Solvability of System of Generalized Vector Quasi-Equilibrium Problems

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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 53, HIKARI Ltd, Solvability of System of Generalized Vector Quasi-Equilibrium Problems Wenxin Zhu College of Basic Science, TianJin Agricultural University, TianJin, , China Yunyan Song College of Science, Tianjin University of Technology, Tianjin, , China Copyright c 2014 Wenxin Zhu and Yunyan Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems, and establish some new existence results of a solution for the system of vector quasi-equilibrium problems. Mathematics Subject Classification: 90C29, 49J40 Keywords: system of generalized vector quasi-equilibrium problems, C(x) convex, C(x) quasiconcave like 1 Introduction In recent years, the equilibrium problem with vector-valued functions and setvalued maps have been studied in [1-5] and the references therein. Very recently, Ansari and Yao [6] generalized the quasi-equilibrium problem and introduced the generalized vector quasi-equilibrium problem. They established some existence results of a solution for the generalized vector quasi-equilibrium problems. The system of vector quasi-equilibrium problems, i.e., a family of quasi-equilibrium problems for vector-valued bifunctions defined on a product

2 2628 Wenxin Zhu and Yunyan Song set, was introduced by Ansari et.al. [2] with applications in Debreu type equilibrium problem for vector-valued function. As generalizations of the above models, we introduce some new types of system of generalized vector quasiequilibrium problems, i.e., a family of quasi-equilibrium problems for set-valued maps defined on a product set, and use a fixed point theorem we derive some new existence results for the system of vector quasi-equilibrium problems. Let I be an index set. For each i I, let X i be a Hausdorff topological vector space. Consider a family of nonemptyconvex subsets {K i } i I with K i in X i. Throughout this paper K = i I K i and X = i I X i. For each i I, let Y i be a topological vector space and let C i : K 2 Y i be a multivalued map such that for each x K, C i (x) is a proper, closed and convex cone with intc i (x), where intc i (x) denote the interior of C i (x). For each i I, let F i : K K i 2 Y i be a set-valued map and A i : K 2 K i be a multivalued map with nonempty values. The following classes of system of generalized vector equilibrium problems are of interest to us: (I) Weak type I system of generalized vector quasi-equilibrium problems (in short, WI-SGVQEP): Find x K such that for each i I, x i A i ( x) : F i ( x, y i ) intc i ( x), for all y i A i ( x). (III) Strong type I system of generalized vector quasi-equilibrium problems (in short, SI-SGVQEP): Find x K such that for each i I, x i A i ( x) : F i ( x, y i ) C i ( x), for all y i A i ( x). (IV)Strong type II system of generalized vector quasi-equilibrium problems (in short, SII-SGVQEP):Find x K such that for each i I, x i A i ( x) : F i ( x, y i ) C i ( x), for all y i A i ( x). The following are some special cases: (a) If the index set I is singleton, then problems above reduce to find x K such that x A( x) and for all y A( x) F ( x, y) intc( x), F( x, y) Y \ ( intc(x)) F ( x, y) C( x), F( x, y) C( x) respectively. These generalized vector equilibrium problems were introduced and studied in [1]. (b) If F i is a single-valued function F i : K K i Y i for each i I, then both the (WI-SGVQEP) and the (WII-SGVQEP) reduce to the weak type system of vector equilibrium problems which is to find x K such that for each i I, x i A i ( x) :F i ( x, y i ) / intc i ( x), y i A i ( x), which is introduced and studied in [2]. (c) If for each i I,F i : K L(X i,y) be a map, C be a proper, closed and convex cone, where L(X i,y) denotes the space of all continuous linear functions from X i to Y. Then both (WI-SGVQEP) and the (WII-SGVQEP)

