear n o r m e d space ~ and an
~: Kv(E)-~, satisfying the universality condition; i.e., if E l is a linear space and ~p : Kv(E)~gl then there exists a linear operator ~ : E--~EI such that the diagram
is a semiliaear mapping,
Kv (E) -~ E E1 is c o m m u t a t i v e . L e t L, E be l i n e a r n o r m e d s p a c e s , l e t U be a n open s u b s e t in L and l e t F : U-~Kv(E)
be an m - m a p p i n g .
1.6.2. D e f i n i t i o n . An m - m a p p i n g F i s s a i d t o b e 7 r - d i f f e r e n t i a b l e at the p o i n t x 0 E U if the m a p p i n g n o F : U-~/~ i s d i f f e r e n t i a b l e at t h i s p o i n t . IS an m - m a p p i n g F is 7 r - d i f f e r e n t i a b l e a t e a c h x EU, t h e n F is s a i d to be v - d i f f e r e n t i a b l e on U. T h u s , F i s ~ - d i f f e r e n t i a b l e at t h e p o i n t x0E U if t h e r e e x i s t s a c o n t i n u o u s l i n e a r m a p p i n g such that
noF (x) -- noF (Xo)-- d F x, (x -- Xo)= o (ll x-
dFx.:L-->F.
x0 I1).
T h e f o l l o w i n g p r o p e r t i e s of ~ r - d i f f e r e n t i a b l e m - m a p p i n g s a r e o b v i o u s . 1.6.3.
If an m - m a p p i n g F is 7 r - d i f f e r e n t i a b l e at t h e p o i n t x 0 E U, t h e n F is c o n t i n u o u s at t h i s p o i n t .
1.6.4.
If F is ~ r - d i f f e r e n t i a b l e a t t h e p o i n t x 0, then
the
differential
dF2c o
i s d e f i n e d in a unique m a n n e r .
We c o n s i d e r s o m e e x a m p l e s of r - d i f f e r e n t i a b l e m - m a p p i n g s . 1.6.5. L e t r : U-~E be a d i f f e r e n t i a b l e m a p p i n g and l e t AeKv(E) is a ~-differentiable m-mapping and
; t h e n , t h e m - m a p p i n g F(x) = r(x) + A
d F x o ( h X ) = ( r'(xo)hX, 0 ) 9 1.6.6. If g : U - ~ R is a s m o o t h f u n c t i o n of a f i x e d s i g n [ i . e . , g ( x ) > _ 0 o r g ( x ) AEKv(E) , t h e n t h e m - m a p p i n g F(x) = g(x)A i s ~ - d i f f e r e n t i a b l e and
_ 0 f o r any x E U ]
and
d F ~ , ( h x ) ~ g ' x o ( A x } ( A, 0 ) . 1.6.7. T H E O R E M . L e t U be a n o p e n s u b s e t of R m and a s s u m e t h a t the m a p p i n g s ~,~ : U-*R s a t i s f y t h e i n e q u a l i t y f ( x ) < ~ ( x ) , x e U . T h e m - m a p p i n g F : U-~Kv(R) d e f i n e d b y F ( x ) = [ / ( x ) ; if(x)] i s v - d i f f e r e n t i a b l e at the p o i n t x 0 if a n d only if the f u n c t i o n s f and ~ a r e d i f f e r e n t i a b l e at t h i s p o i n t . F r o m the r e s u l t s of [1168] t h e r e f o l l o w s t h e f o l l o w i n g s t a t e m e n t . 1.6.8. T H E O R E M . L e t U be an open s u b s e t of R m a n d l e t F : U-~Kv(R ~) be a n - d i f f e r e n t i a b l e m - m a p ping; t h e n , f o r any p o i n t x 0 E U and any p o i n t Y0 E F ( x 0) t h e r e e x i s t s a d i f f e r e n t i a b l e m a p p i n g ~ : U-~R ~ s u c h t h a t f ( x ) E F(x) f o r any x E U and f ( x 0) =Y0" T h u s , t h r o u g h e a c h p o i n t of t h e g r a p h F F of a r - d i f f e r e n t i a b l e m - m a p p i n g t h e r e p a s s e s a d i f f e r e n t i a b l e s e c t i o n . W e n o t e t h a t a n o t h e r t h e o r e m on the e x i s t e n c e of a d i f f e r e n t i a b l e s e c t i o n i s p r o v e d in [302, 303]. B. D. G e l ' m a n h a s p r o v e d a t h e o r e m w h i c h s h o w s t h a t t h e r e a r e s u f f i c i e n t l y m a n y ~ - d i f f e r e n t i a b l e m mappings:
732
1.6.9. THEOREM. Let L be a linear normed space and let U be an open subset of L. Let (I) : U-+Kv(R ~) be an upper semicontinuous m-mapping; then, for any number ~ > 0 there exists a ~-differentiable m-mapping F: U-~I(v(R ~) , such that
F(x)~q~(x),
(1)
(2) r e c U r ( r e )
f o r any
x~U;
, w h e r e U~(rr
is the e - n e i g h b o r h o o d of the set
p r o o f . F o r each xE U we fix ~-----~(x), 0 < ~ < e
r
so that
P.
in the s p a c e
~ (O(x));
UXR".
such a 5 always e x i s t s by
v i r t u e of the u p p e r semieontinuity of the m - m a p p i n g ~. For
"q(x)=lo(x)
we c o n s i d e r the covering {Un(~)(x)}~Ev of the s p a c e U and we e x t r a c t f r o m it a locally
finite s u b c o v e r i n g {Un(~j)(x])}j U. Let {qgj}]~z be a smooth partition of unity, c o n s t r u c t e d o v e r this covering. We define a mapping F : U - + K v ( R ~) by the relation:
F (x)= ~ % (x). u~(r (x j)). j~l
2
The mapping F is the desired one. Indeed, since the covering {U~(xj)(xj)}j U is locally finite, it follows that f o r any point x 0 E U t h e r e exists an open neighborhood V(xo)~U such that only a finite n u m b e r of functions ~oj are different from zero at the points of this neighborhood. Let these functions be ~J1' ~J2 ..... ~Jk; then k
F (Xo)= ~ q% (xo)u~ (r (xh)). i=[
~= g)h (xo)= 1 and xofiUn(x]~)(xj~), we
Since
The m-mapping
F is ~-differentiable
have r
2
(xo)~Ue
r
(q)(xj~)) and, consequently,
at the point x 0 ~ U since in the neighborhood
V(x 0) we have
F (x) = ~ F~ (x),
Fi(x)=el h (x).UL(Cp(x]~))
where
and Fi(x) is ~-differentiable
(see 1.6.6). It is easy to verify that the con-
2
dition (2) of this theorem
also holds.
We note that the original definition of ~-differentiability [376] has been given for the case when E is a reflexive Banaeh space and the m-mapping F acts in the collection of nonempty, closed, bounded subsets of E. Other definitions of differentiability of m-mappings and some applications have been considered 298, 359, 376, 431, 438, 563, 566, 736, 761, 868, 922, 935, 1168, 1179, etc.].
in [206,
We mention, however, that the theory of the differentiation of m-mappings is not sufficiently developed and has not found yet wide applicability in the theory of multivaiued mappings and in their applications. 1.7. A.
Measurable F.
Filippov's
Multivalued Lemma.
Mappings. The
Superposition
The
Integral. Operator
1.7.1. M e a s u r a b l e Multivalued Mappings. Apparently, the concept of a m e a s u r a b l e multivalued mapping, g e n e r a l i z i n g the c l a s s i c a l concept of a m e a s u r a b l e mapping, has been introduced f o r the f i r s t t i m e in [1010]. Subsequently, the p r o p e r t i e s of m e a s u r a b l e multivalued mappings have been investigated in an entire cycle of w o r k s (see, f o r e x a m p l e , [120-122, 358, 367, 368, 462, 472, 474-479, 482, 485, 526, 552, 556, 557, 567, 570, 575, 661, 736, 742, 749, 750, 753, 755, 760, 761, 767, 769, 774, 779, 780, 876, 880, 934, 982, 1010, 1071, 1072, 1074, 1087, 1088, 1099, 1100, 1196, 1197, 1200, 1201, 1203-1206, 1234, 1251, etc.]). We indicate certain fundamental properties of measurable multivalued mappings, without aspiring to a maximal generality. A detailed analysis of the properties of such mappings and some of their applications to convex analysis and to variational problems can be found in [3, 120, 121, 122, 475, 477, 485, 556, 557, 742, 755, 760, 761, 779, 1071, 1201].
L e t A~R be a m e a s u r a b l e subset; let R be endowed with the Lebesgue m e a s u r e #.
733
1.7.2. Definition. An m - m a p p i n g set F + ' (V) is m e a s u r a b l e .
F : A-~K(R n) is s a i d to be m e a s u r a b l e if f o r any open s e t V-~Rn
the
An equivalent condition is the m e a s u r a b i l i t y of F -1 (W) f o r any closed set W-~R~. 1.7.3. THEOREM. The m e a s u r a b i l i t y of an m - m a p p i n g F : A ~ K ( R ~) is equivalent to the m e a s u r a b i l i t y of the set F+1 (W) [F Z~(V)] f o r any closed s e t W-=l(~ [open s e t V~_R~]. Obviously, an upper or a lower semicontinuous m - m a p p i n g is m e a s u r a b l e . 1.7.4. Definition. A s i n g l e - v a l u e d mapping f : A-~R ~ is said to be a m e a s u r a b l e section of the m - m a p p i n g F:A-+K(R~), if f is m e a s u r a b l e and f(t)~F(t) f o r a l m o s t all t~A. The following t h e o r e m d e s c r i b e s the fundamental p r o p e r t i e s of m e a s u r a b l e m - m a p p i n g s . 1.7.5. THEOREM. F o r an m - m a p p i n g F : A-+K(R n) the following s t a t e m e n t s are equivalent: a) F is m e a s u r a b l e ; b) f o r any point x E Rn with rational coordinates, the function (p is the m e t r i c in Rn).
(~ : A-+l(, ~ (t) = p (x, F(t)) is m e a s u r a b l e
c) t h e r e exists a countable s e t {f,~}~1 of m e a s u r a b l e sections of F such that
d) f o r any i5 > 0 t h e r e exists a closed set A 6 ~ alogue of Luzin's C - p r o p e r t y ) .
such that I~(A\z~8)<6 and FI% is continuous (the a n -
The proof of this t h e o r e m and bibliographic indications can be found, f o r example, in [120-122,475, 779]. M e a s u r a b l e sections of m - m a p p i n g s have been investigated in [3, 102, 150-152, 299, 332, 333, 339,368, 372, 373, 392, 404, 475, 479, 482,485, 493, 516, 554, 572, 575, 581, 589,615, 649, 655,661, 681, 682, 739, 746, 747, 773,774, 792, 804, 817, 850-853, 856, 880, 890) 896, 897, 899,900, 992, 1065, 1067, 1074, 1096-1097, 1102, 1103, 1160, 1169, 1201, 1219]. Applying T h e o r e m 1.7.5 to the operations on m - m a p p i n g s , we obtain the following s t a t e m e n t . 1.7.6. COROLLARY. a) If the m - m a p p i n g s {F1}i%1, F y:A-+ K (R n) a r e m e a s u r a b l e and if er
n Fj (0 6 f o r each t E A, then the i n t e r s e c t i o n Fi:A~K(R" ) j--1
is m e a s u r a b l e ;
b) if the m - m a p p i n g s F0, F I : h - + K ( R " ) a r e m e a s u r a b l e , then t h e i r C a r t e s i a n product FoX FI:A'--~K (Rn X R") and t h e i r s u m Fo--k Fa:h-+ K (R") are measurable; c) if the m-mapping
F:A-+K(R")
is m e a s u r a b l e and f : A ~ R
uct
f.F:A~K(R") is measurable; 734
is a m e a s u r a b l e function, then t h e i r p r o d -
d) if the m - m a p p i n g
F:A--+K(Rn) is
m e a s u r a b l e , then the convex closure
"c-oF:A-+K(Rn) is
measurable.
1.7.7. COROLLARY. If the m - m a p p i n g s F0, FI:A-~K(R~) are m e a s u r a b l e , then the functions
p, (t)=p, (Fo(t), F, (t));
p,:A-+R,
H,:A-+R, H , ( t ) = H (Fo(t), Fl(t)) (where p . is the deviation and H is the Hausdorff metric)
are m e a s u r a b l e .
We mention also the following fact. 1.7.8.
THEOREM. If the m - m a p p i n g s
{Fj}7~,, Fj:A--> K(R n) are m e a s u r a b l e and there exists an m - m a p p i n g
r
n)
such that j=1
Fj (t)~r (t)
f o r all t E A, then the m-mapping
U Fj:~-~/((rn),
1=1
U Fj(t)= ~ F~(t)
j=l
i=1
is measurable.
1.7.9. Integral of a Multivalued Function. By the integral of a multivalued mapping the set of all integrals of the summable sections of F. In other words,
F:A-~P(R n) we mean
F(t)dt=(glg= ~ f(t)dt, f~Li(A), f(t)EF(t) VtEA}~ Clearly, f o r the existence of the integral
IF(t)dt
it is sufficient that the m-mapping
A
m e a s u r a b l e and be m a j o r i z e d by some summable function
z : A~R
F : A ~ K ( R ~)
be
, i.e.,
liP(t) [I= supfllf[I I[eF(t)}
THEOREM. The integral
S F(t)dt is a convex set.
If
F:A->K(Rn) is
m e a s u r a b l e and is m a j o r -
n
ized b y a summable function, then
~ F(t) clt=S ~ F (t)dt. n
n
The p r o o f of this statement can be found in [122]. The integral of a multivalued mapping and its applications have been investigated in [162, 237, 291, 357, 367, 438, 462, 485, 529, 553, 556, 557, 570, 715, 736, 760, 761, 876, 924, 948, 985, 1196, 1199, 1200, 1201, 1218, 1220, 1234, etc.]. 1.7.11. C a r a t h 6 o d o r y Conditions and A. F. FilippovVs L e m m a . that the set A~R is compact. Let E be a Banach space. 1.7.12.
Definition.
We
say that an m-mapping
a) for all fixed x E E the m-mapping b) for almost
F : AXE-.-K(R
E v e r y w h e r e in the sequel we shall a s s u m e
n) satisfies the Carath~odory
conditions
if:
F(., x) : A -~K(R ~) is measurable;
all fixed t ~A the m-mapping
F(t, .) : E-~K(R
~) is upper semicontinuous.
735
An m-mapping F satisfies the strong Carath~odory conditions if it satisfies the condition a) and the condition b') for almost all fixed tE 2x the m - m a p p i n g F(t, .) : E ~ K ( R n ) i s continuous. Both of these definitions are natural generalizations of the CarathSodory conditions for single-valued f u n c tions. We give some p r o p e r t i e s of m - m a p p i n g s which satisfy the Carath~odory conditions. 1.7.13. THEOREM [818]. Assume that the m - m a p p i n g F : A• ~) satisfies the strong Carathgodory conditions. Then, f o r any 5 > 0 there exists a c o m p a c t set A ~ A such that
is continuous. We mention that a s i m i l a r statement for m - m a p p i n g s which satisfy the (simple) Carath6odory conditions does not hold (see a counterexample in [210]). The following statement is a generalization of a result which is important for the t h e o r y of controlled s y s t e m s , proved in its initial f o r m by A. F. Filippov [306]. Subsequently, this statement, known as "A. F. Filippovts l e m m a on implicit fanctions, ~ has been g e n e r a l i z e d in a s e r i e s of works (see [188, 189,472, 475, 589, 613, 743, 767, 769, 778]). 1.7.14. THEOREM. Assume that the m-mapping F : A • ~) satisfies the strong Carath~odory conditions and let U: A ~ K ( R ~ ) b e a m e a s u r a b l e m-mapping. Assume that g : A-~Rn is a m e a s u r a b l e mapping such that g(t)eF(t, U(t)) for almost all tEA. Then, there exists a m e a s u r a b l e section u : A ~ R ~ of the m - m a p p i n g U such that g(t) E F(t, u(t)). 1.7.15. Superposition Operator. E v e r y m - m a p p i n g F : A• ~) defines an o p e r a t o r which a s s o c i a t e s to an m-mapping Q : A-*P(E) the m - m a p p i n g qb : A-+P(R ~) according t o t h e rule
r
=r(~, q(t)).
This o p e r a t o r is called the superposition operator. We describe some of its p r o p e r t i e s . 1.7.16. THEOREM [816]. If the m - m a p p i n g F : A • ~) satisfies the strong Carath~odory conditions, then F is superpositionally m e a s u r a b l e ; i.e., f o r any m e a s u r a b l e m-mapping Q : A-+K(R~) the m mapping ~,
(D(t) =F(t, Q(t)) is m e a s u r a b l e . We mention that the p r o p e r t y of superpositional measruability of an m-mapping F is lost when passing f r o m the strong Carath6odory conditions to the usual ones (see [210]). N e v e r t h e l e s s , it turns out that also in the case of the usual Carath6odory conditions, an m - m a p p i n g p o s s e s s e s definite "good n p r o p e r t i e s . Namely, we have the following statement. 1.7.17. THEOREM [473]. Let E be a Banach space and assume that F : A • ~) satisfies the C a r a th~odory conditions. Let q : A-~E be a m e a s u r a b l e mapping. Then, t h e r e .exists a m e a s u r a b l e m - m a p p i n g S : A-+K(Rn) such that
S(t)~F(t,q(t)) for almost all t E A. CHAPTER FIXED
POINTS
OF
MULTIVALUED
2 MAPPINGS
Let X=-Y, and let F : X--.P(Y) be an m - m a p p i n g . A point x E X such that x E F(x) is said to be a fixed point of F. The set of all fixed points of F is denoted by Fix F. The p r o b l e m s on the fixed points of m - m a p p i n g s , important for applications, have been investigated by many authors, starting with yon Neumann [977], Kakutani [801], Wallace [1221], E i l e n b e r g - M o n t g o m e r y [610], etc. In this chapter we p r e s e n t a survey of the fundamental directions in the theory of the fixed points of m mappings. 736
2.1. on
Fixed Partially
Points
of
Ordered
Multivalued
Mappings
on
Continua
and
Sets
We recall some definitions. tinuum is a Hausdorff space which tinuum which does not contain any separated by a third one is called The following theorems
A compact, connected Hausdorfs space is called a continuum. A Peano conis the continuous image of the segment [0, i]. A dendrite is a Peano conclosed Jordan curves. A continuum in which each pair of distinct points is a tree.
are classical results on the fixed points of m-mappings
2.1.1. THEOREM (Wallace into i t s e l f has a fixed point.
[1221]).
Each
upper semicontinuous
continuum-valued
on trees. m-mapping
of a tree
Capel and S t r o t h e r [468] have p r o v e d this t h e o r e m making u s e of the l a t t i c e o r d e r on a t r e e and of the t h e o r e m on continuous s e c t i o n s . 2.1.2. THEOREM (Plunkett [1019]). A P e a n o continuum X has a f i x e d point f o r any continuous m - m a p p i n g F : X ~ C(X) if and only if X is a d e n d r i t e . 2.1.3. THEOREM (Smithson [1150]). E v e r y l o w e r s e m i c o n t i n u o u s m - m a p p i n g of a t r e e into i t s e l f , with c o n n e c t e d i m a g e s , has a fixed point. Many i n v e s t i g a t i o n s have been devoted to the study of v a r i o u s fixed point p r i n c i p l e s f o r m u l t i v a l u e d m a p pings on continua (see, f o r e x a m p l e , [393, 394, 401, 499, 690, 894, 895, 903, 904, 964-966, 975, 976, 1019, 1080, 1107, 1109, 1111, 1139, 1141, 1143, 1145, 1149, 1221-1225]; we a l s o mention the s u r v e y s [970, 1151, 1208, 1226]). In a s e r i e s of p a p e r s one i n v e s t i g a t e s the f i x e d points of m u l t i v a l u e d m a p p i n g s which p r e s e r v e the o r d e r on p a r t i a l l y o r d e r e d s e t s . We give the following t h e o r e m of S m i t h s o n [1148]. 2.1.4. THEOREM. Let (X, >-) be a p a r t i a l l y o r d e r e d s e t such that each nonempty chain in X h a s a l e a s t u p p e r bound. A s s u m e that the m - m a p p i n g F : X--,-P(X) s a t i s f i e s the conditions: 1) if xl, xp~X; x~<xpand yl~F(xl), then t h e r e e x i s t s 2) t h e r e e x i s t
yp~F(xp)such that YI
e~X and yeF(e) such that e
C~X be a chain and a s s u m e that F has a monotone s e c t i o n f on C. If x 0 = sup C, then t h e r e yo~F(xo) such t h a t f(x)
3) l e t exists
Then, FixF=/= ~ . F o r o t h e r r e s u l t s in this d i r e c t i o n s e e [964, 966, 1151, 1152, 1226]. 2.2.
Fixed
Points
of m - M a p p i n g s
of Metric
Spaces
In t h i s s e c t i o n we c o n s i d e r t h e o r e m s on f i x e d points of m - m a p p i n g s , which a r e f o r m u l a t e d in m e t r i c t e r m s without i n t r o d u c i n g a d d i t i o n a l s t r u c t u r e s . A l a r g e n u m b e r of p a p e r s a r e devoted to this q u e s t i o n s i n c e p r a c t i c a l l y e v e r y t h e o r e m on fixed points of s i n g l e - v a l u e d c o n t r a c t i v e m a p p i n g s has a m u l t i v a l u e d analogue. A n a t u r a l g e n e r a l i z a t i o n of the Banaeh p r i n c i p l e is the following s t a t e m e n t . 2.2.1. THEOREM [967]. Let (X, d) be a c o m p l e t e m e t r i c s p a c e , let Cb(X) be the c o l l e c t i o n of n o n e m p t y , closed, bounded s u b s e t s of X and l e t F : X - - - Cb(X) be an m - m a p p i n g such that
H ( F (x), F (y))~
q, O ~ < q < l .
Then, F I x F ~ O .
The p r o o f of this t h e o r e m c o n s i s t s in the f a c t that, s t a r t i n g with an a r b i t r a r y point x 0 E X, one c o n s t r u c t s a s e q u e n c e {xt}~~ such that a)
xt~F ( xi_l), i--~ l, 2 . . . . ;
b) d(xt, xl+~)~~l, q ~ q , < I. This s e q u e n c e is a Cauchy sequence and
x,...~X . C l e a r l y , ~ is a fixed point of the m - m a p p i n g F .
F o r m u l t i v a l u e d mappings one h a s a l o c a l v a r i a n t of Banach~s t h e o r e m . 2.2.2. Definition. A m e t r i c s p a c e (X, d) is s a i d to be e - c o n n e c t e d if f o r any points x,y 6 X t h e r e e x i s t s a sequence x 0 = x , x 1. . . . . Xn = y , s u c h t h a t d(x i, x i + 1) < g, i = 0, 1 . . . . , n - 1. 737
2.2.3. THEOREM [967]. Let (X, d) be a complete e - c o n n e c t e d m e t r i c space and let F : X ~ Cb(X) be an m - m a p p i n g such that
H(F(x), F(y))
< 1. Then, FixF~=~.
In [540], the principle of contraction mappings is g e n e r a l i z e d to multivalued mappings whose images are nonempty, closed subsets of a m e t r i c space. One has obtained a s e r i e s of t h e o r e m s on the fixed points of m - m a p p i n g s in convex m e t r i c s p a c e s . f o r example, the following statement holds.
Thus,
2.2.4. THEOREM [362]. Let (X, d) be a complete, convex m e t r i c space, let M be anonempty, closed set in (X, d) and let F : M ~ C ( M ) be an m - m a p p i n g such that: a) H(F(x), F(y))
H(F(x), F(y))
b) t h e r e exists A E K(X) such that s o m e subsequence of the sequence {P~(A)}~=~ converges in (K(X), H). Then, Fix F~= ~. Various generalizations of the principle of c o n t r a c t i o n mappings are contained in [345, 347, 361,362, 370, 3 7 1 , 4 2 2 , 5 0 0 , 505, 531, 539, 540, 576, 586, 588, 597, 600, 6 0 1 , 6 3 2 , 654, 658, 698, 700, 701, 745, 770, 772,775, 841, 902, 905, 913, 925, 927, 967, 968, 1040-1043, 1046, 1048, 1050, 1051, 1053, 1089, 1094, 1106, 1118, 1120, 1128, 1129, 1147, 1183]; see also the p a p e r s on nonexpanding m - m a p p i n g s : [423, 599, 845, 862, 881, 882, 883, 884, 914, 918, 926, 933, 1049, 1155]. We also mention the s u r v e y s [112, 598, 968]. 2.3.
Approximative
Methods
2.3.1. Rotation of Completely Continuous Multivalued V e c t o r Fields. Let E be a Hausdorff locally convex space (LCS); let X~_E. E v e r y m-mapping F : X-+P(E) defines an m - m a p p i n g r : X-+P(E), r =x--F(x), called the multivalued v e c t o r field (briefly: m.v.-field) c o r r e s p o n d i n g to F. Denoting by i : X---~E the i m bedding mapping, we shall write ~ = i -F.. If G : X• is a family of m - m a p p i n g s , then W : X• W(x,L)=x--F(x, ~.) is called a family of m . v . - f i e l d s . A point xE X s u c h t h a t 0E~(x) is called a singular point of the m . v . - f i e l d ~. Clearly, the singular points of a m . v . - f i e l d 9 = i - F are flexed points of F and the converse is also true. If Fix F = 0 , then 9 = i - F is nondegenerate, i.e., q~ : X~P(E\O). The family W: XXL~P(E\O) is also called nondegenerate. If the m - m a p p i n g F : X ~ P(E) is upper semicontinuous and compact, i.e., F(X) is relatively compact in E, then both F and its corresponding m . v . - f i e l d ~ = i - F are said to be completely continuous. The collection of all completely continuous m . v . - f i e l d s r =i--F : X--~Kv(E) will be denoted by ~(X) The collection of m.v.-fields $ ~ ( X ) , nondegenerate on X ~ X , is denoted by ~(X, X 0. The collection (X, X) will be denoted by ~0 (X). If L is a topological space, then there is a natural definition (replacing X by X x L) for the complete continuity of a family of m - m a p p i n g s G : X • L--~-Kv(E) and the corresponding family of m . v . - f i e l d s W= i--G. 2.3.2. Definition. The m . v . - f i e l d s ~0, ~,c~(X, X~) are said to be homotopic (r162 completely continuous family of m . v . - f i e l d s (deformation) 738
if there exists a
,I~ : x x [o, l]-+I
Let U ~ E be open. The c l o s u r e and the boundary of the s e t U r = U N T in the r e l a t i v e topology of the s p a c e T will be denoted by ~ T and OUT. For a m.v.-field
r (OUr) one defines an i n t e g e r - v a l u e d c h a r a c t e r i s t i c - t h e rotation
r
Vr (O, OUr) relative to T, p o s s e s s i n g the following fundamental p r o p e r t i e s 2.3.3.
If F ( x ) ~ x o f o r all
(see [29, 31, 33, 37, 40]).
xEOUr, then xoEUr.
2.3.4. Homotopy I n v a r i a n c e . If~0,
r i.e.,
~0T(0UT),
O0~Ol, T
O,E[O0lr , then Vr(O0, 0 U r ) = V r ( r 2.3.5. The quantity
~'r (0, OUr) does not depend on the continuation of ff to UT.
2.3.6. Additive Dependence on the Domain.
r
T, and let
OUr).
Let {Ujr}jEs be a f a m i l y of disjoint s u b s e t s of U T, open in
U r \ U U jr). Then the r o t a t i o n s JEJ
Yr(r OUir) a r e different f r o m z e r o only f o r a finite
n u m b e r of indices j and Vr (O, O U r ) = ~ ~r (O, OUjr). JQJ 2.3.7.
P r i n c i p l e of Mapping R e s t r i c t i o n s .
Let T 1 be a convex, c l o s e d s u b s e t of
E, T1cT . If F(OUr)c-T,,
then
Vr(O, OUr)=~r,(O, OUr,). The application of the concept of r e l a t i v e rotation o f m . v . - f i e l d s to the investigation of fixed points is r e l a t e d to the following p r o p e r t y : 2.3.8,
If O = i - - F ~ r ( O r ,
OUr)and 7r(O, OUr)4=O,
then G ~ F t x F ~ U r . We
describe
briefly the scheme
We start with the "absolute" the m.v.-field CE~ 0(X) , where
of the construction
of the indicated characteristic
case, i.e., we set T = E. is closed.
We
consider
(see [37, 40]).
in the finite-dimensional
space E n
X~E,
2.3.9. Definition. A completely continuous single-valued vector field ~ : X-~E,\O is called a singlevalued homotopic approximation (s.h.-approximation) of the m.v.-field OE~0(X), if ~[~].
739
2.3.10. THEOREM. E v e r y m . v . - f i e l d CE~0(X) has a s.h.-approximation. Indeed, such an approximation ~ can be taken to be an acute-angled approximation (by identifying E~ = En) or an e - a p p r o x i m a t i o n for a sufficiently small e > 0. 2.3.11. Definition. By the rotation 7((I), 0U) of any of its s . h . - a p p r o x i m a t i o n s .
of a m . v . - f i e l d CE~o(OU ) we me an the rotation 7(9, 0U)
A s s u m e now that E is infinite dimensional; let X ~ E
be closed.
2.3.12. Definition. An m . v . - f i e l d r = i -- FE, :X -+ Kv (E \ 0 ) is said to be a finite-dimensional homotopic approximation (f.h.-approximation) of the m . v . - f i e l d (I)@~0(X), if 0~, E[q)] and F ~ , ( X ) ~ E ' , where E' is a finite-dimensional subspace of E. Let U ~ be an open subset, let r and let ~ E ' be an a r b i t r a r y f . h . - a p p r o x i m a t i o n of ~. 2.3.13. Definition. By the rotation y(r OU) we mean the rotation i n t h e s p a c e E ' o f t h e r e s t r i e t i o n o f ~ E, to 0 U ' , w h e r e U ' ~ U O E ' :
9v(O, OU)=~(oe,, OU'). def
We turn now to the general case of relative rotation. solutely convex neighborhood of z e r o .
Let ~ = i - - F E ~ r (OUr). A s s u m e that V is an ab-
2.3.14, Definition. A completely continuous m - m a p p i n g Fv:OU~Kv(T) is said to be V - c l o s e to F on T
if F v ( x ) ~ F ( x ) ~ V for all xEOUr. 2.3.15, ping FV.
THEOREM. F o r any absolutely convex neighborhood V, the m - m a p p i n g F has a V - c l o s e m - m a p -
It is easy to see that the set of z e r o such that V [1r (OUr) = ~ .
r (OUr)
is closed. A s s u m e then that V is an absolutely convex neighborhood
2.3.16. Definition. By the relative rotation tion of any V - c l o s e m . v . - f i e l d r
Yr{r OUr) of the m . v . - f i e l d
qJ~i--F
we mean the r o t a -
vr (0, O U r ) : ~ (or, OU). 2.3.17. ttomotopy P r o p e r t i e s of Noncompact Multivalued V e c t o r Fields. P r e s e n t l y , the investigation of the p r o b l e m s , related to the solvability of inclusions with multivalued o p e r a t o r s , has been r e a l i z e d in a significant degree also for the case of noncompact o p e r a t o r s . We describe the theory of rotations f o r the class of fundamentally contractible multivalued v e c t o r fields, including the condensing and the l i m i t - c o m p a c t fields. This will be p r e c e d e d by the investigation of the homotopy p r o p e r t i e s of such fields. Let E be a Hausdorff LCS, let
X~_E and let L be a compact topological space.
2.3.18. Definition (see [104, 207]). A convex closed set
ing F : X--~K(E) (or for its corresponding m . v . - f i e l d r
T~_E is said to be fundamental f o r the m - r e a p if:
1) F(XflT)~T; 2) f r o m xo6~o (F(xo)t3T) there follows that xo6T. We emphasize t h a t t h e definition does not exclude the case T = ~ e v e r y fundamental set of the m-mapping F contains Fix F.
or
XNT=~J . We also mention that
A convex closed set T ~ E is said to be fundamental f o r the family of m . v . - f i e l d s u/= i--G :X• if it is fundamental f o r each m . v . - f i e l d ~F{., k), k@L f r o m the given family. Examples of fundamental sets are the entire space E and the set
~ F(X).
We d e s c r i b e now the c o n s i d e r e d class of m - m a p p i n g s (see [37, 40, 137, 212, 215]). 2.3.19. Definition. If an upper semicontinuous m - m a p p i n g F : X--~K(E) has a fundamental (possibly empty) set T such that the r e s t r i c t i o n of F to XAT is compact, then F and its c o r r e s p o n d i n g m . v . - f i e l d r = i - F a r e called fundamentally contractible (on T). An upper semicontinuous m - m a p p i n g G :X• such that e v e r y m - m a p p i n g G(-, ~), )~EL , is fundamentally contractible on T and the r e s t r i c t i o n of G to (XNT) • is compact is called a family of fundamentally contractible m - m a p p i n g s (on T).
740
It is c l e a r that a completely continuous m - m a p p i n g F: X - K(E) is fundamentally contractible (on-coF(X)). Other examples a r e l i m i t - c o m p a c t and condensing m - m a p p i n g s (see [37, 207, 1004, 1005]). We give s o m e n e c e s s a r y definitions. F o r an m - m a p p i n g F : X~K(F.) we c o n s t r u c t a transfinite sequence
{DJ of sets in the following manner:
a) Do=co F (X); b) D~=-~F(XAD~-I), if a - 1 exists; c) D ~ = • D~,
if a - 1 does not exist.
13
The sequence ( D , } stabilizes; i.e., there exists an ordinal n u m b e r 6 such that D~=D, for ~>~5 (the proof does not differ f r o m that in the single-valued case; see [265]). The limit set D, is called the limit domain of the values of the m - m a p p i n g F and is denoted by D~. 2.3.20. Definition. An upper semicontinuous m - m a p p i n g r e s t r i c t i o n to the set XnD~ is compact.
F : X~K(E)
is said to be l i m i t - c o m p a c t if its
We recall {see, f o r example, [265]) that a mapping X : 2~-~A , where A is a partially o r d e r e d set and 2E is the collection of all subsets of E, is called a m e a s u r e of noncompactness in E if
~(,o~) =~(~) for any ~ 2 E. A measure of noncompactness X is said to be monotone if ~0, Q ~ 2 E, ~ 0 - ~ imply that ~(~20) <%(f~). A m e a s u r e of noncompactaess X is said to be nonsingular if x({a}U~])=X(~) for any a~s ~ 2 ~ . A measure of noncompactness X : 2E-~A is said to be real if A = [0, oo). A real measure of noncompaethess is said to be a) proper if X(~) = 0 is equivalent to the relative compactness of $2; b) semiadditive if X(~0U~1)=max{x(Q0), Z(~)} for any ~0, ~1G2 E. Extended examples of measures of noncompactness in n o r m e d spaces are the Kuratowski measure of noncompacthess: a(~) = i n f { d l d > O , f~ admits a partition into a finite n u m b e r of sets whose diameters are less than d} and the Hausdorff measure of noncompactness: • = inf{ele>0, ~ has in E a finite g-net}. 2.3.21. Definition. A n upper semicontinuous m - m a p p i n g F :X - ~ K ( E ) or a family of upper semieontinuous m - m a p p i n g s G : ~ f X L - ~ K ( E ) is said to be condensing relative to a measure X of noncompaetness (or Xcondensing) if w e have, respectively,
~(~(~)) ~(~), %(G(QxL)) >~Z(tn) f o r any tncX
that is not relatively compact.
A l i m i t - c o m p a c t m - m a p p i n g is fundamentally contractible since all the sets Da are fundamental. An m - m a p p i n g which is condensing relative to a monotone m e a s u r e of n o n c o m p a c t a e s s is l i m i t - c o m p a c t (see [265]). 2.3.22. Definition. A fundamental set T of an m - m a p p i n g s t r i c t i o n FlXnT is compact is called essential.
Y : X--*-K(E) for which
XF1T=/=~ and the r e -
2.3.23. Definition. If an m - m a p p i n g F : X~K(F.) has an essential fundamental set T, then F and also its c o r r e s p o n d i n g m . v . - f i e l d q~=i--F are called completely fundamentally contractible (on T). 2.3.24. LEMMA. If X=-E is closed and the m - m a p p i n g Y : X-+K(s is condensing relative to a m o n o tone n o a s i n g u l a r m e a s u r e X of n o n c o m p a c t n e s s , then F is completely fundamentally contractible. Let (X, X~), X~-X be a p a i r of subsets of E. E v e r y w h e r e in the sequel we assume that X t is closed. We denote by JF(X, X~), J ~ ( X , X~), ~9~(X, X~) the sets of all m . v . - f i e l d s (I) : X-+Kv(E), 0 ~q)(X~) that are, r e s p e c tively, fundamentally contractible, completely fundamentally contractible, and X-condensing. If X~ = X, then we shall write JC(X), #{~(X), ~)z(X), respectively, while if we c o n s i d e r families of m . v . - f i e l d s , then we also mention the set L of values of the p a r a m e t e r s (J{ (X, X~; L~ etc.). Next, the collections of m . v . - f i e l d s that are fundamentally contractible on a fixed fundamental set T a r e denoted by ~ r , czar and T is a s s u m e d to be essential if we consider the class #/~r.
74t
2.3.25. Definition. The m.v.-fields fundamental set T, in symbols
O0, O,6Atr(X, X , ) [ J e t ( x , X~)] are said to be homotopic relative to a 00 N @i, T
if t h e r e exists a family T 6 ~ r ( X , X~; [0, 1]) [respectively, ~EJZ~(X, X,; [0, 1])] such that T ( . , 0)~O0, T ( . , 1)=O1. The family ,I, is called a deformation, joining 4, 0 and O l" The relative homotopy c l a s s e s of the m . v . - f i e l d o E ~ r [ ~ r] are denoted by
[O]r, [O]r r e s p e c t i v e l y .
2.3.26. Definition. The m . v . - f i e l d s O0, OlE~g(X, X1) [~B(X, X~)] are said to be homotopic, in symbols O0~O1 O0~O~l , if t h e r e exist sequences of m . v . - f i e l d s Q~ ~Jg(X, X1)[J~s(X, X~)], o~
T~, O~i..
1) Qo=Oo, Qn=Ol; 2) Q~Q~+I, o..
We denote the corresponding homotopy class of the field OEJ~ [JgB] by [O] [[O]B]. Examples of fundamentally contractible homotopies a r e the homotopies of condensing and completely continuous m . v . - f i e l d s . Assume that X is closed. The m.v.-fields r O~E~z(X, X0 are said to be homotopic (relative to a m e a s u r e X of noncompactness), in symbols O0~ r if t h e r e exists a family W~(X,X~; [0,1]) s u c h t h a t O ~ ) , (X, X~) relative to X is denoted by [q)]~.
W(-, 0) =q)0, W ( . , 1 ) = q h .
The homotopy class of
Now we formulate the following fundamental t h e o r e m . 2.3.27. THEOREM. If ~=i--F~Jf~(X, X~; L), then for any neighborhood V of z e r o and f o r any essential fundamental set T of the family 9 t h e r e exists a family W=i--G~ ./gBr(X, X~; L• [0, 1])such that 1)
G(x,~.,x)~F(x,X)--FV for all (x,~,x)E(XNT)XLX[O, 1];
2) ~I' (x, ~, x ) = ( 1 - - ~ ) O ( x , )~)+xT(x, ~), w h e r e ~ e ~ ( X , X~; L); 3)
G(XXLX{1})~TNE' , w h e r e E ' ~ E is a finite-dimensional subspace.
We formulate an analogue of T h e o r e m 2.3.27 f o r condensing m-mappings. 2.3.28. THEOREM. Let OED~(X,X1; L) , where (X, X1) is a p a i r of closed subsets and X is a p r o p e r semiadditive monotone and nonsingular m e a s u r e of noncompactness. Then t h e r e exists a family W~D~(X, Xl; LX[0, 1]) s u e h t h a t W ( . , . , 0)=q), W ( . , . , 1)e~(X, X1; L ) a n d t h e field W ( . , . , 1) is finite-dimensional. The construction of the deformations of a completely ftmdamentally contractible m . v . - f i e l d and of a c o m pletely continuous one allows us to prove easily the following statement. 2.3.29. LEMMA. Let O~gr0 (X, X1) N ~gr~ (X, X~), where X1 is closed and To N T~ NX-~ ~ . exists a m . v . - f i e l d q~E~(X, X 0 , such that r162 ToN[r
Then t h e r e
Our purpose now is to d e s c r i b e the relation between the homotopy c l a s s e s of completely fundamentally contractible and completely continuous m . v . - f i e l d s . Namely, we give the conditions under which t h e r e exists a bijeetive c o r r e s p o n d e n c e between these c l a s s e s . 2.3.30. Definition. An m . v . - f i e l d ~ ( X , X1) is said to be 1(2)-fundamentally contractible if 9 p o s s e s ses an essential fundamental set containing any p r e v i o u s l y given point x 0 E X (respectively, pair of points x0, x~ E X). We denote the set of l(2)-fundamentally contractible m . v . - f i e l d s by ~ ( X , Clearly, Jg~ (X, X~)~./g~s(X, X~). If X is a monotone nonsingular m e a s u r e of noncompactness, then
~z (X, X ~ ) ~
(X, X~).
