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Approximate Selection Theorems and Their Applications
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
212, 88]97 Ž1997.
AY975466
Approximate Selection Theorems and Their Applications* Xiyin Zheng Department of Mathematics, Yunnan Uni¨ ersity, Kunming, 650091, People’s Republic of China Submitted by Ying-ming Liu Received September 16, 1996
We prove three approximate selection theorems and give an improved version of the Michael selection theorem. As their applications, new fixed point theorems and Q 1997 Academic Press equilibrium theorems for generalized games are established.
1. INTRODUCTION The Michael selection theorem is a useful tool in nonlinear analysis which can be stated as follows: THEOREM A ŽMichael w6x.. Let X be a metric space, Y a Banach space, and let F: X § Y be a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues. Then F has a continuous selection Ž i.e., there is a continuous function f : X ª Y such that f Ž x . g F Ž x . for each x g X .. Recently, several authors w2, 5, 8, 9x investigated selection theorems in topological linear spaces. The following result, proved by Wu and Shen w8x, might be the most general form among existing selection theorems in topological linear spaces. THEOREM B. Let X be a paracompact topological space and Y be a topological linear space. Suppose that S, T : X § Y are set-¨ alued mappings such that Ž1. Ž2. Fx g NŽ x .
For each x g X, SŽ x . is nonempty and co SŽ x . ; T Ž x .. For each x g X, there is an open neighborhood N Ž x . of x such that SŽ x . / B.
Then T has a continuous selection. * This research is supported by the Science Foundation of Yunnan Province, China. 88 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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If X is connected and S is a point-valued mapping, it is easy to verify that the condition Ž2. in Theorem B implies that S is a constant mapping. Hence the supposition in Theorem B is strong. In this paper, we prove an approximate selection theorem under a suitable condition and establish approximate selection theorems with constraint. Moreover, we give an improved version of the famous Michael selection theorem. As their applications, new fixed point theorems and equilibrium theorems of generalized games are proved.
2. APPROXIMATE SELECTION THEOREMS In this paper, all topological spaces are assumed to be Hausdorff, and a subset of a topological space is considered to have the relative topology. Let X and Y be two topological spaces. A set-valued mapping F: X § Y is called upper semicontinuous Žlower semicontinuous. if the set x g X; F Ž x . ; V 4 Ž x g X; F Ž x . l V / B4. is open in X for every open subset V of Y. A set-valued mapping F: X § Y is called closed if graphŽ F . s Ž x, y . g X = Y; y g F Ž x .4 is closed in X = Y. For a set A in a topological linear space, we denote its convex hull and closure by co A and cl A, respectively. DEFINITION 2.1. Let X be a topological space and Y be a topological linear space. A set-valued mapping F: X § Y is called sub-lower semicontinuous if for each x g X and each neighborhood V of 0 in Y there is z g F Ž x . and a neighborhood UŽ x . of x in X such that for each y g UŽ x ., z g F Ž y . q V. It is clear that the sub-lower semicontinuity is weaker than the lower semicontinuity. DEFINITION 2.2. Let X be a topological space, Y be a topological linear space, and let F and T be two set-valued mappings from X into Y. We say F and T are topologically separated if for each x g X there is a neighborhood V of 0 in Y and a neighborhood UŽ x . of x in X such that F ŽUŽ x .. l ŽT Ž x . q V . s B. THEOREM 2.1. Let X be a paracompact topological space and Y be a locally con¨ ex topological linear space, and let F: X § Y be a set-¨ alued mapping with con¨ ex ¨ alues. Then F is sub-lower semicontinuous iff for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V. Proof. Suppose that F is sub-lower semicontinuous. For each neighborhood V of 0 in Y, there is a convex neighborhood V0 of 0 in Y such that
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V0 ; V Žsince Y is a locally convex topological linear space.. For each x g X, let z x g F Ž x . and UŽ x . be an open neighborhood of x in X such that for each y g UŽ x ., z x g F Ž y . q V0 . It is clear that UŽ x .4x g X is an open covering of X. Since X is paracompact, there is a locally finite open refinement Oi 4i g I of UŽ x .4x g X . Hence for each i g I there is x Ž i . g X such that Oi ; UŽ x Ž i ... Let pi 4i g I be a partition of unity subordinated to Oi 4i g I . The function f : X ª Y defined by f Ž x. s
Ý pi Ž x . z xŽ i. ,
x g X,
igI
is continuous since it is locally a finite sum of continuous functions. For each x g X, pi Ž x . / 0 implies x g Oi ; UŽ x Ž i .., and so z xŽ i. g F Ž x . q V0 . Since F Ž x . q V0 is convex, f Ž x . g F Ž x . q V0 ; F Ž x . q V. Conversely, suppose that for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V. Then for each x g X there is a neighborhood UŽ x . of x in X and z x g F Ž x . such that z x g f Ž x . y V and for each y g UŽ x ., f Ž x . g f Ž y . q V. Therefore, for each y g UŽ x ., z x g F Ž y . q V q V y V. Clearly this implies that F is sub-lower semicontinuous. THEOREM 2.2. Let X be a compact topological space and Y be a locally con¨ ex topological linear space. Assume that F, T : X § Y are two set-¨ alued mappings with the following properties: Ži. F and T are topologically separated. Žii. T is upper semicontinuous. Žiii. F is sub-lower semicontinuous, and F Ž x . is a nonempty con¨ ex set for each x g X. Then for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V and f Ž x . f T Ž x .. Proof. By Ži., for each x g X there is a neighborhood UŽ x . of x in X and a balanced convex neighborhood V Ž x . ; V of 0 in Y such that for each y g UŽ x ., F Ž y . l ŽT Ž x . q V Ž x .. s B. Again by Žii., there is an open neighborhood O Ž x . ; UŽ x . of x in X such that for each y g O Ž x ., T Ž y . ; T Ž x . q 12 V Ž x .. Hence for each y g O Ž x ., F Ž y . l ŽT Ž y . q 12 V Ž x .. s B, that is, Ž F Ž y . y 12 V Ž x .. l T Ž y . s B. By the compactness of X, there is n finitely many points in X, say x 1 , . . . , x n , such that X s Dks1 O Ž x k .. Let 1 n W s F ks 1 y 2 V Ž x k ., then W ; V is a neighborhood of 0 in Y and for each x g X, Ž F Ž x . q W . l T Ž x . s B. By Theorem 2.1, there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q W ; F Ž x . q V and f Ž x . f T Ž x ..
