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Some Nonempty Intersection Theorems in Generalized Interval Spaces with Applications
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
199, 787]803 Ž1996.
0176
Some Nonempty Intersection Theorems in Generalized Interval Spaces with ApplicationsU Shih-sen Chang Department of Mathematics, Sichuan Uni¨ ersity, Chengdu, Sichuan 610064, People’s Republic of China
Shi-yi Cao Department of Mathematics, Xinjiang Uni¨ ersity, Wulumuqi, Xinjiang 830046, People’s Republic of China
Xian Wu Department of Mathematics, Yunnan Normal Uni¨ ersity, Kumming, Yunnan 650092 People’s Republic of China
and Da-cheng Wang Department of Mathematics, Chongqing Teacher’s College, Yongchuan, Sichuan 632168, People’s Republic of China Submitted by Dorothy Maharam Stone Received March 29, 1994
In this paper some parametric types of KKM theorems are established in generalized interval spaces. As applications, we utilize these results to obtain some new minimax theorems, section theorems, and existence theorems of solutions for variational inequalities. The results presented in this paper not only include some important earlier results and the famous von Neumann theorems as their special Q 1996 Academic cases, but also improve and extend other corresponding results. Press, Inc.
U
Supported by the National Natural Science Foundation of China. 787 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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1. PRELIMINARIES DEFINITION 1. A topological space X is called a generalized interval space, if there exists a mapping G: X = X ª AŽ X ., where AŽ X . is a family of nonempty connected subsets of X. For any Ž x 1 , x 2 . g X = X we denote G Ž x 1 , x 2 . by G x 1 , x 2 4 and G x 1 , x 2 4 is called a generalized interval associated with x 1 and x 2 . DEFINITION 2. Let X be a generalized interval space, Y be a topological space, and Z be a completely dense linear ordered space. A subset B of X is called T-convex, if for any x 1 , x 2 g B, we have G x 1 , x 2 4 ; B. A mapping f : X ª Z is called T-quasi-concave Žconvex., if for any z g Z, the set x g X: f Ž x . G z 4 Ž x g X: f Ž x . F z 4. is T-convex. Let D be a subset of X. If for any G x 1 , x 2 4 , D l G x 1 , x 2 4 is a closed Žopen. set in G x 1 , x 2 4 , then D is called a generalized interval closed Žopen. set. Remark. From the above definitions, it is easy to see that interval space, H-space, convex space, contractible space, and topological linear space all can be considered as special cases of generalized interval space. Moreover the H-convex set in H-space is T-convex; the convex sets in topological vector space are also T-convex and the quasi-concave Žconvex. mapping is T-quasi-concave Žconvex.. We have to point out that the intersection of T-convex sets is T-convex and each T-convex set can be considered as a generalized interval space. DEFINITION 3 w4x. Let X be a topological space, Z be an order complete and order dense linear ordered space. A mapping f : X ª Z is called upper Žlower. semi-continuous, if the set x g X: f Ž x . G z 4 Ž x g X: f Ž x . F z 4. is a closed set in X for all z g Z. DEFINITION 4 w4x. Let X and Y be two topological spaces and S : Y = X = Y ª R be a function. T : Y ª 2 X is called S-monotone, if for any y, z g Y, for any u g Ty, and for any ¨ g Tz S Ž y, u, z . y S Ž y, ¨ , z . G 0. DEFINITION 5 w12x. Let X and Y be two topological spaces. A mapping G: Y ª 2 X is called transfer closed valued in Y if for any y g Y, x f GŽ y ., then there exists an yX g Y such that x f G Ž yX . Žin the sequel we denote by A the closure of A.. Remark. It is obvious that if G is closed valued in Y, then G is transfer closed valued on Y. In the sequel we denote F Ž Y . [ A ; Y: A is a nonempty finite set4 .
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2. PARAMETRIC TYPE OF KKM THEOREMS IN GENERALIZED INTERVAL SPACES In the sequel we need the following auxiliary lemmas: LEMMA 2.1 w12x. Let X, Y be two topological spaces and G: Y ª 2 X be a multi¨ alued mapping. Then G is transfer closed ¨ alued on Y if and only if F y g YG Ž y . s F y g Y GŽ y .. Proof. If G is transfer closed valued on Y, it is obvious that F y g YG Ž y . > F y g Y GŽ y .. To prove the conclusion we have to prove F y g YG Ž y . ; F y g Y GŽ y .. Suppose the contrary, then there exists x g F y g YG Ž y . and x f F y g Y GŽ y .. Hence there exists y g Y such that x f GŽ y .. Since G is transfer closed valued on Y, there exists yX g Y such that x f G Ž yX . . This contradicts x g F y g YG Ž y . . Conversely, if F y g YG Ž y . s F y g Y GŽ y ., then for any y g Y and any x f GŽ y ., then x f F y g Y GŽ y . s F y g YG Ž y . . Hence there exists yX g Y such that x f G Ž yX . . This implies that G is transfer closed valued on Y. LEMMA 2.2. Let Y be a generalized inter¨ al space, X be a topological space, and F: Y ª 2 X be a mapping. Then for any x g X, Y _ Fy1 Ž x . is a T-con¨ ex set if and only if for any y 1 , y 2 g Y we ha¨ e F Ž y . ; F Ž y1 . j F Ž y 2 .
for all y g G y 1 , y 2 4 .
Proof. Necessity. Suppose the contrary, there exist y 1 , y 2 g Y and an y 0 g G y 1 , y 2 4 such that F Ž y 0 . o F Ž y 1 . j F Ž y 2 .. Hence there exists x 0 g F Ž y 0 . but x 0 f F Ž yi ., i s 1, 2, and so yi f Fy1 Ž x 0 ., i s 1, 2, i.e., y 1 , y 2 4 ; Y _ Fy1 Ž x 0 .. Since Y _ Fy1 Ž x 0 . is T-convex, G y 1 , y 2 4 ; Y _ Fy1 Ž x 0 .. Hence y 0 g Y _ Fy1 Ž x 0 ., and so x 0 f F Ž y 0 .. This contradicts the choice of x 0 . Therefore for any y 1 , y 2 g Y, we have F Ž y . ; F Ž y1 . j F Ž y 2 .
for all y g G y 1 , y 2 4 .
Sufficiency. Since for any y 1 , y 2 g Y, we have F Ž y . ; F Ž y 1 . j F Ž y 2 . for all y g G y 1 , y 2 4 . Hence for any x g X and for any ˆ y1, ˆ y 2 g Y _ Fy1 Ž x ., FŽ ˆ y. ; FŽ ˆ y1 . j F Ž ˆ y 2 . for all ˆ y g G ˆ y1, ˆ y 2 4 . Since ˆ y1, ˆ y 2 f Fy1 Ž x ., i.e., x f FŽ ˆ y 1 . and x f F Ž ˆ y 2 ., and so x f F Ž ˆ y1 . j F Ž ˆ y 2 .. This implies that x f FŽ ˆ y . for all ˆ y g G y 1 , y 2 4 , i.e., for all ˆ y g G y 1 , y 2 4 , y g Y _ Fy1 Ž x .. Therefore G y 1 , y 2 4 ; Y _ Fy1 Ž x .. This shows that Y _ Fy1 Ž x . is a T-convex set for all x g X. This completes the proof. THEOREM 2.3. Let Y be a generalized inter¨ al space, X be a topological space, Z be a linear ordered space, F, G: Y = Z ª 2 X be two mappings such that F has nonempty ¨ alues and G is transfer closed ¨ alued. If the following
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conditions are satisfied: Ži. for any Ž y, z . g Y = Z, F Ž y, z . ; G Ž y, z . and when z1 F z 2 , F Ž y, z 2 . ; F Ž y, z1 .
for all y g Y ;
Žii. for any z g Z, there exists ˆ z g Z such that F Ž y, z . > G Ž y, ˆ z . for all y g Y; Žiii. for any A g F Ž Y . and for any z g Z, F y g A F Ž y, z . is connected; Živ. for any x g X and for any z g Z, the set y g Y: x f F Ž y, z .4 is T-con¨ ex and generalized inter¨ al closed; Žv. for any z g Z and any y 1 , y 2 g Y, there exist yX1 , yX2 g G y 1 , y 2 4 , such that F Ž yX1 , z . ; F Ž y 1 , z ., F Ž yX2 , z . ; F Ž y 2 , z ., then Ž1. G Ž y, z . : y g Y, z g Z 4 has the finite intersection property; Ž2. if there exists Ž y 0 , z 0 . g Y = Z such that G Ž y 0 , z 0 . is compact, then F y g Y , z g Z GŽ y, z . / B. Proof. Since F has nonempty values, by condition Ži., for all Ž y, z . g Y = Z, G Ž y, z . / B. If for any n elements of G Ž y, z . : y g Y, z g Z 4 their intersection is nonempty, next we prove that for any n q 1 elements of G Ž y, z . : y g Y, z g Z 4 their intersection is also nonempty, where n G 2. In fact, if there exist Ž yi , z i . g Y = Z, i s 1, 2, . . . , n q 1, such that nq 1 F is1 G Ž yi , z i . s B. Since Z is a linear ordered space, without loss of generality we can assume that z1 G z 2 G ??? G z n G z nq1 , letting H s nq 1 Ž F is3 F yi , z1 ., then we have H l F Ž y 1 , z1 . l H l F Ž y 2 , z1 . ; H l F Ž y 1 , z1 . l F Ž y 2 , z 2 . nq1
;
F G Ž yi , z i . s B. is1
This implies that H l F Ž y 1 , z1 . and H l F Ž y 2 , z1 . are separated from each other. By condition Žii. and the assumptions of induction, there exists ˆz g Z such that for any y g Y nq1
H l F Ž y, z1 . s
F F Ž yi , z1 . l F Ž y, z1 . is3
nq1
>
F G Ž yi , ˆz . l G Ž y, ˆz . / B. is3
Define a mapping L: Y ª 2 X by LŽ y . [ F Ž y, z1 ., y g Y, then for any x g X, Y _ Ly1 Ž x . s y g Y: x f F Ž y, z1 .4 . By condition Živ. for any x g
NONEMPTY INTERSECTION THEOREMS
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X, Y _ Ly1 Ž x . is T-convex. In view of Lemma 2.2 for any y 1 , y 2 g Y and for any y g G y 1 , y 2 4 we have LŽ y . ; LŽ y 1 . j LŽ y 2 ., i.e., F Ž y, z1 . ; F Ž y 1 , z1 . j F Ž y 2 , z1 .
for all y g G y 1 , y 2 4 .
Hence we have H l F Ž y, z1 . ; Ž H l F Ž y 1 , z1 . . j Ž H l F Ž y 2 , z1 . . for all y g G y 1 , y 2 4 . By condition Žiii., H l F Ž y, z1 . is connected. In addition, since H l F Ž y 1 , z1 . and H l F Ž y 2 , z1 . are separated from each other, for any y g G y 1 , y 2 4 H l F Ž y, z1 . ; H l F Ž y 1 , z1 .
or
H l F Ž y, z1 . ; H l F Ž y 2 , z1 . .
Letting Ei s y g G y 1 , y 2 4 : H l F Ž y, z1 . ; H l F Ž yi , z1 .4 , i s 1, 2, by condition Žv., Ei / B, i s 1, 2, and G y 1 , y 2 4 s E1 j E2 . Since G y 1 , y 2 4 is nonempty connected, at least one of E1 l E2 and E1 l E2 is nonempty. Without loss of generality we can assume that E1 l E2 / B. Taking y 0 g E1 l E2 , we have H l F Ž y 0 , z1 . ; H l F Ž y 1 , z1 . and there exists a net ya 4a g I ; E2 such that ya ª y 0 . Hence we have H l F Ž ya , z1 . ; H l F Ž y 2 , z1 .
for all a g I.
