A Parallel Hybrid Method for Equilibrium Problems, Variational Inequalities and Nonexpansive Mappings in Hilbert Space

Abstract

Abstract. In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. Strong convergence theorem is proved for the sequence generated by the scheme. Finally, a parallel iterative algorithm for two finite families of variational inequalities and nonexpansive mappings is established.

Modified hybrid projection methods for finding common solutions to variational inequality problems

“…Let us denote the solution set of VIP (2) by V I (A, K ). Problem 1 is a generalization of many other problems including: convex feasibility problems, common fixed point problems, common minimizer problems, common saddle-point problems, hierarchical variational inequality problems, variational inequality problems over the intersection of convex sets, etc., see [3][4][5]7,12,21,22]. In this paper, we focus on projection methods, which together with regularization ones are fundamental methods for solving VIPs with monotone and Lipschitz continuous mappings.…”

Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings

“…From (10), (13) and ||z n − x n || ≤ ||z n − x n+1 || + ||x n+1 − x n || we get lim n→∞ ||z n − x n || = 0.…”

“…Moreover, it can be used to solve systems of monotone operator equations in Hilbert spaces or accretive operator equations in Banach spaces. Other parallel algorithms for finding a common solution of a finite family of accretive operator equations in Banach spaces can be found in [2,3,13].…”

Hybrid projection methods for equilibrium problems with non‐Lipschitz type bifunctions

“…H is a nonlinear operator for each i 2 I, the CSEP (1) becomes the problem of finding of common solutions to variational inequality problems, which has been early studied, for instance, in [7][8][9]. The aforementioned formulation extends this formalism to systems of such problems, covering in particular various forms of feasibility problems such as common fixed point problems, common minimizer problems, common saddle point problems, and systems of operator equations, see [6,7,[10][11][12][13][14][15][16] and the references therein. These problems have received much attention because of broad applicability in many areas of applied mathematics [17][18][19][20][21].One of the most popular methods for solving EPs is the proximal point method (PPM).…”