Strong Convergence by a Hybrid Algorithm for Finding a Common Fixed Point of Lipschitz Pseudocontraction and Strict Pseudocontraction in Hilbert Spaces

Kasamsuk Ungchittrakool

doi:10.1155/2011/530683

Abstract

We prove a strong convergence theorem by using a hybrid algorithm in order to find a common fixed point of Lipschitz pseudocontraction and κ-strict pseudocontraction in Hilbert spaces. Our results extend the recent ones announced by Yao et al. (2009) and many others.

Highlights

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H
In order to obtain a strong convergence theorem for the Mann iteration method 1.4 to nonexpansive mapping, Nakajo and Takahashi 18 modified 1.4 by employing two closed convex sets that are created in order to form the sequence via metric projection so that strong convergence is guaranteed
Together with the fact that {xn} is bounded, which guarantees that every weak limit point of {xn} is a fixed point of T

Summary

Introduction

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. In order to obtain a strong convergence theorem for the Mann iteration method 1.4 to nonexpansive mapping, Nakajo and Takahashi 18 modified 1.4 by employing two closed convex sets that are created in order to form the sequence via metric projection so that strong convergence is guaranteed. Later, it is often referred as the hybrid algorithm or the CQ algorithm. Motivated and inspired by the above works, in this paper, we generalize 1.7 to the Ishikawa iterative process in the case of finding the common fixed point of Lipschitz pseudocontraction and κ-strict pseudocontraction. We provide some applications of the main theorem to find the common zero point of the Lipshitz monotone mapping and γ-inverse strongly monotone mapping in Hilbert spaces

Preliminaries

Main Result