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THEOREM 2.3. Let X be a paracompact topological space and Y be a norm linear space. Assume that F, T : X § Y are two set-¨ alued mappings with the following properties: Ž i . F and T are topologically separated. Žii. T is upper semicontinuous. Žiii. F is sub-lower semicontinuous, and F Ž x . is a nonempty con¨ ex set for each x g X. Then for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V and f Ž x . f T Ž x .. Proof. By the assumption, for each x g X there is an open neighborhood UŽ x . of x in X, e Ž x . ) 0, and z x g F Ž x . such that for each y g UŽ x .,
Ž F Ž y . q B Ž 0, e Ž x . . . l T Ž y . s B and z x g F Ž y . q B Ž 0, e Ž x . . , where B Ž0, e Ž x .. s z g Y; 5 z 5 - e Ž x .4 . Without loss of generality, we can assume that for each x g X, B Ž0, e Ž x .. ; V. For each x g X, let
d Ž x . s sup r ) 0; Ž F Ž x . q B Ž 0, r . . l T Ž x . s B and B Ž 0, r . ; V 4 . It is clear that for each x g X, Ž F Ž x . q B Ž0, d Ž x ... l T Ž x . s B, B Ž0, d Ž x .. ; V and for each y g UŽ x ., d Ž y . G e Ž x .. Since X is paracompact and UŽ x .4x g X is an open covering of X, there is a locally finite open refinement Oi 4i g I of UŽ x .4x g X . Hence for each i g I there is x Ž i . g X such that Oi ; UŽ x Ž i ... Let pi 4i g I be a partition of unity subordinated to Oi 4i g I . Define f : X ª Y by f Ž x. s
Ý pi Ž x . z xŽ i. ,
x g X.
igI
Then f is a continuous function. For each x g X, pi Ž x . / 0 implies x g Oi ; UŽ x Ž i .., and so z xŽ i. g F Ž x . q B Ž0, e Ž x Ž i ... ; F Ž x . q B Ž0, d Ž x ... Hence for each x g X, f Ž x . g F Ž x . q B Ž0, d Ž x .. ; F Ž x . q V Žsince F Ž x . q B Ž0, d Ž x .. is convex. and f Ž x . f T Ž x .. The following proposition gives a sufficient condition to guarantee that F: X § Y and T : X § Y are topologically separated.
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PROPOSITION 2.1. Let X be a topological space and Y be a topological linear space. Assume that F, T : X § Y are two set-¨ alued mappings with the following properties: Ž i . F is upper semicontinuous. Žii. For each x g X, F Ž x . is closed, T Ž x . is compact, and F Ž x . l T Ž x . s B. Then F and T are topologically separated. Proof. By Žii. and Theorem 1.10 in w7x, for each x g X there is a neighborhood Vx of 0 in Y such that Ž F Ž x . q Vx . l ŽT Ž x . q Vx . s B. By Ži., for each x g X there is a neighborhood UŽ x . of x in X such that for each y g UŽ x ., F Ž y . ; F Ž x . q Vx , and so F ŽUŽ x .. l ŽT Ž x . q Vx . s B. The following theorem is an improved version of the Michael selection theorem. THEOREM 2.4. Let X be a paracompact topological space, Y a normed linear space, and let F: X § Y be a lower semicontinuous set-¨ alued mapping such that for each x g X, F Ž x . is a nonempty, complete, closed, and con¨ ex set. Then there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x .. Proof. We claim that there is a sequence of continuous functions f n : X ª Y with the following properties: Ža. For each x g X, f nŽ x . g F Ž x . q B Ž0, 2yn .. Žb. For each x g X, 5 f nŽ x . y f ny1Ž x .5 - 2yn q2 . For n s 1, we apply Theorem 2.1 with V s B Ž0, 2y1 .. Assume that we have constructed the mappings f k up to n and let us construct f nq1. Define G: X § Y by GŽ x . s F Ž x . l B Ž f nŽ x ., 2yn ., x g X. It is clear that G is a lower semicontinuous mapping with nonempty convex values. By Theorem 2.1, there is a continuous function f nq 1: X ª Y such that for each x g X, f nq 1 Ž x . g Ž F Ž x . l B Ž f n Ž x . , 2yn . . q B Ž 0, 2yn y1 . . Therefore, for each x g X we have f nq 1 Ž x . g F Ž x . q B Ž 0, 2yn y1 . f nq 1 Ž x . y f n Ž x . - 2yn q 2yny1 - 2ynq1 . This proves our claim. By Ža., we can define a function g n : X § Y such that for each x g X, Žc.
g nŽ x . g F Ž x . and 5 f nŽ x . y g nŽ x .5 - 2yn .
