A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications

TL;DR

This paper introduces a new hybrid iterative scheme in Banach spaces that guarantees strong convergence to a common solution of multiple equilibrium, variational inequality, and fixed point problems, extending existing results.

Contribution

The paper proposes a novel hybrid method that unifies and extends convergence results for solving multiple nonlinear problems simultaneously in Banach spaces.

Findings

The method converges strongly to a common solution.

It generalizes several existing results in the literature.

Applications demonstrate the method's effectiveness.

Abstract

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^*$. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a common solution of a family of variational inequality problems and a common element of fixed points of a family of a general class of nonlinear nonexpansive maps. The sequence of this new method is proved to converge strongly to a common element of the families. Our theorem and its applications complement, generalize, and extend various results in literature.