Strong Convergence of a System of Generalized Mixed Equilibrium Problem, Split Variational Inclusion Problem and Fixed Point Problem in Banach Spaces

Mujahid Abbas,Abdul Rahim Khan,Manuel De La Sen,Yusuf Ibrahim

doi:10.3390/sym11050722

Abstract

The purpose of this paper is to introduce a new algorithm to approximate a common solution for a system of generalized mixed equilibrium problems, split variational inclusion problems of a countable family of multivalued maximal monotone operators, and fixed-point problems of a countable family of left Bregman, strongly asymptotically non-expansive mappings in uniformly convex and uniformly smooth Banach spaces. A strong convergence theorem for the above problems are established. As an application, we solve a generalized mixed equilibrium problem, split Hammerstein integral equations, and a fixed-point problem, and provide a numerical example to support better findings of our result.

Highlights

Introduction and PreliminariesLet E be a real normed space with dual E∗
We show that x ∗ ∈ (∩i∞=1 generalized mixed equilibrium problem (GMEP)(θi, Ci, Gi, gi )
E1 n n and its strong convergence is guaranteed, which solves the problem of a common solution of a system of generalized mixed equilibrium problems, split Hammerstein integral equations, and fixed-point problems for the mappings involved in this algorithm

Summary

Introduction

Introduction and PreliminariesLet E be a real normed space with dual E∗. A map B : E → E∗ is called: monotone if, for each x, y ∈ E, hη − ν, x − yi ≥ 0, ∀ η ∈ Bx, ν ∈ By, where h·, ·i denotes duality pairing,(ii) e-inverse strongly monotone if there exists e > 0, such that h Bx − By, x − yi ≥ e|| Bx − By||2 ,(iii) maximal monotone if B is monotone and the graph of B is not properly contained in the graph of any other monotone operator. In a smooth Banach space E, the Bregman distance 4 p of x to y, with respect to the convex continuous function f : E → R, such that f ( x ) = 1p k x k p , is defined by 4 p ( x, y) = Let E be a real uniformly convex Banach space, K a non-empty closed subset of E, and T : K → K an asymptotically non-expansive mapping.

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