Variational-Like Inequalities and Equilibrium Problems with Generalized Monotonicity in Banach Spaces

N K Mahato,C Nahak

doi:10.1155/2012/648070

Abstract

We introduce the notion of relaxed (ρ-θ)-η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (ρ-θ)-η-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρ-θ)-pseudomonotonicity for bifunctions, and we consider some examples to show that (ρ-θ)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρ-θ)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.

Highlights

Let K be a nonempty subset of a real reflexive Banach space X, and let X∗ be the dual space of X
We introduce the notion of relaxed ρ-θ -η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps
Inspired and motivated by 1, 3, 10, we introduce the concept of relaxed ρ-θ η-invariant pseudomonotone mappings

Summary

Introduction

Let K be a nonempty subset of a real reflexive Banach space X, and let X∗ be the dual space of X. Many authors established the existence of solutions for VIP and VLIP under generalized monotonicity assumptions, such as quasimonotonicity, relaxed monotonicity, densely pseudomonotonicity, relaxed η-α-monotonicity, and relaxed η-α-pseudomonotonicity see 1, 3–6 and the references therein. Fang and Huang 3 obtained the existence of solution for VLIP using relaxed η-α-monotone mappings in the reflexive Banach spaces. In 1 , Bai et al extended the results of 3 with relaxed η-α-pseudomonotone mappings and provided the existence of solution of the variational-like inequalities problems in reflexive Banach spaces. Using the KKM technique, we establish the existence of solutions for Variational-like inequality problems with relaxed ρ-θ -η-invariant pseudomonotone mappings. The existence of solutions of equilibrium problem with ρ-θ -pseudomonotone mappings in reflexive Banach spaces are demonstrated, by using the KKM technique

Preliminaries

Conclusions