Strong convergence of an inertial iterative algorithm for variational inequality problem, generalized equilibrium problem, and fixed point problem in a Banach space

Monairah Alansari,Rehan Ali,Mohammad Farid

doi:10.1186/s13660-020-02313-z

Abstract

We propose and analyze an inertial iterative algorithm to approximate a common solution of generalized equilibrium problem, variational inequality problem, and fixed point problem in the framework of a 2-uniformly convex and uniformly smooth real Banach space. Further, we study the convergence analysis of our proposed iterative method. Finally, we give application and a numerical example to illustrate the applicability of the main algorithm.

Highlights

1 Introduction Let C be a nonempty closed convex subset of a real Banach space X and X∗ be the dual space of X; let the pairing between X and X∗ be denoted by ·, ·
We study the generalized equilibrium problem which was to find x ∈ C such that g(x, y) + b(x, y) – b(x, x) ≥ 0, ∀y ∈ C
In 2009, Takahashi et al [15] introduced and studied the following iterative method and studied strong convergence for a relatively nonexpansive mapping to approximate the common solution of a fixed point problem and an equilibrium problem in Banach space: x0 ∈ C, un = J–1(αnJxn + (1 – αn)JTxn), zn such that g(zn, y)

Summary

Introduction

Let C be a nonempty closed convex subset of a real Banach space X and X∗ be the dual space of X; let the pairing between X and X∗ be denoted by ·, ·. In 2009, Takahashi et al [15] introduced and studied the following iterative method and studied strong convergence for a relatively nonexpansive mapping to approximate the common solution of a fixed point problem and an equilibrium problem in Banach space: x0 ∈ C, un = J–1(αnJxn + (1 – αn)JTxn), zn such that g(zn, y) Inspired and motivated by the work given in [12, 15, 19], we introduce and study a hybrid iterative algorithm for approximating a common solution of GEP(1.2), VIP(1.5), and a fixed point problem for a relatively nonexpansive mapping.

Results

Conclusion