An alternating iteration algorithm for solving the split equality fixed point problem with L-Lipschitz and quasi-pseudo-contractive mappings

Meixia Li,Xueling Zhou,Haitao Che

doi:10.1186/s13660-019-2203-7

Abstract

In this paper, we are concerned with the split equality common fixed point problem. It is a significant generalization of the split feasibility problem, which can be used in various disciplines, such as medicine, military and biology, etc. We propose an alternating iteration algorithm for solving the split equality common fixed point problem with L-Lipschitz and quasi-pseudo-contractive mappings and prove that the sequence generated by the algorithm converges weakly to the solution of this problem. Finally, some numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.

Highlights

Throughout this paper, we always assume that H1, H2 and H3 are real Hilbert spaces with inner product ·, · and induced norm ·
We find that the equilibrium problem (EP) and the split variational inequality problem (SVI) are special cases of split inverse problem (SIP) from [10, 11, 15, 21]
We prove the weak convergence of the iteration sequence generated by the alternating iteration algorithm

Summary

Introduction

Throughout this paper, we always assume that H1, H2 and H3 are real Hilbert spaces with inner product ·, · and induced norm ·. To solve the split equality feasibility problem (1.6), Moudafi [23] proposed the following alternating CQ algorithm:. Proposed the following iterative algorithm: ⎧ ⎨xn+1 = S(xn – λnA∗(Axn – Byn)), ⎩yn+1 = T (yn + βnB∗(Axn+1 – Byn)) He proved the weak convergence of the sequences generated by scheme (1.9) under the condition that S and T are firmly quasi-nonexpansive mappings. Where S: H1 → H1 and T : H2 → H2 are two L-Lipschitz and quasi-pseudo-contractive mappings with L ≥ 1, Fix(T) = ∅ They proved that the sequence {(xn, yn)} generated by the above modification (1.11) converges weakly to a solution of problem (1.8). 3. we prove that the sequence generated by the algorithm weakly converges to a solution of the split equality common fixed point problem (1.8).

Main results

Conclusions