The definition of the homotopy of the m . v . - f i e l d s r r (X, X,) [ ~ (X, X,)l 742
X~) [respectively, ~ ( X ,
X~)].
in the c o r r e s p o n d i n g s e t s differs f r o m Definition 2.3.26 only in one r e s p e c t : we shall a s s u m e that the m . v . fields Q~, O 4 i 4 n , connecting ~0 and ~l, a l s o belong to ~ ( X , X 0 ( ~ ( X , X 0 , r e s p e c t i v e l y ) . The homotopy c l a s s e s of the m . v . - f i e l d ~ in YZ~(X, X~) [/Z~(X, X~)I will be denoted by [r , [O]~ r e s p e c t i v e l y .
2.3.31. LEMMA. Let O0,
O~e~(X,X~)~B(X,X~) and O0~O, . Then O0~O~.
From Lemma 2.3.31 there follows the followingprinciple of bijective correspondence, generalizing the bijection principle of Yu. I. Sapronov for single-valued condensing mappings (see [266]). 2.3.32. THEOREM. The mapping [O]~-+[@] , where (PEJZ~B(X, X,), @@~(X,X~), O@[0]~, [O] is the homotopy class of ~ in ~ (X, X~), is a bij ective correspondence betweenthe sets of homotopyclasses in ~g~(X, X~) and ~(X, X~). -
2
From Theorem 2.3.28 there follows a variant of the principle of bijective correspondence for condensing m.v.-fields: 2.3.33. THEOREM. The mapping [O]z~[O] , w h e r e OE~Dz(X, X~) , X is a p r o p e r s e m i a d d i t i v e monotone and nonsingular m e a s u r e of n o n c o m p a c t n e s s , OE~ (X, X~), OEICJx, [r being the homotopy c l a s s of r in ~ (X, X 0 , is a bijective c o r r e s p o n d e n c e between the s e t s of homotopy c l a s s e s in ~Dx(X, X,) and ~ ( X , X0. 2.3.34. Rotation of Fundamentally Contractible Multivalued V e c t o r F i e l d s . The concept of the rotation of a fundamentally c o n t r a c t i b l e m . v . - f i e l d , d e s c r i b e d in the p r e s e n t subsection, is b a s e d on the t h e o r y of the rotations of c o m p l e t e l y continuous multivalued v e c t o r fields with convex i m a g e s (see 2.3.1).
Let 9 = i - F e ~ ( 9 OU). 2.3.35. Definition. If @E~B(U, OU), then
y(o, U)~0. If O~./ge (U, OU) , then
~(o, 9
ou),
w h e r e OE~ (U, OU) is an a r b i t r a r y c o m p a c t homotopic a p p r o x i m a t i o n of ~. The validity of this definition is justified by the following s t a t e m e n t . 2.3.36. LEMMA [40]. If r fundamental s e t of ~ .
OU) , then 7(0,
0) =Tr(O, OUr)
, where T is an a r b i t r a r y e s s e n t i a l
L e m m a 2.3.36 opens the path to an equivalent definition of the rotation of fundamentally c o n t r a c t i b l e m . v . fields in t e r m s of the r e l a t i v e rotation. In the m a j o r i t y of c a s e s , this definition is m o r e convenient f o r the computation of the rotation of c o n c r e t e m . v . - f i e l d s . 2.3.37. Definition.
Let O ~ ( U ,
6 U ) . If ~ ( ~ ( U , O U ) , t h e n
~(~, 8)=o. def
If oE./KB(O, OU) , then
~(o, e)=~r(o,
nuT),
where T is an a r b i t r a r y e s s e n t i a l fundamental s e t of the m . v . - f i e l d ~ (see [37, 40, 207, 215]). Now we d e s c r i b e the p r i n c i p a l p r o p e r t i e s of the rotation we have introduced. 2.3.38. Homotopy I n v a r i a n c e . If O0, O~E~(U, 0U)
and
O0~O~ , then Jg
~(0o, ~)=v(o,, 9 P r o p e r t y 2.3.38 admits a p a r t i a l c o n v e r s e , the following analogue of Hopf's c l a s s i f i c a t i o n t h e o r e m . 2.3.39. Let U be an absolutely convex open s u b s e t of the m e t r i z a b l e LCS E such that i n t e r s e c t i o n of U with any f i n i t e - d i m e n s i o n a l s u b s p a c e E' is bounded in E ' . If for the m . v . - f i e l d s r O , E J ~ ( U , OU) we have u U) =Y(O1, U) , then OoNOl. ate If, under the a s s u m p t i o n s of 2.3.39, the m . v . - f i e l d s ~0, ~1 a r e condensing, then it would be i n t e r e s t i n g to find out in which c a s e will they be homotopic in the c l a s s of condensing m . v . - f i e l d s . Taking into account T h e o r e m 2.3.28 and also the f a c t that e v e r y c o m p l e t e l y continuous homotopy is condensing r e l a t i v e to a p r o p e r m e a s u r e of n o n c o m p a c t n e s s , we obtain the following s t a t e m e n t . 743
2.3.40. Let U be as in 2.3.39. Let r 0I E~Dz(U, OU) , w h e r e )/ is a p r o p e r semiadditive monotone and nonsingular m e a s u r e of n o n c o m p a c t n e s s and a s s u m e that V(r d ) ~ 7 (r U). Then r 1 6 2 z 2.3.41. Additive Dependence on the Domain. Let {Uj}j~J be a family of open disjoint subsets of U and let r (U, U \ [J U j) 9 Then the rotations ~ (r bYj) a r e different f r o m z e r o only for a finite n u m b e r of inJEs dicesjEJand ~(r162 U j). J~J Now f r o m the p r o p e r t y 2.3,8 and Definition 2.3.37 t h e r e follows the following i m p o r t a n t 2.3.42.
Fixed Point P r i n c i p l e .
If r
J~B(O, OU) and
7(@, 0 ) : ~ 0 , then O = ~ F i x P ~ U .
In conclusion, we c o n s i d e r the p r o b l e m of the dependence of the rotation of a fundamentally contractible m . v . - f i e l d on the values taken by it inside the domain. Since f o r the proof of the validity of the definition of rotation ( L e m m a 2.3.36) one m a k e s use of i n f o r m a t i o n on the m . v . - f i e l d defined on the entire domain ~ , in g e n e r a l the rotation of a fundamentally c o n t r a c t i b l e (as well us l i m i t - c o m p a c t o r condensing) m . v . - f i e l d is defined on U. It turns out that the 2-fundamentally contractible m . v . - f i e l d s f o r m the c l a s s f o r which the r o t a tion can be c o r r e c t l y defined f r o m the values t a k e n only on the boundary 0U of the domain. Indeed, if Cede, 2(OU) and r ~(0U), r [r ,~ is a c o m p a c t homotopic a p p r o x i m a t i o n of r then, by v i r t u e of T h e o r e m 2.3.32 on the bijection of the homotopy c l a s s e s , the rotation 7(r OU) can be c o r r e c t l y defined as 7(~, OU). In the c a s e when C e ~ 2 ( U , 0 U ) , t h i s definition of the rotation coincides with the one given before. This m e a n s , in p a r t i c u l a r , that f o r a 2-fundamentally c o n t r a c t i b l e m . v . - f i e l d r aU), the rotation 7(r 0) does not depend on the b e h a v i o r of the m . v . - f i e l d r on U. This fact, applied to condensing m . v . - f i e l d s , m e a n s that f o r a m . v . - f i e l d , condensing relative to a m o n o tone nonsingular m e a s u r e of n o n c o m p a c t a e s s , the rotation can be defined on the boundary of the domain and does not depend on the continuation to its inside. We mention, however, that the fact that the rotation 7 ( r U) is independent of the continuation can be e s t a b l i s h e d a l s o f o r an m . v . - f i e l d r condensing r e l a t i v e to a monotone s e m i a d d i 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 . The proof of this fact, taking into account that the l i n e a r d e f o r m a tion of X-condensing m . v . - f i e l d s will be a X-condensing f a m i l y , does not differ f r o m the s i n g l e - v a l u e d c a s e (see [265]). 2.3.43. T h e o r e m s on F i x e d Points. The theory of rotations developed in the previous section allows us to give a s i m p l e and g e o m e t r i c a l l y intuitive p r o o f of a s e r i e s of fixed point p r i n c i p l e s for noncompact m - m a p -
pings. E v e r y w h e r e in the sequel U is an open subset of the Hausdorff LCS E. F i r s t we c o n s i d e r the following g e n e r a l i z a t i o n of R o t h e ' s t h e o r e m . 2.3.44. THEOREM. Let U be convex, a s s u m e that the m - m a p p i n g m e n t a l l y c o n t r a c t i b l e and let
F : G-*Kv(E) is completely funda-
P(x)nC+~ f o r all
xoOU. Then
FixF=~;
if FixFfiOU=~, then 7(i~F,
0):~1.
B o r d e r i n g on T h e o r e m 2.3.44 is the following s t a t e m e n t which g e n e r a l i z e s the c l a s s i c a l t h e o r e m of Kakntani [801]-Glicksberg [660]-Ky Fun [623] as well as a s e r i e s of fixed point p r i n c i p l e s f o r condensing and l i m i t c o m p a c t s i n g l e - v a l u e d mappings (see, f o r e x a m p l e , [265,544, 744, 1005]). 2.3.45. THEOREM. If M is a convex closed subset of E and if the m - m a p p i n g F : M-*Kv(M) is c o m p l e t e l y fundamentally contractible, then Fix F=/=;~. 2.3.46. THEOREM. Let the 1-fundamentally contractible m - m a p p i n g F : ~7-*Kv(E) be such that f o r the corresponding m.v.-field r one has at e a c h point x6OU : r (x) ('1Lxa = Q,
w h e r e a~U, Lxa={z]zEE, z ~ ( x - - a ) ,
~<0}. Then FixF:/:~D.
If F i x F N O U = ~ , t h e n v ( q ) ,
U)=l.
2.3.47. COROLLARY (Generalization of S c h a e f e r ' s T h e o r e m ) . Let 0~U and a s s u m e that the 1-fundam e n t a l l y contractible m - m a p p i n g F: U -~ Kv (E) is such that at each point x~OU we have F (x) D kx = | f o r a l l X > 1 . Then F i x F = ~ D . If F i x F O O U ~ , then v ( i - - F , / ~ ) ~ l . 2.3.48. COROLLARY. Let 0~ U, U is convex and a s s u m e that the 1-fundamentally contractible m - m a p p i n g F : tT-*Kv(E) is d i r e c t e d inside 8U, i.e., F ( x ) ~ l ~ (x) =- {yly~E, x+k(y--x)~U f o r s o m e ~ > 0 }. Then FixF:/=~. If FixFflOU=~ , then v(i-- F, ~ = l . 744
Now we formulate the odd field t h e o r e m f o r fundamentally contractible m . v . - f i e l d s . 2.3.49. THEOREM. Let U be a s y m m e t r i c neighborhood of z e r o and a s s u m e that the m - m a p p i n g F : 0-+ Kv(E) is completely fundamentally contractible and p o s s e s s e s an essential fundamental set which is s y m m e t r i c relative to z e r o . A s s u m e that for the m . v . - f i e l d ~ = i - F one has (x) n ~ r ( - - x ) =
f o r all xEOU and 0 _< p <__1. Then ~(09, 0 ) - - 1 (mod 2) and, consequently, ~=/=FixF
f o r all xEOU and 0 _< # __ 1. Then 7(~, U)----1 (mod2) and, consequently, ~=/=FixFcU. 2.3.51. THEOREM (Generalization of B r o w d e r ' s Theorem). Let U0, Ui, U be open subsets of E; OocU~ ~ O 1 ~ U ; U 0 and U are convex. Let O = i - - F E ~ t ~ ( U , 3U~) and assume that f o r some integer m _> 1 the following conditions hold:
1) 2)
U FJ (D~)~U;
l~<j<.ra--1
U
FJ(Uo)~U~;
l~<j~
3)
F~(UI)=Uo.
Then y(r
O,)=1
and, consequently, ~ @ F i x F ~ U 0 .
T h e o r e m 2.3.51 g e n e r a l i z e s the results established in the single-valued case by B r o w d e r and Yu. G. B o r i s o v i c h [30, 441]. We also mention that the t h e o r e m s of this subsection g e n e r a l i z e the fixed point t h e o r e m s f o r condensing, l i m i t - c o m p a c t and fundamentally contractible m - m a p p i n g s , considered in [207, 212,215, 1005]. The complete proofs of the statements of this subsection can be found in [37, 40]. In the s u r v e y [37] one can also find detailed bibliographic indications regarding the development of approximative methods in the t h e o r y of fixed points of m - m a p p i n g s . T h e r e f o r e , we r e s t r i c t ourselves to Table 1 which illustrates the p r o c e s s of the construction of the topological degree (rotation) f o r m.v.-fields with convex images. Lately, one has developed the theory of the degree of F r e d h o l m o p e r a t o r s p e r t u r b e d by a condensing m mapping (see, f o r example, [32, 34, 94, 185]). 2.4.
Homological
Methods
in the
Theory
of Fixed
Points
2.4.1. The Topological Characteristic of Multivalued Mappings. First we make some preliminary rem a r k s . L e t X , Y be a r b i t r a r y sets , let F:X-,-P(Y) be an m - m a p p i n g and let F ~ X x Y be the graph of F. We c o n s i d e r the mappings t : F~-+X, r : F f + Y , which are the r e s t r i c t i o n s to FF of the natural projections Pr~ : X X Y-~-X, Pr, : XX Y-+Y. Obviously, f o r any xE X we have the equality F (x) =r o t -1 (x). Thus, every m - m a p p i n g F :X-+P(Y) defines a quintuple (X, Y, re, t, r) of objects. The c o n v e r s e is also true" if a quintuple (X, Y, Z , f , g) is given, w h e r e X, u and Z are a r b i t r a r y sets and f : Z-+X, g :Z--+.Y are mappings, w h e r e f is s u r j e c t i v e , t h e n t h e equality g(x)---go [<(x) defines an m - m a p p i n g F : X - + P ( Y ) . I r i s easy to see that one and the same m - m a p p i n g can be defined by different quintuplets. The quintuple (X, Y, FF, t, r) is called a canonical r e p r e s e n t a t i o n of the m - m a p p i n g F. A s s u m e now that u is a v e c t o r space, X ~ Y , and F :X -+P(Y) is aa m-mapping. We consider the m . v . field ~ = i - F. Obviously, the field ~ is defined by the quintuple (X, Y, F~, t, t--r) ; we call this r e p r e s e n t a tion canonical f o r the m . v . - f i e l d ~. Let U be a domain in the finite-dimensional Euclidean space E n+l. We consider the m - m a p p i n g
F:U-~P(En+I),
FixF D L = Q, 745
TABLE i. Topological Degree of m-Mappings with Convex Images
~
acc
Clas of
mappings
~
Completely continuous
1
Banach
Frechet space
I zrailevich [116]; Ma [891]
Granas [683, 684]; Bofisovich, Gel'man, Mukhamadiev, Obukhovskii [85, 36]; CelHna, Lasota [495]; Hukuhara [759]
Condensing
Obukhovskii [20~3
Limit-compact
Webb [1282]; Vanderbauwhede [1207]
Petryshyn, Fitzpatrick [1004, 1005]
L = aU. We
Let
P~(F)
and Fz(F)
1
I
Obukhovskii [212];[ Borisovich, Obukhovskii, [ Obukhovskii Gorokhov [215] [ [37, 40]
Fundamentaily contractible
where
HausdorffLCS
be the graphs of the m-mappin4~
F over U and L, respectively.
consider the m.v.-field 9 = i - - F : O -~P (E"+9.
Since the m . v . - f i e l d ~ does not have singular points on L, it follows that t h e r e is defined the mapping of
pairs
t--r:(r~(F), rL(F))-+(E .+', E.+I\0), which induces a h o m o m o r p h i s m (t - r)* of the A l e k s a n d r o v - C e c h cohomology groups of these spaces. We consider the composition ~ of the h o m o m o r p h i s m s : " 6 H n (En+'\O, G) { t - - r ) n+Hn (Pz(F), G)-+H"+I(Fv(F), I'z(F), G),
2.4.2.
where 6 is the connecting h o m o m o r p h i s m of the exact sequence of the pair (F-D(F), Fz (F)), and G is the coefficient group. 2.4.3. Definition. The topological c h a r a c t e r i s t i c of the m . v . - f i e l d @ on the domain ~ is the h o m o m o r p h i s m • = ~o(t
- -
r).*.
We say that the topological c h a r a c t e r i s t i c of an m . v . - f i e l d is equal to z e r o (different f r o m zero) i f ~ is the z e r o (a nonzero) h o m o m o r p h i s m . It is convenient to introduce the notation •215 L/). 2.4.4. THEOREM. If
~((P, LV)=/=0,then ~W=FixF~U.
We give the formulation of certain t h e o r e m s regarding the computation of the topological c h a r a c t e r i s t i c . 2.4.5. Definition. We say that the m . v . - f i e l d r = i - F does not have opposite orientation on L to that of the continuous single-valued field ~ = i - f if
(x) n ( - - ~ ( x ) ) = ~ , f o r any xeL, ~>0. 2.4.6. THEOREM (An Analogue of the P o i n c a r 6 - B o h l T h e o r e m ) . A s s u m e that the m . v . - f i e l d ~ is nondegenerate and does not have opposite orientation on L to that of the continuous v e c t o r field ~ = i - f . Then • ~) = ~ ot.*o ~.*, tPn9
t*n
6
HnCEn+~\O, G)--+Hn(L, G)-~H.crz(F), G) -+/-/~+~(r~CF), rgF), O).
746
F r o m this t h e o r e m we obtain as a c o r o l l a r y a simple s t a t e m e n t which is i m p o r t a n t in the t h e o r y of fixed points. Let B be the closed unit ball in E ~*~, S = OB. 2.4.7. THEOREM [75]. Let F:B-~P(E n+1) be an m - m a p p i n g without fixed points on S and let F(S)~B. In o r d e r that •162 B)~#0 it is n e c e s s a r y and sufficient that the composition 6otn* of the h o m o m o r p h i s m s
H~(S, Q)-+//~(rs(F), O)-,/-/~+I(rB(F), rs(F), 0 ) should not be z e r o . We also have the following analogue of the odd field t h e o r e m . 2.4.8. THEOREM. A s s u m e that the m - m a p p i n g F : B ~ P ( E n+~)is defined by the canonical quintuple (B, E ~+I, PB(F), t, r), and the h o m o m o r p h i s m 6ot,* t*
5
Hn(S, z~)~H~(rs(F), Z2)-+H~*'(r,(F), rs(F), Z2) is not z e r o . Suppose that t h e r e e x i s t s an upper semicontinuous m . v . - f i e l d conditions:
~F:S---~K[En+I\0), satisfying the following
I) there exists a quintuple (S, En+~\0, Z, f, g) defining the field@, and f:Z-* i.e., the set f-l(x) is acyclic for any xES;
2) (i--F) (x)=-T(x)
f o r any xES;
3) T(x)N~T(--x)=D Then ism
S is a Vietoris mapping,
f o r any x~S; X>O.
x(i--F, B)=/=O.
AS we can see, in Theorems 6 ~ tn*.
2.4.7 and 2.4.8 there occurs the condition of the nontriviality of the homorph-
A sufficient condition for this, by virtue of the Vietoris-Begle-Sldyarenko is that t be an n-Vietoris mapping (see 2.4.10 below). Another
theorem
(see [271, 399, 1212]),
condition is given by the following statement.
2.4.9. THEOREM. Let X be a compact metric space and let F:B ~ K(X) be an upper semicontinuous mmapping. If there exists a sequence of numbers {e~}~=~ such that e~0, [ime~-~0, and if for any ei the mi--~oa
mapping
F admits a single-valued
~i-approximation t* n
fi' then the composition
6eta* of homomorphisms
6
~(s, O)-~Hn(r~(F), O)-~H~+'(r~(f), rs(f), 0) is nonzero
for any nontrivial group G.
In conclusion we formulate
some
theorems
on the fixed points of m-mappings
Let X, Y be metric spaces and let f:Y-* X be a continuous points at which the preimages fail to be k-acyclic, i.e.,
mapping.
We
F.
consider the set
A4k(f)~X
of
A4o(/)={xl x~X, I-IO(f-,(x), 0)~0}, M,(y)={xlxEA', Hn(f-'(x), G)v~0}, k > 0 . 2.4.10. Definition [674]. A mapping f : Y ~ X is called an n - V i e t o r i s mapping, n _> 1 (relative to the c o efficient group G) if the following conditions hold: (1) f is p r o p e r and s u r j e c t i v e ; (2) rdx(M~([) )
rdx denotes the relative dimension
2.4.11. THEOREM isfies the conditions:
(An Analogue
in X.
of RotheTs Theorem).
Assume
that the m-mapping
F : B~K(En+O
sat-
1) F(S)~B; 747
2) there exists a quintuple (B, g~+~, Z,
f,
g) such t h a t f
go ~:-~(x) =F (x)
is an n - V i e t o r i s mapping and
for all
x~.B.
Then, FixF=/=~. 2.4.12. THEOREM. Let F : B..*K(E ,~+i) be an upper semicontinuous m - m a p p i n g such that t h e r e exists a sequence of positive n u m b e r s
{eiIi~,
F(S)cB
. If
lim~i=0, such that for any e i the m - m a p p i n g F admits
i-+0o
a single-valued el-approximation, then FixF4:2L 2.4.13. THEOREM. A s s u m e that the upper semieontinuous m - m a p p i n g F : B--,-K(E ~§ lowing conditions:
satisfies the foI-
1) there exists a quintuple (B, E ~+~, Z, f, g) such that f is an n - V i e t o r i s mapping and
g o ~-i(x)~F(x) and-x
for any
x~B;
2) for each point xES there exists a linear functional ~x, defined on E n+l, separating the sets x - F(x) - F(-x). Then FixF=# ~.
2.4.14. THEOREM. A s s u m e that F : B-+K(E .+~) is an upper semicontinuous m-mapping, satisfying the following conditions: 1) for each point xES there exists a linear functional ~x, defined on E n+l, separating the sets x - F(x) and - x - F ( - x ) ; 2) there exists a sequence of positive numbers {et}~=l, lim s~=0, such that for any e i the m-mapping
g admits an el-approximation
fi on B.
Then FixF=/=~, The topological c h a r a c t e r i s t i c of an m . v . - f i e l d has been introduced and investigated in [37, 74, 75]. This concept is generalized to the case of completely continuous m.v.-fields in infinite dimensional spaces (see [37, 76|). We describe briefly the connection between the concept of topological characteristic of the topological degree, which have been developed by a series of authors. Let X, Y be metric spaces; let F:X~K(Y) ages
be an m-mapping.
We
and certain theories
consider the sets of nonacyclic
im-
of F:
M o ( F ) = { x l x e X , ttO(F (x), Z):#Z}, M , ( F ) = { x l x ~ . X , Hk(F(x), Z)0=/=}, /~>1. 2.4.15. Definition. An upper semicontinuous m-mapping F : X ~ t ( ( Y ) rdxM~(F) 0.
is s a i d t o be n - a c y c l i c (n - 1) if
If F is 1 - a c y c l i c , then the image F(x) of each point xEX is acyclic. Such m - m a p p i n g s are called acyclic. Let B be the closed unit bah in En+l, let S = 8B and let F:B --- K(E n+l) be an n - a c y c l i c m-mapping. 2.4.16.
LEMMA. The diagram t'* n
H.(B)~.(S)
6'
-~ H . ' ( B , S)
t* n
~.
is commutative
5
(rB (FI)-~ [t. (rs (F))-~ g"*' (r~ (F), r~ (F))
and all its vertical arrows
are isomorphisms.
The last statement follows f r o m the V i e t o r i s - B e g l e - S k l y a r e n k o t h e o r e m . If Fix F • S = ~ , then there is defined the mapping
748
(t - - r): Ps ( F ) - > E ~+t \ 0.
We consider the composition ~ of the h o m o m o r p h i s m s in the sequence {t--r) +
t*
-t
Hn (En+,/o)--~H,, (rs (F))(tU+H n(S). In Hn(En+l\O) and Hn(S) we choose identical orientations (i.e., g e n e r a t o r s z 1, z 2 of tile groups H n (E~+~\0) and H ~ (S) such that file h o m o m o r p h i s m induced by the imbedding of S in E~+t\0 c a r r i e s z 1 into z2). Then ~(zl)=k.z2 , where keZ. 2.4.17. Definition. The rotation of an n - a e y e l i e m . v . - f i e l d ~ = i - - F
on S is the n u m b e r Y(q),S)--k.
The K r o n e c k e r index mod 2 f o r acyclie m . v . - f i e l d s in a finite-dimensional space has been c o n s t r u c t e d by Granas and J a w o r o w s l d [686]. This p a p e r has been p r e c e d e d by ffaworowstd's p a p e r [787] in which p r o p e r t i e s of a h o m o m o r p h i s m g e n e r a t e d by an m - m a p p i n g have been investigated. In [38] these methods have been developed for m - m a p p i n g s whose images under file r e t r a c t i o n s of E n + , \ 0 to S turn into acyclic sets and one has defined the rotation of the corresponding m . v . - f i e l d s on the boundary of a simply connected domain. The rotation (toplogical degree) of n - a c y c l i c m.v.-fields has been defined and used in [427, 669]. This rotation p o s s e s s e s many usual p r o p e r t i e s : it is p r e s e r v e d under homotopies in the class of n - a c y c l i c m . v . fields, it is the algebraic n u m b e r of the singular points, etc. We d e s c r i b e the relation between the topological c h a r a c t e r i s t i c and the rotation of n - a c y e l i c m . v . - f i e l d s . 2.4.18. THEOREM (see [37]). Let rD=i--F : B-,.K(E n-m) be an n - a c y c l i c m.v.-field, not degenerated on S; let z I be a generating e l e m e n t of the g r o u p H~(E,~+~\O) . Then there exists a generating elarnent z3~H~+'(FB(F), rs(F)) such that • =~(q~,S).z~ We consider a wider class of m . v . - f i e l d s . 2.4.19. Definition. An m - m a p p i n g F:X ~ K(Y) is said to be g e n e r a l i z e d n - a c y c l i c if t h e r e exists a quintuple (X, Y, Z, f , g) such that the following conditions hold: 1) f:Z -+ X is an n - V i e t o r i s mapping, n >_ 1; 2) g(f-i(x))~_f(x)
f o r all xEX.
The rotation of a g e n e r a l i z e d n - a e y e l i c (n-admissible) m . v . - f i e l d is defined as a certain set r(q~, s) of integers (see [459, 674]). Many t h e o r e m s on the rotation of n - a c y c l i c m . v . - f i e l d s are g e n e r a l i z e d in a natural m a n n e r to this concept (see s a m e r e f e r e n c e s ) . 2.4.20. THEOREM (see [37]). If r(q~,s)4={o}, then the h o m o m o r p h i s m x is nonzero. 2.4.21. In the infinite dimensional ease, the definition of an n - a e y c l i e m . v . - f i e l d can be modified s o m e what, obtaining a m o r e convenient f o r m . 2.4.22. Definition. An upper semicontinuous m - m a p p i n g 1) Mk(F) = ~
F:X~K(Y)
is said to be a l m o s t aeyclie if:
for all k>~k0~>0;
2) rnaxrdxMk(F)< c~. O~:k~:ko
The rotation (degree) of a l m o s t acyelic m . v . - f i e l d s has been c o n s t r u c t e d by Bourgin [429, 430]. Also this concept is connected with the topological c h a r a c t e r i s t i c : it is different f r o m z e r o simultaneously with the r o t a tion (see [37]). The rotation of almost acyclic m.v.-fields c o v e r s , in p a r t i c u l a r , also the class of compact aeyclic m . v . fields (for such fields the degree in an infinite dimensional space has been c o n s t r u c t e d by Williams [1241] and F u r i - M a r t e l l i [652]). The relative rotation of almost acyclic m.v.-fields in LCS has been constructed and investigated in [216]. The extension of the degree ~ e o r y to the case of compact g e n e r a l i z e d n - a c y e l i c m.v.-fields in Banach spaces and LCS has been done in [458, 460, 606].
749
TABLE 2. Topological Degree of m - M a p p i n g s with Nonconvex I m a g e s LCS
c ~ e
1
Finite-dimensional space
Banach space
mappings \
completely continuous mappings
fundamentally contractible mappings
i
Aeyelic
Granas-Iaworow- Furi--Martelski [686]; Borisovieh--Gel'man-i li [652] Obukhovskii [38]
Williams [1241]
I
Almost acyclie
G&niewiez [6691; Bourgin [427]
Generalized aImost aeyclic
Bryszewski-G&niewicz [459, 674]
Arbitrary
Gel'man [74]
Bourgin
[429. 430]
Obukhovskii- ObukhovFleidervish skii [214] [216l
Bryszewski-G~'rniewiez [45% 460]
Dzedzej [606]
Gel'man [763
The c o n s t r u c t i o n of the d e g r e e f o r fundamentally contractible (see Sec. 2.3) a l m o s t acyelic m . v . - f i e l d s has been c o m m u n i c a t e d by Obukhovskii [214] (the detailed publication is in print). The d e v e l o p m e n t of the theory of r o t a t i o n s f o r m . v . - f i e l d s with nonconvex i m a g e s is i l l u s t r a t e d in Table 2. F o r m o r e detailed bibliographical indications and p r o o f s , see the s u r v e y [37]. 2.4.23. Lefschetz N u m b e r of m - M a p p i n g s . be an acyclic m - m a p p i n g .
Let X be a c o m p a c t m e t r i c ANR s p a c e and let F : X - - K(X)
The m - m a p p i n g F defines a canonical quintuple (X, X, F F, t, r), w h e r e t is a 1 - V i e t o r i s mapping and, consequently, it induces an i s o m o r p h i s m t.~ : H~(FF, O)-~ttn(X, G) of the cohomology groups for any n >- 0. E i l e n b e r g and Montgomery [610] have defined the Lefschetz n u m b e r of the m - m a p p i n g F in the following mariner: co
~.(F)= ~,i ( -- 1)" tr (r ,,ot~ 1) n=0
and they have p r o v e d the following s t a t e m e n t . 2.4.24.
THEOREM. If ~(F)~=0
, then F i x F = ~ .
Subsequently, the E i l e n b e r g - M o n t g o m e r y t h e o r e m has been g e n e r a l i z e d and developed in various d i r e c tions (see [39, 80, 272, 273, 276,277, 279,400, 461, 630,641, 642, 650, 653, 6 6 7 , 6 6 8 , 6 7 0 , 671,673, 674,676, 788, 843, 844, 993, 996, 1027-1031, 1132, 1137, 1208, etc.]; f o r m o r e detailed i n f o r m a t i o n see [37]). We indicate b r i e f l y one of the g e n e r a l i z a t i o n s of this t h e o r e m , due to G o r n i e w m z and G r a n a s [676]. F i r s t we give the definition of the g e n e r a l i z e d Lefschetz n u m b e r , due to L e r a y . Let E be a v e c t o r s p a c e o v e r the f i e l d Q of rational n u m b e r s and let s e t N (~)=~U Ker ~", E ~ E / N ( ~ )
and we define the induced e n d o m o r p h i s m
~:E-+E ~:s
be an e n d o m o r p h i s m . We
9 If dim/~ < co , then we
define the g e n e r a l i z e d t r a c e T r ( ~ ) = t r (~). Let
E-~{En} be a g r a d e d v e c t o r s p a c e and let ~{~n}:E-~E be a z e r o - d e g r e e e n d o m o r p h i s m .
2.4.25. Definition. The e n d o m o r p h i s m ~ is said to be a L e r a y e n d o m o r p h i s m if the quotient s p a c e E = is of finite type, i.e., dirnEn< oo for all n and E n = 0 f o r all but finitely m a n y n. F o r such e n d o m o r p h i s m s we can define the (generalized) L e f s c h e t z n u m b e r
{&}
750
oo
A ( ~ ) = ~ ( ~ l y Tr ((..). n=0
2.4.26. (a)
THEOREM.
Let X be an ANR s p a c e and let F:X --~ K(X) be a c o m p a c t acyclic m - m a p p i n g . Then
F,={r,otTi}:H,(X, Q)~H.(X, Q)
is a L e r a y e n d o m o r p h i s m (the A l e k s a n d r o v - C e c h homology with c o m p a c t supports); (b) if A(F.)=/=0
, then FixF=/=~.
2.4.27. At the conclusion of this c h a p t e r we mention that v a r i o u s t h e o r e m s on fixed points of m - m a p p i n g s have been obtained also in [85, 161, 171, 172,180, 204, 245, 349, 363, 366,402, 403, 504, 538, 542, 543, 550, 578, 579, 602,616-618, 628,650, 658,699, 701, 702, 716, 717, 721, 722, 762, 774, 791, 795, 805, 833, 861, 874, 877, 878, 885, 917, 929, 962, 988, 989, 1058, 1093, 1151, 1170, 1172, 1190, 1236].
In the direction connected with the approximate determination of the fixed points, we mention 239, 261, 264, 297, 353, 500, 603,604, 607, 608, 664, 758, 846, 860, 937, 1056, 1158, 1191].
[58, 170,
SUPPLEMENT
Literature
on the
Applications
of Multivalued
Mappings
As a l r e a d y mentioned in the introduction, we give only undifferentiated guiding information on the l i t e r ature r e g a r d i n g the applications; we intend to p r e s e n t these questions in detail in the second p a r t of the p r e s e n t survey.
i. Applications to game theory and mathematical economics: [16, 19, 128, 142, 205, 213, 217, 218, 238, 249, 263, 286, 324, 326, 375, 623,660, 697, 798, 803, 977, 1073, 1181, 1193, 1210]. 2. Applications to Be theory of differential and integral inclusions, to the theory of optimal control and to the theory of differential equations: [i, 2, I0, 12-15, 18, 20-29, 50-55, 57, 59, 61, 62, 64-69, 71, 72, 77-79, 86-88, 90-99, 124-127, 130, 134, 138, 142, 143, 147-149, 160, 163-167, 169, 173-177, 179, 181-183, 189, 192, 193, 195, 197, 209, 210, 212, 219-223, 225-236,249-258, 269, 270, 285, 290, 292-296, 299,304-313, 317-319, 324, 326, 337, 344, 351, 352, 354, 364, 390, 391, 395-397, 410-412, 414, 415, 433, 435, 437,455-457, 463, 465467, 473, 481, 483, 484, 491, 497, 498, 501-503, 507-513, 520, 547-549, 555, 558-561, 564, 568, 584, 591, 595, 612, 640, 657, 659, 687-689, 726, 727, 733-739, 740, 754, 781-784, 798-800, 805, 806, 809, 813, 815-821, 823, 824, 835-838, 842, 847, 857-859, 863-866, 871, 906-911, 916, 936, 974, 984, 986, 995, 998, i000, i001, 1007, 1008, 1011-1018, 1032, 1035-1039, 1044, 1045, 1060, 1079, 1083, 1090, 1115, 1122-1124, 1127, 1166, 1167, 11791181, 1187, 1188, 1202, 1209, 1214, 1215, 1228-1231, 1234, 1240, 1246-1250, 1252-1254]. 3. Applications to the theory of generalized dynamical systems: [I, 8, 9, 37, 39, 43-47, 82, 97, III, 113115, 196, 211, 212, 267, 268, 320-323, 325, 340, 343, 365, 398, 517, 714, 793, 827-829, 889, 999, 1082-1086, 1175-1178, 1192, 1193]. 4. Other applications:
[137, 300, 301,440, 519, 577, 665, 794, 830-832, 956, 1073, 1116]. LITERATURE
1. 2. 3. 4. 5. 6. 7.
CITED
Yu. I. Alimov, non the application of Lyapunov's d i r e c t method to differential equations with multivalued r i g h t - h a n d s i d e s , " Avtomat. T e l e m e k h . , 22, No. 7, 817-830 (1961). B . I . Anan'ev, nAn e x i s t e n c e t h e o r e m f o r a differential inclusion with v a r i a b l e lag," Differents. Uravn., 1_!, No. 7, 1155-1158 (1975). V . I . Arkin and V. L. Levin, "Convexity of the values of v e c t o r i n t e g r a l s , t h e o r e m s on m e a s u r a b l e s e l e c tion, and v a r i a t i o n a l p r o b l e m s , " Usp. Mat. Nauk, 2_~7,No. 3, 21-77 (1972). A . V . Arl~hangel'skii, " B i c o m p a c t s e t s and the topology of s p a c e s , " T r . Mosk. Mat. Obshch., 1_~3,3-55 (1965). A . V . A r k h a n g e l ' s k i i and A. Taimanov, non a t h e o r e m of V. P o n o m a r e v , n Dokl. Akad. Nauk SSSR, 135, No. 2, 247-248 (1960). N . A . Askhabov, "On a p r o p e r t y of conjugate cones in locally convex s p a c e s , n in: M a t h e m a t i c s and the Method of Its I n s t r u c t i o n [in Russian], R o s t o v - o n - D o n (1972), pp. 56-66. N . A . Askhabov, nOn the continuity of c e r t a i n multivalued mappings, n in: M a t h e m a t i c s and the Method of Its I n s t r u c t i o n [in Russian], R o s t o v - o n - D o n (1972), pp. 67-73.
751
8.
9. i0. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
35. 36. 37.