3 Solvability of system of generalized vector quasi-equilibrium problems 2629 reduce to the following vector quasi-variational inequality problem, which is to find x K such that for each i I, x A( x) and Σ i I f i ( x),y i x i / intc, y i A i ( x) which is studied in [3]. 2 Preliminary Notes / Materials and Methods First, we give some notions and results which we will apply in the following. Definition 2.1.[4] Let E be a topological space. A subset D of E is said to be compact open (respectively, compact closed) in E if for any nonempty compact subset L of E, D L is open (respectively, closed) in L. Remark 2.1. (a) It is clear from the above definition that every open (respectively, closed) set is compactly open (respectively, compactly closed). (b) The union or intersection of two compactly open (respectively, compactly closed) set is compactly open (respectively, compactly closed). Definition 2.2. [5] Let C : X 2 Z be a set-valued map with nonempty values. Then the set-valued map H : X X 2 Z is called (i)c(x) quasiconvex if for all x, y 1,y 2 X and t [0, 1], we have either or H(x, y 1 ) H(x, ty 1 +(1 t)y 2 )+C(x) H(x, y 2 ) H(x, ty 1 +(1 t)y 2 )+C(x). (ii) C(x) quasiconcave like if for all x, y 1,y 2 X and t [0, 1], we have either H(ty 1 +(1 t)y 2,x) H(y 1,x)+C(x) or H(ty 1 +(1 t)y 2,x) H(y 2,x)+C(x). Lemma 2.1. [6] Let T be a multivalued map of a topological space X into a topological space Y. Then T is lower semicontinuous at x X if and only if for any net {x α } in X converging to x, there is a net y α such that y α T (x α ) for every α and y α converging to y. Theorem 2.1. [7] Let X and Y be Hausdorff topological spaces, T : X 2 Y be a multivalued map. (i) If T : X 2 Y is an upper semicontinuous multivalued map with closed values, then T is closed. (ii) If X is a compact and T : X 2 Y is an upper semicontinuous multivalued map with compact values, then T (X) is compact. We shall use the following fixed point theorem due to Chowdhury and Tan [8]. Theorem 2.2. Let K be a nonempty convex subset of a topological vector space X and S, T : K 2 K multivalued maps.

4 2630 Wenxin Zhu and Yunyan Song Assume that the following conditions hold: (i) for each x K, S(x) T (x) (ii) for each x K, S(x) (iii) for each x K, T(x) is convex (iv) for each y K, S 1 (y) :={x K : y S(x)} is compactly open (v) there exist a nonempty, closed and compact subset D of K and ȳ D such that K \ D S 1 (ȳ) Then there exists x K such that x T ( x). 3 Results Some existence results of a solution for the four types of system of generalized vector quasi-equilibrium problems are shown. Theorem 3.1. For each i I, let X i be a topological vector space, K i a nonempty convex subset of X i,k = i I K i, let A i : K 2 K i be setvalued maps with nonempty and convex values and for all y i K i,a 1 i (y i ) is compactly open in K. We define a multivalued map A : K 2 K by A(x) = i I A i(x) for all x K such that the set Γ = {x K : x i A i (x)} is compactly closed. For each i I, assume that (i) C i : K 2 X i is a set-valued map such that intc i (x) for each x K; (ii) F i : K K i 2 Y i is a set-valued map satisfies: (a)for each x K, y i K, F i (x, y i ) intc i (x) implies F i (y i,x) intc i (x) (b) y i K i, the set {x K : F i (x, y i ) intc i (x)} is open (c) there exist a nonempty, closed and compact subset D of K and ȳ i D such that F i (x, ȳ i ) intc i (x) for all x K \ D (d) for each x K, y F i (y, x) isc x -quasiconcave-like Then WI-SGVQEP has a solution. Proof. For each i I,x K, let us define a st-valued map P i,q i : K 2 K i by P i (x) ={y i K i : F i (x, y i ) intc i (x)} and Q i (x) ={y i K i : F i (y i,x) intc i (x)} Then clearly for each x K, Q i (x) is convex. In fact, let y i1,y i2 Q i (x) and λ [0, 1], then y i1 K i,y i2 K i and F i (y i1,x) intc i (x), F i (y i2,x) intc i (x). Since K i is convex, λy i1 +(1 λ)y i2 K i. We want to show that F i (λy i1 +(1 λ)y i2,x) intc i (x) for all λ [0, 1]. By assumption, y i F i (y i,x)isc x quasiconcave-like for each x K. Either F i (λy i1 +(1 λ)y i2,x) F i (y i1,x)+c i (x) intc i (x)+c i (x) intc i (x)