On the other hand, in the above we have proved that H l F Ž y 0 , z1 . / B. Taking x 0 g H l F Ž y 0 , z1 ., we know that x 0 f H l F Ž y 2 , z1 .. Hence x 0 f H l F Ž ya , z1 .
for all a g I.
Therefore x 0 f F Ž ya , z1 . for all a g I, i.e., ya 4a g I ; Y _ Ly1 Ž x 0 .. Since ya 4a g I ; G y 1 , y 2 4 and Y _ Ly1 Ž x 0 . is generalized interval closed, y 0 g Y _ Ly1 Ž x 0 ., i.e., x 0 f F Ž y 0 , z1 .. This contradicts the choice of x 0 . By this contradiction, we know that G Ž y, z . : y g Y, z g Z 4 has the finite intersection property. If, in addition, there exists Ž y 0 , z 0 . g Y = Z such that G Ž y 0 , z 0 . is compact, then it is easy to prove that l y g Y , F z g Z G Ž y, z . / B. By Lemma 2.1 we have F y g Y , F z g Z GŽ y, z . / B. This completes the proof. COROLLARY 2.4. Let Y be a generalized inter¨ al space, X be a topological space, Z be a linear ordered space, and F and G: Y = Z ª 2 X be two mappings such that F has nonempty ¨ alues and G has closed ¨ alues. If the following conditions are satisfied: ha¨ e
Ži. for any Ž y, z . g Y = Z, F Ž y, z . ; GŽ y, z ., and when z1 F z 2 we F Ž y, z 2 . ; F Ž y, z1 .
for all y g Y ;
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Žii. for any z g Z there exists a ˆ z g Z such that F Ž y, z . > GŽ y, ˆ z . for all y g Y; Žiii. for any A g F Ž Y . and for any z g Z, F y g A F Ž y, z . is connected; Živ. for any x g X and for any z g Z, y g Y: x f F Ž y, z .4 is T-con¨ ex and generalized inter¨ al closed; Žv. for any z g Z and for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that F Ž yX1 , z . ; F Ž y 1 , z ., F Ž yX2 , z . ; F Ž y 2 , z ., then Ž1. GŽ y, z .: y g Y, z g Z 4 has the finite intersection property; Ž2. in addition, if there exists Ž y 0 , z 0 . g Y = Z such that GŽ y 0 , z 0 . is compact, then F y g Y , F z g Z GŽ y, z . / B. Proof. Since G has closed values, G is transfer closed valued on Y = Z. Hence the conclusion follows from Theorem 2.3 immediately.
3. APPLICATIONS TO MINIMAX PROBLEMS In this section we shall use the results presented in Section 2 to study the minimax problems. We have the following results: THEOREM 3.1. Let Y be a generalized inter¨ al space, X be a topological space, Z˜ be an order complete and order dense linear ordered space, and f, g: X = Y ª Z˜ be two functions satisfying the following conditions:
˜ F y g A x g X: f Ž x, y . ) z 4 is Ži. for any A g F Ž Y . and for any z g Z, a connected set; Žii. Ža. for any x g X, y ¬ f Ž x, y . is T-quasi-con¨ ex and is lower-semicontinuous on any generalized inter¨ al of Y; Žb. for any y g Y, x ¬ f Ž x, y . and x ¬ g Ž x, y . are upper semicontinuous; Žiii. there exist z 0 - inf y g Y sup x g X f Ž x, y . and y 0 g Y and a compact subset L ; X such that g Ž x, y 0 . - z 0 for all x g X _ L; Živ. f Ž x, y . F g Ž x, y . for all Ž x, y . g X = Y; Žv. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yiX . F f Ž x, yi ., i s 1, 2, and for any x g X. Then z# [ sup x g X inf y g Y g Ž x, y . G inf y g Y sup x g X f Ž x, y . \ zU .
˜ we know that zU and z# both exist. Proof. By the completeness of Z, U ˜ z - z 4, then Z is a dense linear ordered space. For Letting Z s z g Z: any Ž y, z . g Y = Z, letting F Ž y, z . s x g X: f Ž x, y . ) z 4 , GŽ y, z . s x g X: f Ž x, y . G z 4 , H Ž y, z . s x g X: g Ž x, y . G z 4 , it follows from condition Žii.Žb. that GŽ y, z . and H Ž y, z . both are closed for all Ž y, z . g Y = Z.
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Next we prove that mappings F and G satisfy all conditions of Corollary 2.4. In fact, by the definition of zU , F: Y = Z ª 2 X is nonempty valued. Again by the definition of F and G, it is easy to see that they satisfy condition Ži. in Corollary 2.4. In addition, for any z g Z with z - zU , by ˜ there exists a ˆz g Z˜ such that z - ˆz - zU and so the denseness of Z, ˆz g Z and F Ž y, z . > GŽ y, ˆz . for all y g Y. This implies that condition Žii. in Corollary 2.4 is satisfied. Again for any x g X and for any z g Z
y g Y : x f F Ž y, z . 4 s y g Y : f Ž x, y . F z 4 . By condition Žii.Ža. we know that condition Živ. in Corollary 2.4 is satisfied. By conditions Ži. and Žv. we know that conditions Žiii. and Žv. are satisfied. By conclusion Ž1. of Corollary 2.4, GŽ y, z .: y g Y, z g Z 4 has the finite intersection property. On the other hand, by condition Živ., for any Ž y, z . g Y = Z, GŽ y, z . ; Ž H y, z .. Hence H Ž y, z .: y g Y, z g Z 4 is a family of closed sets having the finite intersection property. By condition Žiii., for any x g X _ L, g Ž x, y 0 . - z 0 , i.e., x g x g X: g Ž x, y 0 . - z 0 4 . Hence x f H Ž y 0 , z 0 . s x g X: g Ž x, y 0 . G z 0 4 and so H Ž y 0 , z 0 . ; L. Since L is compact and H Ž y 0 , z 0 . is closed, H Ž y 0 , z 0 . is compact. Hence we have
F F H Ž y, z . s F F H Ž y, z . l H Ž y 0 , z 0 . / B.
ygY zgZ
ygY zgZ
Taking ˆ x g F y g Y F z g Z H Ž y, z ., we have g Ž ˆ x, y . G z for all y g Y and for all z g Z. Hence z# s sup x g X inf y g Y g Ž x, y . G z for all z g Z. By the denseness of Z˜ we have sup inf g Ž x, y . G zU s inf sup f Ž x, y . .
xgX ygY
ygY xgX
This completes the proof. COROLLARY 3.2. Let Y be a generalized inter¨ al space, X be a topological space, and Z be an order complete and order dense linear ordered space. If f : X = Y ª Z satisfies the following conditions: Ži. for any A g F Ž Y . and for any z g Z, F y g A x g X: f Ž x, y . ) z 4 is connected; Žii. Ža. for any x g X, y ¬ f Ž x, y . is T-quasi-con¨ ex, and is lower semi-continuous on any generalized inter¨ al of Y; Žb. for any y g Y, x ¬ f Ž x, y . is upper semi-continuous; Žiii. there exist z 0 - inf y g Y sup x g X f Ž x, y ., y 0 g Y, and a compact subset L ; X such that f Ž x, y 0 . - z 0 for all x g X _ L; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yXi . F f Ž x, yi . ,
i s 1, 2 and for all x g X ,
then sup x g X inf y g Y f Ž x, y . s inf y g Y sup x g X f Ž x, y ..
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Proof. The conclusion of Corollary 3.2 can be obtained from Theorem 3.1 immediately. Remark. The condition ‘‘there exist z 0 - inf y g Y sup x g X f Ž x, y ., y 0 g Y, and a compact subset L ; X such that f Ž x, y 0 . - z 0 for all x g X _ L’’ is equivalent to the condition ‘‘there exist z 0 - inf y g Y sup x g X f Ž x, y . and y 0 g Y such that x g X: f Ž x, y 0 . G z 0 4 is compact.’’ Therefore Corollary 3.2 includes the main results of Cheng and Lin w3x as its special cases and so it contains the corresponding results of Brezis, Nirenberg, and ´ Stampacchia w2x, Komornik w7x, M. A. Geraghty and Lin w9x, Stacho ´ w11x, and Wu w13x as its special cases. COROLLARY 3.3. Let Y be a compact generalized inter¨ al space, X be a compact topological space, Z be an order complete and order dense linear ordered space, and f : X = Y ª Z be a mapping satisfying the following conditions: Ži. for any A g F Ž Y . and for any z g Z, F y g A x g X: f Ž x, y . ) z 4 is connected or empty; Žii. for any x g X, f Ž x, y . is T-quasi-con¨ ex and lower semi-continuous in y; Žiii. for any y g Y, x ¬ f Ž x, y . is upper semi-continuous; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yX1 . F f Ž x, yi . for all x g X, i s 1, 2. Then f has a saddle Ž ˆ x, ˆ y . g X = Y. Proof. By using Corollary 3.2 and Proposition 1.4.6 and Theorem 3.10.4 of w4x, we can obtain the conclusion of Corollary 3.3 immediately. Remark. Corollary 3.3 contains the famous von Neumann theorem in mathematical economy and game theory. THEOREM 3.4. Let Y be a generalized inter¨ al space, X be a topological space, Z be an order complete and order dense linear ordered space, and f and g: X = Y ª Z be two mappings satisfying f Ž x, y . F g Ž x, y . for all Ž x, y . g X = Y and the following conditions: Ži. for any x g X, y ¬ f Ž x, y . is T-quasi-con¨ ex and lower semi-continuous; Žii. for any y g Y, f Ž x, y . and g Ž x, y . are upper semi-continuous in x; Žiii. there exist a nonempty subset K ; X and a compact subset H ; Y such that inf sup f Ž x, y . F
ygY xgX
inf
sup f Ž x, y . ,
ygY_H xgK
NONEMPTY INTERSECTION THEOREMS
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and for any finite subset F ; X there exists a compact set K Ž F . > K j F, such that for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . ) z 4 is connected; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yiX . F f Ž x, yi . for all x g X, i s 1, 2. Then sup x g X inf y g Y g Ž x, y . G inf y g Y sup x g X f Ž x, y .. Proof. Letting z# s sup x g X inf y g Y g Ž x, y . and zU s inf y g Y sup x g X f Ž x, y ., by the completeness of Z, we know that z# and zU both exist. If z g Z such that z# - ˆ zz# - zU , again by the density of Z, there exists ˆ zU . For any x g X, letting LŽ x . s y g Y: f Ž x, y . F ˆ z 4 , by condition Ži., LŽ x . is a closed set in Y. For any x g X, letting M Ž x . s LŽ x . l ŽF t g K LŽ t .., then M Ž x . is a closed set. Next, we prove that M Ž x . ; H for all x g X. In fact, for any y 0 g Y _ H, by condition Žiii., we know that sup x g K f Ž x, y 0 . G zU ) ˆ z. Therefore there exists a x 0 g K such that f Ž x0 , y0 . ) ˆ z, i.e., y 0 f LŽ x 0 . and so y 0 f F t g K LŽ t .. This implies that F t g K LŽ t . ; H. Hence M Ž x . ; H for all x g X. Finally, we prove that M Ž x .: x g X 4 has the finite intersection property. In fact, for any finite set F ; X, by condition Žiii., there exists a compact subset K Ž F . > K j F such that for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .:: f Ž x, y . ) z 4 is connected. From Theorem 3.1 we have sup
inf g Ž x, y . G inf
xgK Ž F . ygY
sup f Ž x, y . ,
ygY xgK Ž F .