752
E. A. Barbashin, "On the theory of g e n e r a l i z e d d y n a m i c a l s y s t e m s , " Ueh. Zap. Mosk. Gos. Univ., Mat., 135, No. 2, 110-113 (1948). E. A. B a r b a s h i n and Yu. I. Alimov, "On the theory of dynamical s y s t e m s with multivalued and discontinuous c h a r a c t e r i s t i c s , " Dokl. Akad. Nauk SSSR, 140, No. 1, 9-11 (1961). E. A. B a r b a s h i n and Yu. L Alimov, "On the t h e o r y of r e l a y differential equations," Izv. Vyssh. Uehebn. Zaved., Mat., No. 1, 3-13 (1962). A. A. Beida, "Continuous binary r e l a t i o n s " (Redkollegiya Zh. Izv. Akad. Nauk BSSR, S e r . F i z . - M a t . Nauk), Minsk (1973). Manuscript deposited at VIN1TI, Dec. 18, 1973, No. 7620-73 Dep. Yu. A. Belov, "On the convergence of optimal solutions of a f a m i l y of differential inclusions," in: Trudy Kafedry Mat. Anal. Kaliningr. Univ., Kaliningrad (1969), pp. 3-13. Yu. A. Belov, "On the c h a r a c t e r i s t i c p r o p e r t i e s of optimal t i m e , " in: Collection of P a p e r s on the T h e o r y of Optimal P r o c e s s e s , No. 1, Kaliningrad State Univ. (1973), pp. 3-22. V. I. Berdyshev, "The continuity of multivalued mappings, r e l a t e d to the m i n i m i z a t i o n of convex functionals," Dolk. Akad. Nauk SSSR, 243, No. 3, 562-564 (1978). V. I. B e r d y s h e v , "The continuity of a multivalued mapping, r e l a t e d with t h e p r o b l e m of the m i n i m i z a t i o n of a functional," Izv. Akad. Nauk SSSR, Ser. Mat., 4_~4,No. 3, 483-509 (1980). V. V. B e r e s n e v , " N e c e s s a r y conditions f o r an e x t r e m u m f o r m a x i m i n p r o b l e m s , " Kibernetika, No. 2, 99-107 (1973). V. V. B e r e s n e v , "On mappings conjugate to convex multivalued mappings," Kibernetika, No. 5, 79-83 (1973). V. V. B e r e s n e v , "On n e c e s s a r y optimality conditions for s y s t e m s of d i s c r e t e inclusions," Kibernetika, No. 2, 58-64 (1977). C. Berge, Th~orie G~n~rale des J e u x a n P e r s o n n e s , G a u t h i e r - V i l l a r s , P a r i s (1957). V. I. Blagodatskikh, ~On the convexity of the attainability s p h e r e s , " Differents. Uravn., 8, No. 12, 21492155 (1972). V. L Blagodatskikh, "On the local controllability of differential inclusions," Differents. Uravn., 9, No. 2, 361-362 (1973). V. I. Blagodatskikh, "A sufficient optimality condition," Differents. Uravn., 9, No. 3, 416-422 (1973). V. I. Blagodatskikh "On the differentiability of solutions with r e s p e c t to the initial conditions," Differents. Uravn., 9, No. 12, 2136-2140 (1973). V. I. Blagodatskikh " Sufficient o p t i m a l i t y conditions f o r differential inclusions," Izv. Akad. Nauk SSSR, Ser. Mat., 3_~8,No. 3, 615-624 (1974). V. I. Blagodatskikh "Some r e s u l t s in the t h e o r y of differential inclusions," in: P r e c . S u m m e r School on O r d i n a r y Differential Equations "Difford 74", P a r t II, Brno (1975), pp. 29-67. V. I. Blagodatskikh "The h i g h - s p e e d p r o b l e m f o r differential inclusions," in: P r e c . Second Conf. of Young R e s e a r c h e r s Dept. Comput. Math. and C y b e r n e t i c s , Moscow Univ., Moscow (1975), pp. 30-31. V. I. Blagodatskikh "On the theory of sufficient optimality conditions," Dokl. Akad. Nauk SSSR, 231, No. 5, 1041-1044 (1976). V. I. Blagodatskikh "On the theory of sufficient optimality conditions," T r . Mat. Inst. Akad. Nauk SSSR, 142, 78-87 (1976). Yu. G. Borisovich, "On an application of the concept of rotation of a v e c t o r field," Dokl. Akad. Nauk SSSR, 153, No. 1, 12-15 (1963). Yu. G. Borisovich, "On B r o w d e r ' s fixed point t h e o r e m , " T r . Sem. Fankts. Anal., Voronezh. Gos. Univ., No. 7, 17-25 (1963). Yu. G. Borisovich, "On the relative rotation of c o m p a c t v e c t o r fields in l i n e a r s p a c e s , " T r . Sem. Funkts. Anal., Voronezh. Gos. Univ., No. 12, 3-27 (1969). Yu. G. Borisovich, "Topology and nonlinear functional a n a l y s i s , " Usp. Mat. Nauk, 34, No. 6, 14-22 (1979). Yu. G. Borisovieh, "On the topological theory of nonlinear mappings in Banach s p a c e s , " in: Seventh Math. S u m m e r School, 1969, Kiev (1970), pp. 274-282. Yu. G. Borisovich, "On the t h e o r y of topological d e g r e e of nonlinear F r e d h o l m mappings p e r t u r b e d by a multivalued o p e r a t o r , " Voronezhsk. Univ., Voronezh (1980). M a n u s c r i p t deposited at VINYl?I, Nov. 28, 1980, No. 5026-80 Dep. Yu. G. Borisovieh, B. D. Gel'man, ]~. Mukhamadiev, and V. V. Obukhovskii, "On the rotation of m u l t i valued v e c t o r fields," Dokl. Akad. Nauk SSSR, 187, No. 5, 971-973 (1969). Yu. G. Borisovieh, B. D, Gel'man, E. Mukhamadiev, and V. V. Obukhovskii, "On the rotation of m u l t i valued v e c t o r fields," Tr. Sem. Funkts. Anal., Voronezh. Gos. Univ., No. 12, 69-84 (1969). Yu. G. Borisovieh, B. D. Gel'man, A. D. Myshkis, and V. V. Obukhovskii, "Topological methods in the fixed point t h e o r y of multivalued m a p p i n g s , " Usp. Mat. Nauk, 3._~5,No. l, 59-126 (1980).
38. 39. 40.
41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.
Yu. G. Borisovich, B.D. Gel'man, and V. V. Obukhovskii, "On s o m e topological invariants of multivalued mappings with noneonvex i m a g e s , " T r . Sere. Funkts. Anal., Voronezh. Gos. Univ., No. 12, 85-95 (1969). Yu. G. B o r i s o v i c h and Yu. E. Gliklikh, nOn the Lefschetz n u m b e r f o r a class of multivalued mappings," in: Seventh Math. S u m m e r School, 1969, Kiev (1970), pp. 283-294. Yu. G. B o r i s o v i c h and V. V. Obukilovskii, "Homotopie p r o p e r t i e s , theory of rotation and fixed point t h e o r e m s f o r a c l a s s of noncompact multivalued mappings," Voronezhsk. Univ., Voronezh (1980). Manus c r i p t deposited at VINITI, Nov. 28, 1980, No. 5033-80 Dep. Yu. G. B o r i s o v i c h and Yu. I. Sapronov, "On the topological theory of compactly contracting m a p p i n g s , " T r . Sere. Funkts. Anal., Voronezh. Gos. Univ., No. 12, 43-68 (1969). K. B o r s u k and A. Kosinski, " F a m i l i e s of aeyelie c o m p a c t a in an n - d i m e n s i o n a l Euclidean s p a c e , ' Bull. Acad. Polon. Sei. C1. KI, 3, No. 6, 289-293 (1955). I . U . Bronshtein, "On dynamical s y s t e m s without uniqueness, as s e m i g r o u p s of n o n - s i n g l e - v a l u e d m a p pings of a topological s p a c e , " Dokl. Akad. Nauk SSSR, 144, No. 5, 954-957 (1962). I . U . Bronshtein, "On dynamical s y s t e m s without uniqueness, as s e m i g r o u p s of n o n - s i n g l e - v a l u e d m a p pings of a topological space," Izv. Akad. Nauk MSSR, S e t . E s t e s t v . Tekh. Nauk, 1, 3-18 (1963). I . U . Bronshtein, " R e c u r r e n c e , p e r i o d i c i t y and t r a n s i t i v i t y in dynamical s y s t e m s without uniqueness," Dokl. Akad. Nauk SSSR, 151, No. 1, 15-18 (1963). I . U . Bronshtein, " R e c u r r e n t points and m i n i m a l sets in dynamical s y s t e m s without uniqueness," Icy. Akad. Nauk MSSR, Ser. F i z . - T e k h . Mat. Nauk, 7, 14-21 (1965). B . M . Budak, "The concept of motion in a g e n e r a l i z e d dynamical s y s t e m , " Uch. Zap. Mosk. Gos. Univ. Mat., 155, No. 5, 174-194 (1952). A . I . Bulgakov, "On the Cauehy p r o b l e m f o r a differential inclusion," T r . Mosk. Inst. Khim. M a s h i n o s t r . , No. 64, 73-74 (1975). A . I . Bulgakov, ' O n the question of the p r o p e r t i e s of the set of solutions of differential inclusions," Diff e r e n t s . Uravn., 142, No. 6, 971-977 (1976). A . I . Bulgakov, "On a c l a s s of integral equations," in: Functional-Differential Equations and Boundary Value P r o b l e m s of Mathematical P h y s i c s , P e r m (1978), pp. 150-154. A . I . Bulgakov, "On the existence of a g e n e r a l i z e d solution of a f u n c t i o n a l - i n t e g r a l inclusion," Differents. Uravn., 1._~5,No. 3, 514-520 (1979). A . I . Bulgakov, " K n e s e r ' s t h e o r e m f o r a c l a s s of integral inclusions," Differents. Uravn., 16, No. 5, 894900 (1980). A . I . Bulgakov and L. N. Lyapin, " C e r t a i n p r o p e r t i e s of the s e t of solutions of the V o l t e r r a - H a m m e r s t e i n i n t e g r a l inclusion," Differents. Uravn., 14, No. 8, 1465-1472 (1978). A . I . Bulgakov and L. N. Lyapin, non an integral inclusion with a functional o p e r a t o r , n Differents. Uravn., 15, No. 5, 878-884 (1979). ~.. B. Veliev, "On the question of the definition of the concept of solution of equations in contingencies," Dagestan. Gos. Univ., Makhaehkala (1974). Manuscript deposited at VINITI, July 17, 1974, No. 1977-74 Dep. ~. B. Veliev, "The V o l t e r r a integral o p e r a t o r f o r multivalued mappings, ~ Dagestan. Gos. Univ., Makhaehkala (1974). M a n u s c r i p t deposited at VINITI, August 26, 1974, No. 2352-74 Dep. V . I . Venets, "Differential inclusions in convex p r o b l e m s , " Avtomat. T e l e m e l ~ . , No. 9, 5-14 (1979). B . A . V e r t g e i m , "On the a p p r o x i m a t e d e t e r m i n a t i o n of fixed points of continuous mappings," Dokl. Akad. Nauk SSSR, 191, No. 1, 9-11 (1970). E . E . Viktorovskii, nOn the t h e o r y of fields of cones of d i r e c t i o n s , " Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 38-45 (1965). Yu. N. V l a d i m i r s k i i , ' O n the theory of multivalued ~ + - o p e r a t o r s in topological s p a c e s . I," Litov. Mat. Sb. (Liet. Mat. Rintdnys), 10, No. 1, 17-28 (1970). D . R . G a i d a r o v and R. K. Ragimkhanov, "On an integral inclusion of H a m m e r s t e i n , w Sib. Mat. Zh., 21, No. 2, 19-24 (1980). E . A . Gango, "Onthe solutions of differential equations with a multivalued right-hand side," Mezhvuzovskie XXI G e r t s e n o v s k i e Chteniya, Leningrad (1968), Leniagr. Gos. Pedagog. Inst., 77-79. E . A . Gango, "Some fixed point t h e o r e m s of multivalued mappings," Uchebn. Zap. Leningr. Gos. Pedagog. Inst., 464, 137-146 (1970). E . A . Gango, "On the question of the existence of solutions of differential equations with a multivalued r i g h t - h a n d side and r e t a r d e d a r g u m e n t , " II Zonal'n. Konf. Pedinst. S e v . - Z a p . Zony, Novgorod (1970),
3~, 25-28. 65.
E . A . Gango, "On the solutions of differential equations with multivalued r i g h t - h a n d side," Uehebn., Zap. Leningr. Gos. Pedagog. Inst., 404, 335-342 (1970).
753
66. 67. 68. 69.
70. 71. 72. 73. 74. 75. 76. 77. 78. 79.
80. 81. 82. 83.
84. 85. 86. 87. 88. 89. 90. 91. 92.
754
E. A. Gango, nBoundary-value p r o b l e m s f o r differential equations with a multivalued r i g h t - h a n d side," T r . Zonal'n. Ob"edin. Mat. Kafedr Ped. Inst. Sibiri, No. 1, 161-175 (1972). E. A. Gaago and A. I. Povolotskii, "Bounded solutions of differential equations with a multivalued r i g h t hand side," Mat. Zap. K r a s n o y a r s k . Pedagog. Inst., No. 2, 60-67 (1970). E. A. Gango and A. I. Povolotskii, " P e r i o d i c solutions of differential equations with a multivalued r i g h t hand side," Uch. Zap. Leningr. Gos. Pedagog. Inst., 464, 235-242 (1970). E. A. Gango and A. I. Povolotskii, " P r o p e r control functions f o r differential equations with multivalued r i g h t - h a n d side," in: T h e o r y of Functions and Functional Analysis [in Russian], Leningrad (1975), pp. 35-41. V. A. G e i l e r , "On continuous s e l e c t o r s in u n i f o r m s p a c e s , " Dokl. Akad. Nauk SSSR, 195, No. 1, 17-19 (1970). B. D. G e l ' m a n "Integral multivalued o p e r a t o r s and differential equations," Sb. T r . A s p i r a n t o v Mat. Fak. Voronezh. Univ., No. 1, 24-29 (1971). B. D. G e l ' m a n "Multivalued integral o p e r a t o r s and w-periodic solutions," T r . Mat. Fak. Voronezh. Univ., No. 4, 35-44 (1971). B. D. G e l ' m a n nA g e n e r a l i z a t i o n of the Kakutani fixed point t h e o r e m to multivalued mappings," Dokl. Akad. 51auk SSSR, 209, No. 1, 22-24 (1973). B. D. G e l ' m a n "A topological c h a r a c t e r i s t i c of multivalued mappings and t h e o r e m s of Kakutani type," T r . Mat. Fak. Voronezh. Univ., No. 12, 12-21 (1974). B. D. G e l ' m a n "A topological c h a r a c t e r i z a t i o n of multivalued mappings and fixed point t h e o r e m s , " Dokl. Akad. Nauk SSSR, 221, No. 3, 524-527 (1975). B. D. G e l ' m a n "A topological c h a r a c t e r i s t i c of multivalued mappings in a Banach s p a c e , " T r . Mat. Fak. Voronezh. Univ., No. 16, 17-23 (1975). B. D. G e l ' m a n "Topological methods in the theory of multivalued o p e r a t o r equations," Dissertation; V o r o n e z h (1975). B. D. G e l ' m a n and :fu, E. Gliklikh, "A two-point boundary-value p r o b l e m in g e o m e t r i c m e c h a n i c s with discontinuous f o r c e s , " Prild. Mat. Mekh., 44, No. 3, 565-569 (1980). B. D. G e l ' m a n and V. V. Obukhovskii, " P e r i o d i c solutions of differential equations with a multivalued r i g h t - h a n d side," in: P r o c . Sixth All-Union Conference on Topology, M e t s n i e r e b a , T b i l i s i (1972), pp. 33-35. B. D. G e l ' m a n and V. A. Chubarov, "On the topological invariants of multivalued mappings," T r . Mat. Fak. Voronezh. Univ., No. 12, 54-59 (1974). N. L Glebov, "On a g e n e r a l i z a t i o n of Kakutani's fixed point t h e o r e m , " Dokl. Akad. Nauk SSSR, 185, No. 5, 981-983 (1969). u E. Gliklikh, " F i x e d points of multivalued mappings with nonconvex i m a g e s and the rotation of m u l t i valued fields," Sb. T r . A s p i r a n t o v Mat. Fak. Voronezh. Univ., No. 1, 30-38 (1971). V. Gol'shtynskii (W. Holsztynski)and S. Iliadis, "Approximation of multivalued mappings by s i n g l e - v a l u e d ones and s o m e applications," Bull. Acad. Polon. Sci. S e t . Sci. Math. Astron. Phys., 16, No. 10, 765-769 (1968). S. V. Gorlenko, "Differential p r o p e r t i e s of functions and m u l t i v a l u e d m a p p i n g s , " Dokl. Akad. Nauk Ukr. SSR, S e t . A, No. 6, 493-495 (1976). I. IV[. Gul', "Topological incidence f o r m u l a , " Dokl. Akad. Nauk SSSR, 5_~6,No. 9, 895-898 (1947). V. N. Gurov, "On n e c e s s a r y conditions f o r an e x t r e m u m f o r differential inclusions and L. S. P o n t r y a g i n ' s m a x i m u m p r i n c i p l e , " in: The T h e o r y of Optimal Solutions [in Russian], Kiev (1979), pp. 10-14. V. N. Gurov, "On n e c e s s a r y conditions f o r an e x t r e m u m on differential inclusions," Dokl. Akad. Nauk Ukr. SSR, Ser. A, 51o. 9, 757-759 (1979). V. 51. Gurov, "On the e x t r e m a l s on the boundary of the domain of definition of a differential inclusion," in: Theory of Optimal Solutions [in Russian], Kiev (1980), pp. 20-26. T. D. Davydova and V. V. Obukhovskii, "On c e r t a i n g e n r a l i z a t i o n s of the B i r k h o f f - K e l l o g - R o t h e t h e o r e m , " in: Methods of Solving O p e r a t o r Equations [in Russian], Voronezh (1978), pp. 38-41. 51. 51. Dzyuba, " N e c e s s a r y conditions f o r an e x t r e m u m f o r p r o b l e m s with a differential inclusion," in: The Theory of Optimal Solutions [in Russian], Kiev (1974), pp. 12-21. A. P. D m i t r i e v , "Contingent differential equations with a multivalued semieontinuous right-hand side," in: Differential Equations [in Russian], 51o. 5, Ryazan' (1975), pp. 102-117. A. P. D m i t r i e v , "The Cauchy p r o b l e m f o r points with a strong singularity of contingent differential equations with a multivalued semicontinuous r i g h t - h a n d side," in: Differential Equations [in Russian], No. 5, Ryazan' (1975), pp. 118-128.
93. A. P. D m i t i r i e v , "The set of solutions of a differential equation in contingencies with a s e m i c o n t i a u o u s multivalued right-hand side," in: Differential Equations [in Russian], No. 9, R y a z a n ' (1977), pp. 35-44. 94. V. T. Dmitrienko, "The homotopic c l a s s i f i c a t i o n of a c l a s s of multivalued mappings," Voronezh. Univ., V o r o n e z h (1980). Manuscript deposited at VINITI, May 27, 1980, No. 2092-80 Dep, 95. D. T r . Dochev, " I n v a r i a n t sets in a g e n e r a l dynamical s y s t e m , " Godislmik Vyssh. Uchebn. Zaved., P r i lozlma Mat., 14, No. 3, 179-187 (1978(1979)). 90. D. T r . Dochev and D. D. Bainov, "On the existence of solutions of a c e r t a i n c l a s s of multivalued d i f f e r ential equations with deviating a r g u m e n t , " Izv. Akad. Nauk KSSR, S e t . F i z . - M a t . , No. 1, 34-37 (1976). 97. D. Tr. Dochev and D. D. Bainov, "On the existence of solutions of a c l a s s of multivalued differential equations with deviating a r g u m e n t , " Ukr. Mat. Zh., 30, No. 5, 659-664 (1978). 98. D. T r . Dochev and D. D. Bainov, "On the e x i s t e n c e of solutions of a c l a s s of multivalued differential e q u a tions with deviating a r g u m e n t , " Rev. Roumaine Math. P u r e s Appl., 2._~3,No. 10, 1497-1503 (1978). 99. D. Tr. Doehev and D. T. Dochev, "Controllability and optimal control in a g e n e r a l dynamical s y s t e m , " Godislmik Vyssh. Uchebn. Zaved., P r i l o z h n a Mat., 10, No. 2, 25-36 (1974(1975)). 100. G. P. Dybov, "Introduction of rotation f o r a c l a s s of multivalued o p e r a t o r s , " in: Boundary Value P r o b l e m s f o r P a r t i a l Differential Equations [in Russian], Kiev (1978), pp. 57-60. 101. G. Duvaut and J . - L . Lions, Inequalities in Mechanics and P h y s i c s , S p r i n g e r - V e r l a g , Berlin (1976). 102. L V. Evstigneev, " M e a s u r a b l e s e l e c t i o n and the continuum axiom," Dokl. Akad. Nauk SSSR, 238, No. 1, 11-14 (1978). 103. M. P. E r s h o v , "On s e l e c t o r s in a b s t r a c t s p a c e s , " Usp. Mat. Nauk, 3__3,No. 6, 205-206 (1978). 104. P. P. z a b r e i k o , M. A. K r a s n o s e l ' s k i i , and V. V. Strygin, "On the i n v a r i a n c e - o f - r o t a t i o n p r i n e i p l e s , " Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 51-57 (1972). 105. Yu. B. Zelinskii, "On n - a d m i s s i b l e multivalued m a p p i n g s , " in: M e t r i c Questions in the T h e o r y of F u a e tiolls and Mappings [in Russian], No. 7, Naukova Dumka, Kiev (1975), p p . 61-83. 106. Yu. B. Zelinskii, "On the local d e g r e e of multivalued m a p p i n g s , " D o k l . A k a d . N a u k U k r . SSR, S e r . A , No. 10o 872-876 (1975). 107. Yu. B. Zelinsldi, "The degree f o r multivalued m a p p i n g s , " P r e p r i n t I M - 7 6 - 8 , Inst, Mat. Akad. Nauk Ukr. SSR, Kiev (1976). 108. Yu. B. Zelinskii, "A t h e o r e m on continuation and c r i t e r i a f o r the p r e s e r v a t i o n of the domain f o r m u l t i valued m a p p i n g s , " Ukr. Mat. Zh., 2__9,No. 3, 383-387 {1977). 109. Yu. B. Zelinskii, " C h a r a c t e r i s t i c h o m o m o r p h i s m s f o r extensions of multivalued mappings, ~ Dokl. Akad. Nauk Ukr. SSR, Set. A, No. 8, 683-686 (1977). 110. Yu. B. Zelinskii, "On multivalued mappings having r e g u l a r r e s t r i c t i o n s , " in: Questions in the Metric T h e o r y of Mappings and Its Applications. P r o c . of the Fifth Colloquium, Donetsk, 1976, Kiev (1978), pp. 31-36. 111. V. I. Zubov, The Stability of Motion (The Lyapunov Methods and T h e i r Applications) [in Russian], V y s s h a y a Shkola, Moscow (1973). 112. A. A. Ivanov, " F i x e d points of mappings of m e t r i c s p a c e s , " z a p . Nauchn. Sere. Leniagr. Otd. Mat. Inst., 666, 5-102 (1976). 113. M . S . Izman, "Stability of s e t s and a t t r a c t o r s in d i s p e r s i v e dynamical s y s t e m s , " Mat. I s s l e d . , 3, No. 4, 51-77 (1968(1969)). 114. M. S. Izman, "The application of Lyapunov's second method f o r the investigation of stability and a s y m p totic stability of sets in d i s p e r s i v e dynamical s y s t e m s , " Differents. Uravn., 5, No. 7, 1207-1212 (1969). 115. M . S . I z m a n , "On the a s y m p t o t i c stability of s e t s in d i s p e r s i v e dynamical s y s t e m s , " Differents. Uravn., 7, No. 4, 615-621 (1971). 116. Ya. A. I z r a i l e v i c h , "On the concept of r e l a t i v e rotation of a multivalued v e c t o r field," T r . Sere. Ftmkts. Anal. Voronezh. Univ., No. 12, 111-115 (1969). 117. Ya. A. I z r a i l e v i c h and V. V. Obukhovskii, ~On e q u i v a r i a n t multivalued m a p p i n g s , " Dokl. Akad. Nauk SSSR, 205, No. 1, 16-18 (1972). 118. Ya. A. I z r a i l e v i c h and V. V. Obukhovskii, "On the computation of the d e g r e e of equivariant multivalued fields," in: P r o c . Sixth All-Union Topology Conf., M e t s n i e r e b a , Tbilisi (1972), pp. 57-58. 119. Ya. A. I z r a i l e v i c h and V. V. Obukhovskii, "On c e r t a i n topological c h a r a c t e r i s t i c s of e q u i v a r i a n t m u l t i valued m a p p i n g s , " T r . Mat. Fak. Voronezh. Univ., No. 10, 52-61 (1973). 120. A. D. Ioffe and V. L. Levin, "Subdifferentials of convex functions," Tr. Mosk. Mat. Obshch., 2__6, 3-73 (1972). 121. A. D. Ioffe and V. M. T i k h o m i r o v , "The duality of convex functions and e x t r e m a l p r o b l e m s , " Usp. Mat. Nauk, 2_3, No. 6, 51-116 (1968).
755
122. A. D. Ioffe and V. M. Tikhomirov, The Theory of Extremal Problems [in Russian], Nauka, Moscow (1974). 123. D. M. Ipate, "The structural analysis of multivalued mappings," Mat. Issled., No. 53, 75-97 (1979). 124. A. E. Irisov, "On sufficient optimality conditions for the periodic solutions of a differential inclusion," in: Problems of the Contemporary Theory of Periodic Motions [in Russian], No. 2, Izhevsk (1978), pp. 55-58. 125. A. E. I r i s o v and E. L. Tonkov, "Sufficient conditions f o r the existence and optimality of periodic solutions of a differential inclusion," Mat. Fiz. (Kiev), No. 27, 13-19 (1980). 126. A. E. Irisov, V. S. Tonkova, and E. L. Tonkov, "On the existence of periodic solutions of differential inclusions," in: P r o b l e m s of the C o n t e m p o r a r y Theory of P e r i o d i c Motions [in Russian], No. 1 (1977), pp. 17-20. 127. A. E. I r i s o v , V. S. Tonkova, and E. L. Tonkov, " T h e periodic solutions of a differential inclusion," in: Nonlinear Oscillations and T h e o r y of E l a s t i c i t y [in Russian], No. 2, I z h e v s k (1978), pp. 3-15. 128. I t e r a t i v e Methods in Game T h e o r y and P r o g r a m m i n g [in Russian], Nauka, Moscow (1974). 129. A. V. Kaplan, "On the B r o w d e r t h e o r e m in l i n e a r topological s p a c e s , " T r . Mat. Fak. Voronezh. Univ., No. 6, 38-42 {1972). 130. M. R. Karibov, "The solvability of c e r t a i n boundary-value p r o b l e m s for functional-differential inclusions," P e n n . Politekh. Inst., P e r m (1980). M a n u s c r i p t deposited at VIN1TI, July 31, 1980, No. 3374-80 Dep. 131. P. S. Kenderov, "On the continuity of multivalued mappings," Dokl. Bolg., 26, No. 6, 735-738 (1973). 132. P. S. Kenderov, "The continuity of multivalued mappings and the differentiability of convex functions," Godishnik Sofiisk. Univ. Mat. Fak., 6__6, 163-168 (1971-1972 (1974)}. 133. P. S. Kenderov, "Multivalued mappings and t h e i r p r o p e r t i e s s i m i l a r to continuity," Usp. Mat. Nauk, 35, No. 3, 194-196 (1980). 134. S. S. Klimchuk, "The a v e r a g i n g method in l i n e a r boundary-value p r o b l e m s f o r differentia/ inclusions," Odessk. Univ., Odessa (1980). Manuscript deposited at VINITI, June 5, 1980, No. 2221-80 Dep. 135. I. Yu. K o m a r o v , "On points of coincidence of two mappings," Vesta. Mosk. Univ. Ser. I, Mat. Mekh., 3__.1, No. 2, 11-14 (1974). 136. V. L Korneichuk, "The m i n i m i z a t i o n of multivalued functions in a s y s t e m of s e t - t h e o r e t i c o p e r a t i o n s , " Vestn. Kiev. Politekh. Inst. Ser. Avtomat. E l e k t r o p r i b o r o s t r . , No. 8, 46-50 (1971). 137. M. A. K r a s n o s e l ' s k i i and P. P. Zabreiko, G e o m e t r i c Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975). 138. M. A. Krasnosel'skii and A. V. Pokrovskii, "Equations with discontinuous nonlinearities," Dokl. Akad. Nauk SSSR, 248, No. 5, 1056-1059 (1979). 139. S. N. Kuzhen'kin, "Special properties of point-set mappings," in: The Application of Functional Analysis in Approximation Theory [in Russian], Kalinin (1980), pp. 59-65. 140. C. Kuratowski, Topology, Vol. I, Academic Press, New York (1966). 141. C. Kuratowski, Topology, Vol. II, Academic Press, New York (1968). 142. A. B. Kurzhanskii, "Differential approach games with restricted phase coordinates," Dold. Akad. Nauk SSSR, 192, No. 3, 491-494 (1970). 143. A. B. Kurzhanskii, "On the existence of solutions of equations with a f t e r e f f e c t , " Differents. Uravn., 6, No. 10, 1800-1809 {1970). 144. A. G. K u s r a e v , "On the subdifferential mappings of convex o p e r a t o r s , " O p t i m i z a t s i y a , No. 21(38), 36-40 (1978). 145. Yu. K. Lando, "On a controllability c r i t e r i o n , " T r . Mosk. Inst. Khim. M a s h i n o s t r . , No. 48, 73-74 (1973). 146. Yu. K. Lando, "On controllable o p e r a t o r s , " Differents. Uravn., 10, No. 3, 531-536 (1974). 147. A. A. Levakov, "On sufficient optimality conditions," Differents. Uravn., 11, No. 2, 385-387 (1975). 148. A. A. Levakov, " N e c e s s a r y and sufficient conditions f o r optimality," Vestn. B e l o r u s s . Gos. Univ. Ser. I, No. 2, 79-80 (1975). 149. A. A. Levakov, " W e l l - p o s e d n e s s of a c l a s s of optimal control p r o b l e m s , " Vestn. B e l o r u s s . Univ. S e t . I, No. 2, 33-37 (1980). 150. V. L. Levin, "On Borel c r o s s sections of multivalued mappings," Sib. Mat. Zh., 19, No. 3, 617-623 (1978). 151. V. L. Levin, " M e a s u r a b l e c r o s s sections of multivalued mappings and p r o j e c t i o n s of m e a s u r a b l e s e t s , " Funkts. Anal. P r i l o z h e n . , 12, No. 2, 40-45 {1978). 152. V. L. Levin, " M e a s u r a b l e c r o s s sections of multivalued mappings in topological s p a c e s and upper envelopes of Carath6odory integrands," Dokl. Akad. Nauk SSSR, 252, No. 3, 535-539 (1980). 153. R. S. Linichuk, "Multivalued mappings and the continuity of decompositions of topological s p a c e s , " in: Tenth Mathematical School, Kiev (1974), pp. 308-329. 154. R. S. Linichuk, "On the r e p r e s e n t a t i o n of mappings in the f o r m of s u p e r p o s i t i o n s , " Ukr. Mat. Zh., 27, No. 5, 678-682 (1975).
756
155. Yu. 6 . Linke, "Sublinear o p e r a t o r s and continuous s e l e c t o r s , " Sb. T r u d o v Inst. Mat. Sib. Otd. Akad. Nauk SSSR, No. 8(25), 71-76 (1972). 156. Yu. ]~. Linke, "Continuous s e l e c t o r s of mappings acting in a bail of the conjugate s p a c e , " O p t i m i z a t s i y a , No. 17(34), 117-129 (1975). 157. Yu. E. Linke, "Sublinear o p e r a t o r s with values in s p a c e s of continuous functions," Dokl. Akad. Nauk SSSR, 228, No. 3, 540-542 (1976). 158. Yu. E. Linke, "Sublinear o p e r a t o r s and L i n d e n s t r a u s s s p a c e s , " Dokl. Akad. Nauk SSSR, 234, No. 1, 26-29 (1977). 159. zh. L. Lions (J. L. Lions), "On partial differential inequalities," Usp. Mat. Nauk, 2._66,No. 2, 205-263 (1971). 160. L . N . Lyapin, "An integral o p e r a t o r in the s p a c e of multivalued fanctions," Tr. T a m b o v s k . Inst. Khim. M a s h i n o s t r . , No. 4, 33-36 (1970). 161. L . N . Lyapin, "On the existence of g e n e r a l i z e d solutions of equations with multivalued o p e r a t o r s , " T r . T a m b o v s k . Inst. Khim. M a s h i n o s t r . , No. 4, 37-40 (1970). 162. L . N . Lyapin, "On the t h e o r y of the A u m a n n - H u t a t h a r a integral, n T r . T a m b o v s k . Inst. K_him. M a s h i n o s t r . , No. 6, 3-8 (1971). 163. L . N . Lyapin, "The g e n e r M i z e d solution of an equation with a discontinuous o p e r a t o r , " Differents. Uravn., 7, No. 6, 1108-1113 (1971). 164. L.N. Lyapin, "On the existence of a solution of an operator inclusion," Tr. Mosk. Inst. IGn_im. Mashinostr., No. 39, 15-19 (1972). 165. L.N. Lyapin, "Set-valued mappings in the theory of integral equations with a discontinuous operator," Differents. Uravn., 9, No. 8, 1511-1519 (1973). 166. L.N. Lyapin, "Continuous solutions of integral inclusions," Differents. Uravn., I0, No. ii, 2048-2055 (1974). 167. L. ixl. Lyapin, "The fixed point principle for a class of integral inclusions," Tr. Mosk. Inst. Khim. Mashinostr., No. 53, 35-37(1974). 168. N. L. Lyapin, "Certain properties of a multivalued superposition operator," Tr. Mosk. Inst. Khim. Mashinostr., No. 64, 60-63 (1975). 169. L.N. Lyapin, "On inclusions of Hammerstein type," Differents. Uravn., 12, No. 5, 908-915 (1976). 170. I.V. Magarik, "Algorithms for the computation of the approximate values of the fixed points of continuous and point-set mappings," Programs and Algorithms of the Central Econom. Mat. Inst. Akad. Nauk SSSR, No. 83, Moscow (1977). 171. V. E. Matusov, "On fixed points of multivalued mappings in partially ordered spaces," Sb. Nauchn. Trudov Tashkent Univ., No. 573, 37-39 (1978). 172. V. E. Matusov, "Theorems on fixed points for multivalued compressions and expansions of a cone," Dokl. Akad. Nauk UzSSR, No. 9, 6-8 (1979). / 173. E.N. Makhmudov, "The optimality principle for polyhedral differential inclusions and linear problems of optimal control on linear manifolds," Dokl. Akad. Nauk AzSSR, 3__5,No. i, 3-13 (1979). g 174. E.N. Makhmudov, "Polyhedral differential inclusions and linear probleras of optimal control on linear manifolds," Tr. Molod. Uchen. Inst. Kibernet. Baku, 1978, 107-111. Manuscript deposited at VINITI, August 21, 1979, No. 3121-79 Dep. 175. ~. N. Mal~.hmudov, "Necessary conditions for an extremum for no, autonomous differential inclusions ," Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, No. 6, 3-8 (1979). I 176. E. N. Makhmudov and B. N. Pshenichnyi, "The optimality principle for polyhedral differential inclusions," Inst. Kibern. Akad. Nauk Ukr. SSR, Preprint, No. 25, 3-22 (1978). 177. E. N. Makhmudov and B. N. Pshenichnyi, "On the optimality principle for polyhedral differential inclusions," Izv. Akad. lqauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, No. 6, 26-29 (1978). 178. ~. l~I.Makhmudov and B. N. Pshenichnyi, "Polyhedral mappings," Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, I~o. 2, 10-15 (1979). 179. I. B. Medvedovskii, "On a duality theorem in discrete optimal control," Kibern. Vychisl. Tekh., No. 40, 3-8 (1978). 180. P. D. M i l ' m a n , " F i x e d points and the d e g r e e of m a p p i n g s , " Mat. Sb., 87, No. 2, 275-292 (1972). 181. L. I. Minchenko, "A support p r i n c i p l e f o r difference inclusions," Differents. Uravn., 1_33,~qo. 3, 567-569 (1977). 182. L. I. Minchenko, "Sufficient optimality conditions f o r difference inclusions," Differents. Uravn., 13, No. 7, 1320-1324 (1977). 183. V. M. Mikhalevich, "The optimal control p r o b l e m on an a b s t r a c t manifold," in: Slozlm. S i s t e m y Upr., Kiev (1977), pp. 11-22.
757
184. M. Misic, "Two t h e o r e m s on multivalued mappings of 7r-bicompact topological s p a c e s , " in: P r o c . Int. Symp. Topol. and Its Appl., Herceg-blovi, 1968, Belgrade (1969), pp. 258-259. 185. M. Misic, "On p r o p e r t i e s of the b o u n d a r i e s of the image and the p r e i m a g e of points under multivalued mappings," Publ. Inst. Math., 13, 81-88 (1972). 186. M. Misic, "On the p r o p e r t i e s of the image of points under multivalued m a p p i n g s , " in: P r o c . Int. Syrup. Topol. andIts Appl., Budva, 1972, Belgrade (1973), pp. 179-180. 187. M. Misic, "On multivalued mappings of q - s p a c e s , " Math. Balkanica, 4, 431-435 (1974). 188. B. Sh. Morduld~ovich, "On the p r o b l e m of the existence of optimal c o n t r o l s , " Differents. Uravn., 7, No. 12, 2161-2167 (1971). 189. B. Sh. Mordukhovich, " C e r t a i n p r o p e r t i e s of multivalued mappings and differential inclusions with an application to the p r o b l e m of the existence of optimal controls," Redkol. Zh. Izv. Akad. Nauk BSSR, Ser. F i z . - M a t . Nauk, Minsk (1980). M a n u s c r i p t deposited at VINITI, Dec. 12, 1980, No. 5268-80 Dep. 190. M. M r s e v i c , " C e r t a i n p r o p e r t i e s of the s p a c e 2X of a topological R0-space," Usp. Mat. Nauk, 3_~4,No. 6, 166-170 (1979). 191. L . G . Mustafaev and R. B. F e i z u l l a e v , " S e m i h e a p s of multivalued mappings of topological s p a c e s , " Dokl. Akad. Nauk SSSR, 232, No. 4, 777-779 (1977). 192. A. M. Mukhsinov, nOn differential inclusions in Banach s p a c e s , " Dold. Akad. Nauk SSSR, 217, No. 4, 759761 (1974). 193. M . A . Mikhsinov, "On the invariant s e t s of differential inclusions," Dokl. Akad. Nauk TSSR, 2._~2,No. 1, 16-19 (1979). 194. A . D . Myshkis, "On uniqueness domains f o r solutions of s y s t e m s of linear p a r t i a l differential equations," Mat. Sb., 1._99,No. 3 , 4 8 9 - 5 2 2 (1946). 195. A. D. Myshkis, "The g e n e r a l theory of differential equations with r e t a r d e d a r g u m e n t , " Usp. Mat. Nauk, 4, No. 5, 99-141 (1949). 196. A. D. Myshkis, " G e n e r a l i z a t i o n s of the t h e o r e m on the s t a t i o n a r y point of a dynamical s y s t e m inside a closed t r a j e c t o r y , " Mat. Sb., 34, No. 3, 525-540 (1954). 197. A . D . Myshkis, "On differential inequalities with locally bounded d e r i v a t i v e , " Zap. Mekh.-Mat. Fak. Kharkov. Mat. Obshch., 3.0, 152-163 (1964). 198. S. I. Nedev, "Multivalued sections of multivalued m a p p i n g s , " Dokl. Bolg. Akad. Nauk, 27, No. 6, 725-728 (1974). 199. S. I. Nedev, " F o u r t h e o r e m s of E. Michael on continuous s e l e c t i o n s , " Izv. Mat. Inst. Bolg. Akad. Nauk, 1..55, 389-393 (1974). 200. S . I . Nedev, "Selected t h e o r e m s on multivalued sections and extensions," Serdica, 1_, No. 34, 285-294 (1975). 201. S. I. Nedev, "Multivalued selections and topological dimension," in: Mathematics and Mathematical Education, Bolg. Akad. Nauk, Sofia (1976), pp. 195-198. 202. S. I. Nedev and M. M. Choban, " F a c t o r i z a t i o n t h e o r e m s f o r multivalued mappings, multivalued sections and topological dimension," Math. Balkanica, No. 4 , 4 5 7 - 4 6 0 (1974). 203. G. M. Nepomnyashchii, "On selections and continuations of uniformly continuous mappings," Dokl. Akad. Nauk SSSR, 240, No. 6, 1289-1292 (1978). 204. T . V . Nechaeva, " C e r t a i n t h e o r e m s on fixed points," Sb. Trudov Inst. Mat. Sib. Otxl. Akad. Nauk SSSR, No. 4(21), 41-52 (1971). 205. H. Nikaido, Convex S t r u c t u r e s and Economic T h e o r y , Academic P r e s s , New York (1968). 206. E. L. Nurminskii, "On the differentiability of multivalued m a p p i n g s , " Kibernetika, No. 5, 46-48 (1978). 207. V . V . Obukhovskii, "On c e r t a i n fixed point p r i n c i p l e s f o r multivalued condensing o p e r a t o r s , " T r . Mat. Fak. Voronezh. Univ., No. 4, 70-79 (1971). 208. V . V . Obukhovskii, "The rotation of condensing multivalued v e c t o r fields," in: P r o c . Sixth All-Union Conf. on Topology, Tbilisi, M e t s n i e r e b a (1972), pp. 93-94. 209. V . V . Obukhovskii, " P e r i o d i c solutions of controllable s y s t e m s , " T r . Mat. Fak. Voronezh. Univ., No. 7, 68-76 (1972). 210. V . V . Obukhovskii, "On the question of periodic solutions of differential equations with a multivalued r i g h t hand side," T r . Mat. Fak. Voronezh. Univ., No. 10, 74-82 (1973). 211. V . V . Obukhovskii, "Asymptotic fixed point t h e o r e m s and s t a t i o n a r y points of dynamical s y s t e m s without uniqueness," in: Sb. Rabot A s p i r a n t o v po T e o r i i Funktii i Differents. Uravn., Voronezh (1974), pp. 30-38. 212. V. V. Obukhovskii, "Equations with multivalued o p e r a t o r s and some of t h e i r applications," D i s s e r t a t i o n , V o r o n e z h (1974). 213. V . V . Obukhovskii, "On a condition f o r positive equilibrium in a model of competitive economy," in: P r o c . Sixth Winter School on Mathematical P r o g r a m m i n g and Related Questions, Drogobych, 1973, Moscow (1975), pp. 183-189.
