An extragradient algorithm for split generalized equilibrium problem and the set of fixed points of quasi-φ-nonexpansive mappings in Banach spaces

Abstract

In this paper, we study the problem of finding a common solution to split generalized mixed equilibrium problem and fixed point problem for quasiφ -nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach space E1 and a smooth, strictly convex, and reflexive Banach space E2 . An iterative algorithm with Armijo linesearch rule for solving the problem is presented and its strong convergence theorem is established. The convergence result is obtained without using the hybrid method which is mostly used when strong convergence is desired. Finally, numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm in comparison with other existing algorithms in the literature. Our results extend and improve many recent results in this direction.

A Totally Relaxed Self-Adaptive Subgradient Extragradient Scheme for Equilibrium and Fixed Point Problems in a Banach Space

“…Let f : C × C → R ∪ {+∞} be a bifunction such that C ⊂ int(dom(f, •)), then for every x ∈ C, the Equilibrium Problem (EP) (see [3,14]), is to find a point x * ∈ C such that f (x * , y) ≥ 0, for all y ∈ C. (1.1) The EP is a generalization of many important optimization problems, such as Variational Inequality Problem (VIP), Fixed Point Problem (FPP) and so on (see [6,14] and the references therein). In particular, if f (x, y) = ⟨Ax, y − x⟩, where A : C → E * , is a nonlinear mapping, then EP (C, f ) (1.1) reduces to the classical VIP introduced by Stampacchia [47] (see also [36,38,41,52]), which is to find a point x * ∈ C such that ⟨Ax * , y − x * ⟩ ≥ 0, for all y ∈ C. (1.2) There are two important directions of research on EP: These are the existence of solution of EP and other related problems (see [14,29] for more details) and the development of iterative algorithms for approximating the solution of EP, its several generalizations and related optimization problems (see [1,12,13,33,34,[42][43][44] and the references therein).…”

A Strong Convergence Theorem for Split Null Point Problem and Generalized Mixed Equilibrium Problem in Real Hilbert Spaces

“…where F : C × C → R is a bifunction. The EP unify many important problems, such as variational inequalities, fixed point problems, optimization problems, saddle point (minmax) problems, Nash equilibria problems and complimentarity problems [2][3][4][5][6][7]. It also finds applications in other fields of studies like physics, economics, engineering and so on [1,2,[8][9][10].…”

Strong Convergence Results of Split Equilibrium Problems and Fixed Point Problems

“…(i) e split equilibrium problem studied in [29,30] can be formulated to find an element u ∈ C such that u ∈ SEP(C, f), Au ∈ SEP(Q, g).…”