A Quasiconvex Asymptotic Function with Applications in Optimization

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

“…In [16], one of the most interesting results is [16,Proposition 4.2]. Now the extension of this result, for infinite dimensional Banach spaces, is given below.…”

“…At first, we extend the definition of qx-asymptotic function [16], for extended real-valued function that define on an infinite dimensional topological normed space and it has neither lower semicontinuity nor quasi-convexity conditions. Also, we establish some property of this asymptoptic function.…”

“…Notice that, there is no connection between f σg and f ∞ σ or f ∞ q , that was presented in [21]. Indeed, similar to example in [16], we consider the constant real-valued function f (x) = α, for all x ∈ X. We have…”

“…Let f be quasi-convex and sequentially σ-lsc, thus similar to equation ( 12) in [16], by Proposition 1(iv…”

“…In section 2, we have some basic definitions and results. In section 3, we extend the definition of qx-asymptotic function [16], for extended real-valued function that define on an infinite dimensional space. Also, we establish some property of this asymptotic function.…”