Noncoercive and noncontinuous equilibrium problems: existence theorem in infinite-dimensional spaces

Abstract

In this paper, we extend the definition of the qx-asymptotic functions, for an extended real-valued function defined on the infinite-dimensional topological normed spaces without lower semicontinuity or quasi-convexity condition. As the main result, by using some asymptotic conditions, we obtain sufficient optimality conditions for the existence of solutions to equilibrium problems, under weaker assumptions of continuity and convexity, when the feasible set is an unbounded subset of infinite-dimensional space. Also, as a corollary, we obtain necessary and sufficient optimality conditions for the existence of solutions to equilibrium problems with an unbounded feasible set. Finally, as an application, we establish a result for the existence of solutions to minimization problems.