Strong convergence theorem for monotone operators and strict pseudo-nonspreading mapping

Jeremiah Nkwegu Ezeora

doi:10.28919/afpt/2203

Abstract

In this paper, based on the recent results of Osilike et al. [9] and motivated by the results of Liu et al. [10] and Takahashi et al. [13], we introduce an iterative sequence and prove that the sequence converges strongly to a common element of the set of fixed points of strict pseudo-non spreading mapping, T and the set of zeros of sum of an α−inverse strongly monotone mapping A and a maximal monotone operator B in a real Hilbert space. Our results improve and generalize many recent important results.

Highlights

Throughout this paper, we assume that H is a real Hilbert space, C is a nonempty subset of H
We denote by xn x and xn → x weak and strong convergence of a sequence {xn}, respectively and by F(T ) the set of fixed points of a mapping T : C → C
In the case where T : C → C is a nonexpansive mapping, A : C → H is an α− inverse strongly monotone mapping, and B ⊂ H × H is a maximal monotone operator, Takahashi et al [13] proved a strong convergence theorem for finding a point of F(T ) ∩ (A + B)−1(0), where (A + B)−1(0) is the set of zero points of (A + B)

Summary

Introduction

Throughout this paper, we assume that H is a real Hilbert space, C is a nonempty subset of H. In the case where T : C → C is a nonexpansive mapping, A : C → H is an α− inverse strongly monotone mapping, and B ⊂ H × H is a maximal monotone operator, Takahashi et al [13] proved a strong convergence theorem for finding a point of F(T ) ∩ (A + B)−1(0), where (A + B)−1(0) is the set of zero points of (A + B). Motivated by the results of Takahashi et al [13], Liu et al [10], Osilike et al [9], Kurokawa and Takahashi [4], we introduce a new algorithm and prove strong convergence of the sequence of the algorithm to a common element of the set of fixed points of k−strictly pseudo nonspreading mapping and the set of zero points of sum of α−inverse strongly monotone operator A and a maximal operator B in a real Hilbert space. [10] and a host of other recent important results

Preliminaries

Main results