5 Solvability of system of generalized vector quasi-equilibrium problems 2631 or F i (λy i1 +(1 λ)y i2,x) F i (y i2,x)+c i (x) intc i (x)+c i (x) intc i (x) This shows that F i (λy i1 +(1 λ)y i2,x) intc i (x) for all λ [0, 1] and λy i1 +(1 λ)y i2 Q i (x). Therefore, Q i (x) is convex for each i I. By condition (a) of (ii), we have P i (x) Q i (x) for all x K and each i I. By condition (b) in (ii), i I, y i K i, the set P 1 i (y i )={x K : F i (x, y i ) intc i (x)} is open in K. Therefore P 1 i (y i ) is compactly open. For each i I, we also define two other multivalued maps S i,t i : K 2 K i by { Ai (x) P S i (x) = i (x), if x Γ; A i (x), ifx K \ Γ; { Ai (x) Q T i (x) = i (x), if x Γ; A i (x), ifx K \ Γ; Then for all x K, T i (x) is convex and S i (x) T i (x). Since for each i I,y i K i,a 1 i (y i ),P 1 i (y i ) and K \ Γ are compactly open and for all i I,y i K i S 1 i (y i )=(A 1 i (y i ) Pi 1 (y i )) ((K \ Γ) A 1 i (y i )) we have Si 1 (y i ) is compactly open. Assume to the contrary that for each i I,x Γ,A i (x) P i (x). Then for each i I,S i (x). Now define set-valued maps S, T : K 2 K by S(x) = i I S i(x),t(x) = i I T i(x). Since S 1 (y) = i I S 1 i (y i ) and S 1 i (y i ) is compactly open for each i I and for all y i K i we have S 1 (y) is compactly open. Then S, T satisfy all conditions of Theorem 2.1. Therefore there exists x K such that x T (x ). From the definition of Γ and T we have {x K : x T (x)} Γ Therefore x Γ and x A i (x ) Q i (x ) and in particular F i (x,x )=0 intc i (x) which is a contradiction. Hence there exists x Γ such that A i ( x) P i ( x) =, that is x i A i ( x) and F i ( x, y i ) intc i ( x) for all y i A i ( x). Theorem 3.2. Assume that all the hypotheses of Theorem 3.1 are satisfied, except that the condition (ii) is replaced by (ii) F i : K 2 K i satisfies: (a) For each x K, y i K, F i (x, y i ) ( intc i (x)) implies F i (y i,x) ( intc i (x)). (b)for each y i K i, the set {x K : F i (y i,x) ( intc i (x)) } is open