758
214. V. V. Obukhovsldi, "On the d e g r e e of a c l a s s of n o n c o m p a c t m u l t i v a l u e d m a p p i n g s , " in: P r o c . of the School of O p e r a t o r T h e o r y in Functional Spaces [in Russian], Minsk (1978), pp. 102-103. 215. V . V . Obukhovskii and E. V. Gorokhov, "On the d e t e r m i n a t i o n of the rotation of a c l a s s of c o m p a c t l y contractible multivalued v e c t o r fields," T r . Mat. Fak. Voronezh. Univ., No. 12, 45-54 (1974). 216. V. V. Obukhovskii and M. A. F l e i d e r v i s h , "On the relative rotation of a l m o s t acyclic multivalued v e c t o r fields, n Voronezh. Univ., Voronezh (1980). Manuscript deposited at VINtTI, Nov. 28, 1980, No. 5034-80 Dep. 217. V. I. Opoitsev, "The rotation of multivalued v e c t o r fields as a m a t h e m a t i c a l m e a n s f o r the investigation of s y s t e m s , n Avtomat. T e l e m e k h . , No. 9, 144-154 (1976). 218. V. I. Opoitsev, E q u i l i b r i u m and Stability in Models of Collective Behavior [in Russian], Nauka, Moscow (1977). 219. S. Otalmlov, nOn differential equations with multivalued right-hand side and retarded argument," Vopr. Vychisl. Prild. Mat. (Tashkent), No. 52, 12-21 (1978). 220. S. Otaknlov, non the theory of differential equations with multivalued right-hand side and aftereffect," Samarkand. Univ., Samarkand (1978). Manuscript deposited at VINITI, Oct. 26, 1978, No. 3367-78 Dep. 221. S. Otakulov, nThe dependence of the set of solutions and optimal solutions of differential inclusions with aftereffect onthe initial function and parameter," Samarkand. Univ., Samarkand (1979). Manuscript deposited at VINITI, Oct. 25, 1979, No. 3728-79 Dep. 222. S. Otakulov, "On a property of solutions of differential inclusions," Samarkand. Univ., Samarkand (1980). Manuscript deposited at VINITI, May 29, 1980, No. 2148-80 Dep. 223. A. I. Panasyuk and V. I. Panasyuk, "On an equation generated by a differential inclusion," Mat. Zametki, 27, No. 3, 429-437 (1980). 224. N. N. Petrov, "Controllable systems that are similar to systems with a convex vectegram,' Vestn. Leningr. Univ., No. i, 63-69 (1974). 225. V. A. Plotnikov, "The averaging method for differential inclusions and its application to optimal control problems, n Differents. Uravn., 15, No. 8, 1427-1433 (1979). 226. V. A. Plotnikov, "The averaging of differential inclusions," Ukr. Mat. Zh., 31, No. 5, 573-576 (1979). 227. V. A. Plotnikov, "The averaging of differential inclusions," Odessk. Univ., Odessa (1979). Manuscript deposited at VINITI, Feb. 12, 1979, No. 540-79 Dep. 228. V.A. Plotnikov, "The partial averaging of differential inclusions," Mat. Zametki, 2_~7,No. 6, 947-952 (1980). 229. V. A. Plotnikov and V. P. Zheltikov, "The averaging method for differential inclusions with retarded a r gument," Odessk. Univ., Odessa (1980). Manuscript deposited at VINITI, Sep. 22, 1980, No. 4182-80 Dep. 230. A. I. Povolotskii and E. A. Gango, "Bounded solutions of differential equations with multivalued righthand side," Mat. Zap. Krasnoyarsk. Gos. Pedagog. Inst., l~o. 2, 60-67 (1970). 231. A.I. Povolotskii and E. A. Guago, nPeriodic solutions of differential equations with a multivalued righthand side," Uch. Zap. Leningr. Gos. Pedagog. Inst., 464, 235-242 (1970). 232. A. I. Povolotskii and E. A. Gango, "On differential equations with a multivalued right-hand side and with retarded argument," Uch. zap. Leaingr. Gos. Pedagog. Inst., 49__66, 281-293 (1972). 233. A. L Povolotskii and E. A. Gango, nOn the periodic solutions of differential equations with a multivalued right-hand side," Uch. Zap. Leningr. Gos. Pedagog. Inst., 541, 145-154 (1972). 234. A. I. Povolotskii and E. A. Gango, "On bounded solutions of differential equations with a multivalued right-hand side, n in: nXXV]]I Gertsenovsk. Chteniya," Matematika, Leningrad (1975), pp. 56-58. 235. A. I. Povolotskii and E. A. Gango, "Positive periodic solutions of equations with a multivalued righthand side," in: "XXIXGertsenovsk. Chteniya," Matematilm, Leningrad (1976), pp. 52-53. 236. A. I. Povolotskii and E. A. Gango, "The existence of periodic solutions of differential equations with multivalued right-hand side," in: Mathematical Analysis and Function Theory [in Russian], No. 8, Moscow (1977), pp. 106-113. 237. E. S. Polovinldn, "The Riemann integral of a multivalued mapping," in: Proc. Second Conf. of Young Researchers of Dept. Comput. Math. and Cybernetics. Reports. Moscow Univ., Moscow (1975), pp. 44-46. 238. V. M. Polterovich and V. A. Spivak, "Gross interchangeability of multivalued mappings and the structure of equilibrium sets," Preprint. Central Econ.-Mat. Inst. of the Acad. Sci. USSR, 1978. 239. R.A. Polyak, "On the search for a fixed point of a class of point-to-set mappings," Dokl. Akad. Nauk SSSR, 242, No. 6, 1252-1255 (1978). 240. V. I. Ponomarev, "A new space of closed sets and multivalued mappings of bicompacta," Dokl. Akad. Nauk SSSR, 118, No. 6, 1081-1084 (1958). 241. V. I. P o n o m a r e v , "A new s p a c e of closed s e t s and multivalued continuous mappings of b i c o m p a c t a , " Mat. Sb., 48, No. 2, 191-212 (1959). 242. V. I. !~ "On multivalued mappings of topological s p a c e s , " Dold. Akad. Nauk SSSR, 124., No. 2, 268-271 (1959). 759
243. V. I. P o n o m a r e v , "On p r o p e r t i e s of topological s p a c e s , p r e s e r v e d under multivalued continuous mappings," Mat. Sb., 5_1, No. 4, 515-536 (1960). 244. V. I. P o n o m a r e v , "On the continuation of m u l t i v a l u e d mappings of topological s p a c e s to t h e i r b i c o m p a c t extensions, n Mat. Sb., 52, No. 3, 847-862 (1960). 245. V. I. P o n o m a r e v , nOn a c o m m o n fixed set for two continuous multivalued mappings of a b i c o m p a c t u m into itself," Colloq. Math., 10, No. 2, 227-231 (1963). 246. L. S. Pontryagin, V. G. Boltyanskii, R. V. G a m k r e l i d z e , and E. F. Mishchenko, The Mathematical T h e o r y of Optimal P r o c e s s e s , Wiley, New Y o r k (1962). 247. A. S. Potapov, "A r e m a r k on the rotation of multivalued v e c t o r fields," Sb. Rabot A s p i r a n t o v Voronezh. Univ., No. 2, 41-44 (1974). 248. B. N. Pshenichnyi, "Convex multivalued mappings and t h e i r conjugates," Kibernetika, No. 3, 94-102 (1972). 249. B. N. Pshenichnyi, nMultivalued mappings and the theory of differential games," in: Controllable Random Processes and Systems [in Russian], Kiev (1973), pp. 54-73. 250. B. N. Pshenichnyi, "Necessary conditions for the extremum for differential inclusions,~ Kibernetika, No. 6, 60-73 (1976). for the extremum of differential inclusions, ~ in: Prec. All251. B. IN. Pshenichnyi, 'Necessaryeonditions Union Symp. Optimal Control and Differential Games, Tbilisi, 1976, Metsniereba (1977), pp. 233-237. 252. B. N. Pshenichnyi and V. N. Gurov, "Necessary conditions for an extremum for differential inclusions with a no,smooth object functional," in: Theory of Optimal Solutions [in Russian], Kiev (1978), pp. 76-82. 253. B. N. Psheniclmyi and I. B. Medvedovsldi, "Necessary conditions for an extremum for a linear optimal control problem with phase constraints, n in: Theory of Optimal Solutions [in Russian], Kiev (1977), pp. 3-13. 254. B. N. Pshenielmyi and A. P. Yakovleva, " N e c e s s a r y conditions f o r an e x t r e m u m f o r differential and diff e r e n c e inclusions with a f t e r e f f e c t , " Inst. Kibern. Akad. Nauk Ukr. SSR, Kiev (1979). Manuscript deposited at Ukr. VINITI, A p r . 12, 1979, No. 1428. 255. B. N. Pshenichnyi and A. P. Yakovleva, "On n e c e s s a r y conditions f o r an e x t r e m u m f o r a s y s t e m with a f t e r e f f e c t , " Kibernetika, No. 3, 108-109 (1979). 256. B. N. Pshenictmyi and A. P. Yakovleva, " N e c e s s a r y a n d sufficient optimality conditions f o r a s y s t e m with a f t e r e f f e c t in the case of a convex multivalued mapping," Kibernetika, No. 4, 68-69 (1979). 257. R. K. Ragimkhanov, non the theory of differential equations with a multivalued right-hand side," Funkts. Anal. T e o r . Funkts. P r i m e , . , Makhachkala, No. 2, P a r t 2, 89-96 (1975). 258. R. K. Ragimkhanov, "On the question of the existence of the solution of an i n t e g r a l equation with a m u l t i valued right-hand side," Sib. Mat. Zh., 17, No. 3, 697-700 (1976). 259. R. K. Ragimkhanov, "On the question of the d e t e r m i n a t i o n of the multivalued o p e r a t o r of the i n t e r i o r s u p e r p o s i t i o n , " in: Functional-Differential Equations and Boundary Value P r o b l e m s of Mathematical P h y s i c s [in Russian], P e r m (1978), pp. 147-149. 260. D. A. Raikov, "A t w o - s i d e d closed g r a p h t h e o r e m f o r l i n e a r topological s p a c e s , " Sib. Mat. Zh., 7, No. 2, 353-372 (1966). 261. V. A. R a s t r e n i n , "On the approximation of equations in Banach s p a c e with a nonlinear multivalued o p e r a t o r , " Diffrents. Uravn., 15, No. 560-562 (1979). 262. A. M. Rubinov, " P o i n t - t o - s e t mappings defined on a cone," in: Optimal Planning [in Russian], No. 14, I~Iovosibirsk (1969), pp. 96-113. 263. A. M. Rubinov, S u p e r l i n e a r Multivalued Mappings and T h e i r Application to P r o b l e m s of Mathematical E c o n o m i c s [in Russian], Nauka, Leningrad (1980). 264. I. A. Rus, "On the method of s u c c e s s i v e a p p r o x i m a t i o n s , n Rev. Roumaine Math. P u r e s Appl., 17, No. 9, 1433-1437 (1972). 265. B. N. Sadovskii, " L i m i t - c o m p a c t and condensing o p e r a t o r s , n Usp. Mat. Nauk, 2-7, No. 1, 81-146 (1972). 266. Yu. I. Sapronov, "On the homotopy c l a s s i f i c a t i o n of condensing mappings, n T r . Mat. Fak. Voronezh. Univ., No. 6, 78-80 (1972). 267. K. S. Sibirskii and I. A. Chirkova, "Discontinuous d i s p e r s i v e dynamical s y s t e m s , " Mat. Issled., 6, No. 4, 165-175 (1971). 268. K. S. Sibirskii and I. A. Chirkova, non the theory of discontinuous d i s p e r s i v e dynamical s y s t e m s , n Dokl. Akad. Nauk SSSR, 211, No. 6, 1302-1305 (1973). 269. V. M. Simonov, " M a g i s t r a l t h e o r e m s f o r a c l a s s of smooth dynamical s y s t e m s , " in: T r . 18-oi Nauclm. Konf. Mosk. F i z . - T e k h . Inst., 1972, Ser. Aeromeldlanika. P r o t s e s s y Upr., Dolgoprudnyi (1973), pp. 195-202. 270. V. M. Simonov, "A m a x i m u m principle f o r a c l a s s of dynamical s y s t e m s with a f t e r e f f e c t , " Kibernetika, No. 1, 96-103 (1964).
760
271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. 293. 294. 295.
296. 297. 298. 299. 300. 301. 302.
E. G. Sldyarenko, "Some applications of the theory of s h e a v e s in g e n e r a l topology," Usp. Mat. Nauk, 19, I~o. 6, 47-70 (1964). G. S. Skordev, " F i x e d points of multivalued m a p p i n g s , " Dokl. Bolg. Akad. Nauk, 26, No. 5, 595-597 (1973). G. S. Skordev, "On p e r t u r b a t i o n s of v e c t o r fields," Dokl. Bolg. Akad. Nauk, 26, No. 8, 999-I000 (1973). G. S. Skerdev, "Antipodal mappings," Dold. Bolg. Akad. Nauk, 27, No. 2, 159-160 (1974). G. S. Skordev, "Incidence and coincidence of acyclie m a p p i n g s , " Dokl. Bolg. Akad. Nauk, 27, No. 3, 301302 (1974). G. S. Skordev, " F i x e d points of multivalued mappings of b i c o m p a c t AANR' s p a c e s , " Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 22, No. 4 , 4 1 5 - 4 2 0 (1974). G. S. Skordev, "Coincidence and c o - c o i n c i d e n c e of acyclic m a p p i n g s , " S e r d i e a , 1, No. 1, 29-41 (1975). G. S. Skordev, "On Rouch6's t h e o r e m , " Godishn. Sofiisk. Univ. Fak. Mat. Mekh., 67, 1-6 (1972-73(1976)). G. S. Skordev, " F i x e d points of acyclic m a p p i n g s , " Godishn. Sofiisk. Univ. Fak. Mat. Mekh., 67, 7-18 (1972-73(1976)). G. S. Skordev, " E q u i v a r i a n t multivalued mappings of a s p h e r e , " Godishn. Sofiisk. Univ. Fak. Mat. Mekh., 67, 113-137 (1972-73(1976)). G. S. Skordev, "On the degree of an equivariant acyclic mapping of a sphere," Godislm. Sofiisk. Univ. Fak. Mat. Mekh., 6_~7,139-164 (1972-73(1976)). ]VI. Stanojevic, "Two properties of multivalued quotient mappings," Math. Ballmnica, 4, 561-563 (1974). I. A. Sultanov, nControl processes described by equations with incompletely determined functional parameters," Avtomat. Telemekh., No. 10, 30-41 (1980). A. D. Taimanov, "Closed multivalued images of B-sets," Tr. Tbilissk. Mat. Inst. Akad. Nauk GSSR, 2__77, 53-56 (1960). S. P. Tashev, "On the existence of the solution of a differential equation with a multivalued right-hand side, M Serdica, 6, No. I, 98-104 (1980). A. V. Timokhov, "Certain fixed point theorems," in: Methods of Functional Analysis in Mathematical Economics [in Russian], Moscow (1978), pp. 98-110. V. L. Timokhovich, "Multivalued extensions of closed mappings," Dokl. Akad. Nauk BSSR, I_~9,No. i, 14-16 (1975). V. T. Todorov, "On nonexpansive mappings," Godishn. Sofiisk. Univ. Fak. Mat. Mekh., 68, 257-262 (197374(1977)). A. A. Tolstogonov, "The topological structure of continuous multivalued mappings," Sib. Mat. Zh., I_.66, No. 4, 837-852 (1975). A. A. Tolstogonov, "The m a x i m u m principle f o r differential inclusions," Differents. Uravn., 1_~1,No. 10, 1838-1842 (1975). A. A. Tolstogonov, "On the R a d o n - N i k o d y m and A. A. Lyapunov t h e o r e m s f o r a multivalued m e a s u r e , " Dokl. Akad. Nauk SSSR, 225, No. 5, 1023-1-26 (1975). A. A. Tolstogonov, "The c l a s s i c a l solutions of differential inclusions in Banach s p a c e s with a nonconvex r i g h t - h a n d side," Nelinein. Kolebaniya i T e o r i y a Upr. (Izhevsl0, No. 2, 16-23 (1978). A. A. Tolstogonov, "On differential inclusions in Banach s p a c e s and continuous s e l e c t o r s , " Dokl. Akad. NaukSSSR, 244, No. 5, 1080-1092 (1979). A. A. Tolstogonov, "On the p r o p e r t i e s of the solutions of differential inclusions in a Banach s p a c e , " Dokt. Akad. Nauk SSSR, 248, No. 1, 42-46 (1979). A. A. Tolstogonov, "On c o m p a r i s o n t h e o r e m s f o r differential inclusions in a Banach s p a c e with a nonconvex r i g h t - h a n d side. I. Existence of solutions," in: Lyapunov V e c t o r - F u n c t i o n s and T h e i r C o n s t r u c tion [in Russian], N o v o s i b i r s k (1980), pp. 6-34. A. A. Tolstogonov and I. A. Finochenko, "On functional-differential inclusions in a Banach s p a c e with a nonconvex right-hand side," Dokl. Akad. Nauk SSSR, 254, No. 1, 45-49 (1980). Yu. V. Trubnikov and V. V. Yurgelas, "On an iteration method f o r finding the roots of multivalued m o n o tone m a p p i n g s , " in: Applied Analysis [in Russian], Voronezh {1979), pp. 116-119. Yu. N. Tyurin, "The m a t h e m a t i c a l f o r m u l a t i o n of a s i m p l i f i e d model of industrial planning," Ekon. Mat. Metody, 1, No. 3, 391-409 (1965). Faro Khyu Shak, "Controllability in multivalued p r o c e s s e s , " Differents. Uravn., 1-2, No. 3, 484-493 (1976). F a m Khyu Shak, "On the theory of the control p r o c e s s e s with multivalued o p e r a t o r s , " Kibernetika, No. 2, 107-116 (1976). Fan Van Ch'yu (Fan Van Chong), "On a smooth selection theorem and its applications to multivalued integral equations," Mat. Sb., 105, No. 4, 622-637 (1978). M. Sh. F a r b e r , "Differentiable sections of multivalued mappings," in: Questions of M a t h e m a t i c a l C y b e r netics and Applied Mathematics [in Russian], Balm (1978), No. 3, pp. 47-60.
761
303. M. Sh. F a r b e r , ~On smooth sections of the i n t e r s e c t i o n of multivalued mappings, n Izv. Akad. Nauk AzSSR, S e t . F i z . - T e k h . Mat. Nauk, No. 6, 23-28 (1979). 304. R. P. Fedorenko, ~The m a x i m u m principle f o r differential inclusions, ~ Zh Vychisl. Mat. Mat. F i z . , 1_00, No. 6, 1385-1393 (1970). 305. R. P. Fedorenko, nThe m a x i m u m principle f o r differential inclusions (necessity), N Zh. Vychisl. Mat. Mat. Fiz., 11, No. 4, 885-893 (1971). 306. A. F. Filippov, ~On c e r t a i n questions in the t h e o r y of optimal regulation," Vestn. Mosk. Univ. Ser. Mat. Mekh. Astron. Fiz. Khim., No. 2, 25-32 (1959). 307. A. F. Filippov, ~Differential equations with a discontinuous right-hand side," Mat. Sb., 51, No. 1, 99-128 (1960). 308. A. F. Filippov, nDifferential equations with a multivalued discontinuous r i g h t - h a n d side, MDokl. Akad. Nauk SSSR, 151, No. 1, 65-68 (1963). 309. A. F. Filippov, nThe c l a s s i c a l solutions of differential equations with a multivalued r i g h t - h a n d side, n Vestn. Mosk. Univ. Mat. Mekh., No. 3, 16-26 (1967). 310. A. F. Filippov, ~On the existence of solutions of m u l t i v a l u e d differential equations, n Mat. Z a m e t k i , 10, No. 3, 307-313 (1971). 311. A. F. Filippov, ~On conditions f o r the e x i s t e n c e of solutions of multivalued differential equations ,n Diff e r e n t s . Uravn., 1_.33,No. 6, 1070-1078 (1977). 312. A. F. Filippov, nStability f o r differential equations with discontinuous and multivalued right-hand sides,n D i f f e r e n t s . Uravn., 15, No. 6, 1018-1027 (1979). 313. I. A. Finogenko, non the solutions of functional-differential inclusions with a nonconvex right-hand side, n P r o b l . Soy. T e o r i i Periodich. I)vizhenii, Izhevsk, No. 3, 85-88 (1979). 314. Fu P e i - z h e n ' , nOn A r z e l a ' s t h e o r e m f o r multivalued mappings, n Iilin Shida Xuebao, 1, 13-18 (1965). 315. A. Z. Khamishon, nOn the s e m i g r o u p s of m u l t i v a l u e d continuous t r a n s f o r m a t i o n s defining topological s p a c e s , n V e s t s i Akad. Navuk BSSR, S e t . F i z . - M a t . Nauk, Izv. Akad. Nauk BSSR, Ser. F i z . - M a t . 1Nauk, No. 5, 10-16 (1973). 316. A. Z. Khamishon, nContinuous s e m i g r o u p s of multivalued t r a n s f o r m a t i o n s , N Ukr. Mat. Zh., 26, No. 6, 775-783 (1974). 317. V. Z. Tsalyuk, "Conditions f o r the p r e s e r v a t i o n of the exponential stability of differential inclusions," Sb. Nauchn. T r . P e r m . Politekh. Inst., No. 203, 71-72 (1977). 318. V. Z. Tsalyuk, nGeneralized functions a s p e r t u r b a t i o n of exponential inclusions, n in: F u n c t i o n a l - D i f f e r ential Equations and Boundary Value P r o b l e m s of M a t h e m a t i c a l P h y s i c s [in Russian], P e r m (1978), pp. 155-158. 319. V. Z. Tsalyuk, ~On the stability in the f i r s t a p p r o x i m a t i o n of differential inclusions ,H Differents. Uravn., 1..66, No. 2, 258-263 (1980). 320. N. K. Chebaa, n The u n i f o r m a p p r o x i m a t i o n of s e t s in p a r t i a l l y o r d e r e d d i s p e r s i v e dynamical s y s t e m s , " Bul. Akad. Shtiintse RSS Moldav., Izv. Akad. Nauk MSSR, Ser. F i z . - T e k h . Mat. Nauk, No. 8, 87-91 (1967). 321. N. K. Cheban, nlnvariant s e t s in p a r t i a l l y o r d e r e d s e m i g r o u p s of multivalued mappings of the s p a c e , " Mat. I s s l e d . , 5, No. 3, 172-181 (1970). 322. N. K. Cheban, nThe r e s t o r a t i o n of points and domains in p a r t i a l l y o r d e r e d s e m i g r o u p s of multivalued mappings of the space, n in: M a t e r i a l y Konf. po Itogam N a u c h n . - I s s l e d . Raboty P r o f . - P r e p . Sostava Inst. z a 1971 g. (Fiz.-Mat. i Khim. Nauk), Lumina, Kishinev (1973), pp. 3-4. 323. N. K. Chebaa, nMappiags of d i s p e r s i v e dynamical s y s t e m s , n Mat. Issled., No. 55, 123-130 (1980). 324. A. G. Chentsov, nSelectors of multivalued q u a s i s t r a t e g i e s in differential g a m e s , n I n s t . Mat. Mekh. U r a l ' s k . Nauclm. T s e n t r a Akad. Nauk SSSR, S v e r d l o v s k (1978). Manuscript deposited at VINITI, Sep. 26, 1978, No. 3101-78 Dep. 325. I. A. Chirkova, nI4rapunov stability in discontinuous d i s p e r s i v e dynamical s y s t e m s , " Mat. Issled., No. 55, 131-134 (1980). 326. S. V. Chistyakov, nOn the existence of solutions of differential g a m e s without coalitions, n in: Control in D y n a m i c a l S y s t e m s , Leningrad (1979), pp. 128-145. Manuscript deposited at VINITI, July 24, 1979, No. 2794-79 Dep. 327. M. M. Choban, "Multivalued mappings and B o r e l s e t s , n Dokl. Akad. Nauk SSSR, 182, No. 3, 514-517 (1968). 328. M. M. Choban, "Multivalued mappings and s p a c e s with a countable net, n Dokl. Akad. Nauk SSSR, 18_66,No. 5, 1023-1026 (1969). 329. M. M. Choban, nMultivalued mappings and c e r t a i n r e l a t e d questions, n Dokl. Akad. Nauk SSSR, 190, No. 2, 293-296 (1970). 330. M. M. Choban, nMultivalued mappings and B o r e l s e t s . I, n T r . Mosk. Mat. Obshch., 22, 229-250 (1970). 331. M. M. Choban, "Multivalued mappings and B o r e l s e t s . II," T r . Mosk. Mat. Obshch., 2._33,277-301 (1970).
762
332. M. M. Choban, MOrt B - m e a s u r a b l e sections," Dotal. Akad. Nauk SSSR, 207, No. 1, 48-51 (1972). 333. M. M. Choban, "On the descriptive theory of s e t s , " Dokl. Bolg. Akad. Nauk, 2_~6,No. 4, 449-452 (1973). 334. M. M. Choban, ~Reduction t h e o r e m s on the existence of continuous sections. Sections o v e r subsets of f a c t o r spaces of topological groups, ~ Mat. Issled., 8, No. 4(30), 111-156 (1973). 335. M. M. Choban, "General t h e o r e m s on sections and t h e i r applications," Serdica, 4, No. 1, 74-90 (1978). 336. M. M. Choban and V. M. Vylov, ~On a t h e o r e m of Michael on sections," Dokl. Bolg. Akad. Nauk, ~ No. 7, 871-873 (1975). 337. P. I. Chugunov, NOn t h e o r e m s on the approximation of the solutions of differential inclusions, n Probl. S o v r e m . Teorii Periodich. Dvizhenii (Izhevstd, No. 3, 79-84 (1979). 338. I. V. Shragin, "Fixed points of multivalued mappings, ~ T r . Tambovsk. Last. Khim. Mashinostr., No. 4, 27-33 (1970). 339. I. V. Shragin, ~On an application of the t h e o r e m s of Luzin, T i e t z e - U r y s o n and of the t h e o r e m on the m e a surable selection, ~ in: Boundary Value P r o b l e m s [in Russian], P e r m (1979), pp. 171-175. 340. A. S. Shubg, "The connection of a d i s p e r s i v e dynamical s y s t e m on the line with s e m i d y n a m i c a l s y s t e m s , " Mat. Issled., No. 55, 135-147 (1980). 341. M. M. Shui'ts, non the existence and uniqueness of p r e i m a g e s under a multivalued mapping," Uch. Zap. Gor'kov,/ Univ., No. 166, 207-210 (1973). 342. L. Ya. Engel'son, "The point-exponential topology," Uch. Zap. Latv. Univ., 257, No. 2, 121-126 {1976). 343. A. P. Yakovleva, ~ N e c e s s a r y e x t r e m u m conditions f o r d i s c r e t e s y s t e m s with aftereffect," Kibernetika, No. 3, 111-112 (1979). 344. A. P. Yakovleva, "A discrete m a x i m u m principle for s y s t e m s with lag," in: The Theory of Optimal Solutions [in Russian], Kiev (1979), pp. 91-97. 345. J. Achari, ~Generalized multivalued contractions and fixed points," Rev. Roumaine Math. P u r e s Appl., 2_.~4,No. 2, 179-182 (1979). 346. N. U. Ahmed and K. L. Teo, ' C o m m e n t s on a s e l e c t o r t h e o r e m in Banach spaces, n J. Optim. Theory Appl., 1__99,No. 1, 117-118 (1976). 347. A. Alesina, S. Massa, and D. Roux, "Panti uniti di multifanzioni con condizione di tipo B o y d - W o n g , ~ Boll. Un. Mat. Ital., 8, No. 1, 29-34 (1973). 348. W. Alt, Stabilitat m e n g e n w e r t i g e r Abbildungen mit Anwendungen aufnichtlineare Optimiergungsprobleme, n Math. Schr., Bayreuth, No. 3, VII (1979). 349. G. F. Andrus and T. Nishiura, NFixed points of random s e t - v a l u e d maps, ~ Nonlinear AnaL, 3, No. 1, 65-72 (1979). 350. T. S. Angell, "On controllability for nonlinear h e r e d i t a r y s y s t e m s : a fixed point approach," Nonlinear Anal., 4, No. 3, 529-545 (1980). 351. G. Anichini and P. Zecca, ~Problemi ai limiti p e r equazionl differeaziali multivoche su intervalli non compatti," Riv. Mat. Univ. P a r m a , 1_, 199-212 (1975(1977)). 352. G. Anichini and P. Zecca, "Multivalued differential equations in Banach space. An application to control theory," J. Optim. Theory Appt., 2.1, No. 4, 477-486 (1977). 353. R. Ansorge and K. Taubert, "Set-valued collectively c o m p a c t o p e r a t o r s and applications to s e t - v a l u e d differential equations," Computing, 2._~3,No. 4, 333-343 (1979). 354. H. A. Antosiewicz and A. Cellina, "Continuous selections and differential relations, ~ J. Diff. Equations, 1..~9, No. 2, 386-398 (1975). 355. H. A. Antosiewicz and A. Cellina, "Continuous extensions of multifanctions," Ann. Polon. Math., 3..~4, No. 1, 107-111 (1977). 356. G. Aquaro, "Sur les applications multivalentes d'un espace toplogique dans un espace uniforme et compact, ~ C. R. Acad. Sci. P a r i s , 244, No. 2, 155-157 (1957). 357. G. A r o n s s o n , "A new approach to nonlinear controllability,N J. Math. Anal. Appl., 4_.44,No. 3, 763-772 (1973). 358. Z. Artstein, "Set-valued m e a s u r e s , " T r a n s . Am. Math. Soc., 165, 103-125 (1972). 359. Z. Artstein, "On the calculus of closed s e t - v a l u e d functions, ~ Indiana Univ. Math. J., 24_, No. 5, 433-441 (1974). 360. Z. Artstein, "Weak convergence of s e t - v a l u e d functions and control," SIAM J. Control Optim., 1_33,No. 4, 865-878 (1975). 361. N. A. Assad, nFixed point t h e o r e m s f o r set valued t r a n s f o r m a t i o n s on compact s e t s , ' Boll. Un. Mat. Ital., 8, No. 1, 1-7 (1973). 362. N. A. A s s a d and W. A. Kirk, "Fixed point t h e o r e m s f o r s e t - v a l u e d mappings of contractive type, ~ Pac. J. Math., 4.~3, No. 3, 553-562 (1972). 363. J . - P . Aubin, NPropri6t6 de P e r r o n - F r o b e n i u s pour des c o r r e s p o n d a a c e s , " C. R. Acad. Sci. P a r i s , 286., No. 20, A911-A914 (1978).
763
364. J.-P. Aubin, A. Cellina, and J. Nohel, "Monotone trajectories of multivalued dynamical systems," Arm. Mat. Pura Appl., 115, 99-117 (1977). 365. J.-P. Aubin and J. Nohel, "Existence de trajectories monotones de systemes dynamiques multivoques discrets," C. R. Acad. Sci. Paris, 282, No. 5, A267-A270 (1976). 366. J.-P. Aubin and J. Siegel, "Fixed points and stationary points of dissipative multivalued maps," Proc. Am. Math. Soc., 78, No. 3, 391-398 (1980). 367. R. J. Aumann, "Integrals of set-valued functions," J. Math. Anal. Appl., 1__2,No. 1, 1-12 (1965). 368. R. J. Aumarm, "Measurable utility and the measurable choice theorem," Colloq. Integr. Nat. Centre Nat. Rech. Scient., No. 171, 15-26 (1969). 369. M. Avram, "Points fixed communs pour les applications multivoque daas les espaces m~triques," Mathematica (Cluj), 17, No. 2, 153-156 (1975). 370. C. Avramescu, "Fixed point theorems for multivalued contracting mappings in uniform spaces," An. Univ. Craiova, Ser. 4a, 1, 63-67 (1970). 371. C. A v r a m e s e u , " T h ~ o r e m e s de point fixe p o u r les applications c o n t r a c t a n t e s et a n t i c o n t r a c t a n t e s , " Manuscr. Math., 6, No. 4, 405-411 (1972). 372. A. G. Babiker, " M e a s u r a b i l i t y of sections f o r m a p s r e p r e s e n t i n g certain m e a s u r e s , " J. Reine Angew. Math., 315, 115-120 (1980). 373. L. W. Bagett and A. R a m s a y , "A functional analytic proof of a selection l e m m a , " Can. J. Math., 3__~2,No. 2, 441-448 (1980). 374. E. J. B a k e r , "On a g e n e r a l i z e d open-mapping t h e o r e m , " Math. Ann., 172, No. 3 , 2 1 7 - 2 2 1 (1967). 375. Y. Balasko, "The g r a p h of the W a l r a s c o r r e s p o n d e n c e , " E c e n o m e t r i c a , 4__33,No. 5-6, 907-912 {1975). 376. H. T. Banks and M. Q. J a c o b s , "A differential calculus for multifuactions," g. Math. Anal. Appl., 2._99,No. 2, 246-272 (1970). 377. T. Banzaru "On the continuity of multivalued mappings," Bul. Stiint. Tehn. Inst. Politehn. T i m i s o a r a , 16, No. 1, 7-12 (1971). 378. T. Barmaru "On certain p r o p e r t i e s of multivalued mappings," Stud. C e r c . Mat., 2._44,No. 19, 1503-1510 (1972). 379. T. Banzaru "Multivalued mappings and M - p r o d u c t s p a c e s , " Bul. Stiint. Tehn. Inst. Politehn. T i m i s o a r a , 1__7.7,No. 1, 17-23 (1972). 380. T. Banzaru " L a convergence uniforme et quasi uniforme et la continuit5 des applications multivoques," C. R. Acad. Sci. P a r i s , 277, No. 2, A91-A92 (1973). 381. T. Banzaru "The continuity and the uniform continuity of multivalued mappings between uniform s p a c e s , " Bul. Stiint. Tehn. Inst. Politehn. T i m i s o a r a , 188, No. 2, 129-133 (1973). 382. T. Banzaru, "Continuit~ et continuit~ uniforme des applications multivoques e n t r e les e s p a c e s u n i f o r m e s , " C. R. Acad. Sci. P a r i s , 277, No. 3, A161-A162 (1973). 383. T. Banzaru "Die q u a s i g l e i c h m a s s i g e Konvergenz und Stetigkeit d e r mehredeutigen Abbildungen," Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1_.7.7,No. 1, 9-16 (1973). 384. T. Banzaru "On the convergence of g e n e r a l i z e d sequences of multivalued mappings," Bul. Stiint. Tehn. Inst. Politehn. T i m i s o a r a , Ser. Chim., 2__00,No. 1, 13-15 (1975). 385. T. Banzaru "Sur la quasicontinuit~ des applications multivoques," Bul. Stiint. Tehn. Inst. Politehn. T i m i s o a r a , 2__.1,No. 1, 7-8 (1976). 386. T. Banzaru ~On the convergence of g e n e r a l i z e d sequences of multivalued mappings to quasicontinuous multivalued mappings," Stud. C e r e . Mat., 2__99,No. 5, 447-452 (1977). 387. T. Banzaru "Die punktuelle und die g l e i c h m a s s i g e Konvergenz der v e r a l l g e m e i n e r t e n Folgen m e h r deutigen Abbildungen mit W e r t e n in u n i f o r m e n Raumen," An. Univ. T i m i s o a r a , S e t . Stiint. Mat., 12, No. 1, 13-21 (1974(1977)). 388. T. Banzaru, "Uber die Stetigkeit einiger mehrdeutigen Abbildungen die aus einer gegebenen mehrdeutigen Abbildung hergeleitet werden," Rev. Roumaine Math. Pures Appl., 23, No. 6, 843-846 (1978). 389. C. Bardaro and C. C. Gori, "Sull'estensione di un teorema di E. Baiada nile multifunzioni," Atti Sere. Mat. Fis. Univ. Modena, 2__8,No. 1, 1-8 (1979). 390. C. Bardaro and P. Pucci, rUn teorema di esistenza per equazioni contingenti in spazi di Banach," Atti Sere. Mat. Fis. Univ. Modena, 2__7,No. 2, 207-211 (1978). 391. C. Bardaro and P. Pucci, rUn teorema di esistenza per equazioni differenziali generalizzate in spazi di Banach," Atti Sere. Mat. Fis. Univ. Modena, 28, No. 1, 9-14 (1979). 392. B. R. Barmish, "Measurable selection theorems and their application to problems of guaranteed perf o r m a n c e , " IEEE T r a n s . Automat. Control, 2_.~3,No. 4 , 6 8 5 - 6 8 7 (1978). 393. E. N. B a s s a m , "On coincidence producing multifunctions onto t r e e s , " Bull. Acad. Polon. Sci. S~r. Sci. Math. Astron. Phys., 20, No. 12, 1007-1013 (1972).
764
394. E. N. B a s s a m , " F i x e d points f o r multivalued functions of B - s p a c e s , " Bull. Acad. P o l o , . Sci. S6r. Sci. Math. Astron. Phys., 2__.1,No. 6, 523-529 (1973). 395. J. B e b e r n e s , " P o s i t i v e i n v a r i a n c e and a WazewsM t h e o r e m , " in: Lect. Notes Math., No. 415, 35-39 (1974). 396. J. B e b e r n e s and W. Kelley, "Some b o u n d a r y - v a l u e p r o b l e m s f o r g e n e r a l i z e d differential equations," SIAM J. Appl. Math., 25, No. 1, 16-23 (1973). 397. J . B e b e r n e s and J. D. Schuur, "The W a z e w s k i topological method f o r contingent equations," Ann. Mat. P u r a Appl., 8%7, 271-279 (1970). 398. J. B e b e r n e s and J. D. Schuur, "Investigations in the topological method of Wazewski," Atti Accad. Naz. Lincei Rend. C1. Sci. F i s . Mat. Natur., 4_~9,Nos. 1-2, 39-42 (1970). 399. E. G. Bogle, "The V i e t o r i s mapping t h e o r e m for bicompact s p a c e s , " Ann. Math., 51.1, No. 2, 534-543 (1950). 400. E. G. Begle, "A fixed point t h e o r e m f o r b i c o m p a c t s p a c e s , " Ann. Math., 5_!1,No. 2, 544-550 (1950). 401. tI. Bell, "On fixed point p r 6 p e r t i e s of plane continua," T r a n s . Am. Math. Soc., 128, No. 3, 539-548 (1967). 402. H. Bell, "Fixed points, index, and degree f o r s o m e s e t valued functions," Fund. Math., 6_0_9, No. 3, 213-217 (1970). 403. H. Bell, "Fixed points, index, and degree f o r s o m e s e t valued functions," in: Lect. Notes Math., No. 171, 1-7 (1970). 404. M. Benamara, "Sections measurables extr~males d'une multiapplication," C. R. Acad. Sci. Paris, A278, No. 19, 1249-1252 (1974). 405. M. L. Bennati, nUn teorema di controllabilita," Publ. Ist. Mat. Univ. Genova, No. 171 (1976). 406. M. L. Bennati, nun teorema di controllabilita," Boll. Un. Mat. Ital., AI4, No. I, 190-198 (1977). 407. C. B e r g e , Topological Spaces, Macmillan, New Y o r k (1963). 408. A. B e r g e r and K.-H. E l s t e r , " Z u m Begriff d e r halbstetigen Punkt-Menge-Abbildungen in topologischen Strukturen," W i s s . Z. Teehn. I-Iochsch. Ilmenau, 2._~1,No. 1, 39-56 (1975). 409. A. B e r g e r and K.-H. E l s t e r , "Untersuchungen u b e r halbstetige Punkt-Menge-Abbiludungen," W i s s . Z. Techn. Hochsch. Ilmenau, 2..~2,No. 4, 11-29 (1976). 410. H. B e r l i o c c h i and J . - H . L a s r y , " P r i n c i p e de P o n t r y a g i n p o u r des s y s t e m e s r6gis p a r une 6quation diff6rentielle multivoque," C. R. Acad. Sci. P a r i s , 277, No. 22, A1103-A1105 (1973). 411. A. Bielecki, "Extension de la m~thode du r 6 t r a c t e de T. Wazewski aux 6quations au paratingent," Ann. Univ. M a r i a e Curie-Sklodowska, A_~9,No. 1-13, 37-61 (1955(1957)). 412. A. Bielecki, " C e r t a i n e s propriSt6s topologiques des solutions des 6quations du paratingent," Ann. Univ. Marine Curie-Sklodowska, A__~9,No. 1-13, 63-79 {1955(1957)). 413. L. J. Billera, "Topologies f o r 2X; s e t - v a l u e d functions and t h e i r g r a p h s , " T r a n s . Am. Math. Soc., 155, No. 1, 137-147 (1971). 414. V. I. Blagodatskih, "Sufficient conditions of optimality f o r contingent equations," Lect. Notes Comput. Sci., 3, 319-328 (1973). 415. V. I. Blagodatskikh, " T i m e optimal control p r o b l e m f o r differential inclusions," Lect. Notes Comput. Sci., 27, 147-152 (1975). 416. H. F. Bohnenblust and S. Karlin, "On a t h e o r e m of Ville. Contributions to the theory of g a m e s . I," Ann. Math. Studies, P r i n c e t o n (1950), pp. 155-160. 417. G. Bol, "Eigenschaften v e r s c h i e d e n e r Topologien auf Systemen von a b g e s c h l o s s e n e n Teilmengen," Oper. R e s . - V e r f a h r e n , 1_.~8,19-25 {1974). 418. C. J. R. B o r g e s , "A study of multivalued functions," Pac. J. Math., 2_~3,No. 3 , 4 5 1 - 4 6 1 (1967). 419. C. J. R. B o r g e s , non a question of Wojdyslawski and S t r o t h e r , " Duke Math. J . , 3._55,No. 4, 829-833 (1968). 420. K. Borsuk, "On a m e t r i z a t i o n of the h y p e r s p a c e of a m e t r i c s p a c e , " Fund. Math., 9...44,No. 1, 191-207 (1977). 421. J. Borwein, "Multivalued convexity and optimization: a unified a p p r o a c h to inequality and equality cons t r a i n t s , " Math. P r o g r a m m i n g , 1__.3,No. 2, 183-199 (1977). 422. R. K. Bose and R. N. Mukherjee, " C o m m o n fixed points of s o m e multivalued m a p p i n g s , " Tamkang J. Math., 8, No. 2, 245-249 (1977). 423. R. K. Bose and R. N. Mukherjee, nOn fixed points of nonexpansive s e t - v a l u e d m a p p i n g s , " P r o c . Am. Math. Soc., 7.~2, No. 1, 97-98 (1978). 424. M. G. Bouligand, "Sur la semicontinuit6 d'inclusion et quelques sujets cormexes," Enseign. Math., 31, 14-22 (1932). 425. D . G . Bourgin, "On the Vietoris-Begle theorem," Bull. Am. Math. Soc., 7_~6,No. 4, 747-751 (1970). 426. D . G . Bourgin, ~Set valued transformations," Bull. Am. Math. Soc., 78, No. 4, 597-599 (1972). 427. D , G . Bourgin, "Cones and Vietoris-Begle type theorems," Trans. Am. Math. Soc., 174, 155-183 (1972). 428. D . G . Bourgin, "Fixed point and min-max theorems," Pac. J. Math., 4_~5,No. 2, 403-412 (1973). 429. D . G , Bourgin, "A degree for nonacyclic multiple-valued transformations," Bull. Am. Math. S oc., 80, No. i, 59-61 (1974).
