Coincidence and Fixed Points on Product Generalized Convex Spaces

Transcription

1 International Mathematical Forum, 5, 2010, no. 19, Coincidence and Fixed Points on Product Generalized Convex Spaces Yew-Dann Chen Center for General Education Southern Taiwan University 1 Nan-Tai St. Yung-Kang City Tainan Hsien 710, Taiwan chen77@mail.stut.edu.tw Young-Ye Huang Center for General Education Southern Taiwan University 1 Nan-Tai St. Yung-Kang City Tainan Hsien 710, Taiwan yueh@mail.stut.edu.tw Tian-Yuan Kuo Fooyin University 151 Chin-Hsueh Rd. Ta-Liao Hsiang, Kaohsiung Hsien 831, Taiwan sc038@mail.fy.edu.tw Jyh-Chung Jeng Nan-Jeon Institute of Technology Yen-Shui, Tainan Hsien 737, Taiwan jhychung@mail.njtc.edu.tw Abstract In this paper we at first establish some new results of coincidence and fixed points for a family of multimaps on a product G-convex space, and then apply them to the study of system of inequalities, minimax problem, maximal elements and equilibrium points of abstract economy.

2 904 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng Mathematics Subject Classification: 47H10, 54H25 Keywords: coincidence point, fixed point, G-convex space 1. Introduction and Preliminaries Recently, the theory of coincidence and fixed points for a family of multimaps on a product space of topological vector spaces has been investigated by many authors, cf.[1], [2], [7], [8], [12] and the references therein. In this paper we shall make a study of this type of problems on a product G-convex space instead of a product topological vector space. We at first establish some new results of coincidence and fixed points for a family of multimaps on a product G-convex space, and then apply them to the study of system of inequalities, minimax problem, maximal elements and equilibrium points of abstract economy. We begin with recalling some notations and terminology concerned with multimaps which will be used throughout the paper. For a nonempty set Y,2 Y and Y denote the class of all subsets of Y and the class of all nonempty finite subsets of Y respectively. For A Y, A denotes the cardinality of A. A multimap T : X 2 Y is a function from a set X into the power set 2 Y of Y. The notation T : X Y stands for a multimap T : X 2 Y having nonempty values. If B Y, then the inverse image of B under T is T (B) ={x X : T (x) B }. All topological spaces are supposed to be Hausdorff. The closure of a subset A of a topological space is denoted by cl(a). A subset A of a topological space X is compactly open if for every nonempty compact subset K of X, A K is open in K. The compactly interior of A is defined by cf.[3]. cint(a) = {B X : B A and B is compactly open in X}, Definition 1.1.[13]. Let X and Y be two topological spaces. A multimap T : X 2 Y is said to be transfer compactly open valued if for any x X and y T (x) there is an x X such that y cint(t (x )). A function f : X Y R is said to be transfer compactly l.s.c. ( resp. u.s.c.) on Y if for each y Y and each λ R with y {z Y : f(x, z) >λ} ( resp. y {z Y : f(x, z) <λ}) there exists an x X such that y cint({z Y : f(x,z) >λ}) (resp. y cint({z Y : f(x,z) <λ})).

3 Coincidence and fixed points 905 The following lemmas are well-known. Lemma 1.2.[8] Let X and Y be two topological spaces and f : X Y R. Then f is transfer compactly l.s.c. (resp. u.s.c.) on Y if and only if for any λ R, the multimap F : X 2 Y defined by is transfer compactly open valued. F (x) ={y Y : f(x, y) >λ} (resp. F (x) ={y Y : f(x, y) <λ}) Lemma 1.3.[8] Let X and Y be two topological spaces and T : X 2 Y. Then the following two statements are equivalent. (1.3.1) For each x X, T (x) is nonempty and T : Y 2 X is transfer compactly open valued. (1.3.2) X = y Y cint(t (y)). For n 0, Δ n denotes the standard n-simplex of R n+1, that is, { } n Δ n = α =(α 0,,α n ) R n+1 : α i 0 for all i and α i =1 ; and {e 0,, e n }, the standard basis of R n+1, is the set of the vertices of Δ n. The notion of a generalized convex space was first introduced by Park and Kim [10]: Definition 1.4. A generalized convex space or a G-convex space (E;Γ) consists of a topological space E and a map Γ: E E such that (1.4.1) for any A, B E, A B implies Γ(A) Γ(B); and (1.4.2) for each A = {a 0,,a n } E with A = n +1, there exists a continuous function ϕ A :Δ n Γ(A) such that ϕ A (Δ J ) Γ(B) for any B A with B = J, where J {0,,n} and Δ J denotes the face of Δ n corresponding to B. In this paper, we assume that a G-convex space (E;Γ)always satisfies the extra condition: x Γ({x}) for any x E, and the Γ is also denoted by Γ E if it is necessary. A subset K of a G-convex space (E;Γ) is said to be Γ-convex if for any i=0