and so we have inf
sup f Ž x, y . F sup inf g Ž x, y . s zU - ˆ z.
ygY xgK Ž F .
xgX ygY
Now we prove that F x g K Ž F . M Ž x . / B. Suppose the contrary, F x g K Ž F . M Ž x . s B, then we have Y s D x g K Ž F .Ž Y _ M Ž x .., and so for any y g Y, there exists x Ž y . g K Ž F . such that y g Y _ M Ž x Ž y .., i.e., y f M Ž x Ž y .. s LŽ x Ž y .. l ŽF t g K LŽ t ... Hence y f LŽ x Ž y .. or y f F t g K LŽ t .. If y f LŽ x Ž y .., then f Ž x Ž y ., y . ) ˆ z; if y f F t g K LŽ t ., then there exists ˆ x Ž y . g K ; K Ž F . such that y f LŽ ˆ x Ž y .., i.e., f Ž ˆ x Ž y ., y . ) ˆ z. This implies that there exists a mapping ˆ x: Y ª K Ž F . such that f Ž ˆ x Ž y ., y . )ˆ z. Therefore we have sup x g K Ž F . f Ž x, y . ) ˆ z for all y g Y. Hence inf y g Y sup x g K Ž F . f Ž x, y . G ˆ z. This contradicts inf y g Y sup x g K Ž F . f Ž x, y . ˆz. Therefore F x g K Ž F . M Ž x . / B. Since F x g F M Ž x . > F x g K Ž F . M Ž x ., M Ž x .: x g X 4 has the finite intersection property.
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However, since H is compact and M Ž x . ; H for all x g X, F x g X M Ž x . / B. Hence there exists a ˆ y g M Ž x . for all x g X, and so ˆ y g LŽ x . for all x g X, i.e., f Ž x, ˆ y. F ˆ z for all x g X. Hence zU F ˆ z. This contradicts the choice of ˆ z. Therefore z# G zU . This completes the proof. Remark. Theorem 3.4 improves and extends Theorem 2 of Lin and Quan w8x. COROLLARY 3.5. Let Y be a generalized inter¨ al space, X be a topological space, Z be an order complete and order dense linear ordered space, and f : X = Y ª Z be a mapping satisfying the following conditions: Ži. for any x g X, f Ž x, y . is T-quasi-con¨ ex and lower semi-continuous in y; Žii. for any y g Y, x ¬ f Ž x, y . is upper semi-continuous; Žiii. there exist a nonempty subset K ; X and a compact subset H ; Y such that inf sup f Ž x, y . F
ygY xgX
inf
sup f Ž x, y . ,
ygY_H xgK
and for any finite set F ; X there exists a compact subset K Ž F . > K j F such that for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . ) z 4 is connected; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yXi . F f Ž x, yi . for all x g X , i s 1, 2. Then sup x g X inf y g Y f Ž x, y . s inf y g Y sup x g X f Ž x, y .. Proof. It follows from Theorem 3.4 that sup inf f Ž x, y . G inf sup f Ž x, y . .
xgX ygY
ygY xgX
However, it is obvious that inf y g Y sup x g X f Ž x, y . G sup x g X inf y g Y f Ž x, y .. Hence the conclusion is obtained. Remark. Since the condition ‘‘for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . G z 4 is connected’’ implies the condition ‘‘for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . ) z 4 is connected,’’ therefore Corollary 3.5 contains Corollary 1 of Lin and Quan w8x as its special case. COROLLARY 3.6 w6x. Let X and Y be nonempty con¨ ex subsets of Hausdorff topological ¨ ector spaces M and N, respecti¨ ely, and f : X = Y ª R be a function satisfying the following conditions: Ži. for any x g X, y ¬ f Ž x, y . is quasi-con¨ ex and lower semi-continuous;
NONEMPTY INTERSECTION THEOREMS
797
Žii. for any y, x ¬ f Ž x, y . is quasi-conca¨ e and upper semi-continuous; Žiii. there exists a nonempty compact con¨ ex subset K ; X and a compact subset H ; Y such that inf sup f Ž x, y . F
ygY xgX
inf
sup f Ž x, y . .
ygY_H xgK
Then inf y g Y sup x g X f Ž x, y . s sup x g X inf y g Y f Ž x, y .. Proof. For any x 1 , x 2 g X and for any y 1 , y 2 g Y, letting G x 1 , x 2 4 s co x 1 , x 2 4 , G y 1 , y 2 4 s co y 1 , y 2 4 , under the structures like this, X and Y both are generalized interval space. Furthermore for any finite subset F ; X, letting K Ž F . s coŽ K j F ., then all conditions in Corollary 3.5 are satisfied. The conclusion follows from Corollary 3.5 immediately.
4. APPLICATIONS TO SECTION PROBLEMS In this section, we shall use the results presented in Section 2 to study the section problems. We have the following results. THEOREM 4.1. Let Y be a generalized inter¨ al space, X be a topological space, Z be a dense linear ordered space, and B and C be two sets of X = Y = Z satisfying the following conditions: Ži. B ; C; Žii. for any Ž y, z . g Y = Z, the sections BŽ y , z . [ x g X : Ž x, y, z . g B 4 / B, and CŽ y , z . [ x g X : Ž x, y, z . g C 4 are transfer closed ¨ alued and there exists Ž y 0 , z 0 . g Y = Z such that CŽ y 0 , z 0 . is compact; Žiii. if z 2 G z1 , then BŽ y, z . ; BŽ y, z . for all y g Y; and for any z g Z 2 1 there exists ˆ z g Z such that BŽ y, z . > CŽ y, ˆz . for all y g Y; Živ. for any A g F Ž Y . and for any z g Z, F y g A BŽ y, z . is connected; Žv. for any Ž x, z . g X = Z, the set Y _ BŽ x, z . [ y g Y: Ž x, y, z . f B4 is generalized inter¨ al closed and T-con¨ ex in Y; Žvi. for any z g Z and for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that BŽ yX , z . ; BŽ y i , z . , i s 1, 2.
798
CHANG ET AL.
Then there exists an ˆ x g X such that ˆ x 4 = Ž Y = Z . ; C. Proof. Letting F Ž y, z . s BŽ y, z . , GŽ y, z . s CŽ y, z . , it is easy to see that F and G satisfy all conditions in Theorem 2.3. By conclusion Ž2. in Theorem 2.3, F y g Y , F z g Z GŽ y, z . / B. Hence there exists an ˆ x g X such that ˆx g GŽ y, z . for all Ž y, z . g Y = Z, i.e., Ž ˆx, y, z . g C for all Ž y, z . g Y = Z, and so ˆ x 4 = Ž Y = Z . ; C. COROLLARY 4.2. Let Y be a generalized inter¨ al space, X be a topological space, Z be a dense linear ordered space, and B and C be two subsets of X = Y = Z satisfying the following conditions: Ži. B ; C; Žii. for any Ž y, z . g Y = Z, the section BŽ y, z . [ x g X: Ž x, y, z . g B4 / B, and CŽ y, z . [ x g X: Ž x, y, z . g C 4 is closed and there exists Ž x 0 , y 0 . g Y = Z such that CŽ y , z . is compact; 0 0 Žiii. if z1 F z 2 , then BŽ y, z . ; BŽ y, z . for all y g Y and for any z g Z 2 1 there exists a ˆ z g Z such that BŽ y, z . > CŽ y, ˆz . for all y g Y; Živ. for any A g F Ž Y . and for any z g Z, F y g A BŽ y, z . is connected; Žv. for any Ž x, z . g X = Z, Y _ BŽ x, z . [ y g Y: Ž x, y, z . f B4 is generalized inter¨ al closed in Y and T-con¨ ex; Žvi. for any z g Z and for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that BŽ yXi , z . ; BŽ y i , z . , i s 1, 2. Then there exists an ˆ x g X such that ˆ x 4 = Ž Y = Z . ; C.
5. APPLICATIONS TO VARIATIONAL INEQUALITY PROBLEMS In this section we shall use the results presented in Section 2 to study the variational inequality problems in generalized interval spaces. For this purpose, we first give the following lemma. LEMMA 5.1 w4x. Let X be a topological space, Si : X ª R j q`4 be lower Ž upper . semi-continuous, i g I. Then sup i g I Si Žinf i g I Si . is lower Ž upper . semi-continuous. LEMMA 5.2 w1x. Let X and Y be two topological spaces, and W: X = Y ª R and G: Y ª 2 X be two mappings. Let V Ž y . s sup x g GŽ y . W Ž x, y .. Ž1. if W is lower semi-continuous on X = Y and G is lower semi-continuous at y 0 , then V is lower semi-continuous at y 0 ;
799
NONEMPTY INTERSECTION THEOREMS
Ž2. if W is upper semi-continuous on X = Y and G is upper semi-continuous at y 0 g Y and GŽ y 0 . is compact, then V is upper semi-continuous at y0 ; Ž3. if X is compact and W is lower semi-continuous on X = Y, then Ž . M y s inf x g X W Ž x, y . is lower semi-continuous on Y. THEOREM 5.3. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be a lower semi-continuous function, S : Y = X = Y ª R be a function, and T : Y ª 2 X be a upper semi-continuous S-monotone mapping with compact ¨ alues. If the following conditions are satisfied: Ži. y ¬ S Ž y, ¨ , z . is lower semi-continuous on Y, Ž ¨ , z . ¬ S Ž y, ¨ , z . is upper semi-continuous on X = Y, and for any y g Y, there exists ¨ g Ty such that S Ž y, ¨ , y . G 0; Žii. S Ž y, ¨ , z . q hŽ y . is T-quasi-con¨ ex in y on Y; Žiii. there exist y 0 g Y, r 0 - 0, u 0 g Ty 0 and a compact subset L ; Y, such that S Ž y 0 , u 0 , z . y hŽ z . - r 0 y hŽ y 0 . for all z g Y _ L; Živ. for any A g F Ž Y . and for r - 0, the set
F ½ z g Y:
ygA
sup S Ž y, ¨ , z . q h Ž y . y h Ž z . ) r
¨ gTz
5
is connected; Žv. for any y 1 , y 2 g Y, there exist yX1 , yX2 g G y 1 , y 2 4 such that sup S Ž yi , ¨ , z . q h Ž yi . G sup S Ž yiX , ¨ , z . q h Ž yXi .
¨ gTz
¨ gTz
for all z g Y , i s 1, 2; then there exists ˆ z g Y such that inf ug T y S Ž y, u, ˆ z . G hŽ ˆ z . y hŽ y . for all y g Y. Proof. Since T is S-monotone, for any y, z g Y and for any u g Ty,
¨ g Tz, we have S Ž y, u, z . G S Ž y, ¨ , z ., and so
inf S Ž y, u, z . G sup S Ž y, ¨ , z . .
ugTy
¨ gTz
Letting g Ž z, y . s inf S Ž y, u, z . q h Ž y . y h Ž z .
for all y, z g Y ,
f Ž z, y . s sup S Ž y, ¨ , z . q h Ž y . y h Ž z .
for all y, z g Y ,
ugTy
¨ gTz
then f Ž z, y . F g Ž z, y . for all Ž z, y . g Y = Y.
