Extended -Vector Equilibrium Problem

We introduce and study extended -vector equilibrium problem. By using KKM-Fan Theorem as basic tool, we prove existence theorem in the setting of Hausdorff topological vector space and reflexive Banach space. Some examples are also given. 1. Introduction Equilibrium problems have been extensively studied in recent years; the origin of this can be traced back to Blum and Oettli [1] and Noor and Oettli [2]. The equilibrium problem is a generalization of classical variational inequalities and provides us with a systematic framework to study a wide class of problems arising in finance, economics, operations research, and so forth. General equilibrium problems have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. Vector equilibrium problems have attracted increasing interest of many researchers and provide a unified model for several classes of problems, for example, vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems; see [1–4] and references therein. Many existence results for vector equilibrium problems have been established by several eminent researchers; see, for example, [5–13]. The generalized monotonicity plays an important role in the literature of equilibrium problems and variational inequalities. There are a substantial number of papers on existence results for solving equilibrium problems and variational inequalities based on different monotonicity notions such as monotonicity, pseudomonotonicity, and quasimonotonicity. Let and be two Hausdorff topological vector spaces, let be a nonempty, closed, and convex subset of , and let be a pointed, closed, convex cone in with . Given a vector-valued mapping , the vector equilibrium problem consists of finding such that Inspired by the concept of monotonicity, KKM-Fan Theorem, and the other work done in the direction of generalization of vector equilibrium problems (see [14–16]) we introduce and study extended -vector equilibrium problem and prove some existence results in the setting of Hausdorff topological vector spaces and reflexive Banach spaces. 2. Preliminaries The following definitions and concepts are needed to prove the results of this paper. Definition 1. The Hausdorff topological vector space is said to be an ordered space denoted by if ordering relations are defined in by a pointed, closed, convex cone of as follows: If the interior of is , then the weak ordering relations in are also defined as follows: Throughout this paper, unless otherwise specified, we assume