On generalized implicit vector equilibrium problems in Banach spaces

Let X and Y be real Banach spaces, K be a nonempty convex subset of X , and C : K → 2Y be a multifunction such that for each u ∈ K , C(u) is a proper, closed and convex cone with intC(u) 6= ∅, where intC(u) denotes the interior of C(u). Given the mappings T : K → 2L(X,Y ,A : L(X, Y )→ L(X, Y ), f1 : L(X, Y )×K×K → Y , f2 : K×K → 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 f1(As∗, v, g(u)) + f2(v, g(u)) 6∈ −intC(u). By using theKKM technique and thewell-knownNadler’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. © 2009 Elsevier Ltd. All rights reserved.