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This and Žb. imply that for each x g X, 5 g nq 1Ž x . y g nŽ x .5 - 2yn q3 , and so for each natural number p, 5 g nq p Ž x . y g nŽ x .5 - 2yn q4 . Since each F Ž x . is complete, there is a function f : X ª Y such that g n4 converges pointwise to f and f Ž x . g F Ž x . and 5 f Ž x . y g nŽ x .5 F 2yn q4 for all x g X. This and Žc. imply that f n4 uniformly converges to f on X. Hence f is a continuous function. Kakutani and Fan’s fixed point theorem is important in nonlinear analysis which asserts: Let C be a nonempty compact convex subset of a locally convex topological linear space, and let F: C § C be an upper semicontinuous set-valued mapping with nonempty closed convex values. Then F has a fixed point. Himmelberg w4x gave an improved version of Kakutani and Fan’s fixed point theorem which says: Let C be a nonempty convex subset of a locally convex topological linear space, D be a compact subset of C, and let F: C § D be an upper semicontinuous set-valued mapping with nonempty closed convex values. Then F has a fixed point. The following fixed point theorem, in which the set-valued mapping F is lower semicontinuous instead of upper semicontinuous, supplements Kakutani and Fan’s fixed point theorem and Himmelberg’s fixed point theorem. THEOREM 2.5. Let C be a con¨ ex subset of a normed linear space, D a compact subset of C, and let F: C § D be a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues. Then F has a fixed point. Proof. Notice that a metric space is paracompact. By Theorem 2.4, there is a continuous function f : C ª D such that for each x g C, f Ž x . g F Ž x .. By Himmelberg’s fixed point theorem, there is x* g C such that x* s f Ž x*. g F Ž x*.. 3. THE EXISTENCE OF EQUILIBRIUM FOR GENERALIZED GAMES Let I be an index set. For each i g I, let X i be a nonempty subset of a topological linear space Yi . Throughout the section, we denote Ł i g I X i by X. Let A i , Bi , Pi : X § X i be set-valued mappings. A generalized game Žor an abstract economy. G s Ž X i , A i , Bi , Pi . i g I is defined as a family of ordered quadruples Ž X i , A i , Bi , Pi .. For each i g I, let p i be the projection mapping from X onto X i . For each x g X, we denote p i Ž x . by x i . An equilibrium for the generalized game G is a point x* g X such that for each i g I, xUi g cl Bi Ž x*. and A i Ž x*. l Pi Ž x*. s B where cl Bi is a set-valued mapping from X into X i such that for each x g X, cl Bi Ž x . is the closure of Bi Ž x ..
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DEFINITION 3.1. Let Ci be a subset of X. A i : X § X i is said to have the Ž e-SLC.-property on Ci , if for each convex neighborhood V of 0 in Yi , there is a sub-lower semicontinuous set-valued mapping TiV : Ci § X i with nonempty convex values such that Ti and p i are topologically separated on Ci and TiV Ž x . ; A i Ž x . q V for all x g Ci . Clearly, the Ž e-SLC.-property is weaker than the Ž e-CS.-property in w10x. DEFINITION 3.2. Let Ci be a subset of X. A i : X § X i is said to have the ŽLCS.-property on Ci , if there is a lower semicontinuous set-valued mapping Ti : Ci § X i with nonempty closed convex values such that Ti Ž x . ; A i Ž x . and x i f Ti Ž x . for all x g Ci . LEMMA 3.1. Let Yi 4i g I be a family of locally con¨ ex topological linear spaces. Then there is a directed set J such that for each i g I, Yi has a base Vi j 4j g J of neighborhoods of 0 with the following properties: Ž18. Ž28.
for each j g J, Vi j is closed and con¨ ex. Vi j 2 ; Vi j1 whene¨ er j1 F j2 in J.
Proof. For each i g I, let B i Ž0. be the base of all closed and convex neighborhoods of 0 in Yi , and J s Ł i g I Wi ; Wi g B i Ž0. for all i and Wi s Yi but finitely many i4 . Then J is a base of neighborhoods of 0 in Ł i g I Yi with respect to the product topology. Define a order F on J as follows: for arbitrary j1 , j2 g J, j1 F j2 if j2 ; j1. It is clear that J is a directed set with respect to F. For each i g I and j g J, let Vi j s p i Ž j .. It is clear that for each i g I, Vi j 4j g J is a base of neighborhoods of 0 in Yi , Vi j is closed and convex for all j g J, and Vi j 2 ; Vi j1 whenever j1 F j2 in J. LEMMA 3.2. Let X be a topological space, Y be a topological linear space, and let A: X § Y be an upper semicontinuous set-¨ alued mapping with compact ¨ alues. Assume that C ; Y and K ; Y are closed and compact, respecti¨ ely. Then T : X § Y defined by T Ž x . s Ž AŽ x . q C . l K for all x g X is upper semicontinuous. Proof. Let F Ž x . s AŽ x . q C and GŽ x . s K for all x g X. It is clear that T Ž x . s F Ž x . l GŽ x . for all x g X. By Theorem 8 in w1, p. 110x, it suffices to show that F is closed. Suppose that F is not closed, then there is x 0 g X and y 0 g Y such that Ž x 0 , y 0 . g cl GraphŽ F . and y 0 f F Ž x 0 . s AŽ x 0 . q C. Since AŽ x 0 . is compact and C is closed, AŽ x 0 . q C is closed. Hence there is a neighborhood V of 0 in Y such that Ž y 0 q V . l Ž AŽ x 0 . q C q V . s B Žsee Theorem 1.10 in w7x.. Since A is upper semicontinuous, there is a neighborhood U of x 0 in X such that for all x g U, AŽ x . ; AŽ x 0 . q V, and so F Ž x . s AŽ x . q C ; AŽ x 0 . q C q V. By Ž x 0 , y 0 . g cl GraphŽ F ., there is a net xa 4a g D in X and a net ya 4a g D in Y such that xa ª x 0 , ya ª y 0 and for all a g D, ya g F Ž xa .. It follows
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that there is an index a 0 g D such that xa g U whenever a G a 0 . Hence ya g F Ž xa . ; AŽ x 0 . q C q V, and so ya f y 0 q V, whenever a G a 0 . This contradicts ya ª y 0 . THEOREM 3.1. The generalized game G s Ž X i , A i , Bi , Pi . i g I admits an equilibrium if for each i g I: Ži. X i is a nonempty, compact con¨ ex subset of a locally con¨ ex topological linear space Yi . Žii. cl Bi is an upper semicontinuous set-¨ alued mapping with nonempty con¨ ex ¨ alues. Žiii. Bi has the Ž e-SLC .-property on a closed set Ci with x g X; A i Ž x . l Pi Ž x . / B4 ; int X Ž Ci . Ž where int X Ž Ci . is the relati¨ e interior of Ci in X .. Proof. Let J and Vi j 4j g J be as in Lemma 3.1. By Ži. and Tychonoff’s theorem, X s Ł i g I X i is a compact convex set. Hence Ci is compact for all i g I. By Žiii. and Theorem 2.2, for each j g J, there is a continuous function f i j : Ci ª X i such that for each x g Ci f i j Ž x . g Ž Bi Ž x . q Vi j . l X i
and
fi j Ž x . / x i .
