On generalized implicit vector equilibrium problems in Banach spaces

  • Authors:

  • Lu-Chuan Ceng;Sy-Ming Guu;Jen-Chih Yao

  • Affiliations:

  • Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;Department of Business Administration, College of Management, Yuan-Ze University, Chung-Li City, Taoyuan Hsien, 330, Taiwan;Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2009

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Abstract

Let X and Y be real Banach spaces, K be a nonempty convex subset of X, and C:K-2^Y be a multifunction such that for each u@?K, C(u) is a proper, closed and convex cone with intC(u)0@?, where intC(u) denotes the interior of C(u). Given the mappings T:K-2^L^(^X^,^Y^), A:L(X,Y)-L(X,Y), f"1:L(X,Y)xKxK-Y, f"2:KxK-Y, and g:K-K, we introduce and consider the generalized implicit vector equilibrium problem: Find u^*@?K such that for any v@?K, there is s^*@?Tu^* satisfying f"1(As^*,v,g(u^*))+f"2(v,g(u^*))@?-intC(u^*). By using the KKM technique and the well-known Nadler's result, we prove some existence theorems of solutions for this class of generalized implicit vector equilibrium problems. Our theorems extend and improve the corresponding results of several authors.