Noncoercive and Noncontinuous Equilibrium Problems (Existence Theorem in Infinite Dimensional Spaces)

TL;DR

This paper extends the concept of qx-asymptotic functions in infinite dimensional spaces to establish weaker conditions for the existence of solutions to equilibrium problems, including unbounded sets, without requiring lower semicontinuity or quasi-convexity.

Contribution

It introduces a generalized framework for equilibrium problems in infinite dimensional spaces, relaxing traditional continuity and convexity assumptions.

Findings

Established sufficient conditions for equilibrium solutions with weaker assumptions.

Derived necessary and sufficient conditions for unbounded feasible sets.

Applied results to prove existence of solutions in minimization problems.

Abstract

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