Some results on continuous pseudo-contractions in a reflexive Banach space

Songtao Lv,Yan Hao

doi:10.1186/1029-242x-2013-538

Abstract

In this paper, we investigate fixed point problems of a continuous pseudo-contraction based on a viscosity iterative scheme. Strong convergence theorems are established in a reflexive Banach space which also enjoys a weakly continuous duality mapping.

Highlights

1 Introduction and preliminaries Fixed point problems of nonlinear operators, which include many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, and saddle point problems, recently have been studied as an effective and powerful tool for studying many real world problems which arise in economics, finance, medicine, image reconstruction, ecology, transportation, and network; see [ – ] and the references therein
What happens if the mapping f is a strong pseudo-contraction instead of a contraction? Does Theorem ZS still hold? Since we do not know whether the mapping PCf, where f is a strong pseudo-contraction, has a unique fixed point or not, we cannot answer the above question based on Suzuki’s results
Let C be nonempty closed and convex subset of a real Banach space E, and let T : C → C be a continuous pseudo-contraction

Summary

Introduction

Introduction and preliminariesFixed point problems of nonlinear operators, which include many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, and saddle point problems, recently have been studied as an effective and powerful tool for studying many real world problems which arise in economics, finance, medicine, image reconstruction, ecology, transportation, and network; see [ – ] and the references therein. Reich [ ] extended Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth Banach space, xt converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from C onto F(T).

Results

Conclusion