765
430. D. G. Bourgin, "A generalization of the mapping degree, H Can. J. Math., 2_6, No. 5, 1109-1117 (1974). 431. M. Bradley and R. Datko, "Some analytic and measure theoretic properties of set-valued mappings," SIAM J. Control Optim., i__55,No. 4, 625-635 (1977). 432. T. A. Brahana, M. K. Fort, Jr., and W. G. Horstman, "Homotopy for cellular set-valued functions," Proc. Am. Math. Soc., i_~6,No. 3, 455-459 (1965). 433. A. J. Brandao Lopes Pinto, E. S. De Blasi, and F. Iervolino, "Uniqueness and existence theorems for differential equations with compact convex valued solutions," Boll. Un. Mat. Ital., 3, No. i, 47-54 (1970). 434. E. J. Braude, "Perfect open and distinguishable multivalued maps," Trans. Am. Math. Soc., 182, 431-441 (1973). 435. A. Bressan, "On a bang-bang principle for nonlinear systems," Boll. Un. Mat. Ital., Suppl., i, 53-59 (1980). 436. T. F. Bridgland, Jr., nContributions to the theory of generalized differential equations. I,~ Math. Systems Theory, 3, No. i, 17-50 (1969). 437. T. F. Bridgland, Jr., "Contributions to the theory of generalized differential equations. If," Math. Systems Theory, 3, No. 2, 156-165 (1969). 438. T. F. Bridgland, Jr., "Trajectory integrals of set-valued functions," Pac. J. Math., 3__33,No. I, 43-68 (1970). 439. T. F. Bridgland, Jr., "Extreme limits of eompacta valued functions," Trans. Am. Math. Soc., 170, 149-163 (1972). 440. B. Brosowski, nFixed point theorems in approximation theory," in: Theory and Applications of Monotone Operators, Gubbio (1969(1970)), pp. 285-290. 441. F. E. Browder, "On a generalization of the Schauder fixed point theorem," Duke Math. J., 2.6, No. 2, 291303 (1959). 442. F. E. Browder "Multivalued monotone nonlinear mappings in Banach spaces," Trans. Am. Math. Sec., 118, No. 6, 338-351 (1965). 443. F. E. Browder "Nonlinear maximal monotone operators in Banach space, ~ Math. Ann., 175, No. 2, 89-113 (1968). 444. F. E. Browder "Nonlinear monotone and accretive operators in Banach spaces," Proc. Nat. Acad. Sci., U.S.A., 6_~i,No. 2, 388-393 (1968). 445. F. E. Browder "The fixed point theory of multivalued mappings in topological vector spaces," Math. Ann., 17.7, No. 4, 283-301 (1968). 446. F. E. Browder "Nonlinear variational inequalities and maximal monotone mappings in Banach spaces," Math. Ann., 183, No. 3, 213-231 (1969). 447. F. E. Browder ~Generalizations of accretivity for nonlinear mappings of Banach spaces," Bull. Am. Math. Soc., 7__.8,No. i, 93-98 (1972). 448. F. E. Browder Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Amer. Math. Soc., Providence (1976). 449. F. E. Browder "Asymptotically orientation-preserving mappings in Banach spaces," J. Funct. Anal., 2_~5, No. 2, 121-127 (1977). 450. F. E. Browder ~On the range of the sum of nonlinear operators and the Landesmann-Lazer principle," Boll. Un. Mat. Ital., BI6, No. i, 364-376 (1979). 451. F. E. Browder D. G. de Figueirede, and C. P. Gupta, "Maximal monotone operators and nonlinear integral equations of Harnmerstein type," Bull. Am. Math. Soc., 76.6,No. 4, 700-705 (1970). 452. F. E. Browder and P. Hess, ~Nonlinear mappings of monotone type in Banach spaces," J. Funct. Anal., i_~i,No. 3, 251-294 (1972). 453. P. Brunovsky, "On the n e c e s s i t y of a c e r t a i n convexity condition f o r lower c l o s u r e of control p r o b l e m s , " SIAM J. Control, 6, No. 2, 174-185 (1968). 454. P. Brunovsky, " S c o r z a - D r a g o n i ' s t h e o r e m f o r unbounded s e t - v a l u e d functions and its applications to cont r o l p r o b l e m s , " Mat. Cas., 20, No. 3, 205-213 (1970). 455. P. Brunovsky, "On two c o n j e c t u r e s about the c l o s e d - l o o p t i m e - o p t i m a l control," Lect. Notes Comp. Sci., 3, 341-344 (1973). 456. P. Brunovsky, "The closed-loop time-optimal control. I: Optimality," SIAM J. Control, i_~2,No. 4,624634 (1974). 457. P. Brunovsky, "The closed-loop time-optimal control. If: Stability," SIAM J. Control Optim., i_~4,No. I, 156-162 (1976). 458. J. Bryszewski, nOn a class of multivalued vector fields in Banach spaces," Fund. Math., 97, No. 2, 79-94 (1977). 459. J. Bryszewski and L. Gorniewicz, nMultivalued maps of subsets of Euclidean spaces, n Fund. Math., 9__00, No. 3, 233-251 (1976). 460. J. Bryszewskl and L. Gorniewicz, "A Poincar6 type coincidence theorem for multivalued maps," Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys., 24, No. 8, 593'598 (1976). 766
461. 5. B r y s z e w s k i and M. Serbinowski, ~On the minimal displacement under admissible maps, ~ Bull. Acad. Polon. Sci., Set. Sci. Math., 2_~7,No. 2, 201-207 (1979). 462. C. L. Byrne, HRemarks on the s e t - v a l u e d integrals of Debreu and Aumann, n J. Math. Anal. Appl., 6__2, No. 2, 243-246 (1978). 463. O. Caligaris and P. Oliva, ~A variational approach to existence t h e o r e m s f o r time-dependent multivalued differential equations, ~ Publ. Ist. Mat. Univ. Genova, 1978, No. 250. 464. B. D. Calvert, ~The local fixed point index f o r multivalued t r a n s f o r m a t i o n s in a Banach space, n Math. Ann., 190, No. 2, 119-128 (1970). 465. E. Campu, ' ~ q u a t i o n s diff6rentielles au paratingent a r e t a r d e m e n t , dans les e s p a c e s de Banach. Th~or e m e s dTexistence des solutions, n Rev. Roumaine Math. P u r e s Appl., 20_0,No. 6, 631-657 (1975). 466. E. Campu, HApproximation des solutions des 5quations diff~rentielles au paratingent a r e t a r d e m e n t , dans les espaces de Banach, H Rev. Roumaine Math. P u r e s Appl., 25, No. 4, 509-518 (1980). 467. E. Campu and A. Halanay, ~Systemes des c o m m a n d e et ~quations au paratingent a r e t a r d e m e n t , " Colloq. Math., 1_~8, 29-35 (1967). 468. C. E. Capel and W. L. Strother, nMultivalued functions and p a r t i a l o r d e r , ~ Portugal. Math., 1__.7,No. 1-2, 41-47 (1958). 469. M. Capon, NSurjectivit~d'applications multivoques dans les convexes compacts. Complement, ~ Bull. Sci. Math., 97, No. 1, 3-4 (1973). 470. L. Carbone, RUn c r i t e r i o di selezione continua, ~ Bull. Ua. Mat. Ital., 1.~2, No. 1-2, 127-137 (1975). 471. L. Carbone, ~Selezioni continue in spazi non lineari e punti fissi," Rend. Circ. Mat. P a l e r m o , 25, No. 1-2, 101-115 (1976). 472. C. Castaing, ~Quelques p r o b l e m e s de mesurabilit~ li6s a la ~h~orie de la commande, n C. R. Acad. Sci. P a r i s , AB262, No. 7, A409-A411 (1966). 473. C. Castaing, ~Sur les 6quations diff~rentielles multivoques, ~ C. R. Acad. Sci. P a r i s , 263, No. 2, A63-A66 (1966). 474. C. Castaing, ' S u r un th6oreme de reprSsentation integrale li~ a la c o m p a r a i s o n des m e s u r e s , ~ C. R. Acad. Sci. P a r i s , 264, No. 24, A1059-A1062 (1967). 475. C. Castaing, HSur les multiapplications m e s u r a b l e s , MRev. F r a n c . Inform. Rech. Op6rat., 1, No. 1, 91-126 (1967). 476. C. Castaing, Sur les multiapplications m e s u r a b l e s , Doctoral Dissertation, Univ. de Caen, F r a n c e (1967). 477. C. Castaing, Le th4oreme de D u n f o r d - P e t t i s g~n~ralis~, Univ. de Montpellier, No. 43 (multigraph) (1968-1969). 478. C. Castaing, Sur le g r a p h dVune multiapplication souslinienne (mesurable), S e c r e t a r i a t des Math~matiques de la Facult~ des Sciences de Montpellier, 1969-1970, Publication No. 55 bis, Universit6 de Mentpellier (1969). 479. C. Castaing, RUn th~oreme d'existence de sections s~par6ment m e s u r a b l e s et s 6 p a r 6 m e n t absolumeat continues d'une multiapplication s~par6ment m e s u r a b l e et s~par6ment absolument continue, ~ C. R. Acad. Sci. P a r i s , 276, No. 5, A367-370 (1973). 480. C. Castaing, HCompacit~ faible des sections dVune multiapplication dans les e s p a c e s de Kothe g6n~ralis6s, ~ C. R. Acad. Sci. P a r i s , 281, No. 10, A333-A335 (1975). 481. C. Castaing, HSolutiens continues a droite d'une ~quatien diff6rentielles multivoque avec contraint s u r l'~tat, ~ C. R. Acad. Sci. P a r i s , AB288, No. 9, A507-A509 (1979). 482. C. Castaing, ~Compacit~ faible, d6composabilit~ dans L~ et dans certains espaces du type K o t h e - O r l i c z , H C. R. Acad. Sci. P a r i s , AB290, No. 1, A17-A19 (1980). 483. C. Castaing and M. Valadier, H~quations diff~rentielles multivoques dans les e s p a c e s l o c a l e m e n t convexes, ~ C. R. Acad. Sci. P a r i s , 266, No. 19, A985-A987 (1968). 484. C. Castaing and M. Valadier, ~ q u a t i o n s diffSrentielles multivoques dans les espaces v e c t o r i e l s l o c a l e m e n t convexes," Rev. F r a n c . Inform. Rech. Oper., 3, No. 16, 3-16 (1969). 485. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, S p r i n g e r - V e r l a g , Berlin (1977). 486. P. J. Caulfield, ~Multivalued functions in uniform s p a c e s , ~ Doctoral Dissertation, Ohio State University, 1967; D i s s e r t a t i o n A b s t r a c t s , B28, No. 9, 3776 (1968). 487. R. Cauty, nUn th~oreme de s~lection et l ' e s p a c e des r~tractions d'une s u r f a c e , " Am. J. Math., 9_~7,No. 1, 282-290 (1975). 488. A. Cellina, "A t h e o r e m on the approximation of c o m p a c t multivalued mappings, ~ Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 4..~7, No. 6, 429-433 (1969{1970)). 489. A. Cellina, ~Approximation of s e t - v a l u e d functions and fixed point t h e o r e m s , H Ann. Mat. P u r a Appl., 82,
17-24 (1969).
767
490. A. Cellina, "A f u r t h e r r e s u l t on the a p p r o x i m a t i o n of s e t - v a l u e d mappings," Atti Accad. Naz. Lineei Rend. C1. Sci. F i s . Mat. Natur., 4___8,No. 4, 412-416 (1970). 491. A. Cellina, "Multivalued differential equations and o r d i n a r y differential equations," SIAM J. Appl. Math., 1__8, No. 2, 533-538 (1970). 492. A. Cellina, "The role of approximation in the t h e o r y of multivalued mappings," in: Differential G a m e s and Related Topics (Proc. Internat. S u m m e r School, Varenna, 1970), North-Holland, A m s t e r d a m (1971), pp. 209-220. 493. A. Cellina, "A s e l e c t i o n t h e o r e m , " Rend. Sere. Mat. Univ. Padova, 55, 143-149 (1976(1977)). 494. A. Cellina and G. S. Jones, " F i x e d point theory and s e t - v a l u e d mappings," Univ. of Maryland, Report, 1969. 495. A. Cellina and A. Lasota, "A new approach to the definition of topological degree f o r multivalued m a p pings," Atti Accad. Naz. Lincei Rend. C1. Sci. F i s . Mat. Natur., 4_.7.7,No. 6, 434-440 (1969(1970)). 496. L. C e s a r i , " E x i s t e n c e t h e o r e m s f o r optimal solutions in Pontryagin and Lagrange p r o b l e m s , " J. Soc. Industr. Appl. Math., A3, No. 3, 475-498 (1965(1966)). 497. L. C e s a r i and M. B. S u r y a n a r a y a n a , " N e m i t s k y ' s o p e r a t o r s and lower closure t h e o r e m s , " J. Optim. T h e o r y Appl., 19, No. 1, 165-183 (1976). 498. L. C e s a r i and M. B. S u r y a n a r a y a n a , "Upper semicontinuity p r o p e r t i e s of s e t - v a l u e d functions," Nonlinear Anal.: Theory, Methods Appl., 4, No. 4, 639-656 (1980). 499. F. F. Charatonik, "The fixed point p r o p e r t y f o r s e t - v a l u e d mappings of s o m e acyclic c u r v e s , " Colloq. Math., 26, 331-338 (1972). 500. M . - P . Chen and M.-H. Shih, " F i x e d - p o i n t t h e o r e m s f o r point-to-point and p o i n t - t o - s e t m a p s , " J. Math. Anal. AppI., 71, No. 2, 516-524 (1979). 501. S.-N. Chow and J. D. Schuur, " F u n d a m e n t a l t h e o r y of contingent differential equations in Banach s p a c e , " T r a n s . Am. Math. Soc., 179, 133-144 (1973). 502. E. N. Chukwu, "Functional inclusion and controllability of nonlinear neutral functional differential s y s t e m s , " J. Optim. T h e o r y Appl., 2._~9,No. 2, 291-300 (1979). 503. E. N. Chukwu and J. M. Gronski, " A p p r o x i m a t e and complete controllability of nonlinear s y s t e m s to a convex t a r g e t set," J. Math. Anal. Appl., 6._~1,No. 1, 97-112 (1977). 504. Ph. V. Chuong, " V e r s i o n s a l 6 a t o i r e s du th~oreme de point fixe de Katmtani-Ky Fan," C. R. Acad. Sci. P a r i s , AB291, No. 2, 143-146 (1980). 505. L. B. Ciric, "Fixed points f o r g e n e r a l i z e d multivalued contractions," Mat. Vesnik, 9, No. 3, 265-272 (1972). 506. L. B. Ciric, "On some m a p s with a n o , unique fixed point," Publ. Inst. Math., 17, 52-58 {1974). 507. F. H. C l a r k e , "Admissible relaxation in v a r i a t i o n a l and control p r o b l e m s , " J . Math. Anal. Appl., 5__.1, No. 3, 557-576 (1975). 508. F. H. Clarke, "The E u l e r - L a g r a n g e differential inclusion," J. Diff. Equations, 1_~9,No. 1, 80-90 (1975). 509. F. H. C l a r k e , "Optimal solutions to differential inclusions," J. Optim. Theory Appl., 19, No. 3, 469-478 (1976). 510. F. H. Clarke, " N e c e s s a r y conditions for a g e n e r a l control p r o b l e m , " in: Calculus of Variations and Cont r o l Theory, P r o c . Syrup., Madison, Wisconsin, 1975, A c a d e m i c P r e s s , New York (1976), pp. 257-278. 511. F. H. Clarke, "Optimal control and the true Hamiltonian," SIAM Rev., 2_.1, No. 2, 157-166 (1979). 512. F. H. Clarke, "The E r d m a n n condition and Hamiltonian inclusions in optimal control and the calculus of v a r i a t i o n s , " Can. J. Math., 3._~2,No. 2, 494-509 (1980). 513. F. H. Clarke and J . - P . Aubin, "Monotone invariant solutions to differential inclusions ," J. London Math. Soc., 16, No. 2, 357-366 (1977). 514. A. Climescu, " R e m a r q u e s s u r une construction d'applications a p a r t i r d'une application multivoque," BuI. Inst. Politetm. Iasi, Sec. 1, 1.~8, No. 1-2, 1-4 (1972). 515. E. A. Coddington, "Multivalued o p e r a t o r s and boundary-value p r o b l e m s , n in: L e c t u r e Notes Math., S p r i n g e r - V e r l a g , 183., 2-8 (1971). 516. J. K. Cole, "A s e l e c t o r t h e o r e m in Banach s p a c e , " J. Optim. T h e o r y Appl., 7, No. 3, 170-172 (1971). 517. C. C. Conley, "Some a b s t r a c t p r o p e r t i e s of the s e t of invariant s e t s of a flow," Illinois J. Math., 16, No. 4, 663-668 (1972). 518. R. Conaely, "An extension of B r o u w e r ' s fixed-point t h e o r e m to nonacyclic, s e t - v a l u e d functions," P r o c . Am. Math. Soc., 43, No. 1, 214-218 (1974). 519. R. Conti and E. De P a s c a l e , "The n u m e r i c a l range in the nonlinear c a s e , " Boll. Un. Mat. Ital., B15, No. 1, 210-216 (1978). 520. R. Conti, "Recent trends in the theory of boundary value p r o b l e m s f o r o r d i n a r y differential equations," Boll. Un. Mat. Ital., 22, No. 2, 135-178 (1967).
768
521. R. Conti, I. Massabo, and P. Nistri, "Set-valued perturbations of differential equations at resonance," Nonlinear Anal.: Theory, Meth. Appl., 4, No. 6, 1031-1041 (1980). 522. R. Conti and P. Nistri, "A fixed point t h e o r e m for noncompact a e y c l i c - v a l u e d maps, N Rend. Accad. Sci. Fis. Mat. Napoli, 4_22, 510-517 (1975(1976)). 523. R. Conti and P. Nistri, "On surjectivity f o r e-quasibounded multivalued maps, ~ Atti. Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur., 6_.1, No. 3-4, 195-201 (1976(1977)). 524. R. Conti and P. Nistri, "A definition of asymptotic s p e c t r u m f o r multivalued maps in Banach s p a c e s , w Atti Sere. Mat. Fis. Univ. Modena, 2.6, No. 1, 1-14 (1977(1978)). 525. H. Cook, "Upper semicontinuous continuum-valued mappings into circlelike continua," Fund. Math., 60, No. 3, 233-239 (1967). 526. R. R. Cornwall, "Conditions for the graph and the integral of a correspondence to be open," J. Math. Anal. Appl., 3._~9,No. 3, 771-792 (1972). 527. J. M. Coron, "Image de la s o m m e de deux sous-diff6rentiels," Boll. Un. Mat. Ital., A17, No. 1, 161-166 (1980). 528. P. Constantini, "Migliore a p p r o s s i m a z i o n e uniforme mediante polinomi vincolati," Boll. Un. Mat. Ital., A16, No. 3, 531-535 (1979). 529. A. Coste, "La propri6t~ de R a d o n - N i k o d y m en int6gration multivoque," C. R. Acad. Sci. P a r i s , 280, No. 22, A1515-A1518. 530. R. S. Countryman, J r . , "Continuous selections and algebraic closure in function a l g e b r a s , " Doctoral D i s sertation, Univ. of Minnesota, 1966; D i s s e r t . A b s t r a c t s , B27, No. 8, 2774 (1967). 531. H. Covitz and S. B. Nadler, J r . , "Multivalued contraction mappings in generalized m e t r i c s p a c e s , " I s r a e l J. Math., 8_, No. 1, 5-11 (1970). 532. D. E. Cowles, "Upper semicontinuity p r o p e r t i e s of variable sets in optimal control, n J. Optim. Theory Appl., 10, No. 4, 222-236 (1972). 533. M. G. Crandall and A. Pazy, "On the range of accretive o p e r a t o r s , " I s r a e l J. Math., 27, No. 3-4, 235-246 (1977). 534. M. G. Crandall and A. P a z y , "An approximation of integrable functions by step functions with an application," P r o c . Am. Math. Soc., 7_~6,No. 1, 74-80 (1979). 535. G. C r o m b e z , "Topologies on spaces of continuous multifunctions," Publ. Inst. Math., 21, 35-40 (1977). 536. J . - P . Crouzeix, "Continuit6 des applications lin6aires multivoques," Rev. F r a n c . Automat. Inform. Rech. Oper., 7, No. R - l , 62-67 (1973). 537. D. W. Curtis, " H y p e r s p a c e s of noncompact m e t r i c s p a c e s , ~ Compositio Math., 40, No. 2, 139-152 (1980). 538. S. Czerwik, "A fixed point t h e o r e m for a s y s t e m of multivalued t r a n s f o r m a t i o n s , " P r e c . Am. Math. Soc., 5._~5,No. 1, 136-139 (1976). 539, S. Czerwik, "Multivalued contraction mappings in m e t r i c spaces," Aequationes Math., 16, No. 1-2, 188-189 (1977). 540. S. Czerwik, "Multivalued contraction mappings in m e t r i c spaces," Aequationes Math., 1_6,6No. 3, 297-302 (1977). 541. R. Dacic, "On multivalued functions, ~ Publ. Inst. Math., 9, 5-7 (1959). 542. L. Dal Soglio, ~Grado topologico e t e o r e m i di e s i s t e n z a di punti uniti p e r t r a n s f o r m a z i o n i plurivalenti di 3-celle," Rend. Sere. Mat. Univ. Padova, 25, 386-405 (1956). 543. L. Dal Soglio, "Grado topologico e t e o r e m i di e s i s t e n z a di punti uniti p e r t r a n s f o r m a z i o n i plurivalenti di n - c e l l e , " Rend. Sere. Mat. Univ. Padova, Pt. 1, 2_7.7,103-121 (1957). 544. J. Danes, "Some fixed point t h e o r e m s , " Comment. Math. Univ. Carolin., 9, No. 2, 223-235 (1968). 545. J. Danes, "On densifying and related mappings and their application in nonlinear functional analysis," Schriftenz. Zentralinst. Math. Mech., No. 20, 15-56 (1974). 546. T. Dan-i, "Connectors generated by binary relations," Mere. Def. Acad. Jpn., 14, No. 4, 137-153 (1974). 547. P. D'Anna and E. Muselli, "Un t e o r e m a di esistenza per campi di orientatori e applicazioni al p r o b l e m a libero del Calcolo delle Variazione in dimensione infinita," Publ. Inst. Mat. Univ. Genova, No. 248, 1978. 548. P. D'Anna and E. Muselli, "Un t e o r e m a di e s i s t e n z a p e r campi di orientori e applicazioni al p r o b l e m a libero del Calcolo della Variazieni in dimensione infinita," Ann. Mat. P u r a Appl., 122., 245-267 (1979). 549. G. Da P r a t e , M. Iannelli, and L. T u r b a r o , ~Dissipative functions and finite-dimensional stochastic differential equations," J. Math. P u r e s Appl., 5.7, No. 2, 173-190 (1978). 550. C. Darbo, "Grado topologico e t e o r e m e di esistenza di punti uniti p e r t r a n s f o r m a z i o n i plurivalenti di bicelle," Rend. Sere. Mat. Univ. Padova, 1._99,371-395 (1950). 551. C. Darbo, " T e o r i a dell'omologia in una categoria di mappe plurivalenti ponderate," Rend. Sere. Mat. Univ. Padova, 2_~8,No. 1, 188-220 (1958). 552. R. Datko, "Measurability p r o p e r t i e s of s e t - v a l u e d mappings in a Banach space," SIAM J. Control, 8, No. 2, 226-238 (1970). 769
553. R. Datko, "On the integration of s e t - v a l u e d mappings in a Banach s p a c e , " Fund. Math., 78, No. 3, 205-208 (1973). 554. J. P. D a u e r and F. S. Van Vleck, " M e a s u r a b l e s e l e c t o r s of multifunctions and applications," Math. S y s t e m s T h e o r y , 7, No. 4, 367-376 (1974). 555. J . - P . D a u r e s , "Contribution a l'~tude des ~quations diffSrentielles multivoques dans les e s p a c e s des Banach," C. R. Acad. Sei. P a r i s , 270, No. 12, A769-A772 (1970). 556. J . - P . D a u r e s , " V e r s i o n multivoque du t h d o r e m e de Doob," C. R. Acad. Sei. P a r i s , 275, No. 11, A527-A530 (1972). 557. J . - P . D a u r e s , " V e r s i o n multivoque du th~oreme de Doob," Ann. Inst. H. PoIncar~, B9, No. 2, 167-176 (1973). 558. J . - P . D a u r e s , "Un p r o b l e m e d ' e x i s t e n c e de c o m m a n d e s optimales avec liaison s u r l'~tat," C. R. Acad. Sci. P a r i s , A279, No. 13, 511-514 (1974). 559. J. L. Davy, " P r o p e r t i e s of the solution s e t of a g e n e r a l i z e d differential equation," Bull. A u s t r a l . Math. Soc., 6_, No. 3, 379-398 (1972). 560. J. L. Davy, nGeneralized differential equations satisfying Carath~odory type conditions," Bull. Austral. Math. Soc., 1_~1,No. 2, 317-318 (1974). 561. M. Dawidowski, "On s o m e g e n e r a l i z a t i o n of the Bogoliubov a v e r a g i n g t h e o r e m , " Fanet. Approx. (PRL), 7, 55-70 (1979). 562. J. M. Day and K. Kuratowski, "On the nonexistence of a continuous s e l e c t o r f o r a r c s lying in the plane," P r o c . Koninkl. Nederl. Akad. Wet., A69, No. 2, 131-132 (1966); Indag. Math., 2-8, No. 2, 131-132 (1966). 563. F. S. De Blasi, "On the differentiability of multiftmctiens," Pac. J. Math., 6_66, No. 1, 67-81 (1976). 564. F. S. De Blasi, " E x i s t e n c e and stability of solutions f o r autonomous multivalued differential equations in Banach s p a c e , " Atti Accad. Naz. Lincei Rend. C1. Sci. F i s . Mat. Natur., 60, No. 6, 767-774 (1976). 565. F. S. De Blasi, ' S o m e g e n e r i c p r o p e r t i e s in fixed point theory," J. Math. Anal. Appl., 7.1, No. 1, 161-166 (1979). 566. F. S. De Blasi arid F. Iervolino, "Equazioni differenziali con soluzioni a v a l o r e eompatto c o n v e s s o , ' Boll. Un. Mat. Ital., 2, No. 4-5, 491-501 (1969). 567. F. S. De Blasi and A. Lasota, " C h a r a c t e r i z a t i o n of the integral of s e t - v a l u e d functions," Atti Accad. Naz. Lineei Rend. C1. Sci. F i s . Mat. Natur., 4_66,No. 2, 154-157 (1969). 568. F. S. De Blasi and J. Myjak, "The g e n e r i c p r o p e r t y of existence of solutions f o r a c l a s s of multivalued differential equations in Hilbert s p a c e s , " Funkcial. Ekvac., 21, No. 3, 271-278 (1978). 569. F. S. De Blasi and J . Schinas, "Exponential stability of difference equations which cannot be linearized," Atti Accad. Naz. Lincei Rend. C1. Sci. F i s . Mat. Natur., 54, No. 1, 16-21 (1973). 570. G. Debreu, " I n t e g r a t i o n of c o r r e s p o n d e n c e s , " in: P r o c . of Fifth B e r k e l e y Syrup. Math. Star. Probability, Vol. II, P a r t 1, Univ. California P r e s s (1966), pp. 351-372. 571. H. D e b r u n n e r and P. F l o r , "Ein E r w e i t e r u n g s s a t z fur montone Mengen," Arch. Math., 15, No. 6, 445-447 (1964). 572. G. Debs, "S~lection d'une multiapplication a v a l e u r s (~6 ," Bull. C1. Sci. Acad. R. Belg., 6-5, No. 5, 211-216 (1979). 573. D. G. De Figueiredo, "An existence t h e o r e m f o r pseudomonotone o p e r a t o r equations in Banaeh s p a c e s , " J. Math. Anal. Appl., 3-4, No. 1, 151-156 (1971). 574. F. Delbaen and E. V e r g i s e n , "A note on s e l e c t i o n s , " Bull. C1. Sei. Aead. R. Belg., 55, No. 7, 682-686 (1969). 575. C. Delode, O. Arino, and ff.-P. Penot, " C h a m p s m e s u r a b l e s et m u l t i s e c t i o n s , " Ann. Inst. H. Poincar6, B12, No. 1, 11-42 (1976). Del P r e t e I m m a c o l a t a , " T e o r e m i di punto unito p e r applicazioni A - l i p s c h i t z i a n e a pin v a l o r i , " Rend. C i r c . 576. Mat. P a l e r m o , 23, No. 1, 5-17 (1974). 577. A. P. D e m p s t e r , "Upper and lower p r o b a b i l i t i e s induced by a multivalued mapping," Ann. Math. Stat., 3-8, No. 2 , 3 2 5 - 3 3 9 (1967). 578. J . Denel, "Extensions of the continuity of p o i n t - t o - s e t m a p s : applications to fixed point a l g o r i t h m s , " Math. P r o g r a m m i n g Stud., 10, 48-68 (1979). 579. E. De P a s c a l e and R. Guzzardi, ' On boundary conditions and fixed points f o r ~ - n e n e x p a n s i v e multivalued m a p p i n g s , " Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 5-8, I~o. 3, 300-305 (1975). 580. E. De P a s c a l e and R. Guzzardi, "On the boundary values dependence f o r the topological degree of m u l t i valued n o n c o m p a c t m a p s , " Boll. Un. Mat, Ital., 13-A, No. 1, 110-116 (1976). 581. M. De Wilde and C. Sunyach, "Un th~oreme de s~lection pour des applications a graphe bor~lien," C. R. Aead. Sci. P a r i s , 269, No. 6, A273-A274 (1969). 582. J. I. Diaz Diaz, "Solutions with c o m p a c t s u p p o r t f o r s o m e degenerate p a r a b o l i c p r o b l e m s , MNonlinear Anal: T h e o r y , Methods Appl., 3, No. 6, 831-847 (1979).
770
583. Dinh-Nho-Chuong, " P r e c l o s e d multivalued m a p p i n g s , " in: General Toplogy and Its Relations to Modern Analysis and A l g e b r a , Vol. 2, P r a g u e (1967), pp. 108-110. 584. L. Diomeda, "Sulla dipendenza dal p a r a m e t r o delle soluzioni di un'equazione differziale multivoca x' E F(t, ~ , x)," Boll. Un. Mat. Ital., B14, No. 1, 203-221 (1977). 585. S. Z. Ditor, "Averaging o p e r a t o r s in C(S) and l o w e r semicontinuous s e l e c t i o n s of continuous m a p s , " T r a n s . Am. Math. Soc., 175, 195-208 (1973). Do Hong Tan, "Seme r e m a r k s on fixed points and t h e i r continuity," Toan Hoe, 6, No. 1, 15-23 (1978). 586. 587. Do Hong Tan, non l i n e a r multivalued m a p p i n g s , " Tap Chi Toan Hoe, 6, No. 3, 1-6 (1978). 588. Do Hong Tan, " C o m m o n fixed points of a sequence of multivalued m a p p i n g s , " Toan Hoc, 7, No. 1, 1-5 (1979). 589. S. Doleeki, " E x t r e m a l m e a s u r a b l e s e l e c t i o n s , " Bull. Acad. Polon. Sci. Set. Sei. Math. Astron. P h y s . , 25, No. 4, 355-360 (1977). 590. S. Doleeki, " R e m a r k s on semieontinuity," Bull. Acad. Polon. Sei. S e t . Sei. Math. Astron. Phys., 2.~5, No. 9, 863-867 (1977). 591. S. Dolecki, " A b s t r a c t study of optimality conditions," J. Math. Anal. Appl., 7__33,No. 1, 24-48 (1980). 592. S. Dolecki and S. Rolewiez, "A c h a r a c t e r i z a t i o n of s e m i e o n t i n u i t y - p r e s e r v i n g multifunctions," J. Math. Anal. Appl., 6._~5,No. 1, 26-31 (1978). 593. S. Doleeld and S. Rolewiez, " M e t r i c c h a r a c t e r i z a t i o n s of u p p e r semicontinuity," J. Math. Anal. Appl., 69, No. 1, 146-152 (1979). 594. V. Dolezal, " F e e d b a c k s y s t e m s d e s c r i b e d by monotone o p e r a t o r s , " SIAM J. Control Optim., 17, No. 3, 339-364 (1979). 595. V. Dolezal, "Des E x i s t e n z p r o b l e m fur allgemeine d i s k r e t e S y s t e m e , " Z. Angew. Math. Mech., 59, No. 8, 390-392 (1979). 596. R. T. Douglass, "Connectivity multifunctions and the pseudoquotient topology," D o c t o r a l D i s s e r t a t i o n , Univ. of K a n s a s , 1967; D i s s e r t . A b s t r a c t s , B28, No. 7, 2931 (1968). 597. D. Downing and W. A. Kirk, "Fixed point t h e o r e m s f o r s e t - v a l u e d mappings in m e t r i c and Banach s p a c e s , " Math. Jpn., 2._~2,No. 1, 99-112 (1977). 598. E. L. Dozo, "Nonexpansive mappings in Banach s p a c e s , " V a t . Publ. Ser. Mat. Inst. Aarhus Univ., No. 23, 1-9 (1972). 599. E. L. Dozo, "Multivalued nonexpansive mappings and Opial's condition," Proc. Am. Math. Soc., 38, No. 2, 286-292 (1973). 600. L. S. Dube, "A theorem on commonfixed points of multivalued mappings," Ann. Soe. Sci. Bruxelles, Ser. 1, 8_.9, No. 4, 463-468 (1975). 601. L. S. Dube and S. P. Singh, nOn multivalued contraction m a p p i n g s , " Bull. Math. Soc. Sci. Math. R. S. Roumaine, 14, No. 3, 307-310 (1970(1971)). 602. J. Dugundji, " P o s i t i v e - d e f i n i t e functions and coincidences," Fund. Math., 9_0.0,No. 2, 131-142 (1976). 603. J. C. Dunn, " I t e r a t i v c construction of fixed points f o r multivalued o p e r a t o r s of the monotone Wpe," J . Funet. Anal., 2_7.7,No. 1, 38-50 (1978). 604. J. C. ]:)unn, "A r e l a x e d P i e a r d iteration p r o c e s s for s e t - v a l u e d o p e r a t o r s of the monotone type," P r o e . Am. Math. Soe., 7.33, No. 3, 319-327 (1979). 605. R. Dyckheff, "On a t h e o r e m of Kuratowski concerning u p p e r semicontinuity," Bull. Acad. P o l o , . Sci. S e r . Sci. Math. Astron. Phys., 2.1, No. 2, 163-164 (1973). 606. Z. Dzedzej, "Multivalued c o m p a c t v e c t o r fields in locally convex s p a c e s , " Z e s z . Nauk. UG. Mat., No. 4, 19-31 (1978). 607. B. C. E a v e s , "Computing Kalmtani fixed points," SIAM J . Appl. Math., 2_1.1,No. 2, 236-244 (1971). 608. B. C. E a v e s and R. Saigal, " H e m o t o p i e s f o r computation of fixed points on unbounded r e g i o n s , " Math. P r o g r a m m i n g , 3, No. 2, 225-237 (1972). 609. M. Ehrlich, "A t h e o r e m of B o r s u k - U l a m type f o r multifunctions," Fund. Math., 102, No. 3, 203-208 (1979). 610. S. E i l e n b e r g and D. Montgomery, " F i x e d point t h e o r e m s for multivalued t r a n s f o r m a t i o n s , " Am. J. Math., 6_~8, 214-222 (1946). 611. G. E i s e n a c k and C. F e n s k e , Fixpunkttheorie, Bibliographosches Institut, Mannheim (1978). 612. I. Ekeland, " P a r a m 6 t r i s a t i o n en th~orie de la c o m m a n d e , " C. R. Aead. Sci. P a r i s , 267, No. 2, A98-A100
(1968). 613. I. Ekeland and M. V a l a d i e r , " R e p r e s e n t a t i o n of s e t - v a l u e d mappings," J. Math. Anal. Appl., 3__5,No. 3, 621-629 (1971). 614. R. Engelking, "Quelques r e m a r q u e s c o , c e m e n t les operations s u r les functions semicontinues dans les e s p a c e s topologiques," Bull. Acad. Polon. Sci. Set. Sci. Math. Astron. Phys., 1_~1,No. 12, 719-726 (1963).