4 906 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng A K, Γ(A) K. For a nonempty subset Q of E, the Γ-convex hull of Q, denoted by Γ-co(Q), is defined by Γ-co(Q) = {C : Q C E,C is Γ-convex}. Q. It is easy to see that Γ-co(Q) is the smallest Γ-convex subset of E containing Definition 1.5. A G-convex uniform space (E; U, Γ) is a G-convex space so that its topology is induced by a uniformity U. AG-convex uniform space (E; U, Γ) is said to be a locally G-convex uniform space if the uniformity U has a base B consisting of open symmetric entourages such that for each V B and any x E, V [x] :={y X :(x, y) V } is Γ-convex. For details of uniform spaces we refer to Kelley [6]. Tan and Zhang [11] showed that the product of an arbitrary family of G-convex spaces is a G-convex space: Suppose {(E i ;Γ i )} i I is any family of G-convex spaces. Let E =Π i I E i be equipped with product topology. For each i I, let π i : E E i be the projection. Define Γ=Π i I Γ i : E E by Γ(A) =Π i I Γ i (π i (A)) for each A E. Then (E;Γ) is a G-convex space. The following Lemmas 1.6, 1.7 and 1.8 which will be quoted in section 2 can be found in [4] and [5]. Lemma 1.6. Let (E;Γ) = (Π i I E i ;Π i I Γ i ) be the product G-convex space of a family of G-convex spaces (E i ;Γ i ), i I. Then K =Π i I K i is Γ-convex in E provided for each i I, K i is a Γ i -convex subset of E i. Lemma 1.7. Suppose {(E i ; U i, Γ i )} i I is any family of locally G-convex uniform spaces. Let E =Π i I E i be equipped with product topology and U =Π i I U i be the product uniformity on E. Then (E; U, Γ) is a locally G-convex uniform space. Lemma 1.8. If K is a Γ-convex subset of a locally G-convex uniform space (E; U, Γ), then its closure cl(k) is also Γ-convex. 2. Coincidence and Fixed Point Theorems

5 Coincidence and fixed points 907 Throughout this paper, (E;Γ) = (Π i I E i ;Π i I Γ i ) denotes the product G- convex space of a family of G-convex spaces (E i ;Γ i ), i I, and for each i I, let E i =Π j I\{i} E j be equipped with the product convex structure Γ i =Π j I\{i} Γ j. Any point x E is expressed as x =[x i,x i ], where x i E i and x i E i. Following Park [9], we make the following definition. Definition 2.1. Let X be a topological space and (E;Γ) a G-convex space. A multimap T : X 2 E is called a generalized Φ-mapping if there exists a multimap S : X 2 E satisfying that (2.1.1) for any x X and any A S(x), Γ-co(A) T (x); (2.1.2) X = y E cint(s (y)). The mapping S is called a companion mapping of T. We are now able to establish the following coincidence result which generalizes the Theorem 3.1 of Lin [8] from topological vector spaces to G-convex spaces. Theorem 2.2. Suppose for each i I, the two multimaps F i : E i 2 Ei and T i : E i 2 E i satisfy the following conditions: (2.2.1) F i is a generalized Φ-mapping with the companion mapping H i ; (2.2.2) T i is a generalized Φ-mapping with the companion mapping S i ; (2.2.3) There exists a nonempty compact subset M i of E i such that for each Q i E i there is a compact Γ i -convex subset L Q i of E i with and Q i L Q i E i \ M i x i L Q i cint(h i (xi )); (2.2.4) There exists a nonempty compact subset K i of E i such that for each N i E i there is a compact Γ i -convex subset L Ni of E i with N i L Ni and E i \ K i yi L Ni cint(s i (y i)). Then there exist x =( x i ) i I and ȳ =(ȳ i ) i I in E such that for each i I, ȳ i T i ( x i ) and x i F i (ȳ i ). Proof. For each i I, we have by (2.2.1) E i = x i E icint(h i (xi )),

