Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2Y a point-to-set mapping such that for any χ ϵ K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ∗ ϵ K such that f g(χ∗), y) ∉ -int C(χ) for all y ϵ K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.