771
615. R. Engelking, " S e l e c t o r s of the f i r s t Baire c l a s s f o r semicontinuous s e t - v a l u e d functions," Bull. Acad. Polon. Sei. S6r. Sci. Math. Astron. Phys., 16, No. 4, 277-282 (1968). 616. H . W . Engl, "Some random fixed point t h e o r e m s f o r s t r i c t contractions and nonexpansive mappings," Nonlinear Anal.: T h e o r y , Methods Appl., 2, No. 5,619-626 (1978). 617. H . W . Engl, "Random fixed point t h e o r e m s f o r multivalued mappings," Pac. J. Math., 7___6,No. 2, 351-360 (1978). 618. H . W . Engl, "Random fixed point t h e o r e m s , n in: Nonlinear Equations in A b s t r a c t Spaces, Academic P r e s s , New Y o r k (1978), pp. 67-80. 619. M. Fabian, "On s i n g l e - v a l u e d n e s s and (strong) upper semieontinuity of m a x i m a l monotone mappings," C o m m e n t . Math. Univ. Carolin., 1_~8,No. 1, 19-39 (1977). 620. M. Fabian, "Once m o r e on the continuity of m a x i m a l monotone mappings," C o m m e n t . Math. Univ. Carolin., 1_~8, No. 1, 105-114 (1977). 621. M. Fabian, non s i n g l e - v a l u e d n e s s and continuity of monotone mappings," in: G e n e r a l Topology and Its Relations to Modern Analysis and Algebra, Vol. 4, P r o e . 4th P r a g u e Topology Syrup., 1976, P a r t B, Prague (1977), pp. 116-118. 622. H. Fakhoury, "S61ections continues dans les e s p a c e s u n i f o r m e s , " C. R. Acad. Sci. P a r i s , 280, No. 4, A213A216 (1975). 623. K. Fan, " F i x e d - p o i n t and m i n i m a x t h e o r e m s in locally convex topological linear s p a c e s , " P r o c . Nat. Acad. Sci. USA, 3._88,121-126 (1952). 624. K. Fan, "A g e n e r a l i z a t i o n of Tychonoff's fixed point t h e o r e m , n Math. Ann., 142, No. 3, 305-310 (1961). 625. K. Fan, " I n v a r i a n t c r o s s sections and invariant l i n e a r s u b s p a e e s , " I s r a e l J. Math., 2, No. 1, 19-26 (1964). 626. K. Fan, "Extensions of two fixed point t h e o r e m s of F. E. B r o w d e r , " Math. Z., 112, No. 3, 234-240 (1969). 627. O. F e i e h t i n g e r and W. B. Geiger, "Some notes on multifunetions," in: Studies in Topology, A c a d e m i c P r e s s , New York (1975), pp. 135-137. 628. C. C. F e n s k e , " N i c h t - a b s t o s s e n d e Fixpunkte o b e r h a l b s t e t i g e r Abbildungen," B e t . G e s . Math. und Datenv e r a r b . , No. 6, 7-14 (1968). 629. C. C. Fenske and H.-O. Peitgen, "Repulsive fixed points of multivalued t r a n s f o r m a t i o n s and the fixed point index," Math. Ann., 218, No. 1, 9-18 (1976). 630. C. C. Fenske and H.-O. Peitgen, " A t t r a c t o r s and the fixed point index f o r a c l a s s of multivalued m a p pings. I, II," Bull. Acad. Polon. Sei. Ser. Sci. Math. Astron. Phys., 2..~5,No. 5, 477-482, 483-487 (1977). 631. A. S. Finbow, "The fixed point set of r e a l multivalued contraction mappings," Can. Math. Bull., 15, No. 4, 507-511 (1972). 632. B. F i s h e r , " S e t - v a l u e d mappings on bounded m e t r i c s p a c e s , ~ Indian J. Pure Appl. Math., 1__.1,No. 1, 8-12 (1980). 633. P. M. F i t z p a t r i c k , "Surjectivity r e s u l t s f o r nonlinear mappings f r o m a Banach s p a c e to its dual," Math. Ann., 204, No. 3, 177-188 (1973). 634. P. M. F i t z p a t r i c k and W. V. Petryshyn, " F i x e d point t h e o r e m s f o r multivalued noncompact acyclic m a p pings, n P a c . 5. Math., 5..~4, No. 2, 17-23 (1974). 635. P. M. F i t z p a t r i c k and W. V. Petryshyn, "Fixed point t h e o r e m s and the fixed point index f o r multivalued mappings in cones," J. London Math. Soc., 1_.~2,No. 1, 75-85 (1975). 636. P . M . F i t z p a t r i c k and W. V. Petryshyn, " P o s i t i v e eigenvalues f o r nonlinear multivalued noncompaet o p e r a t o r s with applications to differential o p e r a t o r s , " J. Diff. Equations, 22, No. 2, 428-441 (1976). 637. S. P. F i t z p a t r i c k , "Monotone o p e r a t o r s and dentability," Bull. Austral. Math. Soc., 18, No. 1, 77-82 (1978). 638. G. Fodor and A. Mat6, "On the s t r u c t u r e of set mappings and the existence of f r e e s e t s , " Acta Sci. Math., 2_~6, No. 1-2, 2-7 (1965). 639. J. Fontanillas, "Un t e o r e m a de s e l e c c i o n u n i f o r m e , " Rev. Mat. H i s p . - A m . , 3_77,No. 1-2, 3-18 (1977). 640. J. P. Foti, "Asymptotically autonomous multivalued differential equations," T r a n s . Am. Math. Soc., 221, No. 2, 449-452 (1976). 641. G. F o u r n i e r and L. Gorniewicz, "The Lefschetz fixed point t h e o r e m f o r multivalued m a p s of n o n m e t r i z a b l e s p a c e s , " Fund. Math., 92, No. 3, 213-222 (1976). 642. G. F o u r n i e r and L. Gorniewicz, "The Lefschetz fixed point t h e o r e m f o r s o m e n o , compact multivalued m a p s , " Fund. Math., 94, No. 3, 245-254 (1977). 643. G. Fox and P. M o r a l e s , "A multifunction g e n e r a l i z a t i o n of G a l e ' s Aseoli t h e o r e m , " J. A u s t r a l . Math. Soc., 20, No. 2, 165-171 (1975). 644. G. Fox and P. M o r a l e s , "Non-Hausdorff multifunction g e n e r a l i z a t i o n of the K e l l e y - M o r s e Ascoli t h e o r e m , " Pac. J. Math., 6_44, No. 1, 137-143 (1976). 645. S. P. F r a n k l i n and R. H. Sorgenfrey, " C l o s e d and i m a g e - c l o s e d r e l a t i o n s , " Pac. J. Math., 19, No. 3, 433439 (1966).
772
646. R. B. F r a s e r , ' S e q u e n c e s of c o n t r a c t i v e s e t - v a l u e d m a p s , " Lect. Notes Math., No. 171, 24-26 (1970). 647. R. B. F r a s e r and S. B. Nadler J r . , "Sequences of c o n t r a c t i v e m a p s and fixed points," Pac. J. Math., 31, No. 3, 659-667 (1969). 648. O. Frink, "Topology in l a t t i c e s , " T r a n s . Am. Math. Soc., 51, 569-582 (1942). 649. A. F r y s z k o w s k i , " C a r a t h 6 o d o r y type s e l e c t o r s of s e t - v a l u e d m a p s of two v a r i a b l e s , " Bull. Acad. Polon. Sci. S6r. Sci. Math. Astron. Phys., 2_55, No. 1, 41-46 (1977). 650. F. B. F u l l e r , " F i x e d points of m u l t i p l e - v a l u e d t r a n s f o r m a t i o n s , " Bull. Am. Math. Soc., 6__.7,No. 2, 165-169 (1961). 651. M. F u r l and M. Martelli, "Some fixed-point t h e o r e m s f o r multivalued mappings in topological v e c t o r s p a c e s , " Ann. Mat. P u r a Appl., 92, 169-175 {1972). 652. M. F u r i and M. Martelli, "A d e g r e e f o r a c l a s s of a c y c l i c - v a l u e d v e c t o r fields in Banach s p a c e s , " Ann. Scuola N o r m . Super. Pisa, Ser. IV, 1, No. 3-4, 301-310 (1974(1975)). 653. M. F u r l and M. Martelli, "On the minimal d i s p l a c e m e n t under a c y c l i c - v a l u e d m a p s defined on a c l a s s of A N R ' s , " Boll. Un. Mat. Ital., l_!, No. 3, Suppl., 238-246 (1975). 654. G. Garegnani and S. Massa, "Multivalued mappings of c o n t r a c t i v e type," Rend. Ist. L o m b a r d o . Accad. Sci. Lett., Al12, N o . 2 , 2 8 3 - 2 8 8 (1978). 655. F . S. G a r n i r - M o n j o i e , "Sur c e r t a i n e s multifonctions # - m e s u r a b l e s , " Bull. Soc. Roy. Sci. Liege, 4._~2,No. 5-6, 183-194 (1973). 656. F. S. G a r n i r - M o n j o i e , "S61ections continues de multifonctions," Bull. Soc. R. Sci. Liege, 466, No. 1-2, 51-56 (1977). 657. M. Gaultier, "R6solution d'une 6quation nonlineaire," C. R. Acad. Sci. P a r i s , 274, No. 4, A332-A334 (1972). 658. J. H. George, V. M. Sehgal, and R. E. Smithson, "Application of Liapunov's direct method to fixed point t h e o r e m s , " P r o c . Am. Math. Soc., 2._.~8,No. 2, 613-620 (1971). 659. A. G h o u i - l a - H o u r i , "Equations diff6rentielles multivoques," C. R. Acad. Sci. P a r i s , 261, No. 14, 2568-2571 (1965). 660. I. L. Glickberg, 'A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points," Proc. Am. Math. Soc., 3, No. I, 170-174 (1952). 661. Ch. Godet-Thobie, "Sur les multimesures de transition, ~ C. R. Acad. Sci. Paris, A278, No. 21, 1367-1369 (1974). 662. G. Godini, "Set-valued Cauchy functional equations," Rev. Roumaine Math. Pures Appl., 20, No. i0, 11131121 (1975). 663. G. Godini, "On set-valued mappings," Rev. Roumaine Math. Pures Appl., 22, No. i, 53-67 (1977). 664. G. Goldner, "On successive approximations for set valued mappings," in: Proc. Conf. Constructive Theory Funct., Budapest, 1969, Budapest (1972), pp. 189-193. 665. G. S. Goodman, "The duality of convex functions and Cesarits property (Q)," J. Optim. Theory Appl., 1.99, No. i, 17-27 (1976). 666. J. T. Goodykoontz Jr., "C(X) is not necessarily a retract of 2X,, Proc. Am. Math. Soc., 6_7.7,No. i, 177-178 (1977). 667. L. Gorniewlcz, "Fixed point theorem for multivalued mappings of approximative ANR's," Bull. Acad. Polon. Sci. S~r. Sci. Math. Astron. Phys., 18, No. 8, 431-436 (1970). 668. L. Gorniewicz, WFixed point theorems for multivalued mappings of special ANR's," Zesz. Nauk. UG. Matematyka, N9. i, 39-49 (1971(1972)). 669. L. Gorniewlcz, "On nonacyclic multivalued mappings of subsets of Euclidean spaces," Bull. Acad. Polon. Sci. S4r. Sci. Math. Astron. Phys., 20, No. 5, 379-385 (1972). 670. L. Gorniewicz, "Some consequences of the Lefschetz fixed point theorem for multivalued maps," Bull. Acad. Polon. Sci. S~r. Sci. Math. Astron. Phys., 21, No. 2, 165-170 (1973). 671. L. Gorniewicz, "Remarks on the Lefschetz fixed point theorem," Bull. Aead. Polon. Sci. S~r. Sci. Math. Astron. Phys., 21, No. ii, 983-989 (1973). 672. L. Gorniewicz, "On a coincidence theorem," Bull. Acad. Polon. Sei. S~r. Sci. Math. Astron. Phys., 2_~2, No. 2, 147-152 (1974). 673. L. Gorniewicz, "A L e f s c h e t z - t y p e fixed point t h e o r e m , " Fund. Math., 8._.88,No. 2, 103-115 (1975). 674. L. Gorniewicz, Homological methods in fixed-point t h e o r y of multivalued m a p s , Rozpr. Mat., No. 129, 1976. 675. L. Gorniewicz, " F i x e d point t h e o r e m s f o r multivalued m a p s of subsets of Euclidean s p a c e s , " Bull. Acad. Polon. Sci. S6r. Sci. Math., 2__7.7,No. 1, 111-116 (1979). 676. L. Gorniewicz and A. G r a n a s , " F i x e d point t h e o r e m s for multivalued mappings of the absolute n e i g h b o r hood r e t r a c t s , " J . Math. P u r e s Appl., 4._~9,No. 4, 381-395 (1970). 677. J . - P . Gossez, "A note on multivalued monotone o p e r a t o r s , " Michigan Math. J., 1__7,No. 4, 347-350 (1970).
773
678. J . - P . G o s s e z , ~On the range of a c o e r c i v e m a x i m a l monotone o p e r a t o r in a nonreflexive Banach space," P r o c . Am. Math. Soc., 35, No. 1, 88-92 (1972). 679. J . - P . G o s s e z , "On a convexity p r o p e r t y of the range of a m a x i m a l monotone o p e r a t o r , " P r o c . Am. Math. Soc., 55, No. 2, 359-360 (1976). 680. S. Graf, "A selection t h e o r e m f o r Boolean c o r r e s p o n d e n c e s , " J. Reine Angew. Math., 295, 169-186 (1977). 681. S. Graf, "A m e a s u r a b l e selection t h e o r e m f o r c o m p a c t - v a l u e d m a p s , " Manuscr. Math., 27, No. 4, 341-352 (1979). 682. S. Graf, " M e a s u r a b l e weak s e l e c t i o n s , " Lect. Notes Math., 794, 117-140 (1980). 683. A. G r a n a s , "Sur la notion du degr6 topologique p o u r une c e r t a i n e c l a s s e de t r a n s f o r m a t i o n s multivalentes darts les e s p a c e s de Banach, t Bull. Aead. Polon. Sci. S6r. Sci. Math. Astron. Phys., 7, No. 4, 191-194 (1959). 684. A. G r a n a s , " T h e o r e m on antipodes and t h e o r e m s on fixed points f o r a c e r t a i n c l a s s of multivalued m a p pings in Banach s p a c e s , " Bull. Acad. Polon. Sci. S~r. Sci. Math. Astron. Phys., 7, No. 5, 271-275 (1959). 685. A. G r a n a s , "Sur la m~thode de continuit6 de P o i n c a r 6 ," C. R. Acad. Sci. P a r i s , 282, No. 17, A983-A985 (1976). 686. A. Granas and J. W. Jaworowski, "Some theorems on multivalued mappings of subsets of the Euclidean space," Bull. Acad. Polon. Sei. S6r. Sci. Math. Astron. Phys., 7_, No. 5, 277-283 (1959). 687. M. Grandolfi, "Problemi ai limiti per le equazioni differnziali multivoche," Atti Acad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 4.~2,No. 3, 355-360 (1967). 688. M. Granger, "Equations au contingent et problemes d'optimisation," Rev. CETHEDEC, 7_, No. 23, 63-100 (1970). 689. K. A. G r a s s e and P. Z e c c a , " L o c a l controllability of multivalued differential equations," Ann. Mat. P u r a Appl., 124, 281-292 (1980). 690. W. J. G r a y and L. O. Vaughan, " T h e a l m o s t fixed point p r o p e r t y fc~ h e r e d i t a r i l y unieoherent continua," P r o c . Am. Math. Soc., 27, No. 2, 381-386 (1971). 691. D. A. G r e g o r y , "Upper semicontinuity of subdifferential m a p p i n g s , " Can. Math. Bull., 23, No. 1, 11-19 (1980). 692. G. G r i p e , b e r g , "An existence r e s u l t for a nonlinear V o l t e r r a i n t e g r a l equation in a Hilbert s p a c e , " SIAM J. Math. Anal., 9, No. 5, 793-805 (1978). 693. R. Guitart, "D6finitions de l'homotopie entre r e l a t i o n s , " C. R. Acad. Sci. P a r i s , 271, No. 14, A635-A638 (1970). 694. C. P. Gupta, "On an operator equation involving mappings of monotone type," Proc. Am. Math. Soc., 53, No. 1, 143-148 (1975). 695. C. P. Gupta and P. Hess, "Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems," J. Diff. Equations, 22, No. 2, 305-313 (1976). 696. J. Gwinner, "Closed images of convex multivalued mappings in linear topological spaces with applications," J. Math. Anal. Appl., 60, No. 1, 75-86 (1977). 697. C.-W. Ha, "Minimax and fixed point theorems," Math. Aim., 248, l~o. 1, 73-77 (1980). 698. O. Hadzic, Foundations of Fixed Point Theory [in Serbo-Croatian], Inst. za Matem., Novi Sad (1978). 699. O. Hadzic, "Fixed point theorems for multivalued mappings in locally convex spaces," Publ. Inst. Math., 24, 61-66 (1978). 700. O. Hadzic, ~Some t h e o r e m s on the fixed points for multivalued mappings in locally convex s p a c e s , " Bull. Acad. Polon. Sci. S6r. Sci. Math., 2_.7.7,No. 3-4, 277-285 (1979). 701. O. Hadzic, "A fixed point t h e o r e m f o r multivalued mappings in r a n d o m n o r m e d s p a c e s , " Anal. N u m e r . T h e o r . Approx., 8, No. 1, 49-52 (1979). 702. O. Hadzic, " F i x e d point t h e o r e m s f o r multivalued mappings in p r o b a b i l i s t i c m e t r i c s p a c e s , " Mat. Vesnik, 3, No. 2, 125-133 (1979). 703. O. Hadzic, " F i x e d point t h e o r e m s f o r multivalued mappings in topological v e c t o r s p a c e s , " Glas. Mat., 15, No. 1, 113-119 (1980). 704. O. Hadzic,~Fixed point t h e o r e m s f o r multivalued mappings in s o m e c l a s s e s of topological v e c t o r s p a c e s , " in: Topology, 4th Colloq., Budapest, 1978, Vol. 1, A m s t e r d a m (1980), pp. 569-578. 705. M. Hagan, "Continuity of c e r t a i n connected functions and multifunctions," P r o c . Am. Math. Soc., 4-2, No. 1, 295-301 (1974). 706. S. Hahn, "Fixpunktsatze f u r m e n g e n w e r t i g e Abbildungen in lokalkonvexen Raumen," Math. Nachr., 7-3, 269-283 (1976). 707. S. Hahn, " G e b i e t s i n v a r i a n z s a t z und E i g e n w e r t a u s s a g e n fur kenzentrierende Abbildungen," Comment. Math. Univ. Carolin., 1.~8, No. 4, 697-713 (1977). 708. S. Hahn, "A r e m a r k on a fixed point t h e o r e m for condensing s e t - v a l u e d mappings," P r e p r i n t TU D r e s d e n 07-5-77. 774
709. S. Hahn, "Ein e l e m e n t a r e r Zugang z u r L e r a y - S c h a u d e r T h e o r i e , " P r e p r i n t TU D r e s d e n 07-10-77. 710. S. Hahn, r e i n e l e m e n t a r e r Zugang z u r L e r a y - S c h a u d e r T h e o r i e , " C o m m e n t . Math. Univ. Carolin., 109, No. 1, 71-87 (1978). 711. S. Hahn, " E i g e n w e r t a u s s a g e n fur kompakte und kondensierende m e n g e n w e r t i g e Abbildungen in topologischen V e k t o r r a u m e n , ~ C o m m e n t . Math. Univ. Carolin., 20, No. 1, 123-141 (1979). 712. S. Hahn, "Ein StSrungssatz fur positive Eigenwerte k o n d e n s i e r e n d e r m e n g e n w e r t i g e r Abbildangen in lokalkonvexen topologischen V e r k t o r r a u m e n , " C o m m e n t . Math. Univ. Carolin., 20, No. 3, 417-430 (1979). 713. S. Hahn, "Ein H o m o t o p i e e r w e i t e r u n g s s a t z f u r k o n z e n t r i e r e n d e mengenwertige Abbildungen in l o k a l konvexen topologischen V e k t r o r r a u m e n , " Stud. Math., 6__6, No. 2, 107-117 (1979). 714. O. Hajek, t T h e o r y of p r o c e s s e s . I, II," C z e c h o s l o v a k Math. J . , 1_7.7,No. 2 - 3 , 1 5 9 - 1 9 9 , 3 7 2 - 3 9 8 (1967). 715. H. Halkin and E. C. H e n d r i c k s , "Subintegrals of s e t - v a l u e d functions with s e m i a n a l y t i c g r a p h s , " P r o c . Nat. Acad. Sci. USA, 5_.99,No. 2, 365-367 (1968). 716. B. H a l p e r n , " F i x e d points f o r i t e r a t e s , " P a c . J. Math., 2_55,No. 2, 255-275 (1968). 717. B. Halpern, "A g e n e r a l fixed point t h e o r e m , " in: P r o c . Syrup. P u r e Math., 1__88,P a r t 1, Am. Math. Soc., P r o v i d e n c e (1970), pp. 114-131. 718. B. Halpern, " F i x e d point t h e o r e m s f o r s e t - v a l u e d m a p s in infinite-dimensional s p a c e s , " Math. Ann., 189, No. 2, 87-98 (1970). 719. B. H a l p e r n , "Algebraic topology and s e t - v a l u e d m a p s , " Lee,. Notes Math., 171, 27-33 (1970). 720. B. Halpern, " A l m o s t fixed points f o r s u b s e t s of z n , " J. Combin. Theory, Set. A, 1_~1,No. 3, 251-257 (1971). 721. O. Hamilton, "A fixed point t h e o r e m f o r u p p e r semicontinuous t r a n s f o r m a t i o n s on n - c e l l s f o r which the i m a g e s of points a r e nonacyclic continua," Duke Math. J., 1._~4,No. 3, 689-693 (1947). 722. O. Hamilton, "A t h e o r e m of Hamilton: c o r r e c t i o n , " Duke Math. J . , 24, No. 1, 59 (1957). 723. B. H a n s p e t e r , " D e r Satz von H a h n - B a n a c h und Fixpunktsatze in l i m i t i e r t e n V e k t o r r a u m e n , " C o m m e n t . Math. Helv., 4_~5,No. 4, 393-404 (1970). 724. M. Hasumi, "A continuous s e l e c t i o n t h e o r e m f o r e x t r e m a l l y disconnected s p a c e s , " Math. Ann., 179, No. 2, 83-89 (1969). 725. F. Hausdorff, Grundzfige der Mengenlehre, Chelsea, New York (1949). 726. M. L. J. Hautus, "On the uniform boundedness of solutions of contingent equations," J. Opttm. T h e o r y Appl., 2..~5,No. 4, 555-562 (1978). 727. M. L. J. Hantus, M. Heymann, and R. F. Stern, " R e s t point t h e o r e m s f o r autonomous control s y s t e m s , " J. Math. Anal. Appl., 58, No. 1, 98-112 (1977). 728. W. E. H a y e r , "A n e a r - s e l e c t i o n t h e o r e m , " G e n e r a l Topology Appl., 9, No. 2, 117-124 (1978). 729. M. Hegedus, " F i x e d point t h e o r e m s f o r s e t - t o - s e t mappings and a g e n e r a l i z a t i o n of the B r o u w e r fixed point t h e o r e m , " Dept. Math., K. Marx Univ. Econ., Budapest (Publ.), 1976, No. 2. 730. M. Hegedus, rAn extension of K r a s n o s e l s k i i ' s fixed point t h e o r e m f o r multivalued mappings," Dept. Math., K. M a r x Univ. Econ., Budapest (Publ.), 1976, No. 3. 731. D. Henney, " P r o p e r t i e s of s e t - v a l u e d additive functions," Am. Math. Monthly, 75, No. 4 , 3 8 4 - 3 8 6 (1968). 732. D. Heaney, " R e p r e s e n t a t i o n s of s e t - v a l u e d additive functions," Aequationes Math., 3, No. 3, 230-235 (1969). 733. C. Henry, n a n existence t h e o r e m f o r a c l a s s of differential equations with multivalued r i g h t - h a n d side," J. Math. Anal. Appl., 41, No. 1, 179-186 (1973). 734. H. H e r m e s , "Smoothing and approximating control p r o b l e m s , " in: Math. Theory Control, A c a d e m i c P r e s s , New Y o r k (1967), pp. 109-114. 735. H. H e r m e s , "Discontinuous v e c t o r fields and f e e d b a c k control," in: Differential Equations and D y n a m i c a l S y s t e m s , Academic P r e s s , New Y o r k (1967), pp. 155-165. 736. H. H e r m e s , "Calculus of s e t - v a l u e d functions and control," J. Math. Mech., 1-8, No. 1, 47-59 (1968). 737. H. H e r m e s , "The g e n e r a l i z e d differential equation ~E R(t, x)," Adv. Math., 4, No. 2, 149-169 (1970). 738. H. H e r m e s , nOn the s t r u c t u r e of attainable s e t s for g e n e r a l i z e d differential equations and control s y s t e m s , " 5. Diff. Equations, 9, No. 1, 141-154 (1971). 739. H. H e r m e s , "On continuous and m e a s u r a b l e s e l e c t i o n s and the existence of solutions of g e n e r a l i z e d diff e r e n t i a l equations," P r o c . Am. Math. Soc., 29, No. 3, 535-542 (1971). 740. H. H e r m e s and 5. P. La Salle, Functional Analysis and Time Optimal Control, Academic P r e s s , New York (1969). 741. C. 5. H i m m e l b e r g , " F i x e d points of c o m p a c t multifunctions," J. Math. Anal. Appl., 38, No. 1, 205-207 (1972). 742. C. J. H i m m e l b e r g , " M e a s u r a b l e r e l a t i o n s , " Fund. Math., 87, No. 1, 53-72 (1975). 743. C. 5. H i m m e l b e r g , M . Q . 5acobs, and F. S. Van Vleck, " M e a s u r a b l e multifunctioas, s e l e c t o r s and F i l i p p o v ' s implicit functions l e m m a , " 5. Math. Anal. Appl., 2__5,No. 2, 276-284 (1969). 744. C. J. H i m m e l b e r g , J. R. P o r t e r , and F. S. Van Vleck, " F i x e d point t h e o r e m f o r condensing multifunctions," P r o c . Am. Math. Soc., 23, No. 3, 635-641 (1969). 775
745. C. J. H i m m e l b e r g , J. R. P o r t e r , and F. S. Van Vleck, " F i x e d point t h e o r e m s f o r constrictive multifunctions," Boll. Un. Mat. Ital., 5, No. 2, 333-344 (1972). 746. C . J . H i m m e l b e r g and F. S. Van Vleck, "Some selection t h e o r e m s for m e a s u r a b l e multifunctions," Can. J. Math., 21, No.2, 394-399 (1969). 747. C. J. H i m m e l b e r g and F. S. Van Vleck, "Selection and implicit function t h e o r e m s f o r multifunetions with Souslin graph, n Bull. Acad. Polon. Sci. S e t . Math. Astron. Phys., 19, No. 10, 911-916 (1971). 748. C . J . H i m m e l b e r g and F. S. Van Vleck, " F i x e d points of semieondensing multifunetions," Boll. Un. Mat. Ital., 5, No. 2, 187-194 {1972). 749. C. J. H i m m e l b e r g and F. S. Van Vleek, " E x t r e m e points of multifunctions," Indiana Univ. Math. J., 2_~2, No. 8, 791-729 (1973). 750. C. J. H i m m e l b e r g and F. S. Van Vleck, 'Multifunctions on a b s t r a c t m e a s u r a b l e s p a c e s and application to s t o c h a s t i c decision theory," Ann. Mat. P u r a Appl., 101, 229-236 (1974). 751. C. J. H i m m e l b e r g and F. S. Van Vleck, "Multifunctions with values in a s p a c e of probability m e a s u r e s , " J. Math. Anal. Appl., 5._0.0,No. 1, 108-112 (1975). 752. C. J. H i m m e l b e r g and F. S. Van Vleck, "An extension of B r u n o v s k y ' s S e o r z a Dragoni type t h e o r e m f o r unbounded s e t - v a l u e d functions," Math. Slovaea, 26, No. 1, 47-52 (1976). 753. C. J. H i m m e l b e r g and F. S. Van Vleek, " C o r r i g e n d u m and addendum to rMultifunetions on a b s t r a c t m e a s u r a b l e s p a c e s and applications to s t o c h a s t i c decision t h e o r y ' , " Ann. Mat, P u r a Appl., 119, 391-394 (1979). 754. C. J. I-Iimmelberg and F. S. Van Vleck, "On the topological t r i v i a l i t y of solution s e t s , " Rocky Mountain J. Math., 10, No. 1, 247-252 (1980). 755. J . - B . H i r i a r t - U r r u t y , "~tude de quelques p r o p r i 6 t 6 s de la functionelle moyenne et de l'inf-eonvolution continue en analyse convexe stoehastique," C. R. Acad. Sci. P a r i s , 280, No. 3, A129-A132 {1975). 756. T. Hoang, "A s h o r t s e l f - c o n t a i n e d a l g o r i t h m i c proof of B r o u w e r ' s fixed point t h e o r e m , " Tagung F o r t s c h r i t t e in d e r M a t h e m a t i s c h e n Optimierung, Berlin, 29-31 Mai 1978, S e m i n a r b e r . Humboldt-Univ. Berlin. Sek. Math., No. 15, 108-117 (1978). 757. R. H r y c a y , "Noncontinuous multifunctions," Pac. J. Math., 3_~5,No. 1, 141-154 (1970). 758. T. Hu, " F i x e d point t h e o r e m s f o r multivalued m a p p i n g s , " Can. Math. Bull., 2__~3,No. 2, 193-197 (1980). 759. M. Hukuhara, "Sur l'applieation semicontinue dont la v a l e u r est un c o m p a c t convexe," Funkcial. Ekvac., 10, 43-66 (1967). 760. M. Huku_hara, "Integration des application m e a s u r a b l e s dont la v a l e u r e s t un c o m p a c t eonvexe," Funkcial. Ekvac., 1.~0, 205-223 (1967). 761. M. Hukuhara, "Int6gratioa des application m e a s u r a b l e s dont la v a l e u r est un c o m p a c t eonvexe," Publ. Res. Inst. Math. Sei., B, No. 16, 205-223 (1968). 762. S. A. Husain and V. M. Sehgal, "A r e m a r k on a fixed point t h e o r e m of C a r i s t i , " Math. Jpn., 25, No. 1, 27-30 (1980). 763. R. Iarmacei, "The s p e c t r u m f o r nonlinear multivalued m a p s via a p p r o x i m a t i o n s , " Boll. Un. Mat. Ital., B15, No. 5, 527-545 (1978). 764. A. Idzik, "Fixed point t h e o r e m s f o r f a m i l i e s of functions," P r . CO PAN, No. 206, 1975. 765. A. Idzik, " R e m a r k s on H i m m e l b e r g ' s fixed point t h e o r e m s , " Bull. Acad. Polon. Sci. S6r. Sci. Math. A s t r o a . Phys., 2._~6,No. 11, 909-912 (1978(1979)). 766. A. Idzik, " F i x e d point t h e o r e m s f o r f a m i l i e s of functions," Bull. Acad. Polon. Sci. S~r. Sci. Math. Astron. Phys., 2.~6, No. 11, 913-916 (1978(1979)). 767. A. Idzik, "Equivalent definitions f o r the m e a s u r a b i l i t y of a m u l t i v a r i a t e function and Filippov' s le m m a , " Mat. Stos., Ser. 3, 1_.44,63-70 (1979). 768. A. D. Ioffe, " R e p r e s e n t a t i o n t h e o r e m s for multifunctions and analytic s e t s , " Bull. Am. Math. Soc., 8_.i4, No. 1, 142-144 (1978). 769. A . D . Ioffe, " R e p r e s e n t a t i o n t h e o r e m s f o r m e a s u r a b l e multifunctions," Leet. Notes Math., 794, 141-152 (1980). 770. K. Iseki, "Multivalued contraction mappings in complete m e t r i c s p a c e s , " Rend. Sem. Mat. Univ. Padova, 53, 15-19 (1975). 771. V. I. I s t r a t e s c u , Introduction to Fixed Point T h e o r y [in Romanian], Ed. Acad. R. S. Rom., B u c h a r e s t (1973). 772. S. Itoh, "A r a n d o m fixed point t h e o r e m f o r a multivalued contraction mapping," Pac. J. Math., 6._~8,No. 1, 85-90 (1977). 773. S. Itoh, "Nonlinear random equations with monotone o p e r a t o r s in Banach s p a c e s , ~ Math. Ann., 236, No. 2, 133-146 (1978). 774. S. Itoh, " M e a s u r a b l e or condensing multivalued mappings and r a n d o m fixed point t h e o r e m s , " Kodai Math. J . , 2, No. 3, 293-299 (1979). 775. S. Itoh and W. Takahashi, "Single-valued mappings, multivalued mappings and fixed-point t h e o r e m s , " J. Math. Anal. Appl., 5__~9,No. 3, 514-521 (1977). 776
776. S. Itch and W. Takahashi, "The c o m m o n fixed point t h e o r y of s i n g l e - v a l u e d mappings and multivalued mappings, n Pac. J. Math., 7._99,No. 2, 493-508 (1978). 777. P. I v e r s e n , "Mean value of compact convex s e t s , D a y ' s fixed point t h e o r e m and i n v a r i a n t s u b s p a c e s , " J. Math. Anal. Appl., 5.7, No. 1, 1-11 (1977). 778. M . Q . J a c o b s , " R e m a r k s on s o m e r e c e n t extensions of F i l i p p o v ' s implicit functions l e m m a , " SIAM J . Control, 5, No. 4, 622-627 (1967). 779. M . Q . J a c o b s , " M e a s u r a b l e multivalued mappings and L u s i n ' s t h e o r e m , " T r a n s . Am. Math. Soc., 134, No. 3 , 4 7 1 - 4 8 1 (1968). 780. M . Q . J a c o b s , "On the approximation of i n t e g r a l s of multivalued functions," SIAM J. Control, 7, No. 1, 158-177 (1969). 781. T. Janiak, E. L u c z a k - K u m o r e k , and M. Kisielewicz, "Existence t h e o r e m f o r f u n c t i o n a l - d i f f e r e ~ i a l r e l a tions," D e m o n s t r a t i o Math., 1_22,No. 2, 411-420 (1979). 782. J. J a r n i k , "Constructing the m i n i m a l differential relation with p r e s c r i b e d solutions, n Casopis P e s t . Mat., 105, No. 3, 311-315 (1980). 783. J. J a r n i k and J. Kurzweil, "On conditions on right-hand sides of differential r e l a t i o n s , " Casopis P e s t . Mat., 102, No. 4, 334-349 (1977). 784. J . J a r n i k and J . Kurzweil, "Extension of a S e o r z a - D r a g o n i t h e o r e m to differential r e l a t i o n s and funct i o n a l - d i f f e r e n t i a l relations, t in: C o m m e n t a t . Math. T o m . Spec. Honor. L. Orlicz, T. 1, W a r s z a w a (1978), pp. 147-158. 785. J. E. Jayne, " S t r u c t u r e of analytic Hausdorff s p a c e s , " Mathematika, 2_~3,No. 2, 208-211 (1976). 786. J . W . J a w o r o w s k i , " T h e o r e m on antipodes f o r multivalued mappings and a fixed point t h e o r e m , " Bull. Acad. Polon. Sci., C1. 3, No. 4, 187-192 (1956). 787. J . W . J a w o r o w s k i , "Some consequences of the V i e t o r i s mapping t h e o r e m , " Found.Math., 4_.55,No. 3, 261-272 (1958). 788. J.W. Jaworowsld, "Set-valued fixed point theorems for approximative retracts," Lect. Notes Math., 171, 34-39 (1970). 789. J.W. Jaworowski, "Uber Existenzaussagen fur kondensierende Abbildungen," Wiss. Z. Techn. Univ. Dresden, 28, No. 5, 1183-1187 (1979). 790. T. Jerofsky, "Uber Fixpunkte und Eigenwerte nieht azykliseher kondensierender Abbildungen," Wiss. Z. Techn. Univ. Dresden, 29, No. 6, 1305-1309 (1980). 791. J.-h. Jiang, "Essential fixed points of the multivalued mappings," Sci. Sinica, i_~i,No. 3,293-298 (1962). 792. J. A. Johnson, "Extreme measurable selections," Proc. A m . Math. Soc., 44, No. I, 107-112 (1974). 793. G. S. Jones, "On critical sets for nonlinear equations of evolution with nonuniqueness," in: Proc. Symp. Pure Math., Vol. 18, Part i, A m . Math. Sot., Providence (1970), pp. 132-137. 794. J. E. Joseph, "Regularity, normality and multifunctions," Proc. A m . Math. Sot., 70, No. 2, 203-206 (1978). 795. J. E. Joseph, "Multifunctions and graphs," Pac. J. Math., 79, No. 2, 509-529 (1978). 796. J. E. Joseph, "Multfunctions and cluster sets," Proc. A m . Math. Soe., 74, No. 2, 329-337 (1979). 797. J. E. Joseph, "Multifunctions and inverse cluster sets," Can. Math. Bull., 23, No. 2, 161-171 (1980). 798. M. A. Kaaz, "The relevance of multivalued functions in the theory of pursuit games," Kybernetes, _4, No. 2, 97-104 (1975). 799. H. Kaezynsld and C. Olech, "Existence of solutions of orientor fields with noneonvex right-hand side," Ann. Polon. Math., 29, No. i, 61-66 (1974). 800. P. J. Kaiser and M. B. Suryanarayana, "Orientor field equations illBanach spaces," J. Optim. Theory Appl., 19, No. i, 141-164 (1976). 801. S. Kakutani, "A generalization of Brouwer's fixed point theorem," Duke Math. J., _8, 457-459 (1941). 802. T. Kando, "Characterization of topological spaces by some continuous functions," J. Math. Soc. Jpn., 6_ No. I, 45-54 (1954). 803. M. Kaneko, "An extension of the Nash bargaining problem and the Nash social welfare function," Theory Decision, 12, No. 2, 135-148 (1980). 804. J. Kaniewski and R. Pol, "Borel-measurable selectors for compact-valued mappings in the nonseparable case," Bull. Acad. Polon. Sci. Set. Sci. Math. Astron. Phys., 23, No. i0, 1043-1050 (1975(1976)). 805. H. Karkar, "Contribution a l'6tude des 6quations diff6rentiellesmultivoques," C. R. Acad. Sci. Paris, 266, No. 21, AI099-AII02 (1968). 806. H. Karlmr, "~quations differentielles multivoques et optimisation," Rev. CETHEDEC, 7_, No. 23, 1-61 (197 0). 807. G. Kayser, Zur Existenz yon Fixpunkten bei kompakten bzw. konzentrierenden mengenwertigen Abbildungen in lokalkonvexen Raumen, B e r i c h t e d e r JahrestagLmg Num. Mathematik, P o t s d a m - C e c i l i e n h o f , Nov. 25Dec. 1, 1973, T. H. Karl M a r x Stadt, Sektion Math., 1974, pp. 46-53.