800
CHANG ET AL.
For any y g Y and for any r - 0, letting F Ž y, r . s z g Y : f Ž z, y . ) r 4 , G Ž y, r . s z g Y : f Ž z, y . G r 4 , H Ž y, r . s z g Y : g Ž z, y . G r 4 , it is easy to see that F Ž y, r . ; GŽ y, r . ; H Ž y, r . and condition Ži. in Corollary 2.4 is satisfied. For any r - 0, taking r such that r - r - 0 we know that condition Žii. in Corollary 2.4 is also satisfied. Since T is a upper semi-continuous mapping with compact values, h is lower semi-continuous, and by condition Ži. and Lemma 5.2Ž2., z ¬ f Ž z, y . is upper semi-continuous on Y. Hence GŽ y, r . is closed for all y g Y and for all r - 0. Next, by condition Ži. for any y g Y there exists ¨ g Ty such that S Ž y, ¨ , y . G 0, and so sup ¨ g T y S Ž y, ¨ , y . G 0. Hence for any y g Y and for any r - 0, we have y g F Ž y, r .. This implies that F has nonempty values. By condition Živ. for any A g F Ž Y ., F y g A z g Y: sup ¨ g T z S Ž y, ¨ , z . q hŽ y . y hŽ z . ) r 4 is connected, i.e., F y g A F Ž y, r . is connected. Hence condition Žiii. of Corollary 2.4 is also satisfied. By condition Žii., y ¬ f Ž z, y . is T-quasi-convex. Hence for any z g Y and for any r - 0, y g Y: z f F Ž y, r .4 s y g Y: f Ž z, y . F r 4 is T-convex. Since y ¬ S Ž y, ¨ , z . is lower semi-continuous and h is lower semicontinuous, the set y g Y: z f F Ž y, r .4 is closed for all z g Y and for all r - 0. This means that condition Živ. of Corollary 2.4 is satisfied. By condition Žv. we know that condition Žv. in Corollary 2.4 is also satisfied. Hence all conditions in Corollary 2.4 are satisfied. By Corollary 2.4Ž1., GŽ y, r .: y g Y, r - 04 has the finite intersection property, and so H Ž y, r .: y g Y, r - 04 also has the finite intersection property. Furthermore, by condition Ži., z ¬ S Ž y, ¨ , z . is upper semi-continuous, h is lower semicontinuous, and it follows from Lemma 5.1 that g Ž z, y . is upper semi-continuous in z. So H Ž y, r . is a closed set for all y g Y and for all r - 0. By condition Žiii., there exist y 0 g Y, r 0 - 0, u 0 g Ty 0 , and a compact subset L ; Y such that S Ž y 0 , u 0 , z . y hŽ z . - r 0 y hŽ y 0 . for all z g Y _ L. Hence we have inf S Ž y 0 , u, z . q h Ž y 0 . y h Ž z . - r 0
ugTy 0
for all z g Y _ L.
Therefore Y _ L ; z g Y: g Ž z, y 0 . - r 0 4 , i.e., H Ž y 0 , r 0 . ; L. Since L is compact, H Ž y 0 , r 0 . is compact. Hence we have F y g Y F r - 0 H Ž y, r . / B. Thus there exists ˆ z g Y such that ˆ z g H Ž y, r . for all y g Y and for all r - 0, i.e., g Ž ˆ z, y . G r for all r - 0 and for all y g Y. By the arbitrariness of r - 0, we know that g Ž ˆ z, y . G 0 for all y g Y, i.e., inf S Ž y, u, ˆ z . G hŽ ˆ z . y hŽ y .
ugTy
This completes the proof.
for all y g Y .
801
NONEMPTY INTERSECTION THEOREMS
Remark. Theorem 5.3 extends Theorem 2 in Yen w14x to the case of generalized interval spaces. THEOREM 5.4. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be lower semi-continuous, S : Y = X = Y ª R and T : Y ª 2 X be a upper semi-continuous mapping with compact ¨ alues. If conditions Ži., Žii., Živ., Žv. in Theorem 5.3 and the following condition Žiii.X are satisfied: Žiii.X there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that sup S Ž y 0 , ¨ , z . - h Ž z . y h Ž y 0 . q r 0
¨ gTz
for all z g Y _ L,
then there exists a ˆ z g Y such that sup S Ž y, ¨ , ˆ z . G hŽ ˆ z . y hŽ y .
for all y g Y .
¨ gTzˆ
Proof. Letting f Ž z, y . s sup S Ž y, ¨ , z . q h Ž y . y h Ž z . ¨ gTz
for all z, y g Y ,
F Ž y, r . s z g Y : f Ž z, y . ) r 4
for all y g Y and for all r - 0,
G Ž y, r . s z g Y : f Ž z, y . G r 4
for all y g Y and for all r - 0,
by the same methods as in the proof of Theorem 5.3, we can prove that GŽ y, r .: y g Y, r - 04 is a family of closed sets having the finite intersection property. Furthermore by condition Žiii. there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that sup ¨ g T z S Ž y 0 , ¨ , z . - hŽ z . y hŽ y 0 . q r 0 for all z g Y _ L. Hence we have sup S Ž y 0 , ¨ , z . q h Ž y 0 . y h Ž z . - r 0
¨ gTz
for all z g Y _ L.
This implies that Y _ L ; z g Y: sup ¨ g T z S Ž y 0 , ¨ , z . q hŽ y 0 . y hŽ z . r 0 4 . Hence we have G Ž y 0 , r 0 . s z g Y : sup S Ž y 0 , ¨ , z . q h Ž y 0 . y h Ž z . G r 0 ; L.
½
5
¨ gTz
Since L is compact and GŽ y 0 , r 0 . is closed, GŽ y 0 , r 0 . is compact. Hence F y g Y F r - 0 GŽ y, r . / B, and so there exist a ˆ z g GŽ y, r . for all y g Y and for all r - 0. Hence we have f Ž ˆ z, y . G r for all r - 0 and for all y g Y. By the arbitrariness of r - 0, we have f Ž ˆ z, y . G 0 for all y g Y, i.e., sup S Ž y, ¨ , ˆ z . G hŽ ˆ z . y hŽ y . ¨ gTzˆ
This completes the proof.
for all y g Y .
802
CHANG ET AL.
Remark. Theorem 5.4 improves and extends Theorem 3 of Yen w14x. If T is a single-valued mapping, then we have the following results. THEOREM 5.5. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be a lower semi-continuous function, and S : Y = X = Y ª R, T : Y ª X. If S Ž y, Ty, y . G 0 for all y g Y and the following conditions are satisfied: Ži. S Ž y, Ty, z . G S Ž y, Tz, z . for all Ž y, z . g Y = Y; Žii. S Ž y, Ty, z . is upper semi-continuous in z; S Ž y, Tz, z . is lower semi-continuous in y and upper semi-continuous in z; Žiii. S Ž y, Tz, z . q hŽ y . is T-quasi-con¨ ex in y; Živ. there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that S Ž y 0 , Ty 0 , z . y h Ž z . - r 0 y h Ž y 0 . for all z g Y _ L; Žv. for any A g F Ž Y ., F y g A z g Y: S Ž y, Tz, z . q hŽ y . y hŽ z . ) r 4 is connected or empty for all r - 0; Žvi. for any y 1 , y 2 g Y, there exist yX1 , yX2 g G y 1 , y 2 4 such that S Ž yi , Tz , z . q h Ž yi . G S Ž yiX , Tz , z . q h Ž yiX .
for all z g Y , i s 1, 2;
then there exists a ˆ z g Y such that S Ž y, Ty, ˆ z . G hŽ ˆ z . y hŽ y . for all y g Y. Proof. The proof is similar to one in Theorem 5.3, so we omit it here. Similar to the proof of Theorem 5.4, we can prove the following THEOREM 5.6. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be a lower semi-continuous function, and S : Y = X = Y ª R and T : Y ª X be two mappings satisfying S Ž y, Ty, y . G 0 for all y g Y and the following conditions: Ži. S Ž y, Tz, z . is lower semi-continuous in y and upper semi-continuous in z; Žii. S Ž y, Tz, z . q hŽ y . is T-quasi-con¨ ex in y; Žiii. there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that S Ž y 0 , Tz , z . y h Ž z . - r 0 y h Ž y 0 .
for all z g Y _ L;
Živ. for any r - 0 and for any A g F Ž Y ., F y g A z g Y: S Ž y, Tz, z . q hŽ y . y hŽ z . ) r 4 is connected; Žv. for any y 1 y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that S Ž yi , Tz , z . q h Ž yi . G S Ž yiX , Tz , z . q h Ž yiX .
for all y g Y , i s 1, 2.
Then there exists ˆ z g Y such that S Ž y, Tzˆ, ˆ z . G hŽ ˆ z . y hŽ y .
for all y g Y .
NONEMPTY INTERSECTION THEOREMS
803
Remark. Ky Fan’s famous theorem for implicit variational inequalities Žsee w5x. is a special case of Theorem 5.6 in which X s Y, h ' 0, and T is an identity mapping.
REFERENCES 1. J. P. Aubin and I. Ekeland, ‘‘Applied Nonlinear Analysis,’’ Wiley, New York, 1984. 2. H. Brezis, ´ L. Nirenberg, and G. Stampacchia, A remark on Ky Fan’s minimax principle, Boll. Un. Mat. Ital. (4) 6 Ž1972., 293]300. 3. Cao-zong Cheng and You-hao Lin, On the generalization of minimax theorem for the type of fixed points, Acta Math. Sinica 34 Ž1991., 502]507. 4. Shih-sen Chang, ‘‘Variational Inequality and Complementarity Problem Theory with Applications,’’ Shanghai Scientific and Technological Literature Publishing House, 1991. 5. Ky Fan, A minimax inequality and applications, in ‘‘Inequalities III,’’ pp. 103]113, Academic Press, New York, 1972. 6. Chung-wei Ha, A noncompact minimax theorem, Pacific J. Math. 97, No. 1 Ž1981.. 7. V. Komornik, Minimax theorems for upper semi-continuous functions, Acta Math. Acad. Sci. Hungar. 40 Ž1982., 159]163. 8. Bor-luh Lin and Xiu-chi Quan, A non-compact topological minimax theorem, J. Math. Anal. Appl. 161, No. 2 Ž1991., 587]590. 9. M. A. Geraghty and Bor-luh Lin, Topological minimax theorems, Proc. Amer. Math. Soc. 91, No. 3 Ž1984., 377]380. 10. M. Sion, On general minimax theorems, Pacific J. Math. 8 Ž1958., 171]176. 11. L. L. Stacho, ´ Minimax theorems beyond topological vector spaces, Acta Sci. Math. 42 Ž1980., 157]164. 12. Guo-qiang Tian, Generalization of the FKKM theorem and the Ky Fan minimax inequality with applications to maximal elements, Price equilibrium and complementarity, J. Math. Anal. Appl. 170 Ž1992., 457]471. 13. Wen-tsun Wu, A remark on the fundamental theorem in the theory of game, Sci. Record New Ser. 3, No. 6 Ž1959., 229]233. 14. Chi-lin Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97, No. 2 Ž1981., 477]481.
199, 787]803 Ž1996.
