Collectively fixed point and maximal element theorems in topological semilattice spaces

Abstract

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.

Keywords:

• collectively fixed point theorems
• maximal element theorems
• generalized abstract economies
• system of (vector) quasi-equilibrium problems
• system of generalized vector quasi-equilibrium problems
• constrained Nash equilibrium problem
• semi-infinite problems
• topological semilattice spaces

AMS Subject Classifications::

• 49J40
• 47H10
• 90C29
• 47J20
• 91A10

Acknowledgements

S. Al-Homidan and Q.H. Ansari are grateful to the Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia, for providing excellent research facilities to carry out this work. In this research, S. Al-Homidan and Q.H. Ansari were supported by the Fast Track Research Project No. # FT070007 of King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia. J.-C. Yao was supported by the Grant NSC 98-2923-E-110-003-MY3.