6 908 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng and then the compactness of M i implies there is Q i E i such that In a like manner, there is N i E i such that By (2.2.3), we have It follows from (1) and (3) that M i x i Q icint(h i (xi )). (1) K i yi N i cint(s i (y i)). (2) L Ni \ M i E i \ M i x i L Q i cint(h i (xi )). (3) Similarly, (2) and (2.2.4) imply that L Ni x i L Q i cint(h i (xi )). (4) L Q i yi L Ni cint(s i (y i)). (5) Noting that both of L Ni and L Q i are compact, it follows from (4) and (5) that there are {a i 0,,a i m i } L Q i and {bi,0,,b i,li } L Ni such that and L Ni m i j=0 cint(h i (ai j )) (6) L Q i l i j=0 cint(s i (b i,j)) (7) Let A i = cl(γ i ({a i 0,,a i m i })) and B i = cl(γ i ({b i,0,,b i,li })). B i and A i are closed subsets of L Ni and L Q i respectively, and we have from (6) and (7) that ( ) B i = m i j=0 cint(h i (ai j)) B i (8) ( A i = l i j=0 cint(s i (b i,j )) A i). (9) By the definition of a G-convex space, there exists a continuous mapping ϕ i :Δ mi A i such that ϕ i (Δ J ) Γ i (C) A i (10) for any C {a i 0,,a i m i } with C = J, where J {0,,m i }. Meanwhile, since B i is compact, (8) shows that there is a partition of unity {λ i,j } m i j=0 subordinated to { cint ( Hi (ai j) ) } mi B i. For any j {0,,m j=0 i} and any x i B i, we have λ i,j (x i ) 0 x i cint ( H i (ai j )) B i a i j H i(x i ). (11)

7 Coincidence and fixed points 909 Define a mapping ψ i : B i Δ mi by m i ψ i (x i )= λ i,j (x i )e j. j=0 Clearly, ψ i is continuous, and for any x i B i, one has ψ i (x i )= λ i,j (x i )e j Δ J(xi ), j J(x i ) where J(x i )={j {0,,m i } : λ i,j (x i ) 0}. We see from (11) that {a i j : j J(x i )} H i (x i ), so in view of F i being a generalized Φ-mapping with the companion mapping H i, we conclude that Γ i ( {a i j : j J(x i)} ) Γ i -co ( {a i j : j J(x i)} ) F i (x i ). Consequently, the function f i : B i A i defined by f i (x i )=ϕ i ψ i (x i ) has the property that f i (x i )=ϕ i (ψ i (x i )) ϕ ( ) i Δ J(xi ) Γ i (C), where C = {a i j : j J(x i)} F i (x i ), that is, f i is a continuous selection of F i Bi. Reasoning as in the last paragraph, there are continuous functions ϕ i : Δ li B i and ψ i : A i Δ li such that their composition g i = ϕ i ψ i : A i B i is a continuous selection of T i A i, that is, g i (x i ) T i (x i ), x i A i. let Δ = Π i I Δ li, which is a compact convex subset of the locally convex topological space Π i I R l i+1. Define a continuous mapping Ω : Δ Δby Ω(t) =Π i I ( ψ i ϕ i ψ i ϕ i π i ) (t) where π i is the projection of Δ onto Δ li. By Tychonoff s fixed point theorem, there is t =( t i ) i I in Δ such that t =Ω( t) =Π i I ψ i (ϕ i ψ i )(ϕ i ( t i )).

8 910 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng For each i I, let ȳ i = ϕ i ( t i ), x i = f i (ȳ i ), and put x =( x i ) i I, ȳ =(ȳ i ) i I. Then from t =( t i ) i I =Π i I ψ i (f i (ȳ i )) = Π i I ψ i ( x i ) we obtain that and t i = ψ i ( x i ) ( ȳ i = ϕ i ( t i )=ϕ i ψ i ( x i ) ) = g i ( x i ) T i ( x i ) x i = f i (ȳ i ) F i (ȳ i ) for any i I. This completes the proof. Theorem 2.3. For any i I, let (E i ; U i, Γ i ) be a locally G-convex uniform space so that the convex structure Γ i has the property that Γ i -co(c i ) is compact whenever C i is a compact subset of E i, and suppose F i : E i 2 Ei and T i : E i 2 E i satisfy the following conditions: (2.3.1) Both of F i and T i are generalized Φ-mappings with the companion mappings H i and S i respectively. (2.3.2) there exist a nonempty compact subset M i of E i and a Q i E i such that E i \ M i x i Q icint ( H i (xi ) ). (2.3.3) there exist a nonempty compact subset K i of E i and an N i E i such that E i \ K i yi N i cint ( S i (y i) ). Then there exist x =( x i ) i I and ȳ =(ȳ i ) i I in E such that for each i I, ȳ i T i ( x i ) and x i F i (ȳ i ). Proof. For each i I, let L Ni =Γ i -co (M i N i ) L Q i =cl ( Γ i -co(k i Q i ) ). By the assumption on the convex structure Γ i, L Ni subset of E i containing N i so that is a compact Γ i -convex E i \ K i yi L Ni cint ( S i (y i) ). And it follows from Lemmas 1.6, 1.7 and 1.8 that L Q i subset of E i containing Q i so that is a compact Γ i -convex E i \ M i x i L Q i cint ( H i (xi ) ).

