Quasi-equilibrium Problems and Fixed Point Theorems of the Product Mapping of Lower and Upper Semicontinuous Mappings

Abstract

In this paper, we introduce generalized quasi-equilibrium problems. These contain several problems in the optimization theory as special cases. We give sufficient conditions on the existence of their solutions. In particular, we establish several results on the existence of fixed p oints for product mappings of lower and upper semicontinuous mappings. These results generalize some well-known fixed p oint t heorems o btained by p revious a uthors a s F . E . B rowder a nd K y F an, X.

On the upper C-continuity on the solution set of general vector alpha optimization problems in infinite-dimensional spaces

“…Recently, it has attracted much attention. We known that the upper and lower C-continuities of a set-valued mapping in the Hausdorff locally convex topological vector spaces are one of the efficient tools and very useful to establish theorems on the existence of solutions of quasivariational inclusion problems, quasi-equilibrium problems, quasi-optimization problems, general vector alpha optimization problems, general vector alpha quasioptimization problems and the many other problems; see, e.g., [1,2,3,4,5,6] and the references therein.…”

“…There are a lot of results on the solution existence of general vector alpha optimization problems and some other related problems; see, e.g., [2,3,7,8,9,10] and references therein. Tan and Hoa [6] derived the results on the solution existence of quasi-equilibrium problems by using the lower and upper semicontinuous mappings. They also obtained the existence results of quasi-equilibrium problems by using the lower semicontinuous mappings; see [4].…”