Ž 1.
Define a set-valued mapping Fi j : X § X i by Fi j Ž x . s
½
fi j Ž x . 4 , Ž cl Bi Ž x . q Vi j . l X i ,
if x g int X Ž Ci . if x g X _int X Ž Ci . .
Notice that int X Ž Ci . is an open subset of X. By Žii. and Lemma 3.2, it is easy to verify that Fi j is upper semicontinuous. For each j g J, let Fj : X § X such that Fj Ž x . s Ł i g I Fi j Ž x . for all x g X. By Lemma 3 in w3x, Fj is an upper semicontinuous set-valued mapping with closed convex values. By Kakutani and Fan’s fixed point theorem, there is x j g X such that x j g Fj Ž x j .. We claim that x j g X _Di g I int X Ž Ci .. Indeed, if this is not true, then there is i 0 g I such that x j g int X Ž Ci 0 .. By Ž1. and the definition of Fi j , x ij0 s p i 0Ž x j . f f i 0 j Ž x j .4 s Fi 0 j Ž x j .. This implies x j f Fj Ž x j ., a contradiction. Hence for each j g J, xjg
Ł Ž cl Bi Ž x j . q Vi j . .
Ž 2.
igI
Since J is a directed set, x j 4j g J is a net in the compact subset X _Di g I int X Ž Ci . of X. Hence there is a subnet of x j 4j g J to converge to some x* in X _Di g I int X Ž Ci .. Without loss of generality, we can assume lim x j s x*. j
Ž 3.
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XIYIN ZHENG
This is equal to lim x ij s xUi , j
for each i g I.
By Ž2., for each i g I and each j g J, x ij g cl Bi Ž x j . q Vi j . Hence there is yi j g cl Bi Ž x j . and ¨ i j g Vi j such that x ij s yi j q ¨ i j . Since Vi j 4j g J is a base of neighborhoods of 0 in Yi for all i g I, lim j ¨ i j s 0. Hence for each i g I lim yi j s lim x ij y lim ¨ i j s xUi . j
j
j
Ž 4.
By proposition 7 in w1, p. 110x, for each i g I, cl B is closed. By Ž3. and Ž4., xUi g cl B Ž x*. for all i g I. By x* g X _Di g I int X Ž Ci . and Žiii., A i Ž x*. l Pi Ž x*. s B for all i g I. The above proof uses the idea of the proof of Theorem 5 in w10x. It is clear that Theorem 3.1 is a generalization of Theorem 5 in w10x. THEOREM 3.2. The generalized game G s Ž X i , A i , Bi , Pi . i g I admits an equilibrium if for each i g I: Ži. X i is a nonempty con¨ ex subset of a normed linear space Yi . Žii. cl Bi is a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues, and cl Bi Ž X . is contained in a compact subset Hi of X i . Žiii. cl Bi has the Ž LCS .-property on a closed subset Ci of X containing the set x g X; A i Ž x . l Pi Ž x . / B4 . Proof. Since cl Bi has the LCS-property on Ci , there is a lower semicontinuous set-valued mapping Ti : Ci § X i with nonempty closed convex values such that Ti Ž x . ; cl Bi Ž x . and x i f Ti Ž x . for all x g Ci . Define a set-valued mapping Fi from X into X i by Fi Ž x . s
½
Ti Ž x . ,
if x g Ci
cl Bi Ž x . ,
if x g X _Ci .
Since Ci is a closed subset of X, it is easy to verify that Fi is lower semicontinuous and Fi Ž x . is a closed convex set for each x g X. By Theorem 2.5, there is a continuous function f i : X ª X i such that for each xgX f i Ž x . g Fi Ž x . ; cl Bi Ž x . .
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Define a continuous function f from X into a compact subset Ł i g I Hi such that p i Ž f Ž x .. s f i Ž x . for all x g X and all i g I. Since f i is continuous for all i g I, f is continuous. By Himmelberg’s fixed point theorem, there is x* g X such that x* s f Ž x*.. Hence xUi s f i Ž x*. g cl Bi Ž x*.. This and x i f Ti Ž x . and f i Ž x . g Fi Ž x . s Ti Ž x . for all x g Ci imply x* f Ci . By Žiii., A i Ž x*. l Pi Ž x*. s B. In Theorem 3.2, setting Pi Ž x . s B for all i g I and all x g X, we have the following fixed point result. COROLLARY 3.1. For each i g I, let X i be a con¨ ex subset of a normed linear space and Hi be a compact subset of X i , and let Fi : X s Ł i g I X i § Hi be a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues. Then there is x* g X such that xUi g Fi Ž x*. for all i g I. REFERENCES 1. J. P. Aubin and I. Ekeland, ‘‘Applied Nonlinear Analysis,’’ Wiley]Interscience, New York, 1984. 2. X. P. Ding, W. K. Kim, and K. K. Tan, A selection theorem and its applications, Bull. Austral. Math. Soc. 46 Ž1992., 205]212. 3. K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 Ž1952., 121]126. 4. C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 Ž1972., 205]207. 5. T. Husain and E. Tarafdar, A selection theorem and a fixed point theorem and an equilibrium point of an abstract economy, Internat. J. Math. Math. Sci. 18 Ž1995., 179]184. 6. E. Michael, Continuous selections, I, Ann. of Math. 63 Ž1956., 361]382. 7. W. Rudin, ‘‘Functional Analysis,’’ McGraw]Hill, New York, 1973. 8. X. Wu and S. K. Shen, A further generalization of Yannelis-Prabhakar’s continuous selection theorem and its applications, J. Math. Anal. Appl. 197 Ž1996., 61]74. 9. N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 Ž1983., 233]245. 10. J. Zhou, On the existence of equilibrium for abstract economies, J. Math. Anal. Appl. 183 Ž1995., 839]858.