9 Coincidence and fixed points 911 Then, following the same argument as in the proof of the above theorem, the conclusion follows. Essentially the same way of thinking as in the proof of Theorem 2.2, we can prove the following fixed point theorem. Theorem 2.4. Suppose for each i I, T i : E 2 E i satisfies the following conditions: (2.4.1) T i is a generalized Φ-mapping with the companion mapping S i. (2.4.2) There exists a nonempty compact subset K i of E i such that for any N i E i there is a compact Γ i -convex subset L Ni of E i with N i L Ni and E \ K yi L Ni cint ( S i (y i) ), where K =Π i I K i. Then there exists an x =( x i ) i I E such that x i T i ( x) for each i I. Proof. Since K is a compact subset of E and since E = yi E i cint ( S i (y i) ), there is an N i E i such that K yi N i cint ( S i (y i) ). (12) By (2.4.2) there is a compact Γ i -convex subset L Ni and of E i such that N i L Ni E \ K yi L Ni cint ( S i (y i) ). (13) Let N =Π i I N i and L N =Π i I L Ni. L N is a compact Γ-convex subset of E containing N. Noting that L N \ K E \ K, it follows from (13) that L N \ K yi L Ni cint ( S i (y i) ). (14) Since N i L Ni we have from (12) and (14) that L N yi L Ni cint ( S i (y i) ), and hence the compactness of L N implies that there is A i = {y i,0,,y i,li } L Ni with L N = l i j=0 ( cint ( S i (y i,j) ) L N ). (15) By the definition of a G-convex space, there exists a continuous mapping ϕ i :Δ li Γ(A i ) such that ϕ i (Δ J ) Γ(C) Γ(A i ) L Ni (16)

10 912 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng for any C A i with C = J, where J {0,,l i }. Besides, (15) shows that there is a partition of unity {λ i,j } l i j=0 subordinated to { cint ( S i (y i,j) ) } li L N. j=0 For any j {0,,l i } and any x L N, we have y i,j S i (x) provided that λ i,j (x) 0. Define a mapping ψ i : L N Δ li by ψ i (x) = l i j=0 λ i,j (x)e j. Each ψ i is continuous, and for any x L N, one has ψ i (x) = λ i,j (x)e j Δ J(x), j J(x) where J(x) ={j {0,,l i } : λ i,j (x) 0}. Noting that {y i,j : j J(x)} S i (x) and T i is a generalized Φ-mapping with the companion mapping S i, we see from (16) that the function f i : L N L Ni defined by f i (x) =ϕ i ψ i (x) has the property that for any x L N, f i (x) =ϕ i ψ i (x) ( ) ϕ i ΔJ(x) Γ(C), where C = {y i,j : j J(x)} T i (x). Let Δ = Π i I Δ li. Define continuous mappings Ω : Δ L N and Ψ : L N Δ by Ω(t) =Π i I ϕ i (π i (t)) for t Δ, and Ψ(x) =Π k I ψ k (x) for x L N, where π i :Δ Δ li is the projection of Δ onto Δ li. By Tychonoff s fixed point theorem, the continuous function Ψ Ω:Δ Δ has a fixed point t. Let x =Ω( t). Then, x =Ω( t) =Ω(Ψ Ω( t)) =(Ω Ψ) (Ω( t)) =(Ω Ψ) ( x) =Ω(Π k I ψ k ( x)) =Π i I ϕ i (π i (Π k I ψ k ( x)) =Π i I ϕ i (ψ i ( x)) =Π i I (ϕ i ψ i )( x). Thus, x i = ϕ i ψ i ( x) =f i ( x) T i ( x) for each i I.