0176
Some Nonempty Intersection Theorems in Generalized Interval Spaces with ApplicationsU Shih-sen Chang Department of Mathematics, Sichuan Uni¨ ersity, Chengdu, Sichuan 610064, People’s Republic of China
Shi-yi Cao Department of Mathematics, Xinjiang Uni¨ ersity, Wulumuqi, Xinjiang 830046, People’s Republic of China
Xian Wu Department of Mathematics, Yunnan Normal Uni¨ ersity, Kumming, Yunnan 650092 People’s Republic of China
and Da-cheng Wang Department of Mathematics, Chongqing Teacher’s College, Yongchuan, Sichuan 632168, People’s Republic of China Submitted by Dorothy Maharam Stone Received March 29, 1994
In this paper some parametric types of KKM theorems are established in generalized interval spaces. As applications, we utilize these results to obtain some new minimax theorems, section theorems, and existence theorems of solutions for variational inequalities. The results presented in this paper not only include some important earlier results and the famous von Neumann theorems as their special Q 1996 Academic cases, but also improve and extend other corresponding results. Press, Inc.
U
Supported by the National Natural Science Foundation of China. 787 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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CHANG ET AL.
1. PRELIMINARIES DEFINITION 1. A topological space X is called a generalized interval space, if there exists a mapping G: X = X ª AŽ X ., where AŽ X . is a family of nonempty connected subsets of X. For any Ž x 1 , x 2 . g X = X we denote G Ž x 1 , x 2 . by G x 1 , x 2 4 and G x 1 , x 2 4 is called a generalized interval associated with x 1 and x 2 . DEFINITION 2. Let X be a generalized interval space, Y be a topological space, and Z be a completely dense linear ordered space. A subset B of X is called T-convex, if for any x 1 , x 2 g B, we have G x 1 , x 2 4 ; B. A mapping f : X ª Z is called T-quasi-concave Žconvex., if for any z g Z, the set x g X: f Ž x . G z 4 Ž x g X: f Ž x . F z 4. is T-convex. Let D be a subset of X. If for any G x 1 , x 2 4 , D l G x 1 , x 2 4 is a closed Žopen. set in G x 1 , x 2 4 , then D is called a generalized interval closed Žopen. set. Remark. From the above definitions, it is easy to see that interval space, H-space, convex space, contractible space, and topological linear space all can be considered as special cases of generalized interval space. Moreover the H-convex set in H-space is T-convex; the convex sets in topological vector space are also T-convex and the quasi-concave Žconvex. mapping is T-quasi-concave Žconvex.. We have to point out that the intersection of T-convex sets is T-convex and each T-convex set can be considered as a generalized interval space. DEFINITION 3 w4x. Let X be a topological space, Z be an order complete and order dense linear ordered space. A mapping f : X ª Z is called upper Žlower. semi-continuous, if the set x g X: f Ž x . G z 4 Ž x g X: f Ž x . F z 4. is a closed set in X for all z g Z. DEFINITION 4 w4x. Let X and Y be two topological spaces and S : Y = X = Y ª R be a function. T : Y ª 2 X is called S-monotone, if for any y, z g Y, for any u g Ty, and for any ¨ g Tz S Ž y, u, z . y S Ž y, ¨ , z . G 0. DEFINITION 5 w12x. Let X and Y be two topological spaces. A mapping G: Y ª 2 X is called transfer closed valued in Y if for any y g Y, x f GŽ y ., then there exists an yX g Y such that x f G Ž yX . Žin the sequel we denote by A the closure of A.. Remark. It is obvious that if G is closed valued in Y, then G is transfer closed valued on Y. In the sequel we denote F Ž Y . [ A ; Y: A is a nonempty finite set4 .
NONEMPTY INTERSECTION THEOREMS
789
2. PARAMETRIC TYPE OF KKM THEOREMS IN GENERALIZED INTERVAL SPACES In the sequel we need the following auxiliary lemmas: LEMMA 2.1 w12x. Let X, Y be two topological spaces and G: Y ª 2 X be a multi¨ alued mapping. Then G is transfer closed ¨ alued on Y if and only if F y g YG Ž y . s F y g Y GŽ y .. Proof. If G is transfer closed valued on Y, it is obvious that F y g YG Ž y . > F y g Y GŽ y .. To prove the conclusion we have to prove F y g YG Ž y . ; F y g Y GŽ y .. Suppose the contrary, then there exists x g F y g YG Ž y . and x f F y g Y GŽ y .. Hence there exists y g Y such that x f GŽ y .. Since G is transfer closed valued on Y, there exists yX g Y such that x f G Ž yX . . This contradicts x g F y g YG Ž y . . Conversely, if F y g YG Ž y . s F y g Y GŽ y ., then for any y g Y and any x f GŽ y ., then x f F y g Y GŽ y . s F y g YG Ž y . . Hence there exists yX g Y such that x f G Ž yX . . This implies that G is transfer closed valued on Y. LEMMA 2.2. Let Y be a generalized inter¨ al space, X be a topological space, and F: Y ª 2 X be a mapping. Then for any x g X, Y _ Fy1 Ž x . is a T-con¨ ex set if and only if for any y 1 , y 2 g Y we ha¨ e F Ž y . ; F Ž y1 . j F Ž y 2 .
for all y g G y 1 , y 2 4 .
Proof. Necessity. Suppose the contrary, there exist y 1 , y 2 g Y and an y 0 g G y 1 , y 2 4 such that F Ž y 0 . o F Ž y 1 . j F Ž y 2 .. Hence there exists x 0 g F Ž y 0 . but x 0 f F Ž yi ., i s 1, 2, and so yi f Fy1 Ž x 0 ., i s 1, 2, i.e., y 1 , y 2 4 ; Y _ Fy1 Ž x 0 .. Since Y _ Fy1 Ž x 0 . is T-convex, G y 1 , y 2 4 ; Y _ Fy1 Ž x 0 .. Hence y 0 g Y _ Fy1 Ž x 0 ., and so x 0 f F Ž y 0 .. This contradicts the choice of x 0 . Therefore for any y 1 , y 2 g Y, we have F Ž y . ; F Ž y1 . j F Ž y 2 .
for all y g G y 1 , y 2 4 .
Sufficiency. Since for any y 1 , y 2 g Y, we have F Ž y . ; F Ž y 1 . j F Ž y 2 . for all y g G y 1 , y 2 4 . Hence for any x g X and for any ˆ y1, ˆ y 2 g Y _ Fy1 Ž x ., FŽ ˆ y. ; FŽ ˆ y1 . j F Ž ˆ y 2 . for all ˆ y g G ˆ y1, ˆ y 2 4 . Since ˆ y1, ˆ y 2 f Fy1 Ž x ., i.e., x f FŽ ˆ y 1 . and x f F Ž ˆ y 2 ., and so x f F Ž ˆ y1 . j F Ž ˆ y 2 .. This implies that x f FŽ ˆ y . for all ˆ y g G y 1 , y 2 4 , i.e., for all ˆ y g G y 1 , y 2 4 , y g Y _ Fy1 Ž x .. Therefore G y 1 , y 2 4 ; Y _ Fy1 Ž x .. This shows that Y _ Fy1 Ž x . is a T-convex set for all x g X. This completes the proof. THEOREM 2.3. Let Y be a generalized inter¨ al space, X be a topological space, Z be a linear ordered space, F, G: Y = Z ª 2 X be two mappings such that F has nonempty ¨ alues and G is transfer closed ¨ alued. If the following
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conditions are satisfied: Ži. for any Ž y, z . g Y = Z, F Ž y, z . ; G Ž y, z . and when z1 F z 2 , F Ž y, z 2 . ; F Ž y, z1 .
for all y g Y ;
Žii. for any z g Z, there exists ˆ z g Z such that F Ž y, z . > G Ž y, ˆ z . for all y g Y; Žiii. for any A g F Ž Y . and for any z g Z, F y g A F Ž y, z . is connected; Živ. for any x g X and for any z g Z, the set y g Y: x f F Ž y, z .4 is T-con¨ ex and generalized inter¨ al closed; Žv. for any z g Z and any y 1 , y 2 g Y, there exist yX1 , yX2 g G y 1 , y 2 4 , such that F Ž yX1 , z . ; F Ž y 1 , z ., F Ž yX2 , z . ; F Ž y 2 , z ., then Ž1. G Ž y, z . : y g Y, z g Z 4 has the finite intersection property; Ž2. if there exists Ž y 0 , z 0 . g Y = Z such that G Ž y 0 , z 0 . is compact, then F y g Y , z g Z GŽ y, z . / B. Proof. Since F has nonempty values, by condition Ži., for all Ž y, z . g Y = Z, G Ž y, z . / B. If for any n elements of G Ž y, z . : y g Y, z g Z 4 their intersection is nonempty, next we prove that for any n q 1 elements of G Ž y, z . : y g Y, z g Z 4 their intersection is also nonempty, where n G 2. In fact, if there exist Ž yi , z i . g Y = Z, i s 1, 2, . . . , n q 1, such that nq 1 F is1 G Ž yi , z i . s B. Since Z is a linear ordered space, without loss of generality we can assume that z1 G z 2 G ??? G z n G z nq1 , letting H s nq 1 Ž F is3 F yi , z1 ., then we have H l F Ž y 1 , z1 . l H l F Ž y 2 , z1 . ; H l F Ž y 1 , z1 . l F Ž y 2 , z 2 . nq1
;
F G Ž yi , z i . s B. is1
This implies that H l F Ž y 1 , z1 . and H l F Ž y 2 , z1 . are separated from each other. By condition Žii. and the assumptions of induction, there exists ˆz g Z such that for any y g Y nq1
H l F Ž y, z1 . s
F F Ž yi , z1 . l F Ž y, z1 . is3
nq1
>
F G Ž yi , ˆz . l G Ž y, ˆz . / B. is3
Define a mapping L: Y ª 2 X by LŽ y . [ F Ž y, z1 ., y g Y, then for any x g X, Y _ Ly1 Ž x . s y g Y: x f F Ž y, z1 .4 . By condition Živ. for any x g
NONEMPTY INTERSECTION THEOREMS
791
X, Y _ Ly1 Ž x . is T-convex. In view of Lemma 2.2 for any y 1 , y 2 g Y and for any y g G y 1 , y 2 4 we have LŽ y . ; LŽ y 1 . j LŽ y 2 ., i.e., F Ž y, z1 . ; F Ž y 1 , z1 . j F Ž y 2 , z1 .
for all y g G y 1 , y 2 4 .
Hence we have H l F Ž y, z1 . ; Ž H l F Ž y 1 , z1 . . j Ž H l F Ž y 2 , z1 . . for all y g G y 1 , y 2 4 . By condition Žiii., H l F Ž y, z1 . is connected. In addition, since H l F Ž y 1 , z1 . and H l F Ž y 2 , z1 . are separated from each other, for any y g G y 1 , y 2 4 H l F Ž y, z1 . ; H l F Ž y 1 , z1 .
or
H l F Ž y, z1 . ; H l F Ž y 2 , z1 . .
Letting Ei s y g G y 1 , y 2 4 : H l F Ž y, z1 . ; H l F Ž yi , z1 .4 , i s 1, 2, by condition Žv., Ei / B, i s 1, 2, and G y 1 , y 2 4 s E1 j E2 . Since G y 1 , y 2 4 is nonempty connected, at least one of E1 l E2 and E1 l E2 is nonempty. Without loss of generality we can assume that E1 l E2 / B. Taking y 0 g E1 l E2 , we have H l F Ž y 0 , z1 . ; H l F Ž y 1 , z1 . and there exists a net ya 4a g I ; E2 such that ya ª y 0 . Hence we have H l F Ž ya , z1 . ; H l F Ž y 2 , z1 .
for all a g I.
