Multiple-Set Split Feasibility Problems for Asymptotically Strict Pseudocontractions

Shih-Sen Chang,W B Zhang,Yeol Je Cho,J K Kim,L Yang

doi:10.1155/2012/491760

Abstract

In this paper, we introduce an iterative method for solving the multiple-set split feasibility problems for asymptotically strict pseudocontractions in infinite-dimensional Hilbert spaces, and, by using the proposed iterative method, we improve and extend some recent results given by some authors.

Highlights

The split feasibility problem SFP in finite dimensional spaces was first introduced by Censor and Elfving 1 for modeling inverse problems which arise from phase retrievals and in medical image reconstruction 2
For solving the multiple-set split feasibility problem 1.1, let us assume that the following conditions are satisfied: C1 H1 and H2 are two real Hilbert spaces, A : H1 → H2 is a bounded linear operator; C2 Si : H1 → H1, i 1, 2, . . . , M, is a uniformly Li-Lipschitzian and βi, {ki,n} asymptotically strict pseudocontraction, and Ti : H2 → H2, i 1, 2, . . . , M, is a uniformly Li-Lipschitzian and μi, {ki,n} -asymptotically strict pseudocontraction satisfying the following conditions: a C: F
We prove that xn x∗ and un x∗, which is a solution of the problem MSSFP

Summary

Introduction

The split feasibility problem SFP in finite dimensional spaces was first introduced by Censor and Elfving 1 for modeling inverse problems which arise from phase retrievals and in medical image reconstruction 2. The split feasibility problem in an infinite dimensional Hilbert space can be found in 2, 4, 6–8. Throughout this paper, we always assume that H1, H2 are real Hilbert spaces, “ → ”, “ ” are denoted by strong and weak convergence, respectively. The purpose of this paper is to introduce and study the following multiple-set split feasibility problem for asymptotically strict pseudocontraction MSSFP in the framework of infinite-dimensional Hilbert spaces. {yi ∈ H2 : Tiyi yi} denote the sets of fixed points of Si and Ti, respectively. We use Γ to denote the set of solutions of the problem MSSFP , that is,.

Preliminaries

Multiple-Set Split Feasibility Problem