777
808. G. Kayser, "A fixed point t h e o r e m for noncompact s e t - v a l u e d mappings," Abh. Akad. Wiss. DDR, No. 3, 407-411 (1977). 809. W. G. Kelley, " P e r i o d i c solutions of g e n e r a l i z e d differential equations," SIAM J. Appl. Math., 30, No. 1, 70-74 (1976). 810. P. S. Kenderov, "Points of singlevaluedaess of multivalued m e t r i c projections," Dokl. Bolg. Akad. Nauk, 2_99,No. 6, 773-775 (1976). 811. P . S . Kenderov, "Semicontinuity of s e t - v a l u e d mappings with r e s p e c t to two topologies,' Dokl. Bolg. Akad. Nauk, 29, No. 11, 1563-1565 (1976). 812. N. Kenmochi, " R e m a r k s on the m - a c c r e t i v e n e s s of nonlinear o p e r a t o r s , " H i r o s h i m a Math. J., 3, No. 1, 61-68 (1973). 813. N. Kenmochi and S. Oharu, "Difference approximation of nonlinear evolution equations and semigroups of nonlinear o p e r a t o r s , " Publ. Res. Inst. Math. Sci., 1._~0,No. 1, 147-207 (1974). 814. S. Kieltyka and W. Sobieszek, "Sufficient conditions f o r the semicontinuity f r o m below of a certain pointset mapping," Zesz. Nauk PSI, No. 441, 147-153 (1976), 815. N. Kil~chi, " E x i s t e n c e of optimal c o n t r o l s , " RIMS-20, R e s e a r c h Inst. Math. Sci., Kyoto Univ., 1967. 816. N. Kikuchi, "Control p r o b l e m s of contingent equation," Publ. Res. Inst. Math. Sci., Ser. A, 3, No. 1, 85-99 (1967). 817. N. Kikuchi, "On some fundamental t h e o r e m s of contingent equations in connection withthe control p r o b l e m s , " Publ. Res. Inst. Math. Sci., A3, No. 2, 177-201 (1967). 818. N. Kikuchi, "On contingent equations satisfying the Carath6odory type conditions," Publ. Res. Inst. Math. Sci., A3, No. 3,361-371 (1968). 819. N. Kikuchi, "On control problems f o r functional-differential equations," Funkcial. Ekvac., 14, No. 1, 1-23 (1971). 820. N. Kil~chi, "On contingent equations," Lect. Notes Math., 243, 169-181 (1971). 821. N. Kikuchi and Y. Tomita, "On the absolute continuity of multifunctions and orientor fields," Funkcial. Ekvac., 14, No. 3, 161-170 (1971). 822. M. Kisielewicz, "Fixed point theorem for multivalued noncompact mappings," Zeszyty blauk. WSJ w Zielonej Gozze, No. 3, 5-7 (1975). 823. M. Kisielewicz, "Controllability of generalized systems," in: Funct. Differ. Syst. and Related Topics, Proc. ist Internat. Conf., Btazejewko, 1979, Zielona Gora (1980), pp. 161-167. 824. M. Kisielewicz and T. Janiak, "Existence theorem for functional-differential contingent equations," Ann. Polon. Math., 35, No. 2, 159-166 (1977). 825. T. Kita, "A common fixed point theorem for multivalued mappings," Math. Jpn., 22, No. I, 113-116 (1977). 826. V. Klee, ~Stability of the fixed point property," Colloq. Math., 8, No. I, 43-46 (1961). 827. P. Kloeden, "General control systems without backwards extension," in: Differential Games and Control Theory, New York (1974), pp. 49-58. 828. P. Kloeden, "Topologies for general semidynamical systems, ~ J. Diff. Equations, 27, No. i, 92-105 (1978). 829. P. Kloeden, "The funnel boundary of multivalued dynamical systems," J. Austral. Math. Soc., A27, No. I, 108-124 (1979). 830. R. Kluge, "Fixpunktbifurkation fur parameterabhangige vieldeutige vollstetige Abbildungen," Monatsber. Dtsch. Akad. Wiss. Berlin, i_~.I,No. 2, 89-95 (1969). 831. R. Kluge, "Differenzierbare mehrdeutige Abbildungen und Fixpunktbifurkation," Monatsber. Dtsch. Akad. Wiss. Berlin, I_~.I,No. 7, 469-480 (1969). 832. R. Kluge, "Fixpunktbifurkation mehrdeutigen Abbildungen," Schrztenif. Inst. Math. Dtsch. Akad. Wiss. Berlin, A, No. 7, 53-55 (1970). 833. H.-M. Ko, nFixed point theorems for point-to-set mappings and set of fixed points," Pac. J. Math., 4_~2, No. 2, 369-379 (1972). 834. Y. Kodama, "Multivalued maps and shape," Glas. Mat., Ser. III, i_~2,No. i, 133-142 (1977). 835. u Komura, "Differentiability of nonlinear semigroups," J. Math. Soc. Jpn., 21, No. 3, 375-402 (1969). 836. P. Krbec, "Weak stability of multivalueddifferential equations," Czechoslovak Math. J,, 26, No. 3, 470-476 (1976). 837. P. Krbec and J. Kurzweil, "Kneserls theorem for multivalued differential delay equations," Casopis Pest. Mat., 104, I~o. I, 1-8 (1979). 838. B. Krzyzowa, "~quations au paratingent a argument retardS," Ann. Univ. Mariae Curie-Sklodowska, Sec. A, i_.~7,No. i, 7-18 (1963). 839. I. Kubiaczyk, "Common fixed point theorems in metric spaces," Demenstratio Math., 9, No. 2, 301-306 (1976). 840. I. Kubiaczyk, "Some fixed point theorems," Demonstratio Math., 9_, No. 3, 507-515 (1976).
778
841. I. Kubiaczyk and B. Rzepectd, "A note on S m a r t t h e o r e m , n Bull. Math. Soc. Sci. Math. R. S. Roumaine, 2_.00, No. 3-4, 265-267 (1976(1977)). 842. J. K u c e r a , non the a c c e s s i b i l i t y of control s y s t e m ~ EQ(x)," C z e c h o s l o v a k Math. J., 20, No. 1, 122-129 (1970). 843. Z. Kucharski, nA coincidence index," Bull. Acad. Polon. Sci. S e t . Sci. Math. Astron. Phys., 24, No. 4, 245-252 (1976). 844. Z. Kucharski, ',Two consequences of the coincidence index," Bull. Acad. Polon. Sci. Ser~ Sci. Math. Astron. Phys., 24, No. 6, 437-444 (1976). 845. P. K. F. Kuhfittig, ~Fixed points of locally c o n t r a c t i v e and nonexpansive s e t - v a l u e d m a p p i n g s , " Pac. J . Math., 65, No. 2, 399-403 (1976). 846. P. Kuhfittig, ~The m e a n - v a l u e iteration f o r s e t - v a l u e d m a p p i n g s , " P r o e . Am. Math. Soe., 8_~0,No. 3, 401-405 (1980). 847. J. Kupka, ~Some p r o b l e m s in a e e e s s i b i l i t y t h e o r y , " A s t e r i s q u e , No. 75-76, 167-176 (1980). 848. K. Kuratowski, " L o s fonctions s e m i c o n t i n u e s dans l ' e s p a e e des e n s e m b l e s fe~m~s, ~ Fund. Math., 1__.8, 148-159 (1931). 849. K. Kuratowski, "Operations on semicontinuous s e t - v a l u e d mappings, ~ in: Sere. 1962-1963 Analisi, A l g e b r a , G e o m e t r i e a et Topol., Vol. 2, R o m a (1965), pp. 449-461. 850. K. Kuratowski, "On the s e l e c t o r p r o b l e m s for the partitions of Polish s p a c e s and f o r the c o m p a c t - v a l u e d m a p p i n g s , " Ann. Polen. Math., 29, 421-427 (1974). 851. K. Kuratowski, "On the s e l e c t o r p r o b l e m s f o r the partitions of Polish s p a c e s and f o r the c o m p a c t - v a l u e d mappings, ' Ann. Polon. Math., 2._99,No. 4, 421-427 (1975). 852. K. Kuratowski, "The ~ - a l g e b r a g e n e r a t e d by Souslin s e t s and its applications to s e t - v a l u e d mappings and to s e l e c t o r p r o b l e m s , " Bull. Un. Mat. Ital., 11, No. 3, Suppl., 258-259 (1975). 853. K. Kuratowski, " S e l e c t o r s in topology and m e a s u r e t h e o r y , " Wiadom. Mat., Set.2, 19, No. 1, 3-9 (1975). 854. K. Kuratowski, ~Some r e m a r k s on the relation of c l a s s i c a l s e t - v a l u e d mappings to the B a i r e c l a s s i f i c a tion," Colloq. Math., 4_22, 273-277 (1979). 855. K. Kuratowski and A. Maitra, "Some t h e o r e m s on s e l e c t o r s and t h e i r applications to semicontinuous d e c o m p o s i t i o n s , " Bull. Acad. Polen. Sci. S e t . Sci. Math. A s t r o n . Phys., 22, No. 9, 877-881 (1974). 856. K. Kuratowski and C. R y l l - N a r d z e w s k i , ~A g e n e r a l t h e o r e m on s e l e c t o r s , ~ Bull. Acad. Polen. Sol. S e t . Sci. Math. Astron. Phys., 1_~3,No. 6, 397-403 (1965). 857. 5. Kurzweil, "On differential relations and on F i l i p p o v ' s concept of differential equations, n in: Differential Equations, Stockholm (1977), pp. 109-113. 858. J. Kurzweil and J . J a r n i k , "Reducing differential inclusions," Abh. Akad, W i s s . DDR, No. 3, 477-479 (1977). 859. J. Kurzweil and J. J a r n i k , "Right-hand sides of differential inclusions which cannot be reduced," in: P r o b l e m s of the A s y m p t o t i c T h e o r y of Nonlinear Oscillations [in Russian], Kiev {1977), pp. 122-132. 860. G. van d e r Laan and A. J. ft. Talman, ~A r e s t a r t a l g o r i t h m f o r computing fixed points without an e x t r a dimension," Math. P r o g r a m m i n g , 1__7.7,No. 1, 74-84 (1979). 861. E. L a m i Dozo, "Multiapplications P - c o m p a c t e s , n Bull. Soe. Math. Belg., 2._33,No. 4, 416-422 (1971). 862. E. Lami Dozo, " C e n t r e s a s y m p t o t i q u e s dans c e r t a i n s F - e s p a c e s , ~ Boll. Un. Mat. Ital., B17, No. 2, 740-747 (1980). 863. A. L a s o t a and C. Olech, ~On the c l o s e d n e s s of the s e t of t r a j e c t o r i e s of control s y s t e m s , " Bull. Aead. Polon. Sci. Set. Sci. Math. Astron. Phys., 1_~4,No. 11, 615-621 (1966). 864. A. L a s o t a and C. Olech, ~On C e s a r i ' s semicontinuity condition f o r s e t - v a l u e d m a p p i n g s , " Bull. Acad. Polon. Sei. Ser. Sci. Math. Astron. Phys., 1._~6,No. 9, 711-716 (1968). 865. A. L a s o t a and Z. Opial, nAn application of the K a k n t a n i - K y Fan t h e o r e m in the theory of o r d i n a r y differential equations," Bull. Aead. Polon. Sci. Ser. Sci. Math. Astren. Phys., 1_.33,No. 11-12, 781-786 (1965). 866. A. L a s o t a and Z. Opial, " F i x e d - p o i n t t h e o r e m s f o r multivalued mappings and o p t i m a l control p r o b l e m s , " Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 16, No. 8, 645-649 (1968). 867. A. L a s o t a and Z. Opial, "An a p p r o x i m a t i o n t h e o r e m f o r multivalued mappings, ~ Podstawy Sterowania, 1_, No. 1, 71-75 (1971). 868. A. L a s o t a and A. S t r a u s s , " A s y m p t o t i c b e h a v i o r f o r differential equations which cannot be locally l i n e a r ized, n J . Diff. Equations, 1_.00,No. 1, 152-172 (1971). 869. J . - M . L a s r y and R. Robert, "Degr6 p o u r les fenctiens multivoques et applications," C. R. Acad. Sci. P a r i s , 28._00,No. 21, A1435-A1438 (1975). 870. J . - M . L a s r y and R. Robert, "Degr~ et t h 6 o r e m e s de point fixe pour les fonctions multivoques: a p p l i c a tions," S~minaire G o u l a o u i c - L i o n s - S c h w a r t z , 1974-1975: ~quations aux d e r i v 6 e s p a r t i e l l e s lin6aires et non lin~aires. Exp. 14. Centre Math. ]~cole Polytech., P a r i s , 1975.
779
871. J . - M . L a s r y and R. Robert, "Acyclicit6 de l ' e n s e m b l e des solutions de c e r t a i n e s 6qua,ions fonetionelles," C. R. Acad. Sci. Paris, 282, No. 22, A1283-1286 (1976). 872. J.-M. Lasry and R. Robert, "Degr6 topologique pour certains couples de fonctions et applications dilf6rentielles multivoques," C. R. Acad. Sci. Paris, 283, No. 4, AI63-AI66 (1976). K.-S. Lau, "A note on lower semicontinuous set-valued maps," Bull. Acad. Polon. Sci. Ser. Sci. Math. 873. Astron. Phys., 24, No. 4, 271-273 (1976). 874. S. Leader, "A fixed point principle for locally expansive multifunctions," Fund. Math., 106, No. 2, 99-104 (1980). 875. A. Lechieki, nOn bounded and subcontinuous multifunctions," Pac. J. Math., 75, No. 1, 191-197 (1978). 876. A. Lechicki, "On continuous and m e a s u r a b l e multifunctions, ~ Rocz. Polon. Tow. Mat., Ser. 1, 2_~1,No. 1, 141-156 (1980). 877. C.-M. Lee, "An existence t h e o r e m via an intuitive idea and fixed-point t h e o r e m s , " C o m m e n t . Math. Univ. Carolin., 199, No. 4, 715-725 (1978). 878. C.-M. Lee and K.-K. Tan, ~A note on m e t r i c a l l y inward mappings," Comment. Math. Univ. Carolin., 18_, No. 2, 259-263 {1977). 879. C.-M. Lee and K.-K. Tan, nOn F a n ' s extensions of B r o w d e r ' s fixed point t h e o r e m s f o r multivalued inw a r d mappings," J. A u s t r a l . Math. See., A26., No. 2, 169-174 (1974). 880. S. J. L e e s e , "Multifunctions of Souslin type," Bull. A u s t r a l . Math. See,, 1_~.1,No. 3, 395-411 (1974). 881. T . - C h . Lira, "A fixed point t h e o r e m for multivalued nonexpansive mappings on a u n i f o r m l y convex Banach space," Bull. Am. Math. See., 80, No. 6, 1123-1126 (1974). 882. T.-Ch. Lira, " R e m a r k s on s o m e fixed point t h e o r e m s , " P r e c . Am. Math. Soc., 60, 179-182 (1976). 883. T.-Ch. Lira, "Fixed point t h e o r e m s f o r mappings o f n o n e x p a n s i v e t y p e , " P r e c . Am. Math. See., 66...~ No. 1, 69-74 (1977). 884. T . - C h . Lira, "On a s y m p t o t i c c e n t e r s and fixed points of nonexpansive m a p p i n g s , " Can. J. Math., 32, No. 2, 421-430 (1980). 885. T.-Ch. Lira, "Fixed point properties and inverse limit spaces," Pac. J. Math., 2.~5,No. I, 117-122 (1968). 886. Y.-F. Lin, "Tychonoff's theorem for the space of multifunetions, ~ Am. Math. Monthly, 74, No. 4, 399-400 (1967). 887. Y.-F. Lin and D. A. Rose, "Ascoli's theorem for spaces of multifunetions," Pac. J. Math., 34, No. 3, 741747 (1970). 888. J. Lindenstrauss, "A selection theorem," Israel J. Math., 2, No. 3, 201-204 (1964). dinamicos discretos sobre an espacio de Hilbert," An. See. Ciene. Col. Univ. 889. F. E. Llorens, "Semisistemas Gerona, 2, No. 2, 1-12 (1977). 890. V. Losert, "A eounterexample on measurable selections and strong lifting," Leer. Notes Math., 794., 153159 (1980). 891. T . - W . Ma, "Topological d e g r e e s of s e t - v a l u e d compact fields in locally convex s p a c e s , " Rozprawy Mat., No. 92, 1-47 (1972). 892. T.-W. Ma, "Nonsingular s e t - v a l u e d c o m p a c t fields in locally convex s p a c e s , " Fund. Math. 75, No. 3, 249259 (1972). 893. T . - W . Ma, " D e g r e e theory f o r s e t - v a l u e d c o m p a c t p e r t u r b a t i o n s , " Bull. Acad. Felon. Sci. Ser. Sci. Math. Astron. Phys., 20, No. 2, 169-175 (1972). 894. T. Mackowiak, "Continua having the fixed point p r o p e r t y f o r s e t - v a l u e d mappings," Bull. Acad. Polon. Sei. Ser. Sci. Math. Astron. Phys., 2.~3, No. 4 , 4 2 1 - 4 2 4 (1975). 895. T. Mackowiak, "An embedding t h e o r e m f o r t r e e l i k e continua," in: Topology, 4th Colloq., Budapest, 1978, Vol. 2, A m s t e r d a m (1980), pp. 803-813. 896. G. Magerl, "Zu den Schnittsatzen yon Michael and Kuratowski and R y l l - N a r d z e w s k i , " Diss. Doktorgrad. F a c h b e r e i c h . F r i e d e r i c h - A l e x a n d e r - U n i v . N a t u r w i s s . E r l a n g e n - N u r n b e r g , 1977. 897. G. Magerl, "A unified a p p r o a c h to m e a s u r a b l e and continuous s e l e c t i o n s , " T r a n s . Am. Math. See., 24._~5, 443-452 (1978). 898. G. Magerl, " M e t r i z a b i l i t y of c o m p a c t s e t s and continuous s e l e c t i o n s , " P r o c . Am. Math. Soc., 7_~2,No. 3, 607-612 (1978), 899. A. M a i t r a and B. V. Rao, "Selection t h e o r e m s and the reduction p r i n c i p l e , " T r a n s . Am. Math. Soe., 202, 57-66 (1975). 900. A. M a i t r a and B. V. Rao, " G e n e r a l i z a t i o n s of C a s t a i n g ' s t h e o r e m on s e l e c t o r s , " Colloq. Math., 42, 295300 (1979). 901. V. J. Mancuso, "An Aseoli t h e o r e m f o r multivalued functions," J. A u s t r a l . Math. See., 1-2, No. 4, 466-472 (1971).
780
902. R. Manka, "Two fixed point t h e o r e m s on contractions," Bull. Acad. P o l o , . Sci. Ser. Sei. Math. Astron. Phys., 266, No. 1, 41-48 (1978). 903. R. Manka, "Association and fixed points," Fund. Math., 9.1, No. 2, 105-121 {1976). 904. R. Manka, "Fixed point t h e o r e m s for k - d e n d r o i d s , " Fund. Math., 108, No. 2, 119-128 (1980). 905. C. D. Maravall, "Applicaciones U - c o n t r a c t i v a s en los espacios uniformes. T e o r e m a del punto fijo," Rev. Mat. H i s p . - A m . , 2.__55,No. 3, 106-111 (1965). 906. A. Marchaud, "Sur les champs de demidroites et les 6quations diff6rentielles du p r e m i e r o r d r e , " C. R. Acad. Sci. P a r i s , 197, No. 21, 1176-1178 (1933). 907. A. Marchaud, wSur les champs continus de d e m i - c o n e s eonvexes et leurs int~grales," C. R. Acad. Sci. P a r i s , 199, No. 23, 1278-1280 (1934). 908. A. Marchaud, "Sur les champs de d e m i - c o n e s et 6quations diff~rentielles du p r e m i e r o r d r e , " Bull. Soc. Math. F r a n c e , 62, 1-38 (1934). 909. A. Marchaud, "Sur les champs continus de d e m i - c o n e s c o , vexes et leurs int6grales," Compositio Math., 3, No. 1, 89-127 (1936). 910. A. Marchaud, "Sur les champs de d e m i - c o n e s c o , v e x e s , " Bull. Sci. Math., 62, 229-240 (1938). 911. M. V. Marchi, MInsiemi invarianti," Rend. Istit. Mat. Univ. Trieste, 10, No. 1-2, 123-127 (1978). 912. M. M. Marjanovic and M. M. Dresevic, "A note on A s c o l i ' s theorem f o r spaces of multifunctions,' Publ. Inst. Math., No. 14, 111-114 (1972). 913. J. T. Markin, "Fixed point t h e o r e m s f o r set valued contractions," Doctoral Dissertation, Univ. South. California, 1968; D i s s e r t . A b s t r a c t s , B29, No. 3, 1040-1091 (1968). 914. g. T. Markin, "A fixed point t h e o r e m f o r set-valued mappings," Bull. Am. Math. Soe., 73, No. 4, 639-640 (1968). 915. J. T. Markin, "Continuous dependence of fixed point sets," Proc. Am. Math. Soc., 3__88,No. 3, 545-547 (1973). 916. J. T. Markin, "Stability of solution sets for generalized differential equations," J. Math. Anal. Appl., 4__66, No. 2, 289-291 (1974). 917. J. T. Markin, "A fixed point stability theorem for no, expansive set valued mappings," J. Math. Anal. Appl., 54, No. 2, 441-443 (1976). 918. J. T. Markin, "Existence of a fixed point for no, expansive mappings with closed values," Proc. Am. Math. Soc., 5__99,No. 2, 255-259 (1976). 919. M. Martelli, "Some results concerning multivalued mappings defined in Banach spaces," Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur., 54, No. 6, 865-871 (1973(1974)). 920. M. Martelli, "A Rothe's type theorem for noncompact acyclic-valued maps," Boll. Un. Mat. Ital., 11, No. 3, Suppl., 70-76 (1975). 921. M. Martelli, "No,repulsive fixed point theorems and applications to fixed point theory," Riv. Mat. Univ. Parma, 5, No. 1, 43-53 (1979). 922. M. Martelli and A. Vignoli, "On differentiability of multivalued maps," Boll. Un. Mat. Ital., I0, No. 3, 701-712 (1974). 923. M. Martelli and A. Vignoli, "Some surjectivity results for noneompact multivalued maps," Rend. Accad. Sci. Fis. Mat. Napoli, 41, 57-66 (1974). 924. T. Maruyama, "Integration of correspondences and variational problem with operator constraint," Proc. Jpn. Acad., A56, No. 5, 225-230 (1980). 925. S. Massa, "A remark about multivalued mappings," Rend. Istit. LombardoAccad. Sei. Left., All2, No. i, 176-180 (1978). 926. S. Massa and D. Roux, "On the minimal displacement under generalized no,expansive multivalued mappings," Rend. Mat., 1-2, No. 3-4, 577-585 (1979). 927. J. Massabo, "A remark about fixed-point theorems for multivalued mappings," Publ. Istit. Mat. Univ. Genova, No. 124 (1974). 928. J. Massabo and P. Nistri, "A topological degree for multivalued A-proper maps in Banach spaces," Boll. Un. Mat. Ital., BI3, No. 3, 672-685 (1976). 929. Ch. N. Maxwell, "Fixed points of symmetric product mappings," Proc. Am. Math. Soc., 8, No. 4, 808-815 (1957). 930. Ch. N. Maxwe11, "The degree of multiple-valued maps of spheres," Leer. Notes Math., 664, 123-141 (1978). 931. B. L. McAllister, "Hyperspaces and multifunctions. The first half century (1900-1950)," Nieuw Arch. Wisk., 2__66,No. 2, 309-329 (1978). 932. L. F. McAuley, "Concerning open selections," Leer. Notes Math., 171, 49-53 (1970). 933. H. Meister, "Zum Fixpunktverhalten nichtexpansiver Abbildungen," Comment. Math. Univ. Carolin., 21, No. 2, 319-331 (1980).
781
934. S. Menou, " F a m i l l e m e s u r a b l e d ' o p 6 r a t e u r s m a x i m a u x monotones," C. R. Acad. Sci. P a r i s , AB290, No. 16, A741-A744 (1980). 935. H. Methlouthi, "Sur le calcul differentiel multivoque," C. R. Acad. Sci. P a r i s , 284, No. 3, A141-A144 (1977). 936. H. Methlouthi, "t~quations diff~rentielles multivoques s u r une graphe l o c a l e m e n t c o m p a c t , " C. R. Acad. Sei. P a r i s , 28.4, No. 20, A1287-A1290 (1977). 937. G . G . L . M e y e r , ~A s y m p t o t i c p r o p e r t i e s of sequences i t e r a t i v e l y g e n e r a t e d by p o i n t - t o - s e t m a p s , " Math. P r o g r a m m i n g Stud., 10, 115-127 (1979). 938. E. A. Michael, "Topologies on s p a c e s of s u b s e t s , " T r a n s . Am. Math. Soc., 7.1, 152-183 (1951). 939. E . A . Michael, "Selection t h e o r e m s f o r continuous functions," P r o c . Int. Congr. Math., 1954, Vol. 2, A m s t e r d a m (1954), pp. 241-242. 940. E. A. Michael, "Continuous s e l e c t i o n s . I," Ann. Math., 6..~3,No. 2, 361-381 (1956). 941. E. A. Michael, "Continuous selections. II," Ann. Math., 6._~4,No. 3, 562-580 (1956). 942. E. A. Michael, "Continuous s e l e c t i o n s . III," Ann. Math., 6__55,No. 2, 375-390 (1957). 943. E. A. Michael, "A t h e o r e m on semicontinuous s e t - v a l u e d functions," Dul~e Math. J . , 2._66,No. 4, 647-651 (1959). 944. E. A. Michael, "A s e l e c t i o n t h e o r e m , " P r o c . Am. Math. Soc., 17, No. 6, 1404-1406 (1966). 945. E. A. Michael, "A s u r v e y of continuous s e l e c t i o n s , " Leer. Notes Math., 171, 54-58 (1970). 946. E. A. Michael, "Some r e s u l t s on continuous s e l e c t i o n s , " Math. Centre T r a c t s , A m s t e r d a m , No. 115, 161163 (1979). 947. E. A. Michael, "Continuous selections: s o m e old and s o m e new r e s u l t s , " Topology, 4th Colloq., Budapest, 1978, Vol. 2, A m s t e r d a m (1980), pp. 849-851. 948. H. Millington, " P r o d u c t s of g r o u p - v a l u e d m e a s u r e s , " Stud. Math., 5_~4,No. 1, 7-27 (1975). 949. P . S . Milojevic, "Some g e n e r a l i z a t i o n s of the f i r s t F r e d h o l m t h e o r e m s to multivalued condensing and A p r o p e r mappings," Boll. Un. Mat. Ital., B13, No. 3, 619-633 (1976). 950. P. S. Milojevic, "A g e n e r a l i z a t i o n of L e r a y - S c h a u d e r t h e o r e m and s u r j e c t i v i t y r e s u l t s for multivalued A - p r o p e r and pseudo A - p r o p e r mappings," Nonlinear Analysis: T h e o r y Meth. Appl., 1, No. 3, 263-276 (1977). 951. P. S. Milojevic, "Some g e n e r a l i z a t i o n s of the f i r s t F r e d h o l m t h e o r e m to multivalued A - p r o p e r mappings with applications to nonlinear elliptic equations," J . Math. Anal. Appl., 6_~5,No. 2, 468-502 (1973). 952. P~ S. Milojevic, " F i x e d point t h e o r e m s f o r multivalued a p p r o x i m a t e mappings," P r o c . Am. Math. Soe., 73, No. 1, 65-72 (1979). 953. P . S . Milojevic, " F r e d h o l m a l t e r n a t i v e s and s u r j e c t i v i t y r e s u l t s f o r multivalued A - p r o p e r and condensing mappings with applications to nonlinear integral and differential equations," C z e c h o s l o v a k Math. J . , 30, No. 3, 387-417 (1980). 954. P . S . Milo]evic and W. V. P e t r y s h y n , "Continuation t h e o r e m s and the a p p r o x i m a t i o n - s o l v a b i l i t y of e q u a tions involving multivalued A - p r o p e r mappings," J . Math. Anal. Appl., 60, No. 3, 658-692 (1977). 955. J. Mioduszewski, "On continuous s e l e c t i o n s of multivalued functions on d e n d r i t e s , " P r a c e Mat., 5, 73-77 (1961). 956. S. Mirica, "A note on the g e n e r a l i z e d differentiability of m a p p i n g s , " Nonlinear Analysis: Theory, Meth. Appl., 4, No. 3, 567-575 (1980). 957. M. Misic, " E x a m p l e s c~ multivalued mappings of ~ - b i c o m p a c t topological s p a c e s , " Mat. Vesnik, 7, No. 2, 179-181 (1970). 958. P. Momal, " T h ~ o r e m e s de m a x i m u m , " C. R. Acad. Sci. P a r i s , A278, No. 13,905-907 (1974). 959. P . M o r a l e s , " N o n - H a u s d o r f f Ascoli t h e o r y , " D i s s e r t a t i o n e s Math., No. 119 (1974). 960. P. M o r a l e s , "A n o n - H a u s d o r f f multffanction Ascoli t h e o r e m f o r k3-spaces," Can. J. Math., 27_._,No. 4, 893-900 (1975). 961. J . - J . Moreau, "S~l~ctions de multiapplications a r e t r a c t i o n finie," C. R. Acad. Sci. P a r i s , 276, No. 4, A265-A268 (1973). 962. T. B. M u e n z e n b e r g e r , "On the a p p r o x i m a t e fixed-point p r o p e r t y f o r multifunctions," Colloq. Math., 19_., No. 2, 245-250 (1968). 963. T. B. M u e n z e n b e r g e r and R. E. Smithson, " F i x e d points and p r o x i m a t e fixed points," Fund. Math., 63, No. 3, 321-326 (1968). 964. T. B. M u e n z e n b e r g e r and R. E. Smithson, " F i x e d point s t r u c t u r e s , " T r a n s . Am. Math. Soc., 184, 153-173 (1973). 965. T. B. M u e n z e n b e r g e r and R. E. Smithson, "On the equivalence of c e r t a i n coincidence t h e o r e m s and fixed point t h e o r e m s , " Fund. Math., 84, No. 2, 93-100 (1974). 966. T. B. M u e n z e n b e r g e r and R. E. Smithson, "Refluent multifunctions on s e m i t r e e s , " P r o c . Am. Math. Soc., 4..~4, No. 1, 189-195 (1974). 782
967. S. B. Nadler, "Multivalued contraction m a p p i n g s , " Pac. J. Math., 30, No. 2, 475-488 (1969). 968. S. B. Nadler, "Some r e s u l t s on multivalued c o n t r a c t i o n m a p p i n g s , " Lect. N o t e s M a t h . , 171, 64-69 (1970). 969. S. B. N a d l e r and L. E. Ward, J r . , "Concerning continuous s e l e c t i a n s , " P r o c . A m . Math. Soc., 25__,No. 2, 369-374 {1970). 970. S. B. Nadler, H y p e r s p a c e s of Sets. A Text with R e s e a r c h Questions, Dekker, New York (1978). 971. J . Nagata, ' O n multivalued mappings and g e n e r a l i z e d m e t r i c s p a c e s , ' P r o c . Jpn. Acad., 4_6, No. 9, 936940 (1970). 972. S. N a k a g i r i and H. M u r a k a m i , "Some p r o p e r t i e s of s e t - v a l u e d o p e r a t o r s defined by solution f a m i l i e s of V o l t e r r a integral equations," Mere. Fac. Engrg. Kobe Univ., No. 21, 113-130 (1975). 973. H. Nakano and K. Nakano, "Connector t h e o r y , " Pac. J . Math,, 56, No. 195-213 (1975). 974. I. N e c s u l e s c u , "On c e r t a i n sufficient conditions f o r optimization," Bul. Inst. P e t r o l si Gaze, 5, 99-101 (1976(1977)). 975. J. Nelson, J r . , "A g e n e r a l i z a t i o n of fixed p o i n t t h e o r e m s of Wallace and Ward," Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 24.4, No. 6, 445-448 (1976). 976. J . Nelson, J r . , "A c o m m o n fixed point s t r u c t u r e , " Rocky Mountain J . Math., 8, No. 4, 675-682 (1978). 977. J. Von Neumann, " U b e r ein 6 k o n o m i s c h e s G l e i c h u n g s s y s t e m und eine V e r a l l g e m e i n e r u n g des B r o u w e r s c h e n F i x p u n k t s a t z e s , " in: E r g e b n i s s e eines M a t h e m a t i s c h e n Kolloquiums, 8, 1935-1936, L e i p z i g - W i e n (1937), pp. 73-83. 978. H. Nikaido, " Z u m Beweis der V e r a l l g e m e i n e r u n g des F i x p n n k t s a t z e s , " Kodai Math. Sere. Rep., No. 1, 1316 (1953). 979. G. N u r n b e r g e r , "Schnitte fur die m e t r i s c h e P r o j e k t i o n , " J. Approx. Theory, 2_00,No. 2, 196-219 (1977). 980. A. Okuyama, " O n m u l t i v a l u e d monotone closed mappings,,, P r o c . Jpn. Acad., 3_66,No. 3, 106-110 (1960). 981. C. Olano "Sobre s e l e c c i o n e s , " in: Actas 10a Reun. Anu. Mat. E s p . Laguna, 1969, Madrid (1972), pp. 85113. 982. C. Olech "A note concerning s e t valued m e a s u r a b l e functions," Bull. Acad. Polon. Sci. S e r . Sci. Math. Astron. Phys., 1_33,No. 4, 317-321 (1965). 983. C. Olech, " A p p r o x i m a t i o n of s e t - v a l u e d functions by continuous functions," Colloq. Math., 1_99,No. 2, 285293 (1968). 984. C. Olech " E x i s t e n c e of solutions of nonconvex o r i e n t o r fields," Boll. Un. Mat. Ital., 11__.,No. 3, Suppl., 189-1970975). 985. C. Olech " E x i s t e n c e t h e o r y in optimal control," in: Control T h e o r y and Top. Funct. Anal., Vol. 1, Vienna (1976), pp. 291-328. 986. C. Olech "The a c h i e v e m e n t s of Tadeusz Wazewski in the m a t h e m a t i c a l t h e o r y of optimal c o n t r o l , " Wiadom. ~ a t . , Set. 2, 20, No. 1, 66-69 (1976). 987. B. O'Neill, "Induced homology h o m o m o r p h i s m s f o r s e t - v a l u e d m a p s , " P a e . J . Math., 7, No. 2, 1179-1184 (1957). 988. B. O'Neill, "A fixed point t h e o r e m for multivalued functions," Duke Math. J., 24, No. 1, 61-62 (1957). 989. B. O'Neill and E. G. Straus, "A fixedpointtheorem," Proc. Am. Math. Soc., 8, No. 6, 1148-1151 (1957). 990. D. N. O'Steen, "Spaces of set-valued functions," Trans. Am. Math. Soc., 169, 307-316 (1972). 991. H. Otani and Y. Yamada, "On the Navier-Stokes equations in noncyclindrical domains: An approach by the subdifferential operator theory," J. Fae. Sei. Univ. Tokyo, Sec. IA, 25, No. 2, 185-204 (1978). 992. T. Parthasarathy, Selection Theorems and Their Applications, Springer-Verlag, Berlin (1972). 993. S.-N. Patnaik, "Fixed points of multivalued transformations," Fund. Math., 65, No. 3, 345-349 (1969). 994. S.-N. Patnaik, "On the Kakutani fixed point theorem and its relationships withthe selection problem. I," Lect. Notes Math., 171, 70-73 (1970). 995. N. H. Pavel and I. I. Vrabie, 'Equations d'~volution multivoques dans les espaces de Banach," C. R. Acad. Sci. Paris, AB278, No. 5, A315-317 (1978). 996. J. Pejsachowicz, "A Lefschetz fixed point theorem for muftivalued weighted mappings,' Boll. Un. Mat. Ital., 14A, No. 2, 391-397 (1977). 997. J. Pejsachowicz, "The homotopy theory of weighted mappings," Boll. Un. Mat. Ital., BI4, No. 3, 702-721 (1977). 998. J. Pelant, "Relations between g e n e r a l i z e d solutions of o r d i n a r y differential equations," Casopis P e s t . Mat., 101, No. 2, 159-175 (1976). 999. A. P e l c z a r , "On s o m e extensions of the r e t r a c t t h e o r e m of T. Wazewski, V," Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 21, No. 5, 399-404 (1973). 1000. T. K. C. Peng, " I n v a r i a n c e and stability f o r bounded uncertain s y s t e m s , " SIAM J. Control, 1.00, No. 4, 679690 (1972). 1001. J. P. Penot, "A r e m a r k on the d i r e c t method of calculus of v a r i a t i o n s , " P r o c . Am. Math. Soc., 6_.77,No. 1, 135-141 (1977). 783
1002. J. P. Penot and M. Th~ra, "Semicontnuit~ 1003. 1004. 1005. 1006. 1007. 1008. 1009. 1010. 1011. 1012. 1013. 1014. 1015. 1016. 1017. 1018. 1019.
des applications et des multiapplieations," C. R. Acad. Sci. Paris, AB288, No. 4, A241-244 (1979). W. V. Petryshyn, "Existence of nonzero fixed points for noncompact mappings in wedges and cones," J. Funet. Anal., 3_33,No. I, 36-46 (1974). W. V. Petryshyn and P. M. Fitzpatrick, "Degree theory for noneompact multivalued vector fields," Bull. Am. Math. Soc., 7__99,No. 3, 609-613 (1973). W. V. Petryshyn and P. M. Fitzpatrick, "A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings," Trans. Am. Math. Soc., 194, 1-25 (1974). W. V. Petryshyn and P. M. Fitzpatrick, "Fixed point theorem for multivalued noneompact inward maps," J. Math. Anal. Appl., 4_~6,No. 3, 766-767 (1974). G. Pianigiani, 'On the fundamental theory of multivalued differential equations," J. Diff. Equations, 2__55, No. i, 30-38 (1977). M. Pierre, "Un th~oreme gSn6rale de g~n4ration de semigroupes non lin6aires," Israel J. Math., 2_3, No. 3-4, 189-199 (1976). C. P. Pixley, "An example concerning continuous selections on infinite-dimensional spaces," Proc. Am. Math. Soc., 43, No. i, 237-244 (1974). A. Plis, "Remark on measurable set-valued functions," Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 9, No. 12, 857-859 (1961). A. Plis, "Trajectories and quasitrajectories of an orientor field," Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., I_~I,369-370 (1963). A. Plis, "Measurable orientor fields," Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 13, No. 8, 565-569 (1965). A. Plis, "On trajectories of orientor fields,' Bull. Acad. Polon. Sci. Ser. Sei. Math. Astron. Phys., I_~3, No. 8, 571-573 (1965). A. Plis, "Emission zones for orientor fields," Colloq. Math., i_~6,133-135 (1967). A. Plis, "Emission zones in control theory," in: Math. Theory Control, Academic Press, New York (1967), pp. 198-202. A. Plis, "Generalized ordinary differential equations and control theory," Math. Balkanica, 3, 407-410 (1973). A. Plis, "Uniqueness of optimal trajectories for nonlinear control systems," Ann. Polon. Math., 299, No. 4, 397-401 (1975). A. Plis, "Accessible sets in control theory," in: Internat. Conf. Differential Equations, 1974, New York (1975), pp. 646-650. R. L. Plunkett, "A fixed point theorem for continuous multivalued transformations," Proe. Am. Math. Soe., 7, No. i, 160-163 (1956). D. Pompeiu, "Sur la continuit& des fonctions de variables complexes," Ann. Sci. Univ. Toulouse, 2 S~rie, 7, 265-312 (1905). V. Popa, "On a decomposition of the quasicontinuity of multifunctions," Stud. Cerc. Mat., 27, No. 3, 323328 (1975). V. Popa, "Multifunctions and bitopological spaces," Bull. Math. Soc. Sci. Math. R. S. Roumaine, 1__9, No. 1-2, 147-152 (1975(1976)). V. Popa, "Weakly continuous multifunctions," Boll. Un. Mat. Ital., AI5, No. 2, 379-388 (1978). V. Popa, wOn certain properties of quasicontinuous and almost continuous multifanctions," Stud. Cerc. Mat., 30, No. 4, 441-446 (1978). V. Popa, "A common fixed point theorem for a sequence of multifunctions," Studia Univ. Babes-Bolyai Math., 2_~4,No. 2, 39-41 (1979). V. Popa, "H-almost continuous multifunctions," Stud. Cerc. Mat., 32, No. I, 103-109 (1980). M. J. Powers, "Multivalued mappings and fixed point theorems," Doctoral Dissertation, Indiana Univ., 1968; Dissertations Abstracts, B29, No. 2, 689-690 (1968). M. J. Powers, "Multivalued mappings and Lefschetz fixed point theorems," Proc. Cambridge Phil. Soc., 68, No. 3, 619-630 (1970). M. J. Powers, "Lefschetz fixed point theorems for multivalued maps," Loci. Notes Math., 171, 74-81 (1970). M. J. Powers, "Lefsehetz fixed point theorems for a new class of multivalued maps," Pac. J. Math., 4_~2, No. i, 211-220 (1972). M. J. Powers, "Fixed point theorems for noncompaet approximative ANR's," Fund. Math., 75, No. I, 61-68 (1972). T. Pruszko, "An operational equation and differential equations with multivalued nonconvex right-hand side," Demonstratio Math., 1.3, No. I, 33-44 (1980). m
1020. 1021. 1022. 1023. 1024. 1025. 1026. 1027. 1028. 1029. 1030.
1031. 1032.
784
1033. 1034. 1035. 1036. 1037. 1038. 1039. 1040. 1041. 1042. 1043. 1044. 1045. 1046 1047. 1048. 1049. 1050. 1051. 1052. 1053. 1054. 1055. 1056. 1057. 1058. 1059. 1060. 1061. 1062. 1063. 1064. 1065. 1066.
