Collectively fixed point and maximal element theorems in topological semilattice spaces

In this article, we establish a collectively fixed point theorem and a maximal element theorem for a family of multivalued maps in the setting of topological semilattice spaces. As an application of our maximal element theorem, we prove the existence of solutions of generalized abstract economies with two constraint correspondences. We consider the system of (vector) quasi-equilibrium problems (in short, (S(V)QEP)) and system of generalized vector quasi-equilibrium problems (in short, (SGVQEP)). We first derive the existence result for a solution of (SQEP) and then by using this result, we prove the existence of a solution of system of a generalized implicit quasi-equilibrium problems. By using existence result for a solution of (SQEP) and weighted sum method, we derive an existence result for solutions of (SVQEP). By using our maximal element theorem, we also establish some existence results for the solutions of (SGVQEP). Some applications of our results to constrained Nash equilibrium problem for vector-valued functions with infinite number of players and to semi-infinite problems are also given.