Levitin-Polyak Well-Posedness of an Equilibrium-Like Problem in Banach Spaces

Ru-Liang Deng

doi:10.1155/2014/368098

Abstract

The concept of Levitin-Polyak well-posedness of an equilibrium-like problem in Banach spaces is introduced. Under suitable conditions, some characterizations of its Levitin-Polyak well-posedness are established. Some conditions under which an equilibrium-like problem in Banach spaces is Levitin-Polyak well-posed are also derived.

Highlights

In 1966, Tykhonov [1] first established the well-posedness of a minimization problem, which has been known as Tykhonov well-posedness
ELP(F, φ, K) is strongly LP α-well-posed in the generalized sense if ELP(F, φ, K) has nonempty solution set S and every LP α-approximating sequence has a subsequence which converges strongly to some point of S
We show that μ (Ωα (ε)) 󳨀→ 0 as ε 󳨀→ 0

Summary

Introduction

In 1966, Tykhonov [1] first established the well-posedness of a minimization problem, which has been known as Tykhonov well-posedness. Ceng and Yao [9] got some results for the well-posedness of the generalized mixed variational inequality, the corresponding inclusion problem, and the corresponding fixed point problem. Li and Xia [14] considered the Levitin-Polyak well-posedness of a generalized variational inequality in Banach space. They showed that the Levitin-Polyak well-posedness of a generalized variational inequality is equivalent to the uniqueness and existence of its solutions. Motivated and inspired by the research work going on in this field, in this paper, we extend the notion of Levitin-Polyak well-posedness to an equilibrium-like problem in Banach spaces and give some metric characterizations of its LevitinPolyak well-posedness. We derive some conditions under which an equilibrium-like problem is Levitin-Polyak well-posed

Preliminaries

Conditions for Levitin-Polyak Well-Posedness