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 → 2^Y a point-to-set mapping such that for any x ∈ K, C(x) is a pointed, closed and convex cone in Y with apex at the origin and intC(x) ≠φ . 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 x ∈ K such that for all y ∈ K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem.In the first part of this paper, we introduce many other authors' work on equilibrium problem (EP), and show that equilibrium theory has been applied widely. Second, we recall some definitions and lemmas which needed in the main results of this paper. In this paper, 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 characteriza-tions of the set of solutions for (IVEP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity and strict C-pseudomonotonicity, respectively. Under these assumptions and additional conditions, we conclude that' the sets of solutions for (IVEP) are nonempty, closed and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.