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AY975466
Approximate Selection Theorems and Their Applications* Xiyin Zheng Department of Mathematics, Yunnan Uni¨ ersity, Kunming, 650091, People’s Republic of China Submitted by Ying-ming Liu Received September 16, 1996
We prove three approximate selection theorems and give an improved version of the Michael selection theorem. As their applications, new fixed point theorems and Q 1997 Academic Press equilibrium theorems for generalized games are established.
1. INTRODUCTION The Michael selection theorem is a useful tool in nonlinear analysis which can be stated as follows: THEOREM A ŽMichael w6x.. Let X be a metric space, Y a Banach space, and let F: X § Y be a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues. Then F has a continuous selection Ž i.e., there is a continuous function f : X ª Y such that f Ž x . g F Ž x . for each x g X .. Recently, several authors w2, 5, 8, 9x investigated selection theorems in topological linear spaces. The following result, proved by Wu and Shen w8x, might be the most general form among existing selection theorems in topological linear spaces. THEOREM B. Let X be a paracompact topological space and Y be a topological linear space. Suppose that S, T : X § Y are set-¨ alued mappings such that Ž1. Ž2. Fx g NŽ x .
For each x g X, SŽ x . is nonempty and co SŽ x . ; T Ž x .. For each x g X, there is an open neighborhood N Ž x . of x such that SŽ x . / B.
Then T has a continuous selection. * This research is supported by the Science Foundation of Yunnan Province, China. 88 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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If X is connected and S is a point-valued mapping, it is easy to verify that the condition Ž2. in Theorem B implies that S is a constant mapping. Hence the supposition in Theorem B is strong. In this paper, we prove an approximate selection theorem under a suitable condition and establish approximate selection theorems with constraint. Moreover, we give an improved version of the famous Michael selection theorem. As their applications, new fixed point theorems and equilibrium theorems of generalized games are proved.
2. APPROXIMATE SELECTION THEOREMS In this paper, all topological spaces are assumed to be Hausdorff, and a subset of a topological space is considered to have the relative topology. Let X and Y be two topological spaces. A set-valued mapping F: X § Y is called upper semicontinuous Žlower semicontinuous. if the set x g X; F Ž x . ; V 4 Ž x g X; F Ž x . l V / B4. is open in X for every open subset V of Y. A set-valued mapping F: X § Y is called closed if graphŽ F . s Ž x, y . g X = Y; y g F Ž x .4 is closed in X = Y. For a set A in a topological linear space, we denote its convex hull and closure by co A and cl A, respectively. DEFINITION 2.1. Let X be a topological space and Y be a topological linear space. A set-valued mapping F: X § Y is called sub-lower semicontinuous if for each x g X and each neighborhood V of 0 in Y there is z g F Ž x . and a neighborhood UŽ x . of x in X such that for each y g UŽ x ., z g F Ž y . q V. It is clear that the sub-lower semicontinuity is weaker than the lower semicontinuity. DEFINITION 2.2. Let X be a topological space, Y be a topological linear space, and let F and T be two set-valued mappings from X into Y. We say F and T are topologically separated if for each x g X there is a neighborhood V of 0 in Y and a neighborhood UŽ x . of x in X such that F ŽUŽ x .. l ŽT Ž x . q V . s B. THEOREM 2.1. Let X be a paracompact topological space and Y be a locally con¨ ex topological linear space, and let F: X § Y be a set-¨ alued mapping with con¨ ex ¨ alues. Then F is sub-lower semicontinuous iff for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V. Proof. Suppose that F is sub-lower semicontinuous. For each neighborhood V of 0 in Y, there is a convex neighborhood V0 of 0 in Y such that
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V0 ; V Žsince Y is a locally convex topological linear space.. For each x g X, let z x g F Ž x . and UŽ x . be an open neighborhood of x in X such that for each y g UŽ x ., z x g F Ž y . q V0 . It is clear that UŽ x .4x g X is an open covering of X. Since X is paracompact, there is a locally finite open refinement Oi 4i g I of UŽ x .4x g X . Hence for each i g I there is x Ž i . g X such that Oi ; UŽ x Ž i ... Let pi 4i g I be a partition of unity subordinated to Oi 4i g I . The function f : X ª Y defined by f Ž x. s
Ý pi Ž x . z xŽ i. ,
x g X,
igI
is continuous since it is locally a finite sum of continuous functions. For each x g X, pi Ž x . / 0 implies x g Oi ; UŽ x Ž i .., and so z xŽ i. g F Ž x . q V0 . Since F Ž x . q V0 is convex, f Ž x . g F Ž x . q V0 ; F Ž x . q V. Conversely, suppose that for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V. Then for each x g X there is a neighborhood UŽ x . of x in X and z x g F Ž x . such that z x g f Ž x . y V and for each y g UŽ x ., f Ž x . g f Ž y . q V. Therefore, for each y g UŽ x ., z x g F Ž y . q V q V y V. Clearly this implies that F is sub-lower semicontinuous. THEOREM 2.2. Let X be a compact topological space and Y be a locally con¨ ex topological linear space. Assume that F, T : X § Y are two set-¨ alued mappings with the following properties: Ži. F and T are topologically separated. Žii. T is upper semicontinuous. Žiii. F is sub-lower semicontinuous, and F Ž x . is a nonempty con¨ ex set for each x g X. Then for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V and f Ž x . f T Ž x .. Proof. By Ži., for each x g X there is a neighborhood UŽ x . of x in X and a balanced convex neighborhood V Ž x . ; V of 0 in Y such that for each y g UŽ x ., F Ž y . l ŽT Ž x . q V Ž x .. s B. Again by Žii., there is an open neighborhood O Ž x . ; UŽ x . of x in X such that for each y g O Ž x ., T Ž y . ; T Ž x . q 12 V Ž x .. Hence for each y g O Ž x ., F Ž y . l ŽT Ž y . q 12 V Ž x .. s B, that is, Ž F Ž y . y 12 V Ž x .. l T Ž y . s B. By the compactness of X, there is n finitely many points in X, say x 1 , . . . , x n , such that X s Dks1 O Ž x k .. Let 1 n W s F ks 1 y 2 V Ž x k ., then W ; V is a neighborhood of 0 in Y and for each x g X, Ž F Ž x . q W . l T Ž x . s B. By Theorem 2.1, there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q W ; F Ž x . q V and f Ž x . f T Ž x ..
