Weak and Strong Convergence Theorems for the Multiple-Set Split Equality Common Fixed-Point Problems of Demicontractive Mappings

Yaqin Wang,Xiaoli Fang,Tae-Hwa Kim

doi:10.1155/2017/5306802

Yaqin Wang, Xiaoli Fang + Show 1 more

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https://doi.org/10.1155/2017/5306802

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Abstract

We consider mixed parallel and cyclic iterative algorithms in this paper to solve the multiple-set split equality common fixed-point problem which is a generalization of the split equality problem and the split feasibility problem for the demicontractive mappings without prior knowledge of operator norms in real Hilbert spaces. Some weak and strong convergence results are established. The results obtained in this paper generalize and improve the recent ones announced by many others.

Highlights

For modeling inverse problems which arise from phase retrieval and in medical image reconstruction, in 1994, Censor and Elfving [1] firstly introduced the following split feasibility problem (SFP) in finite-dimensional Hilbert spaces.Let C and Q be nonempty closed convex sets of the Hilbert spaces H1 and H2, respectively, and let A : H1 → H2 be a bounded linear operator
We prove weak and strong convergence results of such algorithms for solving multiple-set split equality common fixedpoint problem (MSECFP) (6)
We study two mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings where the step sizes do not depend on the operator norms ‖A‖ and ‖B‖, and we prove the weak and strong convergence of such algorithms

Summary

Introduction

For modeling inverse problems which arise from phase retrieval and in medical image reconstruction, in 1994, Censor and Elfving [1] firstly introduced the following split feasibility problem (SFP) in finite-dimensional Hilbert spaces.Let C and Q be nonempty closed convex sets of the Hilbert spaces H1 and H2, respectively, and let A : H1 → H2 be a bounded linear operator. They introduced the following two mixed iterative algorithms for solving the MSECFP (6) of quasi-nonexpansive mappings: uk = xk − γkA∗ (Axk − Byk) , xk+1 = αk0uk + αk1U1 (uk) + ⋅ ⋅ ⋅ + αkpUp (uk) , (7)

Results

Conclusion