Strong convergence theorem for strict pseudo-contractions in Hilbert spaces

Giuseppe Marino,Bruno Scardamaglia,Erdal Karapinar

doi:10.1186/s13660-016-1072-6

Abstract

In this paper, inspired by Hussain et al. (Fixed Point Theory Appl. 2015:17, 2015), we study a modified Mann method to approximate strongly fixed points of strict pseudo-contractive mappings. In (Hussain et al. in Fixed Point Theory Appl. 2015:17, 2015) it is shown that the same algorithm converges strongly to a fixed point of a nonexpansive mapping under suitable hypotheses on the coefficients. Here the assumptions on the coefficients are different, as well as the techniques of the proof.

Highlights

Let H be a real Hilbert space with the inner product ·, ·, which induces the norm · .Let C be a nonempty, closed, and convex subset of H
Let T be a nonlinear mapping of C into itself; we denote with Fix(T) the set of fixed points of T, that is, Fix(T) = {z ∈ C : Tz = z}
Note that the class of strict pseudo-contractions includes the class of nonexpansive mappings, which are mappings T on C such that

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · , which induces the norm · .Let C be a nonempty, closed, and convex subset of H. If T : C → C is a nonexpansive mapping with a fixed point in a closed and convex subset of a uniformly convex Banach space with a Frechét differentiable norm, and if the control sequence (αn)n∈N is chosen so that ∞, the sequence (xn) generated by Mann’s algorithm converges weakly to a fixed point of T [ ].

Objectives

Results

Conclusion