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THEOREM 2.3. Let X be a paracompact topological space and Y be a norm linear space. Assume that F, T : X § Y are two set-¨ alued mappings with the following properties: Ž i . F and T are topologically separated. Žii. T is upper semicontinuous. Žiii. F is sub-lower semicontinuous, and F Ž x . is a nonempty con¨ ex set for each x g X. Then for each neighborhood V of 0 in Y there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x . q V and f Ž x . f T Ž x .. Proof. By the assumption, for each x g X there is an open neighborhood UŽ x . of x in X, e Ž x . ) 0, and z x g F Ž x . such that for each y g UŽ x .,
Ž F Ž y . q B Ž 0, e Ž x . . . l T Ž y . s B and z x g F Ž y . q B Ž 0, e Ž x . . , where B Ž0, e Ž x .. s z g Y; 5 z 5 - e Ž x .4 . Without loss of generality, we can assume that for each x g X, B Ž0, e Ž x .. ; V. For each x g X, let
d Ž x . s sup r ) 0; Ž F Ž x . q B Ž 0, r . . l T Ž x . s B and B Ž 0, r . ; V 4 . It is clear that for each x g X, Ž F Ž x . q B Ž0, d Ž x ... l T Ž x . s B, B Ž0, d Ž x .. ; V and for each y g UŽ x ., d Ž y . G e Ž x .. Since X is paracompact and UŽ x .4x g X is an open covering of X, there is a locally finite open refinement Oi 4i g I of UŽ x .4x g X . Hence for each i g I there is x Ž i . g X such that Oi ; UŽ x Ž i ... Let pi 4i g I be a partition of unity subordinated to Oi 4i g I . Define f : X ª Y by f Ž x. s
Ý pi Ž x . z xŽ i. ,
x g X.
igI
Then f is a continuous function. For each x g X, pi Ž x . / 0 implies x g Oi ; UŽ x Ž i .., and so z xŽ i. g F Ž x . q B Ž0, e Ž x Ž i ... ; F Ž x . q B Ž0, d Ž x ... Hence for each x g X, f Ž x . g F Ž x . q B Ž0, d Ž x .. ; F Ž x . q V Žsince F Ž x . q B Ž0, d Ž x .. is convex. and f Ž x . f T Ž x .. The following proposition gives a sufficient condition to guarantee that F: X § Y and T : X § Y are topologically separated.
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PROPOSITION 2.1. Let X be a topological space and Y be a topological linear space. Assume that F, T : X § Y are two set-¨ alued mappings with the following properties: Ž i . F is upper semicontinuous. Žii. For each x g X, F Ž x . is closed, T Ž x . is compact, and F Ž x . l T Ž x . s B. Then F and T are topologically separated. Proof. By Žii. and Theorem 1.10 in w7x, for each x g X there is a neighborhood Vx of 0 in Y such that Ž F Ž x . q Vx . l ŽT Ž x . q Vx . s B. By Ži., for each x g X there is a neighborhood UŽ x . of x in X such that for each y g UŽ x ., F Ž y . ; F Ž x . q Vx , and so F ŽUŽ x .. l ŽT Ž x . q Vx . s B. The following theorem is an improved version of the Michael selection theorem. THEOREM 2.4. Let X be a paracompact topological space, Y a normed linear space, and let F: X § Y be a lower semicontinuous set-¨ alued mapping such that for each x g X, F Ž x . is a nonempty, complete, closed, and con¨ ex set. Then there is a continuous function f : X ª Y such that for each x g X, f Ž x . g F Ž x .. Proof. We claim that there is a sequence of continuous functions f n : X ª Y with the following properties: Ža. For each x g X, f nŽ x . g F Ž x . q B Ž0, 2yn .. Žb. For each x g X, 5 f nŽ x . y f ny1Ž x .5 - 2yn q2 . For n s 1, we apply Theorem 2.1 with V s B Ž0, 2y1 .. Assume that we have constructed the mappings f k up to n and let us construct f nq1. Define G: X § Y by GŽ x . s F Ž x . l B Ž f nŽ x ., 2yn ., x g X. It is clear that G is a lower semicontinuous mapping with nonempty convex values. By Theorem 2.1, there is a continuous function f nq 1: X ª Y such that for each x g X, f nq 1 Ž x . g Ž F Ž x . l B Ž f n Ž x . , 2yn . . q B Ž 0, 2yn y1 . . Therefore, for each x g X we have f nq 1 Ž x . g F Ž x . q B Ž 0, 2yn y1 . f nq 1 Ž x . y f n Ž x . - 2yn q 2yny1 - 2ynq1 . This proves our claim. By Ža., we can define a function g n : X § Y such that for each x g X, Žc.
g nŽ x . g F Ž x . and 5 f nŽ x . y g nŽ x .5 - 2yn .
