Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption

Jinzuo Chen,Li-Jun Zhu,Mihai Postolache

doi:10.3390/math7090777

Abstract

Two iterative algorithms are suggested for approximating a solution of the split common fixed point problem involved in pseudo-contractive operators without Lipschitz assumption. We prove that the sequence generated by the first algorithm converges weakly to a solution of the split common fixed point problem and the second one converges strongly. Moreover, the sequence { x n } generated by Algorithm 3 strongly converges to z = proj S 0 , which is the minimum-norm solution of problem (1). Numerical examples are included.

Highlights

The split common fixed point problem was investigated in 2009 by Censor Y. and Segal A. [1].Further research on this problem discussed in works by the authors of [2,3,4,5,6,7,8,9,10,11,12]
Let x ∈ H1 be a solution of split feasibility problem satisfying x ∈ C and Ax ∈ Q, (2)
We extend a previous author’s results from the demicontractive operators

Summary

Introduction

The split common fixed point problem was investigated in 2009 by Censor Y. and Segal A. [1]. The split common fixed point problem was investigated in 2009 by Censor Y. and Segal A. Let x ∈ H1 be a solution of split common fixed point problem satisfying x ∈ F (U ) and Ax ∈ F ( T ). Let x ∈ H1 be a solution of split feasibility problem satisfying x ∈ C and Ax ∈ Q, These two problems ((1) and (2)) have received much attention, and have been extensively investigated due to applications in signal processing, image reconstruction, [14], and intensity modulated radiation therapy [20]. Wang J. et al [22] study the linear convergence of CQ algorithm for solving the problem (2) and investigate an application in gene regulatory network inference. Wang obtained the weak convergence of algorithm (4). Weak and strong convergence of the proposed algorithms are obtained

Preliminaries

Weak Convergence Theorem

Strong Convergence Theorem

Numerical Example

Conclusions