On generalized implicit vector equilibrium problems in Banach spaces

Let XX and YY be real Banach spaces, KK be a nonempty convex subset of XX, and C:K→2YC:K→2Y be a multifunction such that for each u∈Ku∈K, C(u)C(u) is a proper, closed and convex cone with intC(u)≠0̸, where intC(u) denotes the interior of C(u)C(u). Given the mappings T:K→2L(X,Y)T:K→2L(X,Y), A:L(X,Y)→L(X,Y)A:L(X,Y)→L(X,Y), f1:L(X,Y)×K×K→Yf1:L(X,Y)×K×K→Y, f2:K×K→Yf2:K×K→Y, and g:K→Kg:K→K, we introduce and consider the generalized implicit vector equilibrium problem: Find u∗∈Ku∗∈K such that for any v∈Kv∈K, there is s∗∈Tu∗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.