On generalized implicit vector equilibrium problems in Banach spaces

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 int C ( u ) ≠ 0̸ , where int C ( 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 ) × K × K → Y , f 2 : 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 ∗ ∈ T u ∗ satisfying f 1 ( A s ∗ , v , g ( u ∗ ) ) + f 2 ( v , g ( u ∗ ) ) ∉ − int C ( 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.