6 2632 Wenxin Zhu and Yunyan Song (c)there exist a nonempty, closed and compact subset D of K and ȳ i D such that F i (x, ȳ i ) ( intc i (x)) =. (d) y F i (y, x) isc(x) quasiconcave-like Then, there exists x K such that for each i I, F i ( x, y i ) Y i \ ( intc i ( x)). That is the problem (WII-SGVQEP) has a solution. Theorem 3.3. Assume that all the hypotheses of Theorem 3.1 are satisfied, except that the condition (ii) is replaced by (ii) F i : K 2 K i satisfies: (a) For each x K, y i K, F i (x, y i ) C i (x) implies F i (y i,x) C i (x). (b)for each y i K i, the set {x K : F i (y i,x) C i (x)} is open (c)there exist a nonempty, closed and compact subset D of K and ȳ i D such that F i (x, ȳ i ) C i (x) for all x K \ D (d) y F i (y, x) isc(x) quasiconcave-like Then, there exists x K such that for each i I, F i ( x, y i ) C i ( x). That is the problem (SI-SGVQEP) has a solution. Theorem 3.4. Assume that all the hypotheses of Theorem 3.1 are satisfied, except that the condition (ii) is replaced by (ii) F i : K 2 K i satisfies: (a)for each x K, y i K, F i (x, y i ) C i (x) implies F i (y i,x) C i (x). (b)for each y i K i, the set {x K : F i (y i,x) C i (x) } is open (c)there exist a nonempty, closed and compact subset D of K and ȳ i D such that F i (x, ȳ i ) C i (x) =. (d) y F i (y, x) isc(x) quasiconcave-like Then, there exists x K such that for each i I, F i ( x, y i ) C i ( x). That is the problem (SII-SGVQEP) has a solution. Remark 3.1. The condition (b) in (ii) of Theorem 3.1 is satisfied if the following conditions hold for each i I: (1) M i (x) =Y i \ ( intc i (x)) : K 2 Y i is upper semicontinuous, (2) for all y i K i, the multivalued map x F i (x, y i ) is upper semicontinuous on K with compact values. In fact, we can prove P i (y i )={x K : F i (x, y i ) intc i (x)} is closed for all y i K i. Consider a net {x t } P i (y i ) such that x t x K. Since x t P i (y i ), there exists u t F i (x t,y i ) such that u t / intc i (x). From the upper semicontinuity and compact values of F i on K and Theorem 2.1, it suffices to find a subset of {u tj } which converges to some u F i (x, y i ) where

7 Solvability of system of generalized vector quasi-equilibrium problems 2633 u tj F i (x tj,y i ). Since (x tj,u tj ) (x, u), by proposition 7 in [9] and the upper semicontinuity of M i, it follows that u intc i (x) and hence x P i (y i ),P i (y i ) is closed. Remark 3.2. The condition (b) in (ii) of Theorem 3.2 is satisfied if the following conditions hold for each i I: (1) M i (x) =Y i \ ( intc i (x)) : K 2 Y i is upper semicontinuous, (2) for all y i K i, the multivalued map x F i (x, y i ) is lower semicontinuous on K. References [1] L.J.Lin, Existence Results for Primal and Dual Generalized Vector Equilibrium Problems With Applications to Generalized Semi-Infinite Programming, Journal of Global Optimization, 33 (1985), [2] Q.H.Ansari, W.K.Chan, and X.Q.Yang, The System of Vector Quasiequilibrium Problems with Applications, Journal of Global Optimization, 29 (2004), [3] Q.H.Ansari, S.Schaible, and J.C.Yao, Generalized Vector Quasivariational Inequality Problems Over Product Sets, Journal of Global Optimization, 32 (2005), [4] X.P.Ding, Existence of Solutions for Quasi-Eauilibrium Problems in Noncompact Topological Spaces, Computer and Mathematics With Applications, 39 (2000), [5] I.V.Konnov and J.C.Yao, Existence Solutions for Generalized Vector Equilibrium Problems, Journal Of Mathematical Analysis and Applications, 223 (1999), [6] N.X. Tan, Quasi-variational Inequalities in Topological Lineart Locally convex Hausdorff spaces, Mathematical Nach, 122 (1985), [7] J.P.Aubin and A. Cellina, Differential Inclusions, Springer-verlag, Berlin, Heidelberg, Germany, (1999) [8] M.S.R.Chowdhury and K.K.Tan, Generalized Variational Inequalities and there Applications, Bulletin of the Polish Academy of Sciences, Mathematics, 76 (2000), [9] L. Zhi and J.Yu, The Existence of Solutions for the System of Generalized Vector Quasi-Equilibrium problems, Applied Mathematics Letters, 18 (2005), Received: March 15, 2014