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This and Žb. imply that for each x g X, 5 g nq 1Ž x . y g nŽ x .5 - 2yn q3 , and so for each natural number p, 5 g nq p Ž x . y g nŽ x .5 - 2yn q4 . Since each F Ž x . is complete, there is a function f : X ª Y such that g n4 converges pointwise to f and f Ž x . g F Ž x . and 5 f Ž x . y g nŽ x .5 F 2yn q4 for all x g X. This and Žc. imply that f n4 uniformly converges to f on X. Hence f is a continuous function. Kakutani and Fan’s fixed point theorem is important in nonlinear analysis which asserts: Let C be a nonempty compact convex subset of a locally convex topological linear space, and let F: C § C be an upper semicontinuous set-valued mapping with nonempty closed convex values. Then F has a fixed point. Himmelberg w4x gave an improved version of Kakutani and Fan’s fixed point theorem which says: Let C be a nonempty convex subset of a locally convex topological linear space, D be a compact subset of C, and let F: C § D be an upper semicontinuous set-valued mapping with nonempty closed convex values. Then F has a fixed point. The following fixed point theorem, in which the set-valued mapping F is lower semicontinuous instead of upper semicontinuous, supplements Kakutani and Fan’s fixed point theorem and Himmelberg’s fixed point theorem. THEOREM 2.5. Let C be a con¨ ex subset of a normed linear space, D a compact subset of C, and let F: C § D be a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues. Then F has a fixed point. Proof. Notice that a metric space is paracompact. By Theorem 2.4, there is a continuous function f : C ª D such that for each x g C, f Ž x . g F Ž x .. By Himmelberg’s fixed point theorem, there is x* g C such that x* s f Ž x*. g F Ž x*.. 3. THE EXISTENCE OF EQUILIBRIUM FOR GENERALIZED GAMES Let I be an index set. For each i g I, let X i be a nonempty subset of a topological linear space Yi . Throughout the section, we denote Ł i g I X i by X. Let A i , Bi , Pi : X § X i be set-valued mappings. A generalized game Žor an abstract economy. G s Ž X i , A i , Bi , Pi . i g I is defined as a family of ordered quadruples Ž X i , A i , Bi , Pi .. For each i g I, let p i be the projection mapping from X onto X i . For each x g X, we denote p i Ž x . by x i . An equilibrium for the generalized game G is a point x* g X such that for each i g I, xUi g cl Bi Ž x*. and A i Ž x*. l Pi Ž x*. s B where cl Bi is a set-valued mapping from X into X i such that for each x g X, cl Bi Ž x . is the closure of Bi Ž x ..
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DEFINITION 3.1. Let Ci be a subset of X. A i : X § X i is said to have the Ž e-SLC.-property on Ci , if for each convex neighborhood V of 0 in Yi , there is a sub-lower semicontinuous set-valued mapping TiV : Ci § X i with nonempty convex values such that Ti and p i are topologically separated on Ci and TiV Ž x . ; A i Ž x . q V for all x g Ci . Clearly, the Ž e-SLC.-property is weaker than the Ž e-CS.-property in w10x. DEFINITION 3.2. Let Ci be a subset of X. A i : X § X i is said to have the ŽLCS.-property on Ci , if there is a lower semicontinuous set-valued mapping Ti : Ci § X i with nonempty closed convex values such that Ti Ž x . ; A i Ž x . and x i f Ti Ž x . for all x g Ci . LEMMA 3.1. Let Yi 4i g I be a family of locally con¨ ex topological linear spaces. Then there is a directed set J such that for each i g I, Yi has a base Vi j 4j g J of neighborhoods of 0 with the following properties: Ž18. Ž28.
for each j g J, Vi j is closed and con¨ ex. Vi j 2 ; Vi j1 whene¨ er j1 F j2 in J.
Proof. For each i g I, let B i Ž0. be the base of all closed and convex neighborhoods of 0 in Yi , and J s Ł i g I Wi ; Wi g B i Ž0. for all i and Wi s Yi but finitely many i4 . Then J is a base of neighborhoods of 0 in Ł i g I Yi with respect to the product topology. Define a order F on J as follows: for arbitrary j1 , j2 g J, j1 F j2 if j2 ; j1. It is clear that J is a directed set with respect to F. For each i g I and j g J, let Vi j s p i Ž j .. It is clear that for each i g I, Vi j 4j g J is a base of neighborhoods of 0 in Yi , Vi j is closed and convex for all j g J, and Vi j 2 ; Vi j1 whenever j1 F j2 in J. LEMMA 3.2. Let X be a topological space, Y be a topological linear space, and let A: X § Y be an upper semicontinuous set-¨ alued mapping with compact ¨ alues. Assume that C ; Y and K ; Y are closed and compact, respecti¨ ely. Then T : X § Y defined by T Ž x . s Ž AŽ x . q C . l K for all x g X is upper semicontinuous. Proof. Let F Ž x . s AŽ x . q C and GŽ x . s K for all x g X. It is clear that T Ž x . s F Ž x . l GŽ x . for all x g X. By Theorem 8 in w1, p. 110x, it suffices to show that F is closed. Suppose that F is not closed, then there is x 0 g X and y 0 g Y such that Ž x 0 , y 0 . g cl GraphŽ F . and y 0 f F Ž x 0 . s AŽ x 0 . q C. Since AŽ x 0 . is compact and C is closed, AŽ x 0 . q C is closed. Hence there is a neighborhood V of 0 in Y such that Ž y 0 q V . l Ž AŽ x 0 . q C q V . s B Žsee Theorem 1.10 in w7x.. Since A is upper semicontinuous, there is a neighborhood U of x 0 in X such that for all x g U, AŽ x . ; AŽ x 0 . q V, and so F Ž x . s AŽ x . q C ; AŽ x 0 . q C q V. By Ž x 0 , y 0 . g cl GraphŽ F ., there is a net xa 4a g D in X and a net ya 4a g D in Y such that xa ª x 0 , ya ª y 0 and for all a g D, ya g F Ž xa .. It follows
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that there is an index a 0 g D such that xa g U whenever a G a 0 . Hence ya g F Ž xa . ; AŽ x 0 . q C q V, and so ya f y 0 q V, whenever a G a 0 . This contradicts ya ª y 0 . THEOREM 3.1. The generalized game G s Ž X i , A i , Bi , Pi . i g I admits an equilibrium if for each i g I: Ži. X i is a nonempty, compact con¨ ex subset of a locally con¨ ex topological linear space Yi . Žii. cl Bi is an upper semicontinuous set-¨ alued mapping with nonempty con¨ ex ¨ alues. Žiii. Bi has the Ž e-SLC .-property on a closed set Ci with x g X; A i Ž x . l Pi Ž x . / B4 ; int X Ž Ci . Ž where int X Ž Ci . is the relati¨ e interior of Ci in X .. Proof. Let J and Vi j 4j g J be as in Lemma 3.1. By Ži. and Tychonoff’s theorem, X s Ł i g I X i is a compact convex set. Hence Ci is compact for all i g I. By Žiii. and Theorem 2.2, for each j g J, there is a continuous function f i j : Ci ª X i such that for each x g Ci f i j Ž x . g Ž Bi Ž x . q Vi j . l X i
and
fi j Ž x . / x i .