On the other hand, in the above we have proved that H l F Ž y 0 , z1 . / B. Taking x 0 g H l F Ž y 0 , z1 ., we know that x 0 f H l F Ž y 2 , z1 .. Hence x 0 f H l F Ž ya , z1 .
for all a g I.
Therefore x 0 f F Ž ya , z1 . for all a g I, i.e., ya 4a g I ; Y _ Ly1 Ž x 0 .. Since ya 4a g I ; G y 1 , y 2 4 and Y _ Ly1 Ž x 0 . is generalized interval closed, y 0 g Y _ Ly1 Ž x 0 ., i.e., x 0 f F Ž y 0 , z1 .. This contradicts the choice of x 0 . By this contradiction, we know that G Ž y, z . : y g Y, z g Z 4 has the finite intersection property. If, in addition, there exists Ž y 0 , z 0 . g Y = Z such that G Ž y 0 , z 0 . is compact, then it is easy to prove that l y g Y , F z g Z G Ž y, z . / B. By Lemma 2.1 we have F y g Y , F z g Z GŽ y, z . / B. This completes the proof. COROLLARY 2.4. Let Y be a generalized inter¨ al space, X be a topological space, Z be a linear ordered space, and F and G: Y = Z ª 2 X be two mappings such that F has nonempty ¨ alues and G has closed ¨ alues. If the following conditions are satisfied: ha¨ e
Ži. for any Ž y, z . g Y = Z, F Ž y, z . ; GŽ y, z ., and when z1 F z 2 we F Ž y, z 2 . ; F Ž y, z1 .
for all y g Y ;
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CHANG ET AL.
Žii. for any z g Z there exists a ˆ z g Z such that F Ž y, z . > GŽ y, ˆ z . for all y g Y; Žiii. for any A g F Ž Y . and for any z g Z, F y g A F Ž y, z . is connected; Živ. for any x g X and for any z g Z, y g Y: x f F Ž y, z .4 is T-con¨ ex and generalized inter¨ al closed; Žv. for any z g Z and for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that F Ž yX1 , z . ; F Ž y 1 , z ., F Ž yX2 , z . ; F Ž y 2 , z ., then Ž1. GŽ y, z .: y g Y, z g Z 4 has the finite intersection property; Ž2. in addition, if there exists Ž y 0 , z 0 . g Y = Z such that GŽ y 0 , z 0 . is compact, then F y g Y , F z g Z GŽ y, z . / B. Proof. Since G has closed values, G is transfer closed valued on Y = Z. Hence the conclusion follows from Theorem 2.3 immediately.
3. APPLICATIONS TO MINIMAX PROBLEMS In this section we shall use the results presented in Section 2 to study the minimax problems. We have the following results: THEOREM 3.1. Let Y be a generalized inter¨ al space, X be a topological space, Z˜ be an order complete and order dense linear ordered space, and f, g: X = Y ª Z˜ be two functions satisfying the following conditions:
˜ F y g A x g X: f Ž x, y . ) z 4 is Ži. for any A g F Ž Y . and for any z g Z, a connected set; Žii. Ža. for any x g X, y ¬ f Ž x, y . is T-quasi-con¨ ex and is lower-semicontinuous on any generalized inter¨ al of Y; Žb. for any y g Y, x ¬ f Ž x, y . and x ¬ g Ž x, y . are upper semicontinuous; Žiii. there exist z 0 - inf y g Y sup x g X f Ž x, y . and y 0 g Y and a compact subset L ; X such that g Ž x, y 0 . - z 0 for all x g X _ L; Živ. f Ž x, y . F g Ž x, y . for all Ž x, y . g X = Y; Žv. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yiX . F f Ž x, yi ., i s 1, 2, and for any x g X. Then z# [ sup x g X inf y g Y g Ž x, y . G inf y g Y sup x g X f Ž x, y . \ zU .
˜ we know that zU and z# both exist. Proof. By the completeness of Z, U ˜ z - z 4, then Z is a dense linear ordered space. For Letting Z s z g Z: any Ž y, z . g Y = Z, letting F Ž y, z . s x g X: f Ž x, y . ) z 4 , GŽ y, z . s x g X: f Ž x, y . G z 4 , H Ž y, z . s x g X: g Ž x, y . G z 4 , it follows from condition Žii.Žb. that GŽ y, z . and H Ž y, z . both are closed for all Ž y, z . g Y = Z.
NONEMPTY INTERSECTION THEOREMS
793
Next we prove that mappings F and G satisfy all conditions of Corollary 2.4. In fact, by the definition of zU , F: Y = Z ª 2 X is nonempty valued. Again by the definition of F and G, it is easy to see that they satisfy condition Ži. in Corollary 2.4. In addition, for any z g Z with z - zU , by ˜ there exists a ˆz g Z˜ such that z - ˆz - zU and so the denseness of Z, ˆz g Z and F Ž y, z . > GŽ y, ˆz . for all y g Y. This implies that condition Žii. in Corollary 2.4 is satisfied. Again for any x g X and for any z g Z
y g Y : x f F Ž y, z . 4 s y g Y : f Ž x, y . F z 4 . By condition Žii.Ža. we know that condition Živ. in Corollary 2.4 is satisfied. By conditions Ži. and Žv. we know that conditions Žiii. and Žv. are satisfied. By conclusion Ž1. of Corollary 2.4, GŽ y, z .: y g Y, z g Z 4 has the finite intersection property. On the other hand, by condition Živ., for any Ž y, z . g Y = Z, GŽ y, z . ; Ž H y, z .. Hence H Ž y, z .: y g Y, z g Z 4 is a family of closed sets having the finite intersection property. By condition Žiii., for any x g X _ L, g Ž x, y 0 . - z 0 , i.e., x g x g X: g Ž x, y 0 . - z 0 4 . Hence x f H Ž y 0 , z 0 . s x g X: g Ž x, y 0 . G z 0 4 and so H Ž y 0 , z 0 . ; L. Since L is compact and H Ž y 0 , z 0 . is closed, H Ž y 0 , z 0 . is compact. Hence we have
F F H Ž y, z . s F F H Ž y, z . l H Ž y 0 , z 0 . / B.
ygY zgZ
ygY zgZ
Taking ˆ x g F y g Y F z g Z H Ž y, z ., we have g Ž ˆ x, y . G z for all y g Y and for all z g Z. Hence z# s sup x g X inf y g Y g Ž x, y . G z for all z g Z. By the denseness of Z˜ we have sup inf g Ž x, y . G zU s inf sup f Ž x, y . .
xgX ygY
ygY xgX
This completes the proof. COROLLARY 3.2. Let Y be a generalized inter¨ al space, X be a topological space, and Z be an order complete and order dense linear ordered space. If f : X = Y ª Z satisfies the following conditions: Ži. for any A g F Ž Y . and for any z g Z, F y g A x g X: f Ž x, y . ) z 4 is connected; Žii. Ža. for any x g X, y ¬ f Ž x, y . is T-quasi-con¨ ex, and is lower semi-continuous on any generalized inter¨ al of Y; Žb. for any y g Y, x ¬ f Ž x, y . is upper semi-continuous; Žiii. there exist z 0 - inf y g Y sup x g X f Ž x, y ., y 0 g Y, and a compact subset L ; X such that f Ž x, y 0 . - z 0 for all x g X _ L; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yXi . F f Ž x, yi . ,
i s 1, 2 and for all x g X ,
then sup x g X inf y g Y f Ž x, y . s inf y g Y sup x g X f Ž x, y ..
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CHANG ET AL.
Proof. The conclusion of Corollary 3.2 can be obtained from Theorem 3.1 immediately. Remark. The condition ‘‘there exist z 0 - inf y g Y sup x g X f Ž x, y ., y 0 g Y, and a compact subset L ; X such that f Ž x, y 0 . - z 0 for all x g X _ L’’ is equivalent to the condition ‘‘there exist z 0 - inf y g Y sup x g X f Ž x, y . and y 0 g Y such that x g X: f Ž x, y 0 . G z 0 4 is compact.’’ Therefore Corollary 3.2 includes the main results of Cheng and Lin w3x as its special cases and so it contains the corresponding results of Brezis, Nirenberg, and ´ Stampacchia w2x, Komornik w7x, M. A. Geraghty and Lin w9x, Stacho ´ w11x, and Wu w13x as its special cases. COROLLARY 3.3. Let Y be a compact generalized inter¨ al space, X be a compact topological space, Z be an order complete and order dense linear ordered space, and f : X = Y ª Z be a mapping satisfying the following conditions: Ži. for any A g F Ž Y . and for any z g Z, F y g A x g X: f Ž x, y . ) z 4 is connected or empty; Žii. for any x g X, f Ž x, y . is T-quasi-con¨ ex and lower semi-continuous in y; Žiii. for any y g Y, x ¬ f Ž x, y . is upper semi-continuous; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yX1 . F f Ž x, yi . for all x g X, i s 1, 2. Then f has a saddle Ž ˆ x, ˆ y . g X = Y. Proof. By using Corollary 3.2 and Proposition 1.4.6 and Theorem 3.10.4 of w4x, we can obtain the conclusion of Corollary 3.3 immediately. Remark. Corollary 3.3 contains the famous von Neumann theorem in mathematical economy and game theory. THEOREM 3.4. Let Y be a generalized inter¨ al space, X be a topological space, Z be an order complete and order dense linear ordered space, and f and g: X = Y ª Z be two mappings satisfying f Ž x, y . F g Ž x, y . for all Ž x, y . g X = Y and the following conditions: Ži. for any x g X, y ¬ f Ž x, y . is T-quasi-con¨ ex and lower semi-continuous; Žii. for any y g Y, f Ž x, y . and g Ž x, y . are upper semi-continuous in x; Žiii. there exist a nonempty subset K ; X and a compact subset H ; Y such that inf sup f Ž x, y . F
ygY xgX
inf
sup f Ž x, y . ,
ygY_H xgK
NONEMPTY INTERSECTION THEOREMS
795
and for any finite subset F ; X there exists a compact set K Ž F . > K j F, such that for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . ) z 4 is connected; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yiX . F f Ž x, yi . for all x g X, i s 1, 2. Then sup x g X inf y g Y g Ž x, y . G inf y g Y sup x g X f Ž x, y .. Proof. Letting z# s sup x g X inf y g Y g Ž x, y . and zU s inf y g Y sup x g X f Ž x, y ., by the completeness of Z, we know that z# and zU both exist. If z g Z such that z# - ˆ zz# - zU , again by the density of Z, there exists ˆ zU . For any x g X, letting LŽ x . s y g Y: f Ž x, y . F ˆ z 4 , by condition Ži., LŽ x . is a closed set in Y. For any x g X, letting M Ž x . s LŽ x . l ŽF t g K LŽ t .., then M Ž x . is a closed set. Next, we prove that M Ž x . ; H for all x g X. In fact, for any y 0 g Y _ H, by condition Žiii., we know that sup x g K f Ž x, y 0 . G zU ) ˆ z. Therefore there exists a x 0 g K such that f Ž x0 , y0 . ) ˆ z, i.e., y 0 f LŽ x 0 . and so y 0 f F t g K LŽ t .. This implies that F t g K LŽ t . ; H. Hence M Ž x . ; H for all x g X. Finally, we prove that M Ž x .: x g X 4 has the finite intersection property. In fact, for any finite set F ; X, by condition Žiii., there exists a compact subset K Ž F . > K j F such that for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .:: f Ž x, y . ) z 4 is connected. From Theorem 3.1 we have sup
inf g Ž x, y . G inf
xgK Ž F . ygY
sup f Ž x, y . ,
ygY xgK Ž F .