11 Coincidence and fixed points 913 Just as Theorem 2.3, the following fixed point result holds. Theorem 2.5. For any i I, let (E i ; U i, Γ i ) be a locally G-convex uniform space so that the convex structure Γ i has the property that Γ i -co(c i ) is compact whenever C i is a compact subset of E i, and suppose T i : E 2 E i satisfy the following conditions: (2.5.1) T i is a generalized Φ-mapping with the companion mapping S i. (2.5.2) There exist a nonempty compact subset K i of E i and an N i E i such that E \ K yi N i cint ( S i (y i) ), where K =Π i I K i. Then there exists an x =( x i ) i I E such that x i T i ( x) for each i I. 3. System of Inequalities and Minimax Theorems In this section, we give some immediate applications of our coincidence results to the existence of solution for a system of inequalities. Definition 3.1. Let X be a nonempty set, (Y ;Γ) a G-convex space, and f,g : X Y R. Then (3.1.1) g is said to be f-quasiconcave on Y if for any x X and any A Y, min f(x, y) g(x, z), z Γ-co(A). y A (3.1.2) g is said to be f-quasiconvex on Y if for any x X and any A Y, max f(x, y) g(x, z), z Γ-co(A). y A Theorem 3.2. For any i I, let f i,g i,p i,q i : E i E i R be functions and {a i } i I, {b i } i I be two families of real numbers. Suppose for each i I, the following conditions hold: (3.2.1) For each x i E i, x i g i (x i,x i ) is f i -quasiconcave on E i, and for each x i E i, x i q i (x i,x i ) is p i -quasiconvex on E i. (3.2.2) For each x i E i, x i f i (x i,x i ) is transfer compactly l.s.c. on E i, and for each x i E i, x i p i (x i,x i ) is transfer compactly u.s.c. on E i. (3.2.3) For each x i E i, there is an x i E i such that f i (x i,x i ) >a i. (3.2.4) For each x i E i, there is an x i E i such that p i (x i,x i ) <b i. (3.2.5) There exists a nonempty compact subset M i of E i such that for each Q i E i there is a compact Γ i -convex subset L Q i of E i so that Q i L Q i and

12 914 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng for each y i E i \ M i there exists x i L Q i with y i cint ( {v i E i : p i (x i,v i ) <b i } ). (3.2.6) There exists a nonempty compact subset K i of E i such that for each N i E i there is a compact Γ i -convex subset L Ni of E i so that N i L Ni and for each x i E i \ K i there exists y i L Ni with x i cint ( {u i E i : f i (u i,y i ) >a i } ). Then there are x =( x i ) i I and ȳ =(ȳ i ) i I in E such that g i ( x i, ȳ i ) >a i and q i ( x i, ȳ i ) <b i for all i I. Proof. For each i I, define S i,t i : E i 2 E i and H i,f i : E i 2 Ei by S i (x i )={y i E i : f i (x i,y i ) >a i } T i (x i )={y i E i : g i (x i,y i ) >a i } H i (x i )={u i E i : p i (u i,x i ) <b i } F i (x i )={u i E i : q i (u i,x i ) <b i }. Firstly, we show that T i is a generalized Φ-mapping with the companion mapping S i. To see this we have to show that (i) Γ i -co(a) T i (x i ) for any x i E i and any A S i (x i ), and (ii) E i = yi E i cint ( S i (y i) ). For any x i E i and any A S i (x i ), since x i g i (x i,x i )isf i -quasiconcave, we have a i < min y A f i(x i,y) g i (x i,z), z Γ i -co(a). This completes the proof of (i). Next, the map S i : E i 2 Ei defined by S i (x i)= { x i E i : f i (x i,x i ) >a i } is transfer compactly open valued by (3.2.2) and Lemma 1.2, which in conjunction with (3.2.3) and Lemma 1.3 shows that (ii) holds. Similarly, F i is a generalized Φ-mapping with the companion mapping H i. Moreover, conditions (3.2.5) and (3.2.6) imply conditions (2.2.3) and (2.2.4) respectively. Therefore, all the requirements of Theorem 2.2 are satisfied, and thus the conclusion follows. Theorem 3.3. For each i I, suppose f i,g i,p i,q i : E i E i R are functions satisfying the following conditions: (3.3.1) f i (x) g i (x) p i (x) q i (x) for any x E. (3.3.2) For each x i E i, x i g i (x i,x i ) is f i -quasiconcave on E i, and for each

