$\left(\varphi_1, \varphi_2\right)-$Variational principle

TL;DR

This paper introduces a new variational principle in Banach spaces, showing that any lower semi-continuous bounded below function can be approximated by a small convex perturbation to attain a strong minimum, extending classical principles.

Contribution

It extends existing variational principles by establishing the existence of a small convex perturbation that ensures the attainment of a strong minimum in Banach spaces.

Findings

Extension of Ekeland's variational principle

Introduction of (, )-convex functions

Guarantee of strong minima for perturbed functions

Abstract

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $\left(\varphi_1, \varphi_2\right)-$convex function $g, $ with arbitrarily small norm, such that $f + g $ attains its strong minimum on $X. $ This result extends some of the well-known varitional principles as that of Ekeland [18], that of Borwein-Preiss [6] and that of Deville-Godefroy-Zizler [14, 15].