and so we have inf
sup f Ž x, y . F sup inf g Ž x, y . s zU - ˆ z.
ygY xgK Ž F .
xgX ygY
Now we prove that F x g K Ž F . M Ž x . / B. Suppose the contrary, F x g K Ž F . M Ž x . s B, then we have Y s D x g K Ž F .Ž Y _ M Ž x .., and so for any y g Y, there exists x Ž y . g K Ž F . such that y g Y _ M Ž x Ž y .., i.e., y f M Ž x Ž y .. s LŽ x Ž y .. l ŽF t g K LŽ t ... Hence y f LŽ x Ž y .. or y f F t g K LŽ t .. If y f LŽ x Ž y .., then f Ž x Ž y ., y . ) ˆ z; if y f F t g K LŽ t ., then there exists ˆ x Ž y . g K ; K Ž F . such that y f LŽ ˆ x Ž y .., i.e., f Ž ˆ x Ž y ., y . ) ˆ z. This implies that there exists a mapping ˆ x: Y ª K Ž F . such that f Ž ˆ x Ž y ., y . )ˆ z. Therefore we have sup x g K Ž F . f Ž x, y . ) ˆ z for all y g Y. Hence inf y g Y sup x g K Ž F . f Ž x, y . G ˆ z. This contradicts inf y g Y sup x g K Ž F . f Ž x, y . ˆz. Therefore F x g K Ž F . M Ž x . / B. Since F x g F M Ž x . > F x g K Ž F . M Ž x ., M Ž x .: x g X 4 has the finite intersection property.
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However, since H is compact and M Ž x . ; H for all x g X, F x g X M Ž x . / B. Hence there exists a ˆ y g M Ž x . for all x g X, and so ˆ y g LŽ x . for all x g X, i.e., f Ž x, ˆ y. F ˆ z for all x g X. Hence zU F ˆ z. This contradicts the choice of ˆ z. Therefore z# G zU . This completes the proof. Remark. Theorem 3.4 improves and extends Theorem 2 of Lin and Quan w8x. COROLLARY 3.5. Let Y be a generalized inter¨ al space, X be a topological space, Z be an order complete and order dense linear ordered space, and f : X = Y ª Z be a mapping satisfying the following conditions: Ži. for any x g X, f Ž x, y . is T-quasi-con¨ ex and lower semi-continuous in y; Žii. for any y g Y, x ¬ f Ž x, y . is upper semi-continuous; Žiii. there exist a nonempty subset K ; X and a compact subset H ; Y such that inf sup f Ž x, y . F
ygY xgX
inf
sup f Ž x, y . ,
ygY_H xgK
and for any finite set F ; X there exists a compact subset K Ž F . > K j F such that for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . ) z 4 is connected; Živ. for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that f Ž x, yXi . F f Ž x, yi . for all x g X , i s 1, 2. Then sup x g X inf y g Y f Ž x, y . s inf y g Y sup x g X f Ž x, y .. Proof. It follows from Theorem 3.4 that sup inf f Ž x, y . G inf sup f Ž x, y . .
xgX ygY
ygY xgX
However, it is obvious that inf y g Y sup x g X f Ž x, y . G sup x g X inf y g Y f Ž x, y .. Hence the conclusion is obtained. Remark. Since the condition ‘‘for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . G z 4 is connected’’ implies the condition ‘‘for any A g F Ž Y . and for any z g Z, F y g A x g K Ž F .: f Ž x, y . ) z 4 is connected,’’ therefore Corollary 3.5 contains Corollary 1 of Lin and Quan w8x as its special case. COROLLARY 3.6 w6x. Let X and Y be nonempty con¨ ex subsets of Hausdorff topological ¨ ector spaces M and N, respecti¨ ely, and f : X = Y ª R be a function satisfying the following conditions: Ži. for any x g X, y ¬ f Ž x, y . is quasi-con¨ ex and lower semi-continuous;
NONEMPTY INTERSECTION THEOREMS
797
Žii. for any y, x ¬ f Ž x, y . is quasi-conca¨ e and upper semi-continuous; Žiii. there exists a nonempty compact con¨ ex subset K ; X and a compact subset H ; Y such that inf sup f Ž x, y . F
ygY xgX
inf
sup f Ž x, y . .
ygY_H xgK
Then inf y g Y sup x g X f Ž x, y . s sup x g X inf y g Y f Ž x, y .. Proof. For any x 1 , x 2 g X and for any y 1 , y 2 g Y, letting G x 1 , x 2 4 s co x 1 , x 2 4 , G y 1 , y 2 4 s co y 1 , y 2 4 , under the structures like this, X and Y both are generalized interval space. Furthermore for any finite subset F ; X, letting K Ž F . s coŽ K j F ., then all conditions in Corollary 3.5 are satisfied. The conclusion follows from Corollary 3.5 immediately.
4. APPLICATIONS TO SECTION PROBLEMS In this section, we shall use the results presented in Section 2 to study the section problems. We have the following results. THEOREM 4.1. Let Y be a generalized inter¨ al space, X be a topological space, Z be a dense linear ordered space, and B and C be two sets of X = Y = Z satisfying the following conditions: Ži. B ; C; Žii. for any Ž y, z . g Y = Z, the sections BŽ y , z . [ x g X : Ž x, y, z . g B 4 / B, and CŽ y , z . [ x g X : Ž x, y, z . g C 4 are transfer closed ¨ alued and there exists Ž y 0 , z 0 . g Y = Z such that CŽ y 0 , z 0 . is compact; Žiii. if z 2 G z1 , then BŽ y, z . ; BŽ y, z . for all y g Y; and for any z g Z 2 1 there exists ˆ z g Z such that BŽ y, z . > CŽ y, ˆz . for all y g Y; Živ. for any A g F Ž Y . and for any z g Z, F y g A BŽ y, z . is connected; Žv. for any Ž x, z . g X = Z, the set Y _ BŽ x, z . [ y g Y: Ž x, y, z . f B4 is generalized inter¨ al closed and T-con¨ ex in Y; Žvi. for any z g Z and for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that BŽ yX , z . ; BŽ y i , z . , i s 1, 2.
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CHANG ET AL.
Then there exists an ˆ x g X such that ˆ x 4 = Ž Y = Z . ; C. Proof. Letting F Ž y, z . s BŽ y, z . , GŽ y, z . s CŽ y, z . , it is easy to see that F and G satisfy all conditions in Theorem 2.3. By conclusion Ž2. in Theorem 2.3, F y g Y , F z g Z GŽ y, z . / B. Hence there exists an ˆ x g X such that ˆx g GŽ y, z . for all Ž y, z . g Y = Z, i.e., Ž ˆx, y, z . g C for all Ž y, z . g Y = Z, and so ˆ x 4 = Ž Y = Z . ; C. COROLLARY 4.2. Let Y be a generalized inter¨ al space, X be a topological space, Z be a dense linear ordered space, and B and C be two subsets of X = Y = Z satisfying the following conditions: Ži. B ; C; Žii. for any Ž y, z . g Y = Z, the section BŽ y, z . [ x g X: Ž x, y, z . g B4 / B, and CŽ y, z . [ x g X: Ž x, y, z . g C 4 is closed and there exists Ž x 0 , y 0 . g Y = Z such that CŽ y , z . is compact; 0 0 Žiii. if z1 F z 2 , then BŽ y, z . ; BŽ y, z . for all y g Y and for any z g Z 2 1 there exists a ˆ z g Z such that BŽ y, z . > CŽ y, ˆz . for all y g Y; Živ. for any A g F Ž Y . and for any z g Z, F y g A BŽ y, z . is connected; Žv. for any Ž x, z . g X = Z, Y _ BŽ x, z . [ y g Y: Ž x, y, z . f B4 is generalized inter¨ al closed in Y and T-con¨ ex; Žvi. for any z g Z and for any y 1 , y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that BŽ yXi , z . ; BŽ y i , z . , i s 1, 2. Then there exists an ˆ x g X such that ˆ x 4 = Ž Y = Z . ; C.
5. APPLICATIONS TO VARIATIONAL INEQUALITY PROBLEMS In this section we shall use the results presented in Section 2 to study the variational inequality problems in generalized interval spaces. For this purpose, we first give the following lemma. LEMMA 5.1 w4x. Let X be a topological space, Si : X ª R j q`4 be lower Ž upper . semi-continuous, i g I. Then sup i g I Si Žinf i g I Si . is lower Ž upper . semi-continuous. LEMMA 5.2 w1x. Let X and Y be two topological spaces, and W: X = Y ª R and G: Y ª 2 X be two mappings. Let V Ž y . s sup x g GŽ y . W Ž x, y .. Ž1. if W is lower semi-continuous on X = Y and G is lower semi-continuous at y 0 , then V is lower semi-continuous at y 0 ;
799
NONEMPTY INTERSECTION THEOREMS
Ž2. if W is upper semi-continuous on X = Y and G is upper semi-continuous at y 0 g Y and GŽ y 0 . is compact, then V is upper semi-continuous at y0 ; Ž3. if X is compact and W is lower semi-continuous on X = Y, then Ž . M y s inf x g X W Ž x, y . is lower semi-continuous on Y. THEOREM 5.3. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be a lower semi-continuous function, S : Y = X = Y ª R be a function, and T : Y ª 2 X be a upper semi-continuous S-monotone mapping with compact ¨ alues. If the following conditions are satisfied: Ži. y ¬ S Ž y, ¨ , z . is lower semi-continuous on Y, Ž ¨ , z . ¬ S Ž y, ¨ , z . is upper semi-continuous on X = Y, and for any y g Y, there exists ¨ g Ty such that S Ž y, ¨ , y . G 0; Žii. S Ž y, ¨ , z . q hŽ y . is T-quasi-con¨ ex in y on Y; Žiii. there exist y 0 g Y, r 0 - 0, u 0 g Ty 0 and a compact subset L ; Y, such that S Ž y 0 , u 0 , z . y hŽ z . - r 0 y hŽ y 0 . for all z g Y _ L; Živ. for any A g F Ž Y . and for r - 0, the set
F ½ z g Y:
ygA
sup S Ž y, ¨ , z . q h Ž y . y h Ž z . ) r
¨ gTz
5
is connected; Žv. for any y 1 , y 2 g Y, there exist yX1 , yX2 g G y 1 , y 2 4 such that sup S Ž yi , ¨ , z . q h Ž yi . G sup S Ž yiX , ¨ , z . q h Ž yXi .
¨ gTz
¨ gTz
for all z g Y , i s 1, 2; then there exists ˆ z g Y such that inf ug T y S Ž y, u, ˆ z . G hŽ ˆ z . y hŽ y . for all y g Y. Proof. Since T is S-monotone, for any y, z g Y and for any u g Ty,
¨ g Tz, we have S Ž y, u, z . G S Ž y, ¨ , z ., and so
inf S Ž y, u, z . G sup S Ž y, ¨ , z . .
ugTy
¨ gTz
Letting g Ž z, y . s inf S Ž y, u, z . q h Ž y . y h Ž z .
for all y, z g Y ,
f Ž z, y . s sup S Ž y, ¨ , z . q h Ž y . y h Ž z .
for all y, z g Y ,
ugTy
¨ gTz
then f Ž z, y . F g Ž z, y . for all Ž z, y . g Y = Y.