13 Coincidence and fixed points 915 x i E i, x i q i (x i,x i ) is p i -quasiconvex on E i. (3.3.3) For each x i E i, x i f i (x i,x i ) is transfer compactly l.s.c. on E i, and for each x i E i, x i p i (x i,x i ) is transfer compactly u.s.c. on E i. (3.3.4) There exists a nonempty compact subset M i of E i such that for each Q i E i there is a compact Γ i -convex subset L Q i of E i so that Q i L Q i and for each y i E i \ M i there exists x i L Q i with ( ) y i cint {v i E i : p i (x i,v i ) sup inf p i(u i,v i )}. u i E i v i E i (3.3.5) There exists a nonempty compact subset K i of E i such that for each N i E i there is a compact Γ i -convex subset L Ni of E i so that N i L Ni and for each x i E i \ K i there exists y i L Ni with ( ) x i cint {u i E i : f i (u i,y i ) inf sup f i (u i,v i )}. u i E i v i E i Then for any i I. inf u i E i sup f i (u i,v i ) sup inf q i(u i,v i ) v i E i u i E i v i E i Proof. Without loss of generality, we may assume that for any i I, inf u i E i sup f i (u i,v i ) > and sup inf q i (u i,v i ) <. v i E i u i E i v i E i For any i I, let a i and b i be any real numbers satisfying that a i < inf u i E i sup f i (u i,v i ) and b i > sup inf q i(u i,v i ) sup inf p i(u i,v i ). v i E i u i E i u i E i v i E i v i E i Then for each x i E i, there exists an x i E i such that f i (x i,x i ) > a i, and for any x i E i, there exists an x i E i such that p i (x i,x i ) <b i. So conditions (3.2.3) and (3.2.4) are satisfied. Moreover, conditions (3.3.4) and (3.3.5) imply conditions (3.2.5) and (3.2.6) respectively. Consequently, all the requirements of Theorem 3.2 are satisfied, and so there exist x =( x i ) i I and ȳ =(ȳ i ) i I in E such that g i ( x i, ȳ i ) >a i and q i ( x i, ȳ i ) <b i for all i I. From q i ( x i, ȳ i ) g i ( x i, ȳ i ), we obtain that b i >a i. Since both of a i and b i with a i < inf u i E i sup v i E i f i (u i,v i ) and b i > sup vi E i inf u i E i q i(u i,v i ) are arbitrary, we conclude that inf u i E i sup f i (u i,v i ) sup inf q i (u i,v i ) v i E i u i E i v i E i

14 916 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng Corollary 3.4. In Theorem 3.3, if f i = g i = p i = q i for all i I, then sup inf f i (u i,v i ) = inf u i E i v i E i u i E i sup f i (u i,v i ). v i E i To end this section, we like to remark that in the seting of locally G-convex uniform spaces if for each i I, the convex structure Γ i has the property that Γ i -co(c i ) is compact whenever C i is a compact subset of E i, then conditions (3.2.5), (3.2.6), (3.3.4) and (3.3.5) can be replaced with the following conditions (3.2.5),(3.2.6),(3.3.4) and (3.3.5) respectively. (3.2.5) There exist a nonempty compact subset M i of E i and an Q i E i such that for each y i E i \ M i there is x i Q i with y i cint ( {v i E i : p i (x i,v i ) <b i } ). (3.2.6) There exist a nonempty compact subset K i of E i and an N i E i such that for each x i E i \ K i there is y i N i with x i cint ( {u i E i : f i (u i,y i ) >a i } ). (3.3.4) There exist a nonempty compact subset M i of E i and an Q i E i such that ) E i \ M i x i Q ({v icint i E i : p i (x i,v i ) sup inf p i (u i,v i )}. u i E i v i E i (3.3.5) There exist a nonempty compact subset K i of E i and an N i E i such that ( ) E i \ K i yi N i cint {u i E i : f i (u i,y i ) inf sup f i (u i,v i )}. u i E i v i E i 4. Maximal Elements and Equilibrium Points In this section we shall apply our fixed point results in section 2 to obtain some theorems about maximal elements, nonempty intersection, and equilibrium points for an abstract economy. An abstract economy is a family E =((E i ;Γ i ),A i,b i,p i ) i I, where I is a finite or infinite set of agents and for each i I, each commodity space