B. N. Pshenichnyi, "Convex multivalued mappings and t h e i r conjugates," in: Math. Mod. Econ., A m s t e r dam (1974), pp. 333-349. H. W. Pu, "Another Ascoli t h e o r e m f o r multivalued functions," Rocz. Pol. Tow. Mat., Set. 1, 1.7.7,No. 2, 445-450 (1974). P. Pucci, nun metodo di a p p r o s s i m a z i o n e p e r la determinazione di tutte le soluzioni di un campo di p r i e n t o r i , " Atti Sem. Mat. Fis. Univ. Mode,a, 25, No. 1, 64-78 (1976(1977)). P. Pucci, ~Sulla determinatione di tutte le soluzioni di un campo di orientori nonconvesso," Boll. Un. Mat. Ital., B14, No. 2, 383-401 (1977). P. Pucei, "I1 p r o b l e m a di d e t e r m i n a r e tutte le soluzioni di an eampo di orientori," Rend. C i r c . Mat. P a l e r m o , 266, No. 1-3, 156-164 (1977). S. R a c z y n s k i , ' O n n o n c e , v e x control in a separable Banach space," Probl. Upr. Teorii Inf., 9, No. 2, 121-132 (1980). S. Raczynsld, nOn optimal control and reachable sets in a Banach space," Lect. Notes Contr. Inf. Sci., 2__~2,439-448 (1980). B. K. Ray, "On simultaneous fixed points of multivalued maps," Monatsh. Math., 76, No. 5, 448-454 (1972). B. K. Ray, "A note on multivalued contraction m a p p i n g s , ' Atti Aecad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 5.~6, No. 4, 500-503 (1974). B. K. Ray, "Some fixed point t h e o r e m s , " Fund. Math., 9..~2, No. 1, 79-90 (1976). B. K. Ray, " O n C i r i c ' s fixed point theorem," Fund. Math., 9_.4.4,No. 3, 221-229 (1977). R. Redheffer, "Matrix differential e q u a t i o n s , ' Bull. Am. Math. Soc., 81, No. 2, 485-488 (1975). R. Redheffer and P. Volkrnann, "Differential equations in o p e r a t o r algebras. I. Invariance of the Siegel disk," Math. Ann., 239, No. 2, 97-110 (1979). S. Reich, "Kannan's fixed point theorem," Boll. Un. Mat. Ital., 4, No. 1, 1-11 (1971). S. Reich, "Fixed points on multivalued functions," Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 5__.1,No. 1-2, 32-35 (1971). S. Reich, "Fixed points in complete m e t r i c spaces," Atti Aecad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 5_~1,No. 5, 270-273 (1971(1972)). S. Reich, " R e m a r k s on fixed points," Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 52, No. 5, 689-697 (1972). S. Reich, "Fixed points of contractive functions," Boll. Un. Mat. Ital., 5, No. i, 26-42 (1972). S. Reich, "A fixed point theorem for locally contractive multivalued functions," Re,. Roumaine Math. Pures Appl., i_~7,No. 4, 569-572 (1972). S. Reich, "Fixed points in locally convex spaces," Math. Z., 125, No. i, 17-31 (1972). S. Reich, "Remarks on fixed points. II," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 53, No. 3-4, 250-254 (1972(1973)). S. Reich, "A r e m a r k on s e t - v a l u e d mappings that satisfy the L e r a y - S c h a u d e r condition," Atti Accad. Naz. Lincei Rend. C1. Sei. Fis. Mat. Natur., 6._1.1,No. 3-4, 193-194 {1976(1977)). S. Reich, "A minimal displacement problem," Comment. Math. Univ. St. Pauli, 26, No. 2, 131-135 (1978). S. Reich, "An iterative p r o c e d u r e for constructing z e r o s of a c c r e t i v e sets in Banach spaces," Nonlinear Anal., 2, No. i, 85-92 (1978). S. Reich, "Approximate selections, best approximations, fixed points and invariant sets," J. Math. Anal. Appl., 6_.~2,No. 1, 104-113 (1978). S. Reich, "A random fixed point t h e o r e m for s e t - v a l u e d mappings," Atti Aecad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur., 64, No. 1, 65-66 (1978). S. Reich, "Fixed point t h e o r e m s f o r s e t - v a l u e d mappings," J. Math. Anal. Appl., 69, No. 2, 353-358 {1979). Ch. Rennolet, "Existence and boandedness of solutions of a b s t r a c t nonlinear integrodifferential equations of n o , convolution t y p e , ' J. Math. Anal. Appl., 70, No. 1, 42-60 (1979). C. 5. Rhee, "I-Iomotopy functors d e t e r m i n e d by set valued maps," D o c t o r a l D i s s e r t a t i o n , Univ. of Georgia, 1965; D i s s e r t . A b s t r a c t s , B28, No. 3, 1018 (1967). C. J. Rhee, "Homotopy groups f o r cellular s e t - v a l u e d functions," P r e c . Am. Math. Soc., 1___9,No. 4, 874876 (1968). C. J. Rhee, "Homotopy functions determined by set-valued maps," Math. Z., 113, No. 2, 154-158 (1970). C. J. Rhee, "On a class of multivalued mappings in Banach spaces," Can. Math. Bull., 15, No. 3, 387-393 (1972). U. Rieder, "Measurable selection theorems for optimization problems," Manuscripta Math., 24, No. I, 115-131 (1978). R. Robert, "Points de continuit5 de multiapplicatioas semicontinues s u p 6 r i e u r e m e n t , " C. R. Acad. Sci. P a r i s , A278, No. 6, 413-415 (1974).
785
1067. A. P. Robertson, "On m e a s u r a b l e s e l e c t i o n s , " P r o c . R. Soc. Edinburgh, A72, No. 1, 1-7 (1974). 1068. S. M. Robinson, "An i n v e r s e - f u n c t i o n t h e o r e m f o r a c l a s s of multivalued functions," P r o c . A m . Math. Soc., 4_~/, No. 1, 211-218 (1973). 1069. S. M. Robins on and R. R. Meyer, "Lowe r semicontinuity of multivalue d l i n e a r i z a t i o n mappings," SIAM J. Control, 1.1, No. 3, 525-533 (1973). 1070. R. T. Rockafellar, " C h a r a c t e r i z a t i o n of the subdifferentials of convex functions," Pac. J. Math., 17, No. 3, 497-510 (1966). 1071. R. T. R o c k a f e l l a r , " I n t e g r a l s which are convex fanctionals. I," Pac. J. Math., 24._, No. 3, 525-539 (1968). 1072. R. T. R o c k a f e l l a r , " M e a s u r a b l e dependence of convex s e t s and functions of p a r a m e t e r s , " J. Math. Anal. Appl., 28, No. 1, 4-25 (1969). 1073. R. T. R o c k a f e l l a r , UConvex a l g e b r a and duality in dynamic models of production," in: Math. Mod. Econ., A m s t e r d a m (1974), pp. 351-378. 1074. R. T. Rockafellar, " i n t e g r a l functionals, n o r m a l integrands and m e a s u r a b l e s e l e c t i o n s , " Lect. Notes Math., 543, 157-207 (1976). 1075. M. Rogalski, "Surjectivite d'applieatiens multivoques dans les c o " v e x e s c o m p a c t s , " Bull. Sci. Math., 96, 83-87 (1972). 1076. J. T. R o g e r s , J r . , "Applications of a V i e t o r i s - B e g l e t h e o r e m f o r multivalued m a p s to the cohomology of h y p e r s p a c e s , " Michigan Math. J., 22, No. 4, 315-319 (1976). 1077. S. Rolewicz, "On y - p a r a c o n v e x multifunctions," Math. Jpn., 24, No. 3, 293-300 (1979). 1078. S. Rolewicz, "On i n t e r s e c t i o n s multifunctions," Math. O p e r a t i o n s f o r s c h . Statist. Ser. Optim., 11, No. 1, 3-11 (1980). 1079. A. J. Roques, " L o c a l evolution s y s t e m s in g e n e r a l Baaach s p a c e s , " Pac. J. Math., 6-2, No. 1, 197-217 (1976). 1080. R. H. Rosen, " F i x e d points f o r multivalued functions on snakelike continua," P r o c . Am. Math. Soc., 10, No. 1, 167-173 (1959). 1081. E. Roxin, " T h e e x i s t e n c e of optimal c o n t r o l s , " Michigan Math. g., 9, No. 2, t09-119 (1962). 1082. E. Roxin, "Stability in g e n e r a l control s y s t e m s , " J. Diff. Equations, 1_2 No. 2, 115-150 (1965). 1083. E. Rexin, "On g e n e r a l i z e d dynamical s y s t e m s defined by contingent equations," J . Diff. Equations, ]_., No. 2, 188-205 (1965). 1084. E. Roxin, " L o c a l definition of g e n e r a l i z e d control s y s t e m s , " Michigan Math. J . , 1-2, No. 1, 91-96 (1966). 1085. E. Roxin, "On stability in control s y s t e m s , " J. Soc. Industr. Appl. Math., A_.33,No. 3, 357-372 (1965(1966)). 1086. E. Roxin, "On a s y m p t o t i c stability in control s y s t e m s , " Rend. C i r c . Mat. P a l e r m o , 1.~5, No. 2, 193-208 (1966). 1087. E. Rupniewska, "On m e a s u r a b l e multivalued mappings," Z e s z . Nauk PSI, No. 624, 79-82 (1979). 1088. W. Rupp, " R i e s z - p r e s e n t a t i e n of additive and a - a d d i t i v e s e t - v a l u e d m e a s u r e s , " Math. Ann., 239, No. 2, 111-118 (1979). 1089. I. A. Rus, "Fixed point t h e o r e m s f o r multivalued mappings in complete m e t r i c s p a c e s , " Math. Jpn., 20, Special I s s u e , 21-24 (1975). 1090. G. Rzezuchowska, "Differential inclusion with dissipative o p e r a t o r s , " D e m o n s t r a t i o Math., 12, No. 4, 1059-1070 (1979). 1091. A. Sabadis, "A propos d'un t h e o r e m e de point fixe pour des applications multivoques dans les e s p a c e s v e c t o r i e l s topologiques," Bul. Inst. Politehn. Iasi, Sec. 1, 1-8, No. 3-4, 61-63 (1972). 1092. A. Sabadis, "On the continuity of multifunctions," Bul. Inst. Politehn. Iasi, Sec. 1, 1__.9,No. 1-2, 67-70 (1973). 1093. A. Sabadis, "The set of fixed points f o r l i n e a r multifunctiens," Bul. Inst. Politehn. Iasi, Sec. 1, 1..~9,No. 3-4, 41-43 (1973). 1094. A. Sabadis and Gh. F l o r e a , " F i x e d point t h e o r e m s f o r c e r t a i n c l a s s e s of multifunctions," Bul. Inst. P o l i telm. Iasi, Sec. 1, 20, No. 3-4, 23-26 (1974). 1095. A. Sabadis and Gh. F l o r e a , "A r e m a r k on S c h a u d e r ' s t h e o r e m s , " Bul. Inst. Politelm. Iasi, Sec. 1, 20, No. 1-2, 35-39 (1974). 1096. M . - F . Sainte-Beuve, "Sur la g6n6ralisation d'un t h 6 o r e m e de section m e s u r a b l e de yon N e u m a n n - A u m a n n et applications a un t h 6 o r e m e de fenctions i m p l i c i t e s m e s u r a b l e s e t a un t h 6 o r e m e de r e p r 6 s e n t a t i o n int6grale," C. R. Acad. P a r i s , 276, No. 19, A1297-A1300 (1973). 1097. M.-F. Sainte-Beuve, "On the extension of yon Neumann-Aumann's theorem," J. Funct. Anal., 17, No. i, 112-129 (1974). 1098. J. Saint-Pierre, "G6n6ration monotone d'une multiapplication bor~lienne a valeurs co,vexes," C. R. Acad. Sci. Paris, 281, No. 10, A323-A326 (1975).
786
J. S a i n t - P i e r r e , WMultiapplications bor61iennes a v a l e u r s eonvexes de Rn," Ann. Inst. H. P o i n c a r e , B12, No. 2, 163-180 (1976). ii00. J. S a i n t - P i e r r e , " B o r e l convex-valued multifunctions," Lect. Notes Econ. Math. Syst., 144, 180-190 (1977). ii 01. J. S a i n t - P i e r r e , "Points expos6s dtune multiapplication a v a l e u r s c o , vexes de Rn," C. R. Acad. Sci. P a r i s , AB288, No. 2, A121-A123 (1979). 1102. H. Sarbadhikari, nA s e l e c t i o n t h e o r e m for m u l t i f u n c t i o n s , ' P r o c . Am. Math. Soc., 7.1, No. 2, 285-288 (1978). 1103. H. S a r b a d h i k a r i and S. M. S r i v a s t a v a , ' P a r a m e t r i z a t i o n s of G 5 - v a l u e d multifunetions," T r a n s . Am. Math. Soc., 258, No. 2, 457-466 (1980). 1104. R. C. S c a r p a r o , "Un t e o r e m a de s e l e c c i o n p a r a p o r t a d o r e s s i m u l t a n e o s i n v e r s o s continuous," Math. Notae, 2_~2,No. 1-2, 101-105 (1970-1971(1972)). 1105. R. C. S c a r p a r o , "Selecciones p a r a p o r t a d o r e s con dimension finita," Math. Notae, 2_.~3,No. 1-2, 93-98 (1972-1973). 1106. H. S c h i r m e r , " P r o p e r t i e s of the fixed point set of c o n t r a c t i v e multifunctions, n Can. Math. Bull., 13, No. 2, 169-173 (1970). 1107. H. S c h i r m e r , ~Coincidences and fixed points of multifunctions into t r e e s , " P a c . J. Math., 34, No. 3, 759767 (1970). 1108. H. S c h i r m e r , "A Kakutani type coincidence t h e o r e m , " Fund. Math., 6._~9,No. 3, 219-226 (1970). 1109. H. S c h i r m e r , "Biconnected multifunetions of t r e e s which have an end point as fixed point or coincidence," Can. J. Math., 2-3, No. 3, 461-467 (1971). iii0. H. S c h i r m e r , "6-Continuous s e l e c t i o n s of s m a l l multifunctions," Can. J. Math., 2-4, No. 4, 631-635 (1972). IIii. H. S c h i r m e r , " F i x e d point and coincidence s e t s of biconnected multifunctions on t r e e s , " P a c . / I . Math., 40, No. 2, 445-449 (1972). 1112. H. S c h i r m e r , "Homotopy f o r s m a l l multifunctions," Fund. Math., 7.8, No. 2, 169-176 (1973). 1113. H. S c h i r m e r , "Simplicial a p p r o x i m a t i o n of s m a l l multifunctions," Fund. Math., 8._!4,No. 2, 121-126 (1974). 1114. H. S c h i r m e r , "A Nielsen n u m b e r f o r fixed points and n e a r points of s m a l l multifunctions," Fund. Math., 88, No. 2, 145-156 (1975). 1115. W. Schlee, V e r a l l g e m e i n e r t e Differentialgleichungen zur Losang yon Optimierungsaufgaben, Diss. Dokt. N a t u r w i s s . Fak. Allg. W i s s . Techn. Univ. Munchen (1970). 1116. K. Schmitt and H. L. Smith, "On eigenvalue p r o b l e m s f o r nondifferentiable mappings," J. Diff. Equations, 3-3, No. 3, 294-319 (1979). 1117. R. SchSneberg, " L e r a y - S c h a u d e r p r i n c i p l e s f o r condensing multivalued mappings in topological l i n e a r s p a c e s , " P r o c . Am. Math. Soc., 7_.~2,No. 2, 268-270 (1978). 1118. V. M. Sehgal, "Some fixed point t h e o r e m s f o r s e t valued directional contraction m a p p i n g s , " Int. J . Math. Math. Sci., 3, No. 3, 455-460 (1980). 1119. V. M. Sehgal and E. M o r r i s o n , ~A fixed point t h e o r e m f o r multifunctions in a locally convex s p a c e , " P r o c . Am. Math. Soc., 38, No. 3, 643-646 (1973). 1120. V. M. Sehgal and R. E. Smithson, ~A fixed point t h e o r e m f o r w e a k directional contraction multifunctions," Math. Jpn., 25, No. 3, 345-348 (1980). 1121. M. Sekanina, "Topologies on s y s t e m s of s u b s e t s , " in: G e n e r a l Topology and Its Relations to Modern Ana l y s i s and Algebra, IV, P r o c . 4th P r a g u e Topology Syrup., 1976, P a r t B, P r a g u e (1977), pp. 420-424. 1122. R. Sentis, " C o n v e r g e n c e de solutions d' ~quations diff~rentielles multivoques," C. R. Acad. Sci. P a r i s , A278, No. 26, 1623-1626 (1974). 1123. R. Sentis, "Equations diff6rentielles a s e c o n d m e m b r e m e s u r a b l e , " C. R. Acad. Sci. P a r i s , 284, No. 2, Al13-A116 (1977). 1124. R. Sentis, "~quations diff~rentielles a second m e m b r e m e s u r a b l e , " Boll. Un. Mat. Ital., B15, No. 3, 724-742 (1978). 1125. W. M. F l e i s c h m a n (ed.), Set-Valued Mappings, Selections and Topological P r o p e r t i e s of 2X, L e c t u r e Notes in Math., No. 171, S p r i n g e r - V e r l a g , Berlin (1970). 1126. A. K. S h a r m a , "Common fixed points of s e t - v a l u e d m a p s , " Bull. Acad. P o l o , . Sci. S e t . Sci. Math., 27, No. 5, 407-412 (1979). 1127. Wei-Hwa Shaw, " B o u n d a r y - v a l u e p r o b l e m s and periodic solutions f o r contingent differential equations, n J. Math. Anal. Appl., 5-7, No. 3, 610-624 (1977). 1128. Ch. Shiau, K.-K. Tan, and Ch. S. Wong, "Quasinonexpansive multivalued m a p s and s e l e c t i o n s , " Fund. Math., 8-7, No. 2, 109-119 (1975). 1129. Ch. Shiau, K.-K. Tan, and Ch. S. Wong, "A c l a s s c~ quasinonexpansive multivalued m a p s , " Can. Math. Bull., 1-8, No. 5, 709-714 (1975). 1099.
787
1130. 1131.
1132. 1133. 1134. 1135. 1136. 1137. 1138. 1139. 1140. 1141. 1142. 1143. 1144. 1145. 1146. 1147. 1148. 1149. 1150. 1151. 1152. 1153. 1154. 1155. 1156. 1157.
1158. 1159. 1160. 1161. 1162. 1163.
788
H. W. Siegberg and G. S. Skordev, "Fixed point index and chain approximations, ~ Univ. Bremen, Report No. 35 (1981). I. Singer, "Duality theorems for constrained convex optimizations," Control Cybernet., 9, No. 1-2, 37-42 (1980). G. S. Skordev, "Fixed-point theorem for multivalued acyclic mappings," Dold. Belg. Akad. Nauk, 2_~7, No. i0, 1319-1321 (1974). G. S. Skordev, "On the invariance of the domain," Dold. Belg. Akad. Nauk, 27, No. ii, 1471-1472 (1974). G. S. Skordev, "On a theorem of Bourgin-Yang," Serdica, I, No. 2, 183-198 (1975). G. S. Skordev, "On a theorem of H. Hopf," Serdica, i, No. 34, 317-325 (1975(1976)). G. S. Skordev, "On the fixed point theorem of Schauder," Serdica,2, No. 2, 122-125 (1976). G. S. Skordev, "A coincidence theorem for Q-simplieial spaces," Dokl. Belg. Akad. Nauk, 32, No. 3, 279280 (1979). D. R. Smart, Fixed Point Theorems, Cambridge Univ. Press, London (1974). C. M. Smith, "An almost fixed point theorem for refluent multifunctions," Bull. Acad. Polo,. Sci. Ser. Sci. Math. Astron. Phys., 23, No. 3, 277-280 (1975). R. E. Smithson, "Some general properties of multivalued functions," Pac. J. Math., 15, No. 2, 681-703 (1965). R. E. Smithson, "Changes of topology and fixed points for multivalued functions," Proc. Am. Math. Soc., 16, No. 3, 448-454 (1965). R. E. Smithson, "On criteria for continuity, w Nieuw Arch. Wisk., 14, No. 2, 89-92 (1966). R. E. Smithson, "A fixed point theorem for connected multivalued functions," Am. Math. Monthly, 73, No. 4, 351-355 (1966). R. E. Smithson, "A characterization of lower semicontmulty, " " " Am. Math. Monthly, 75, No. 5, 505 (1968). R. E. Smithson, "A note on 6 -continuity and p r o x i m a t e fixed points f o r multivalued functions," P r e c . Am. Math. Soc., 2__33,No. 2, 256-260 (1969). R. E. Smithson, "Uniform convergence f o r multifunctions," Pac. J. Math., 39, No. 1, 253-259 (1971). R. E. Smithson, " F i x e d points f o r c o n t r a c t i v e multifunctions," P r e c . Am. Math. See., 27.7, No. 1, 192-194 (1971). R. E. Smithson, " F i x e d points of o r d e r p r e s e r v i n g multifunctions," P r e c . Am. Math. See., 28, No. 1, 304-310 (1971). R. E. Smithson, " F i x e d point t h e o r e m s f o r pseudomonotone multifunctions," Colloq. Math., 24, No. 2, 157162 (1972). R. E. Smithson, "Fixed point theorems for certain classes of multifunctions," Prec. Am. Math. Soc., 31, No. 2, 595-600 (1972). R. E. Smithson, "Multifunctions," Nieuw Arch. Wisk., 20, No. i, 31-53 (1972). R. E. Smithson, "Fixed points in partially ordered sets," Pac. J. Math., 4_~5,No. i, 363-367 (1973). R. E. Smithson, "Subcontinuity for multifunctions," Pac. J. Math., 6_~I,No. i, 283-288 (1975). R. E. Smithson, "Connected and connectivity multifunctions," Prec. Am. Math. Soc., 64, No. I, 146-148 (1977). P. Soardi, "Un'osservazione su un teorema di punto unite per multifunzioni non espansive," Rend. Istit. Lombardo, Aeead. Sci. Lett., A, 106, No. 4, 911-915 (1972). W. Sobieszek, "On the pcint-to-set mappings and functions maximum related to them," Demonstratio Math., 7, No. 4, 483-494 (1974). W. Sobieszek and P. Kowalski, "On the different definitions of the lower semicontinuity, upper semicontinuity, upper semieompacity, closity and continuity of the point-to-set maps," Demonstratio Math., ii, No. 4, 1053-1063 (1978). D. Solow, "Decomposability in fixed point computation with applications and acceleration techniques, ~ J. Math. Anal. Appl., 71, No. 2, 558-579 (1979). W. Spears, "A generalization of the Stone-Weierstrass theorem to multivalued functions," J. Natur. Sei. Math., 9, 201-208 (1969). S. M. Srivastava, "A representation theorem for closed valued multifunetions," Bull. Aead. Polo,. Sci. S6r. Sei. Math., 27, No. 7-8, 511-514 (1979(1980)). S. M. Srivastava, "Selection theorems for G6-valued multifunctions," Trans. Am. Math. Soc., 254, 283293 (1979). S. M. Srivastava, "A representation theorem for G6 -valued multifunetions," Am. J. Math., 102, No. I, 165-178 (1980). S. M. Srivastava, "A representation theorem for closed valued multifunetions," Bull. Aead. Polo,. Sei. S~r. Sci. Math., 27_.~No. 7-8, 511-514 (1979(1980)).
1164. 1165. 1166. 1167.
1168. 1169. 1170. 1171. 1172. 1173. 1174. 1175. 1176. 1177. 1178. 1179. 1180. 1181. 1182. 1183. 1184. 1185. 1186. 1187. 1188. 1189. 1190. 1191. 1192. 1193. 1194.
M. Stanojevic, "On two mappings a s s o c i a t e d with multivalued and p o i n t - v a l u e d mappings,~ in: P r o c . Internat. Syrup. Topology and Appl., Budva, 1972, Belgrade (1973), pp. 229-232. M. Stanojevic, "On multivalued quotient m a p p i n g s , " Publ. Inst. Math., 1_.~7,155-161 {1974). G. I. Stassinopoulos, "Continuous dependence of c o n s t r a i n e d solutions of l i n e a r differential inclusions," J. Inst. Math. Appl., 25, No. 2, 161-175 {1980). G. I. Stassinopoulos and R. B. Vinter, "Continuous dependence of solutions of a differential inclusion on the r i g h t - h a n d side with applications to stability of optimal control p r o b l e m s , " SIAM J . Control Optim., 1_~7, No. 3, 432-449 (1979). G. Stefani and 1~. Zecca, " P r o p e r t i e s of convex s e t s with application to differential theory of multivalued functions," Nonlinear Anal., 2, No. 5, 583-595 (1978). Stetige und m e s s b a r e Selektionen, T a g u n g s b e r . Math. F o r s c h u n g s i n s t . Oberwolfach, M a r c h 14-17, 1979, No. 12, pp. 1-4. W. L. S t r o t h e r , non an open question concerning fixed points, ' Am. Math. Soc., 4, No. 6, 988-992 (1953). W. L. S t r o t h e r , "Multihomotopy, w Duke Math. J . , 2__2,No. 2, 281-285 (1955). W. L. S t r o t h e r , " F i x e d points, fixed sets and M - r e t r a c t s , " Duke Math. J . , 2..~2,No. 4, 551-556 (1955). C. H. Su and V. M. Sehgal, "Some fixed point t h e o r e m s f o r condensing multifunctions in locally convex s p a c e s , " P r o c . Am. Math. Soc., 50, 150-154 (1975). A. Suszycki, nOn extensions of multivalued m a p s , " Bull. Acad. P o l o , . Sci. S6r. Sci. Math., 2.7, No. 2, 183-188 (1979). G. P. Szego, " C l a s s i f i c a t i o n t h e o r e m s f o r dynamical p o l y s y s t e m s , " in: P r o c . 1st Int. Conf. Theory and Appl. Differential G a m e s , A m h e r s t , M a s s . , 1969, S. 1, V I / 1 - V I / 6 . G. P. Szego and G. T r e c c a n i , "Flow without uniqueness n e a r a c o m p a c t s t r o n g l y invariant set," Boll. Un. Mat. Ital., 2, No. 1, 113-124 (1969). G. P. Szego and G. T r e c c a n i , " S e m i g r u p p i di t r a s f o r m a z i e n i multivoche," Lect. Notes Math., 101 (1969). G. P. Szego and G. T r e c c a n i , " S e m i g r u p p i di t r a s f o r m a z i e n i multivoche in I+," in: Syrup. Math., Vol. 6, L o n d o n - N e w York (1971), pp. 287-307. S. Tagawa, Optimierung mit m e n g e n w e r t i g e n Abbildungen, in: HI Syrup. Operat. R e s e a r c h , Sect. I, Operat. R e s e a r c h V e r f a h r e n , 3__.1,Hain, M e i s e n h e i m (1979), pp. 619-629. P. Tallos, "A note on the existence of solutions to multivalued differential equations," Dept. Math., K. M a r x . Univ. Econ., Budapest, 1977, No. 4. P. Tallos, "On Nash e q u i l i b r i u m t r a j e c t o r i e s of multivalued differential equations," Ann. Univ. Sci. Budapest, Sect. Math., 22-23, 235-242 (1979-1980). L . A . T a l m a n , " F i x e d points f o r condensing multifunctions in m e t r i c s p a c e s with convex s t r u c t u r e , " Kodai Math. Sere. Rep., 2__~9,No. 1-2, 62-70 (1977). K.-K. Tan and C.-S. Wong, "On s o m e topological p r o b l e m s a r i s i n g f r o m m a p s of contractive type," P r o c . Am. Math. Soc., 6-5, No. 1, 167-172 (1977). E. T a r a f d a r and T. Husain, "Duality in fixed point theory of multivalued mappings with applications," J. Math. Anal. Appl., 6__33,No. 2, 371-376 (1978). E. T a r a f d a r and Suat Khoh Teo, nOn the e x i s t e n c e of solutions of the equation LxENx and a coincidence d e g r e e t h e o r y , " J. A u s t r a l . Math. Soc., A28, No. 2, 139-173 (1979). E. T a r a f d a r and R. Vyborny, " F i x e d point t h e o r e m s f o r condensing multivalued mappings on a locally convex topological s p a c e , " Bull. A u s t r a l . Math. Soc., 1..22, No. 2, 161-170 (1975). S. A. T a r s i , "Sull' estensione della t e o r i a di C a r a t h e o d o r y ai s i s t e m i differenziali multivochi," Rend. Istit. L o m b a r d o , Accad. Sci. Lett., A105, No. 3, 563-570 {1971). S. A. T a r s i , "Sull' e s t e n s i o n s della t e o r i a di C a r a t h e o d o r y ai s i s t e m i differenziali multivoehi, con condizioni ai limiti non l i n e a r i , ' Rend. Istit. L o m b a r d o , Accad. Sci. Lett., A106, No. 2, 326-341 (1972). F. T e r k e l s e n , "A s h o r t proof of F a n ' s fixed point t h e o r e m , " P r o c . Am. Math. Soc., 42, No. 2, 643-644 {1974). J. T h o m a s , " ~ - C o h e r e n t multifunctions," Boll. Un. Mat. Ital., 4, No. 2, 232-238 (1971). M. J. Todd, "The computation of fixed points and applications," Lect. Notes Econ. Math. Syst., 124, V]/ (1976). G. Treccani, "Stabilita asintotica dei semigruppi di transformazioni multivoche," Boll. Un. Mat. Ital., 3, No. 4, 646-655 (1970). G. Treccani, "Proprieta delle fanzioni reali come applicazione della teoria dei semigruppi di trasformazioni multivoche, ~ Boll. Un. Mat. Ital., 4, No. i, 123-131 (1971). M. Tucci, "Su un problema di selezione continua di un'applicazione plurivoca ad n valori non tutti distinti, ~ Rend. Accad. Sci. Fis. Mat. Soc. Naz. Sci. Lett. Arti Napoli, 4-5, 292-301 (1978(1979)).
789
1195. M. Tucci, " T e o r e m i di selezione continua p e r applicazione plurivoche i cui valori sono unione di uno s t e s s o n u m e r o n di i n s i e m i eehiusi e e o n v e s s i , " Rend. Aecad. Sci. F i s . Mat. Soc. Naz. Sci. Left. Arti Napoli, 46, 547-561 (1979(1980)). 1196. Tu Le Van, "On multifunctions and m u l t i m e a s u r e s , " Bull. A u s t r a l . Math. Soc., 18, No. 1, 145-146 (1978). 1197. R. Urbanski, nOn s e t valued m e a s u r e s and s e t c o r r e s p o n d e n c e s , " Funct. A p p r o x i m a t i o C o m m e n t . Math., 3, 259-269 (1976). 1198. C. U r s e s c u , "Multifunctions with convex c l o s e d g r a p h , " C z e c h o s l o v a k Math. J., 25, No. 3, 438-441 (1975). 1199. M. V a l a d i e r , "Sur 1'integration d ' e n s e m b l e s convexes c o m p a c t s en dimension infinie," C. R. Acad. Sci. P a r i s , 266, No. 1, A14-A16 (1968). 1200. M. Valadier, " I n t e g r a t i o n de c o , vexes f e r m ~ s n o t a m m e n t d ' ~ p i g r a p h e s . Inf-convolution continue," Rev. F r a n c a i s e I n f o r m a t . R e c h e r c h e Op~rationelle, Ser. R-2, 4, 57-73 (1970). 1201. M. Valadier, "Multiapplications m e s u r a b l e s a v a l e u r s convexes c o m p a c t e s , " ft. Math. P u r e s Appl., 50, No. 3, 265-297 (1971). 1202. M. V a l a d i e r , " E x i s t e n c e globale p o u r Ies 6quations diff~rentielles multivoques," C. R. Acad. Sci. P a r i s , 272, No. 7, A474-477 (1971). 1203. M. Valadier, "Int6grandes s u r les l o c a l e m e n t c o , v e x e s sousliniens," C. R. Acad. Sci. P a r i s , 276, No. 9, A693-A695 (1973). 1204. M. Valadier, "Sur l ' e s p ~ r a n c e conditionelle multivoque non convexe," Ann. Inst. H. Poincar~, B16, No. 2, 109-116 (1980). 1205. B. Van Cuhsem, " E s p ~ r a n c e conditionelle d'une multiapplication a v a l e u r s c o , v e x e s c o m p a c t e s , " C. R. Acad. Sci. P a r i s , 269, No. 4, A211-A214 (1969). 1206. B. Van Cntsem, " M a r t i n g a l e s de multiapplications a v a l e u r s convexes c o m p a c t e s , " C. R. Acad. Sci. P a r i s , 269., No. 11, A429-A432 (1969). 1207. A. Vanderbauwhede, "On a modified d e g r e e theory f o r multivalued mappings," Simon Stevin, 50, No. 2, 65-86 (1976-1977). 1208. T . V a n d e r W a l t , " F i x e d and a l m o s t fixed points," Matak. Centre T r a c t s , No. 1, A m s t e r d a m (1963). 1209. P. P. V a r a i y a , "On the t r a j e c t o r i e s of a differential s y s t e m , " in: M a t h e m a t i c a l Theory of Control, Academic P r e s s , New York (1967), pp. 115-128. 1210. P. P. V a r a i y a , "On the existence of solutions to a differential g a m e , " SIAM J. Control, 5, No. 1, 153-162 (1967). 1211. L. V i e t o r i s , Monat. Math. Ph., 31, 173-204 (1921). 1212. L. V i e t o r i s , " U b e r den hoheren Z u s a m m e n h a n g k o m p a k t e r Raume und eine K l a s s e yon z u s a m m e n h a n g s t r e u e n Abbildangen," Math. Ann., 97, 454-472 (1927). 1213. R. Voreadou, "Sur c e r t a i n e s propri~t~s des fonctions quelconques e n t r e ~spaces topologiques," Math. Balkanica, 2, 300-313 (1972). 1214. I. I. V r a b i e , wLe p r o b l e m e du t e m p s optimal p o u r des 6quations diff~rentielles multivoques," C. R. Acad. Sci. P a r i s , 285, No. 14, A907-A909 (1977). 1215. I. I. V r a b i e , " T i m e optimal control f o r contingent equations in H i l b e r t s p a c e s , " An. Stiint. Univ. "A1. I. Cuza," Sec. I a, 24, No. 1, 125-133 (1978). 1216. I. Vrkoc, "A new definition and s o m e modifications of Filippov cone, w Lect. Notes Math., No. 703, 433441 (1979). 1217. I. Vrkoe, " S c o r z a - D r a g o n i p r o p e r t y of Filippov mappings," Tohoku Math. J . , 32, No. 2, 309-316 (1980). 1218. D. H. Wagner, " I n t e g r a l of a convex-hull-valued function," J. Math. Anal. Appl., 50, No. 3, 548-559 (1975). 1219. D. H. Wagner, "Survey of m e a s u r a b l e s e l e c t i o n t h e o r e m s : an update," Lect. Notes Math., No. 794, 176219 (1980). 1220. D. H. Wagner, " I n t e g r a l of a s e t - v a l u e d function with s e m i c l o s e d values," J. Math. Anal. Appl., 5__~5,No. 3, 616-633 (1976). 1221. A. D. Wallace, "A fixed point t h e o r e m f o r t r e e s , " Bull. Am. Math. Soc., 4-7, 757-760 (1941). 1222. L. E. Ward, fir., "A fixed point t h e o r e m , " Am. Math. Monthly, 6-5, No. 4, 271-272 (1958). 1223. L. E. Ward, J r . , ' A fixed point t h e o r e m f o r multivalued functions,,, P a c . J. Math., 8, No. 4, 921-927 (1958). 1224. L. E. Ward, J r . , " C h a r a c t e r i z a t i o n of the fixed point p r o p e r t y for a c l a s s of s e t - v a l u e d mappings,,, Fund. Math., 50, No. 2, 159-164 (1961). 1225. L. E. Ward, J r . , "A g e n e r a l fixed point t h e o r e m , " Colloq. Math., 1._~5,No. 2, 243-251 (1966). 1226. L. E. Ward~ J r . , " S e t - v a l u e d m a p p i n g s on p a r t i a l l y o r d e r e d s p a c e s , " Lect. Notes Math., 171, 88-99 (1970). 1227. ft. Warga, "An implicit function t h e o r e m without differemtiability, Pl~oc. Am. Math. Soc., 69, No. 1, 65-69 (1978). 1228. T. Wazewski, " S y s t e m e s de c o m m o d e et 6quations au contingent," Bull. Acad. P o l o , . Sei. S6r. Sci. Math. A s t r o n o m . Phys., 9, No. 3, 151-155 (1961).
790
1229. T. Wazewski, "Sur une condition ~quivalente a l'~quation au contingent,' Bull. Acad. P o l o , . Sci. S~r. Sci Math. Astron. Phys., 9, No. 12, 865-867 (1961). 1230. T. Wazewski, "Sur une g~n6ralisation de la notion des solutions d'uae 6quation au c o n t i n g e n t , " Bull. Acad. P o l o , . Sci. S6r. Sci. Math. Astron. Phys., 10, No. 1, 11-15 (1962). 1231. T. Wazewski, "On an optimal control p r o b l e m , ' in: P r o c . Conf. Differential Equations and T h e i r Applications, P r a g u e (1962), pp. 229-242. 1232. J. R. L. Webb, "On d e g r e e t h e o r y f o r multivalued mappings and applications," Boll. Un. Mat. Ital., 9, No. 1, 137-158 (1974). 1233. J. R. L. Webb, "On uniqueness of topological degree f o r set valued mappings," P r o c . R. Soc. Edinburgh, A74, 225-229 (1976). 1234. R. Wegmann, " D e r W e r t e b e r e i c h yon V e k t o r i n t e g r a l e n , " Z. W a h r s c h . Verw. Gebiete, 14, No. 3, 203-238 (1970). 1235. S. W e r t h e i m e r , " Q u a s i c o m p a e t n e s s and decompositions f o r a r b i t r a r y r e l a t i o n s , " P a c . J . Math., 3.7, No. 1, 253-263 (1971). 1236. P. O. Wheatley, " I n v e r s e l i m i t s and the fixed point p r o p e r t y f o r s e t - v a l u e d m a p s , " Notices Am. Math. Soc., 18, No. 3, 546 (1971) (Abstract 685-G3). 1237. G. T. Whyburn, "Continuity of multiftmctions," P r o c . Nat. Acad. Sci. U.S.A., 5-4, No. 6, 1494-1501 (1965). 1238. G. T. Whyburn, " R e t r a c t i n g multffunctions," P r o c . Nat. Acad. Sci. U.S.A., 59, No. 2, 343-348 (1968). 1239. R. K. W i l l i a m s , "Some r e s u l t s on expansive mappings," P r o c . Am. Math. Soc., 26, No. 4, 655-663 (1970). 1240. R. K. W i l l i a m s , " F u r t h e r r e s u l t s on expansive m a p p i n g s , " P r o c . Am. Math. Soc., 5..~8, 284-288 (1976). 1241. S. A. W i l l i a m s , "An index f o r s e t - v a l u e d m a p s in infinite-dimensional s p a c e s , " P r o c . Am. Math. Soc., 31, No. 2, 557-563 (1972). 1242. O. Wyler, "A c h a r a c t e r i z a t i o n of r e g u l a r i t y in topology," P r o c . Am. Math. Soc., 2.~9, No. 3, 588-590 (1972). 1243. S. Y a m a s h i t a , WSome t h e o r e m s on c l u s t e r s e t s of s e t m a p p i n g s , " P r o c . Jpn. Acad., 46, No. 1, 30-32 (1970). 1244. Chi-Lin Yen, NOn F. E. B r o w d e r ' s t h e o r e m , " Tamkang J. Math., 9, No. 2, 223-227 (1978). 1245. G. Young, "Continuous s e l e c t i o n s on locally c o m p a c t s e p a r a b l e m e t r i c s p a c e s , " L e c t . Notes Math., 17.1., 102-110 (1970). 1246. S. K. Z a r e m b a , "Sur une extension de la notion dTequatioa differentielle, tl C. R. Acad. Sci. P a r i s , 199, No. 10, 545-548 (1934). 1247. S. K. Z a r e m b a , "O r o w n a m a c h p a r a t y n g e n s o w y c h , " Dodatek do Rocznika Polskiego T o w a z z y s t w a M a t e m a t y c z n e g o , 9 (1935). 1248. S. K. Z a r e m b a , ' S u r les equations au p a r a t i n g e n t , " Bull. Sci. Math., 6__9.0,No. 2, 139-160 (1936). 1249. P. Zecca, TVSoluzioni periodiche p e r unVequazione dVevoluzione multivoca, ~ Boll. Un. Mat. Ital., A15, No. 1, 140-146 (]978). 1250. P. Z e c c a and P. Zezza, "Nonlinear boundary-value p r o b l e m s in Banach s p a c e s f o r multivalued d i f f e r ential equations on a n o , c o m p a c t interval," Nonlinear Anal., 3, No. 3, 347-352 (1979). 1251. T. Zolezzi, "Well p o s e d optimization p r o b l e m s f o r integral functionals," Publ. Istit. Mat. Univ. Genova, No. 241 (1978). 1252. W. Zygmunt, "On the full solution of the paratingent equations," Ann. UMCS, A26, 103-108 (1972). 1253. W. Zygmunt, "On a c e r t a i n paratingent equation with a deviated a r g u m e n t , " Ann. UMCS, A28, 127-135 (1974). 1254. W. Zygmunt, "On s o m e p r o p e r t i e s of a c e r t a i n f a m i l y of solutions of the p a r a t i n g e n t equation, ~ Ann. UMCS, A2_~8, 137-141 (1974).
791