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CHANG ET AL.
For any y g Y and for any r - 0, letting F Ž y, r . s z g Y : f Ž z, y . ) r 4 , G Ž y, r . s z g Y : f Ž z, y . G r 4 , H Ž y, r . s z g Y : g Ž z, y . G r 4 , it is easy to see that F Ž y, r . ; GŽ y, r . ; H Ž y, r . and condition Ži. in Corollary 2.4 is satisfied. For any r - 0, taking r such that r - r - 0 we know that condition Žii. in Corollary 2.4 is also satisfied. Since T is a upper semi-continuous mapping with compact values, h is lower semi-continuous, and by condition Ži. and Lemma 5.2Ž2., z ¬ f Ž z, y . is upper semi-continuous on Y. Hence GŽ y, r . is closed for all y g Y and for all r - 0. Next, by condition Ži. for any y g Y there exists ¨ g Ty such that S Ž y, ¨ , y . G 0, and so sup ¨ g T y S Ž y, ¨ , y . G 0. Hence for any y g Y and for any r - 0, we have y g F Ž y, r .. This implies that F has nonempty values. By condition Živ. for any A g F Ž Y ., F y g A z g Y: sup ¨ g T z S Ž y, ¨ , z . q hŽ y . y hŽ z . ) r 4 is connected, i.e., F y g A F Ž y, r . is connected. Hence condition Žiii. of Corollary 2.4 is also satisfied. By condition Žii., y ¬ f Ž z, y . is T-quasi-convex. Hence for any z g Y and for any r - 0, y g Y: z f F Ž y, r .4 s y g Y: f Ž z, y . F r 4 is T-convex. Since y ¬ S Ž y, ¨ , z . is lower semi-continuous and h is lower semicontinuous, the set y g Y: z f F Ž y, r .4 is closed for all z g Y and for all r - 0. This means that condition Živ. of Corollary 2.4 is satisfied. By condition Žv. we know that condition Žv. in Corollary 2.4 is also satisfied. Hence all conditions in Corollary 2.4 are satisfied. By Corollary 2.4Ž1., GŽ y, r .: y g Y, r - 04 has the finite intersection property, and so H Ž y, r .: y g Y, r - 04 also has the finite intersection property. Furthermore, by condition Ži., z ¬ S Ž y, ¨ , z . is upper semi-continuous, h is lower semicontinuous, and it follows from Lemma 5.1 that g Ž z, y . is upper semi-continuous in z. So H Ž y, r . is a closed set for all y g Y and for all r - 0. By condition Žiii., there exist y 0 g Y, r 0 - 0, u 0 g Ty 0 , and a compact subset L ; Y such that S Ž y 0 , u 0 , z . y hŽ z . - r 0 y hŽ y 0 . for all z g Y _ L. Hence we have inf S Ž y 0 , u, z . q h Ž y 0 . y h Ž z . - r 0
ugTy 0
for all z g Y _ L.
Therefore Y _ L ; z g Y: g Ž z, y 0 . - r 0 4 , i.e., H Ž y 0 , r 0 . ; L. Since L is compact, H Ž y 0 , r 0 . is compact. Hence we have F y g Y F r - 0 H Ž y, r . / B. Thus there exists ˆ z g Y such that ˆ z g H Ž y, r . for all y g Y and for all r - 0, i.e., g Ž ˆ z, y . G r for all r - 0 and for all y g Y. By the arbitrariness of r - 0, we know that g Ž ˆ z, y . G 0 for all y g Y, i.e., inf S Ž y, u, ˆ z . G hŽ ˆ z . y hŽ y .
ugTy
This completes the proof.
for all y g Y .
801
NONEMPTY INTERSECTION THEOREMS
Remark. Theorem 5.3 extends Theorem 2 in Yen w14x to the case of generalized interval spaces. THEOREM 5.4. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be lower semi-continuous, S : Y = X = Y ª R and T : Y ª 2 X be a upper semi-continuous mapping with compact ¨ alues. If conditions Ži., Žii., Živ., Žv. in Theorem 5.3 and the following condition Žiii.X are satisfied: Žiii.X there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that sup S Ž y 0 , ¨ , z . - h Ž z . y h Ž y 0 . q r 0
¨ gTz
for all z g Y _ L,
then there exists a ˆ z g Y such that sup S Ž y, ¨ , ˆ z . G hŽ ˆ z . y hŽ y .
for all y g Y .
¨ gTzˆ
Proof. Letting f Ž z, y . s sup S Ž y, ¨ , z . q h Ž y . y h Ž z . ¨ gTz
for all z, y g Y ,
F Ž y, r . s z g Y : f Ž z, y . ) r 4
for all y g Y and for all r - 0,
G Ž y, r . s z g Y : f Ž z, y . G r 4
for all y g Y and for all r - 0,
by the same methods as in the proof of Theorem 5.3, we can prove that GŽ y, r .: y g Y, r - 04 is a family of closed sets having the finite intersection property. Furthermore by condition Žiii. there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that sup ¨ g T z S Ž y 0 , ¨ , z . - hŽ z . y hŽ y 0 . q r 0 for all z g Y _ L. Hence we have sup S Ž y 0 , ¨ , z . q h Ž y 0 . y h Ž z . - r 0
¨ gTz
for all z g Y _ L.
This implies that Y _ L ; z g Y: sup ¨ g T z S Ž y 0 , ¨ , z . q hŽ y 0 . y hŽ z . r 0 4 . Hence we have G Ž y 0 , r 0 . s z g Y : sup S Ž y 0 , ¨ , z . q h Ž y 0 . y h Ž z . G r 0 ; L.
½
5
¨ gTz
Since L is compact and GŽ y 0 , r 0 . is closed, GŽ y 0 , r 0 . is compact. Hence F y g Y F r - 0 GŽ y, r . / B, and so there exist a ˆ z g GŽ y, r . for all y g Y and for all r - 0. Hence we have f Ž ˆ z, y . G r for all r - 0 and for all y g Y. By the arbitrariness of r - 0, we have f Ž ˆ z, y . G 0 for all y g Y, i.e., sup S Ž y, ¨ , ˆ z . G hŽ ˆ z . y hŽ y . ¨ gTzˆ
This completes the proof.
for all y g Y .
802
CHANG ET AL.
Remark. Theorem 5.4 improves and extends Theorem 3 of Yen w14x. If T is a single-valued mapping, then we have the following results. THEOREM 5.5. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be a lower semi-continuous function, and S : Y = X = Y ª R, T : Y ª X. If S Ž y, Ty, y . G 0 for all y g Y and the following conditions are satisfied: Ži. S Ž y, Ty, z . G S Ž y, Tz, z . for all Ž y, z . g Y = Y; Žii. S Ž y, Ty, z . is upper semi-continuous in z; S Ž y, Tz, z . is lower semi-continuous in y and upper semi-continuous in z; Žiii. S Ž y, Tz, z . q hŽ y . is T-quasi-con¨ ex in y; Živ. there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that S Ž y 0 , Ty 0 , z . y h Ž z . - r 0 y h Ž y 0 . for all z g Y _ L; Žv. for any A g F Ž Y ., F y g A z g Y: S Ž y, Tz, z . q hŽ y . y hŽ z . ) r 4 is connected or empty for all r - 0; Žvi. for any y 1 , y 2 g Y, there exist yX1 , yX2 g G y 1 , y 2 4 such that S Ž yi , Tz , z . q h Ž yi . G S Ž yiX , Tz , z . q h Ž yiX .
for all z g Y , i s 1, 2;
then there exists a ˆ z g Y such that S Ž y, Ty, ˆ z . G hŽ ˆ z . y hŽ y . for all y g Y. Proof. The proof is similar to one in Theorem 5.3, so we omit it here. Similar to the proof of Theorem 5.4, we can prove the following THEOREM 5.6. Let Y be a generalized inter¨ al space, X be a topological space, h: Y ª R be a lower semi-continuous function, and S : Y = X = Y ª R and T : Y ª X be two mappings satisfying S Ž y, Ty, y . G 0 for all y g Y and the following conditions: Ži. S Ž y, Tz, z . is lower semi-continuous in y and upper semi-continuous in z; Žii. S Ž y, Tz, z . q hŽ y . is T-quasi-con¨ ex in y; Žiii. there exist y 0 g Y, r 0 - 0, and a compact subset L ; Y such that S Ž y 0 , Tz , z . y h Ž z . - r 0 y h Ž y 0 .
for all z g Y _ L;
Živ. for any r - 0 and for any A g F Ž Y ., F y g A z g Y: S Ž y, Tz, z . q hŽ y . y hŽ z . ) r 4 is connected; Žv. for any y 1 y 2 g Y there exist yX1 , yX2 g G y 1 , y 2 4 such that S Ž yi , Tz , z . q h Ž yi . G S Ž yiX , Tz , z . q h Ž yiX .
for all y g Y , i s 1, 2.
Then there exists ˆ z g Y such that S Ž y, Tzˆ, ˆ z . G hŽ ˆ z . y hŽ y .
for all y g Y .
NONEMPTY INTERSECTION THEOREMS
803
Remark. Ky Fan’s famous theorem for implicit variational inequalities Žsee w5x. is a special case of Theorem 5.6 in which X s Y, h ' 0, and T is an identity mapping.
REFERENCES 1. J. P. Aubin and I. Ekeland, ‘‘Applied Nonlinear Analysis,’’ Wiley, New York, 1984. 2. H. Brezis, ´ L. Nirenberg, and G. Stampacchia, A remark on Ky Fan’s minimax principle, Boll. Un. Mat. Ital. (4) 6 Ž1972., 293]300. 3. Cao-zong Cheng and You-hao Lin, On the generalization of minimax theorem for the type of fixed points, Acta Math. Sinica 34 Ž1991., 502]507. 4. Shih-sen Chang, ‘‘Variational Inequality and Complementarity Problem Theory with Applications,’’ Shanghai Scientific and Technological Literature Publishing House, 1991. 5. Ky Fan, A minimax inequality and applications, in ‘‘Inequalities III,’’ pp. 103]113, Academic Press, New York, 1972. 6. Chung-wei Ha, A noncompact minimax theorem, Pacific J. Math. 97, No. 1 Ž1981.. 7. V. Komornik, Minimax theorems for upper semi-continuous functions, Acta Math. Acad. Sci. Hungar. 40 Ž1982., 159]163. 8. Bor-luh Lin and Xiu-chi Quan, A non-compact topological minimax theorem, J. Math. Anal. Appl. 161, No. 2 Ž1991., 587]590. 9. M. A. Geraghty and Bor-luh Lin, Topological minimax theorems, Proc. Amer. Math. Soc. 91, No. 3 Ž1984., 377]380. 10. M. Sion, On general minimax theorems, Pacific J. Math. 8 Ž1958., 171]176. 11. L. L. Stacho, ´ Minimax theorems beyond topological vector spaces, Acta Sci. Math. 42 Ž1980., 157]164. 12. Guo-qiang Tian, Generalization of the FKKM theorem and the Ky Fan minimax inequality with applications to maximal elements, Price equilibrium and complementarity, J. Math. Anal. Appl. 170 Ž1992., 457]471. 13. Wen-tsun Wu, A remark on the fundamental theorem in the theory of game, Sci. Record New Ser. 3, No. 6 Ž1959., 229]233. 14. Chi-lin Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97, No. 2 Ž1981., 477]481.