MIXED VECTOR EQUILIBRIUM-LIKE PROBLEMS IN BANACH SPACES
Byung-Soo Lee, Salahuddin Salahuddin, Mohammad Kalimuddin Ahmad · East Asian Mathematical Journal · 2009
Abstract. In this paper, we consider a new class of generalized mixedvector equilibrium-like problems in Banach spaces. By using Fan-KKMTheorem and Nadler’s Theorem, we prove the existence theorem of solu-tion for this class of generalized mixed vector equilibrium-like problems. 1. Introduction and preliminaries(Vector) equilibrium problem is a uni ed model of several problems, includ-ing optimization problems, xed point problems, variational inequality prob-lems, Nash equilibria problems, saddle point problems, complementarity prob-lems as special cases and provides a natural uni ed framework for studyingmany problems in nance, economics, network analysis, transportation andelasticity [1-3, 5, 8, 9, 14]. We consider a new class of generalized mixed vec-tor equilibrium-like problem and by using Fan-KKM Theorem [4] and Nadler’sTheorem [12] we prove some existence theorems of solutions for the class ofgeneralized mixed vector equilibrium-like problems. Furthermore these exis-tence theorems can be applied to derive some existence results of solutions forgeneralized mixed vector variational-like problem. It is worth pointing out thatthere are no assumptions of pseudo monotonicity in our existence results.De nition 1.1. Let Dbe a nonempty subset of a vector space X. Then a mul-tifunction T: D!2