Strong convergence theorems for generalized nonexpansive mappings on star-shaped set with applications

Eskandar Naraghirad,Lai-Jiu Lin

doi:10.1186/1687-1812-2014-72

Abstract

In this paper, we study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify Halpern’s iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to an open problem posed by Akashi and Takahashi in a recent work of (Appl. Math. Comput. 219(4):2035-2040, 2012) for nonexpansive and nonspreading mappings. Furthermore, we study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iteration scheme for these mappings. As applications of our results, we study multiple sets split monotone inclusion problems for inverse strongly monotone mappings, multiple sets split optimization problems, and multiple sets split feasibility problems. Our results contain many original results on multiple sets split feasibility problem in the literature. Our results also improve and generalize many well-known results in the current literature. MSC:47H10, 37C25.

Highlights

Throughout this paper, we denote the set of real numbers and the set of positive integers by R and N, respectively
We study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space
We study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iterative scheme for these mappings

Summary

Introduction

Throughout this paper, we denote the set of real numbers and the set of positive integers by R and N, respectively. Akashi and Takahashi [ ] proved the following strongly convergence attractive point theorem for nonexpansive mappings on a star-shaped set C of a Hilbert space. We study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify the Halpern iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to open Question. Let C be a nonempty subset of a real Hilbert space H and T : C → C be a nonexpansive mapping. Let C be a nonempty subset of a Banach space X and T : C → C be a mapping which satisfies condition (Eμ) for some μ ≥.

Thus we obtain

We see that the sequence xn converges strongly to