System of Operator Quasi Equilibrium Problems

We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive some existence theorems for solutions to these problems with and without involving -condensing mappings. 1. Introduction In 2002, Domokos and Kolumbán [1] gave an interesting interpretation of variational inequality and vector variational inequalities (for short, VVI) in Banach space settings in terms of variational inequalities with operator solutions (for short, OVVI). The notion and viewpoint of OVVI due to Domokos and Kolumbán [1] look new and interesting even though it has a limitation in application to VVI. Recently, Kazmi and Raouf [2] introduced the operator equilibrium problem which generalizes the notion of OVVI to operator vector equilibrium problems (for short, OVEP) using the operator solution. They derived some existence theorems of solution of OVEP with pseudomonotonicity, without pseudomonotonicity, and with -pseudomonotonicity. However, they dealt with only the single-valued case of the bioperator. It is very natural and useful to extend a single-valued case to a corresponding set-valued one from both theoretical and practical points of view. The system of vector equilibrium problems and the system of vector quasi equilibrium problems were introduced and studied by Ansari et al. [3, 4]. Inspired by above cited work, in this paper, we consider a system of operator quasi equilibrium problems (for short, SOQEP) in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings according to [5] as basic tool, we derive some existence theorems for solutions to SOQEP with and without involving -condensing mappings. Further, we consider a system of generalized quasi operator equilibrium problems (for short, SGQOEP) in topological vector spaces and give some of its special cases and derive some existence theorems for solutions to SOQEP with and without involving -condensing mappings by using well-known maximal element theorem [5] for a family of set-valued mappings, and, consequently, we also get some existence theorems for solutions to a system of operator equilibrium problems. 2. Preliminaries Let be an index set, for each , and let be a Hausdorff topological vector space. We denote , the space of all continuous linear operators from into , where is topological vector space for each . Consider a family of nonempty convex subsets with in . Let Let be a set-valued mapping