Ž 1.
Define a set-valued mapping Fi j : X § X i by Fi j Ž x . s
½
fi j Ž x . 4 , Ž cl Bi Ž x . q Vi j . l X i ,
if x g int X Ž Ci . if x g X _int X Ž Ci . .
Notice that int X Ž Ci . is an open subset of X. By Žii. and Lemma 3.2, it is easy to verify that Fi j is upper semicontinuous. For each j g J, let Fj : X § X such that Fj Ž x . s Ł i g I Fi j Ž x . for all x g X. By Lemma 3 in w3x, Fj is an upper semicontinuous set-valued mapping with closed convex values. By Kakutani and Fan’s fixed point theorem, there is x j g X such that x j g Fj Ž x j .. We claim that x j g X _Di g I int X Ž Ci .. Indeed, if this is not true, then there is i 0 g I such that x j g int X Ž Ci 0 .. By Ž1. and the definition of Fi j , x ij0 s p i 0Ž x j . f f i 0 j Ž x j .4 s Fi 0 j Ž x j .. This implies x j f Fj Ž x j ., a contradiction. Hence for each j g J, xjg
Ł Ž cl Bi Ž x j . q Vi j . .
Ž 2.
igI
Since J is a directed set, x j 4j g J is a net in the compact subset X _Di g I int X Ž Ci . of X. Hence there is a subnet of x j 4j g J to converge to some x* in X _Di g I int X Ž Ci .. Without loss of generality, we can assume lim x j s x*. j
Ž 3.
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This is equal to lim x ij s xUi , j
for each i g I.
By Ž2., for each i g I and each j g J, x ij g cl Bi Ž x j . q Vi j . Hence there is yi j g cl Bi Ž x j . and ¨ i j g Vi j such that x ij s yi j q ¨ i j . Since Vi j 4j g J is a base of neighborhoods of 0 in Yi for all i g I, lim j ¨ i j s 0. Hence for each i g I lim yi j s lim x ij y lim ¨ i j s xUi . j
j
j
Ž 4.
By proposition 7 in w1, p. 110x, for each i g I, cl B is closed. By Ž3. and Ž4., xUi g cl B Ž x*. for all i g I. By x* g X _Di g I int X Ž Ci . and Žiii., A i Ž x*. l Pi Ž x*. s B for all i g I. The above proof uses the idea of the proof of Theorem 5 in w10x. It is clear that Theorem 3.1 is a generalization of Theorem 5 in w10x. THEOREM 3.2. The generalized game G s Ž X i , A i , Bi , Pi . i g I admits an equilibrium if for each i g I: Ži. X i is a nonempty con¨ ex subset of a normed linear space Yi . Žii. cl Bi is a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues, and cl Bi Ž X . is contained in a compact subset Hi of X i . Žiii. cl Bi has the Ž LCS .-property on a closed subset Ci of X containing the set x g X; A i Ž x . l Pi Ž x . / B4 . Proof. Since cl Bi has the LCS-property on Ci , there is a lower semicontinuous set-valued mapping Ti : Ci § X i with nonempty closed convex values such that Ti Ž x . ; cl Bi Ž x . and x i f Ti Ž x . for all x g Ci . Define a set-valued mapping Fi from X into X i by Fi Ž x . s
½
Ti Ž x . ,
if x g Ci
cl Bi Ž x . ,
if x g X _Ci .
Since Ci is a closed subset of X, it is easy to verify that Fi is lower semicontinuous and Fi Ž x . is a closed convex set for each x g X. By Theorem 2.5, there is a continuous function f i : X ª X i such that for each xgX f i Ž x . g Fi Ž x . ; cl Bi Ž x . .
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Define a continuous function f from X into a compact subset Ł i g I Hi such that p i Ž f Ž x .. s f i Ž x . for all x g X and all i g I. Since f i is continuous for all i g I, f is continuous. By Himmelberg’s fixed point theorem, there is x* g X such that x* s f Ž x*.. Hence xUi s f i Ž x*. g cl Bi Ž x*.. This and x i f Ti Ž x . and f i Ž x . g Fi Ž x . s Ti Ž x . for all x g Ci imply x* f Ci . By Žiii., A i Ž x*. l Pi Ž x*. s B. In Theorem 3.2, setting Pi Ž x . s B for all i g I and all x g X, we have the following fixed point result. COROLLARY 3.1. For each i g I, let X i be a con¨ ex subset of a normed linear space and Hi be a compact subset of X i , and let Fi : X s Ł i g I X i § Hi be a lower semicontinuous set-¨ alued mapping with nonempty closed con¨ ex ¨ alues. Then there is x* g X such that xUi g Fi Ž x*. for all i g I. REFERENCES 1. J. P. Aubin and I. Ekeland, ‘‘Applied Nonlinear Analysis,’’ Wiley]Interscience, New York, 1984. 2. X. P. Ding, W. K. Kim, and K. K. Tan, A selection theorem and its applications, Bull. Austral. Math. Soc. 46 Ž1992., 205]212. 3. K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 Ž1952., 121]126. 4. C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 Ž1972., 205]207. 5. T. Husain and E. Tarafdar, A selection theorem and a fixed point theorem and an equilibrium point of an abstract economy, Internat. J. Math. Math. Sci. 18 Ž1995., 179]184. 6. E. Michael, Continuous selections, I, Ann. of Math. 63 Ž1956., 361]382. 7. W. Rudin, ‘‘Functional Analysis,’’ McGraw]Hill, New York, 1973. 8. X. Wu and S. K. Shen, A further generalization of Yannelis-Prabhakar’s continuous selection theorem and its applications, J. Math. Anal. Appl. 197 Ž1996., 61]74. 9. N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 Ž1983., 233]245. 10. J. Zhou, On the existence of equilibrium for abstract economies, J. Math. Anal. Appl. 183 Ž1995., 839]858.