Abstract
In this paper, we introduce a Halpern algorithm and a nonconvex combination algorithm to approximate a solution of the split common fixed problem of quasi- ϕ -nonexpansive mappings in Banach space. In our algorithms, the norm of linear bounded operator does not need to be known in advance. As the application, we solve a split equilibrium problem in Banach space. Finally, some numerical examples are given to illustrate the main results in this paper and compare the computed results with other ones in the literature. Our results extend and improve some recent ones in the literature.
Highlights
Let H1 be a Hilbert space, and let C be the nonempty closed convex subset of H1
Few authors continue to study the algorithm of Censor and Elfving since the difficulty of computing A− 1, even if it exists. Another algorithm solving split feasibility problem (SFP) (1) is more popular which is called CQ algorithm given by Byrne [6, 7]. e CQ algorithm of Byrne is a gradient projection method in convex minimization
The computations of PC and PQ are difficult if these projections did not have the closed-form expressions which is such that the CQ algorithm of Byrne [6, 7] is not easy to implement in this case
Summary
Let H1 be a Hilbert space, and let C be the nonempty closed convex subset of H1. Let H2 be a real Hilbert space, and let Q be the nonempty closed convex subset of H2. In 2018, Hieu and Strodiot [26] introduced a new iterative algorithm for solving pseudomonotone equilibrium problem involving the fixed point problem for quasi-φ-nonexpansive mapping in Banach space without using the hybrid or shrinking projection methods. In this paper, motivated by the work of [22, 26, 27], we introduce some algorithms to solve a split common fixed point problem for two families of quasi-φ-nonexpansive mappings in Banach spaces and prove the strong convergence for the proposed algorithms. Our results extend the one of Ma et al [22] from one quasi-nonexpansive mapping to two quasi-nonexpansive mappings and [27] from Hilbert space to Banach space
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