Parallel hybrid iterative methods for variational inequalities, equilibrium problems and common fixed point problems

TL;DR

This paper introduces two parallel hybrid iterative algorithms that efficiently find common solutions across fixed points, variational inequalities, and equilibrium problems in smooth, convex Banach spaces, with demonstrated numerical effectiveness.

Contribution

The paper presents novel strongly convergent parallel hybrid iterative methods for solving multiple complex problems simultaneously in Banach spaces, extending existing approaches.

Findings

Algorithms converge strongly in Banach spaces.

Numerical experiments confirm efficiency.

Methods handle multiple problem types simultaneously.

Abstract

In this paper we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of quasi $\phi$-asymptotically nonexpansive mappings $\{F(S_j)\}_{j=1}^N$, the set of solutions of variational inequalities $\{VI(A_i,C)\}_{i=1}^M$ and the set of solutions of equilibrium problems $\{EP(f_k)\}_{k=1}^K$ in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.