15 Coincidence and fixed points 917 (E i ;Γ i )isag-convex space, A i,b i : E =Π i I E i 2 E i are the constraint correspondences, and P i : E 2 E i is the preference correspondence. A point x E is said to be an equilibrium point of an abstract economy E = ((E i ;Γ i ),A i,b i,p i ) i I if x i B i ( x) and A i ( x) P i ( x) = for each i I. Theorem 4.1. Let E =((E i ;Γ i ),A i,b i,p i ) i I be an abstract economy. Suppose the following conditions hold: (4.1.1) For each x E, A i (x), and for each N A i (x), Γ i -co(n) B i (x). (4.1.2) For each x =(x i ) i I E, x i / Γ i -co (P i (x)) for every i I. (4.1.3) For each y i E i,allofa i (y i), P i (y i ) and [( C i = {x ) E : P i (x) A i (x) = } are compactly open such that E = yi E i P i (y i ) C i A i (y i) ]. (4.1.4) There exists a nonempty compact subset K i of E i such that for any N i E i there is compact Γ i -convex subset L Ni of E i with N i L Ni and where K =Π i I K i. Then E has an equilibrium point. E \ K yi L Ni [( P i (y i ) C i ) A i (y i ) ], Proof. For each i I, let D i = E \ C i and define S i,t i : E 2 E i by S i (x) = { Pi (x) A i (x), if x D i ; A i (x), if x C i, and T i (x) = { Γi -co (P i (x)) B i (x), if x D i ; B i (x), if x C i. For each x E and each N S i (x), it is easy to check that Γ i -co(n) T i (x). Also, for each y i E i, we see from S i (y i)= ( P i (y i ) A i (y ) ( i) D i A i (y ) i) C i = ( P i (y i ) A i (y i) ) ( A i (y ) i) C i = ( ) P i (y i ) C i A i (y i) and (4.1.3) that that S i (y i) is compactly open and E = yi E i S i (y i). Thus we have shown that T i is generalized Φ-mapping with the companion mapping S i, which together with (4.1.4) shows that all the requirements of Theorem 2.4

16 918 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng are satisfied. Thus, there is an x E such that x i T i ( x) for all i I. Now, (4.1.2) implies that x C i, hence we infer that x i B i ( x) and P i ( x) A i ( x) = for all i I. This completes the proof. Another immediate consequence of Theorem 2.4 is the following existence theorem of maximal elements. Theorem 4.2 Let (E;Γ) be a G-convex space and S, T : E 2 E satisfy the following conditions: (4.2.1) For each x E, Γ-co(S(x)) T (x). (4.2.2) S is transfer compactly open valued. (4.2.3) There exist a nonempty compact subset K and a nonempty compact Γ-convex subset C of E such that E \ K y C cint (S (y)). (4.2.4) T does not have a fixed point on E. Then S has a maximal element, that is, there is an x E such that S( x) =. Proof. If S(x) for any x E, then by Lemma 1.3 E = y E cint ( S (y) ), which in combining with (4.2.1) and (4.2.2) gives us that T is a generalized Φ- mapping with the companion mapping S. Then in view of (4.2.3) we infer from Theorem 2.4 that T has a fixed point, which contradicts (4.2.4). Therefore S must have a maximal element. Theorem 4.2 can be further extended as: Theorem 4.3. Suppose I is a finite index set, and for each i I, S i,t i : E 2 E i satisfy the following conditions: (4.3.1) For all x E, Γ i -co (S i (x)) T i (x). (4.3.2) S i is transfer compactly open valued. (4.3.3) For all x E, x i / T i (x). (4.3.4) There exist a nonempty compact subset K of E and a nonempty compact Γ i -convex subset C i of E i such that for all x E \ K and for all i I with S i (x) there is ŷ i C i with x cint ( S i (ŷ i) ). Then there exists x E such that S i ( x) = for all i I. Proof. For each x E, let I(x) ={i I : S i (x) }. We have to prove that I(x) = for some x E. On the contrary, assume that I(x) for all x X. Fori I and x E, let A i (x) =E i S i (x) and B i (x) =E i T i (x),

17 Coincidence and fixed points 919 and define F, G : E 2 E by and F (x) = i I(x) A i (x) G(x) = i I(x) B i (x). It follows from (4.3.1) that for each x E, Γ-co (F (x)) = Γ-co [ ( i I(x) E i S i (x) )] ( i I(x) E i Γ-co(S i (x)) ) i I(x) ( E i T i (x) ) = G(x), and (4.3.3) leads that x / G(x). We now show that F is transfer compactly open valued. Let y E and x F (y). Noting that y F (x) = i I(x) A i (x) = i I(x) (E i S i (x)), we see that y i S i (x) for all i I(x), that is, x S i (y i) for all i I(x). Since S i is transfer compactly open valued, there is ŷ i E i such that x cint ( S i (ŷ i) ) (17) For each i I \ I(x) choose z i E i, and define ỹ E by { ŷi, if i I(x); ỹ i = z i, if i I \ I(x). (18) Since z A i (ỹ) ȳ A i (z) =E i S i (x) ŷ i S i (z) z S i (ŷ i), we conclude that A i (ỹ) =S i (ŷ i), and so by (17) for all i I(x). Thus, x cint ( A i (ỹ)) x i I(x) cint ( A i (ȳ)) cint ( i I(x) A i (ȳ)) = cint ( F (ȳ) ) once we notice that I(x) is a finite set. This shows that F is transfer compactly open valued. By (4.3.4), for all x E \ K and for each i I(x), there

