On generalized implicit vector equilibrium problems in Banach spaces

AbstractLet 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)≠0̸, 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∗))∉−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