Multiple-sets split feasibility problem and split equality fixed point problem for firmly quasi-nonexpansive or nonexpansive mappings

Tongxin Xu,Luoyi Shi

doi:10.1186/s13660-021-02656-1

Abstract

In this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem (MSSFP for short) and the split equality fixed point problem (SEFPP for short) with firmly quasi-nonexpansive operators or nonexpansive operators in real Hilbert spaces. Under mild conditions, we prove strong convergence theorems for the algorithm by using the projection method and the properties of projection operators. The result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

Highlights

Introduction and preliminaries LetH1, H2, and H3 be three real Hilbert spaces with inner product ·, · and induce norm·
The split feasibility problem (SFP) in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [6] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [3]
We introduce a multiple-sets split feasibility problem (MSSFP) and a split equality fixed point problem (SEFPP), the multiple-set split feasibility problem (MSSFP) is to find a pair (x, y) such that t1

Summary

L with

Lemma 1.4 ([13, 14]) Let X be a Banach space, C be a closed convex subset of X, and T : C → C be a nonexpansive mapping with Fix(T) = ∅. We can get that hα is strictly convex, coercive, and differentiable with gradient hα(w) = Mw + Nw + G∗Gw + αw It follows from Lemma 1.7 that wα is characterized by the inequality h(wα) + αwα, w – wα ≥ 0, ∀w ∈ Fix(T). Theorem 3.2 The sequence {wn} generated by Algorithm 3.1 converges strongly to the minimum-norm solution wof MSSFP and SEFPP. The demiclosedness principle ensures that each weak limit point of {wn} is a fixed point of the nonexpansive mapping R = TS, that is, a point of the solution set of MSSFP and SEFPP.

Numerical experiments

Conclusions