18 920 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng is a ŷ i C i such that x cint ( S i (ŷ i) ) = cint (A i (ỹ)), where ỹ is defined as in (18). Let C =Π i I C i, which is a compact Γ-convex subset of E. Then x i I(x) cint ( A i (ỹ)) cint ( i I(x) A i (ỹ)) = cint(f (ỹ)), that is, E \ K y C cint(f (ỹ)). By means of Theorem 4.2, there is ẑ E such that F (ẑ) =. Since I(ẑ), S i (ẑ) and A i (ẑ) for all i I(ẑ). Hence, F (ẑ) = i I(ẑ) A i (ẑ), which contradicts the fact that F (ẑ) =. This completes the proof. Finally, we deduce a nonempty intersection theorem. Theorem 4.4. Suppose {A i } i I and {B i } i I are two families of nonempty subsets of the product G-convex space E =Π i I E i satisfing that: (4.4.1) For each x i E i and each N A i [x i ], Γ i -co(n) B i [x i ] and there exists a point y i E i such that x i cint (A i [y i ]), where A i [x i ]={y i E i :[x i,y i ] A i } B i [x i ]={y i E i :[x i,y i ] B i } A i [y i ]={x i E i :[x i,y i ] A i }. (4.4.2) There exists a nonempty compact subset K i of E i such that for any N i E i there is a compact Γ i -convex subset L Ni with N i L Ni and where K =Π i I K i. Then i I B i. E \ K yi L Ni (cint(a i [y i ]) E i ), Proof. For each i I, define S i,t i : E 2 E i by S i (x) =A i [x i ] and T i (x) = B i [x i ] for all x E. It follows from (4.4.1) that Γ i -co(n) T i (x) for any x E and any N S i (x). Besides, for each y i E i, we have S i (y i)={x =[x i,x i ] E : y i S i (x)} = {x =[x i,x i ] E : y i A i [x i ]} = {x =[x i,x i ] E : x i A i [y i ]} = A i [y i ] E i cint (A i [y i ]) E i,

19 Coincidence and fixed points 921 hence cint ( S i (y i) ) cint (A i [y i ]) E i for each y i E i. By (4.4.1), for each x E, there is a y i E i such that x =[x i,x i ] cint (A i [y i ]) E i cint ( S i (y i) ). This shows that E = yi E i cint ( S i (y i) ). And thus T i is a generalized Φ- mapping with the companion mapping S i, which together with (4.4.2) leads to the existence of a point x in E so that x i T i ( x) =B i ( x i ) for each i I. In other words, x i I B i. References 1. Q.H. Ansari, A. Idzik and J.C. Yao, Coincidence and fixed point theorems with applications, Topol. Methods Nonlinear Anal. 15(2000) X.P. Ding and J.Y. Park, Collectively fixed point theorem and abstract economy in G-convex spaces, Numer. Funct. Anal. Optimi. 23(2002) X.P. Ding, New H-KKM theorems and their applications to geometric property, coincidence theorems, minimax inequality and maximal elements, Indian J. Pure Appl. Math. 26(1995) Y.Y. Huang, T.Y. Kuo and J.C. Jeng, Fixed point theorems for condensing multimaps on locally G-convex spaces, Nonlinear Analysis, 67(2007) T.Y. Kuo, J.C. Jeng and Y.Y. Huang, Fixed point theorems for compactmultimaps on almost Γ-convex sets in generalized convex spaces, Nonlinear Analysis, 66(2007) J.L. Kelley, General Topology, Van Nostrand, Princeton, NJ, L.J. Lin and Q.H. Ansari, Collective fixed points and maximal elements with applications to abstract economies, J. Math. Anal. Appl. 296(2004) L.J. Lin, System of coincidence theorems with applications, J. Math. Anal Appl. 285(2003) S. Park, Continuous selection theorems in generalized convex spaces, Numer. Funct. Anal. Optimi. 20(1999) S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces. J. Math. Anal. Appl. 197(1996)

20 922 Yew-Dann Chen, Young-Ye Huang, Tian-Yuan Kuo and Jyh-Chung Jeng 11. K.K. Tan and X.L. Zhang, Fixed point theorems on G-convex spaces and applications, in: Proceedings of Nonlinear Functional Analysis and Applications, Vol. 1, Kyungnam University, Masan, Korea, 1996, pp E. Tarafdar, Fixed point theorems in H-spaces and equilibrium points of abstract economies, J. Austral. Math. Soc.(Series A) 53(1992) G. Tian and J. Zhou, Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization, J. Math. Econom. 24(1995) Received: October, 2009