Fixed point theorems for some new nonlinear mappings in Hilbert spaces

Lai-Jiu Lin,Zenn-Tsun Yu,Chih-Sheng Chuang

doi:10.1186/1687-1812-2011-51

Abstract

AbstractIn this paper, we introduced two new classes of nonlinear mappings in Hilbert spaces. These two classes of nonlinear mappings contain some important classes of nonlinear mappings, like nonexpansive mappings and nonspreading mappings. We prove fixed point theorems, ergodic theorems, demiclosed principles, and Ray's type theorem for these nonlinear mappings.Next, we prove weak convergence theorems for Moudafi's iteration process for these nonlinear mappings. Finally, we give some important examples for these new nonlinear mappings.

Highlights

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H.a mapping T : C ® C is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y Î C
We observe the following fixed point theorems for nonexpansive mappings in Hilbert spaces
In 1971, Pazy [2] gave the following fixed point theorems for nonexpansive mappings in Hilbert spaces

Summary

Introduction

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. In 2007, Moudafi [11] studied weak convergence theorems for two nonexpansive mappings T1, T2 of C into itself, where C is a closed convex subset of a Hilbert space H Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C ® C be an asymptotic nonspreading mapping. Let C be a bounded closed convex subset of a real Hilbert space H, and let T : C ® C be an asymptotic nonspreading mapping (respectively, asymptotic TJ mapping). Let C be a nonempty bounded closed convex subset of a real Hilbert space H, and let T : C ® C be a nonexpansive mapping.

3.2: Demiclosed principles

Common fixed point theorems

Authors’ contributions LJL