A strongly convergent algorithm for solving multiple set split equality equilibrium and fixed point problems in Banach spaces

Abstract

In this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly smooth. For this purpose, we design an algorithm so that it does not depend on prior estimates of the Lipschitz-type constants for the pseudomonotone bifunctions. Furthermore, we present an application of our study for finding a common element of the set of solutions of multiple set split equality variational inequality problems and fixed point sets for two finite families of Bregman quasi-nonexpansive mappings. Finally, we conclude with two numerical experiments to support our proposed algorithm.

An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

“…For a sequence {x n } in E, we denote the strong convergence of {x n } to x ∈ E by x n → x and the weak convergence by x n ⇀ x. We briefly introduce some geometry of Banach spaces that are relevant to this work; for details, see [26,27]. An element z ∈ E is called a weak cluster point of {x n } if there exists a subsequence {x n 𝑗 } of {x n } converging weakly to z.…”

“…For a sequence $$$ \left\{{x}_n\right\} $$$ in $$$ E $$$, we denote the strong convergence of $$$ \left\{{x}_n\right\} $$$ to $$$ x\in E $$$ by $$$ {x}_n\to x $$$ and the weak convergence by $$$ {x}_n\rightharpoonup x $$$. We briefly introduce some geometry of Banach spaces that are relevant to this work; for details, see [26, 27]. An element $$$ z\in E $$$ is called a weak cluster point of $$$ \left\{{x}_n\right\} $$$ if there exists a subsequence $$$ \left\{{x}_{n_j}\right\} $$$ of $$$ \left\{{x}_n\right\} $$$ converging weakly to $$$ z $$$.…”

On fixed point algorithms for solving split inverse problems with applications

“…Split variational inequality problem with multiple output sets. Let H be a Hilbert space and A : C → H be a nonlinear mapping, where C is a nonempty, closed, and convex subset of H. The variational inequality problem is formulated as follows: Find x * ∈ C such that y − x * , Ax * ≥ 0 for all y ∈ C. We denote its solution set by VI(C, A), see [37]. Now, we recall the indicator function of C defined by…”