Hybrid viscosity methods for equilibrium problems, variational inequalities, and fixed point problems

Abstract

In this paper, we introduce a hybrid viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of common fixed points of one finite family of nonexpansive mappings, and another infinite family of nonexpansive mappings in a real Hilbert space. This hybrid viscosity iterative algorithm is based on viscosity approximation method, Mann's iterative method, projection method, strongly positive bounded linear Operator, and W -mapping approaches. We study the strong convergence of the proposed algorithm to a common element under appropriate assumptions, which also solves an optimization problem. The result presented in this paper improves and extends some known results in the literature.Keywords: hybrid viscosity iterative algorithm; general mixed equilibrium problem; general system of variational inequalities; nonexpansive mapping; inverse strongly monotone mapping

Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

“…Let T i : C → C be a nonexpansive mapping for each i = 1, 2, ... such that F := ∞ i=1 Fix(T i ) ∩ GSVI(C, A, B) ∅, and f ∈ Ω with contractive constant ∈ (0, 1). Let {V k } ∞ k=1 be defined by (7) and (8).…”

“…Let T i : C → C be a nonexpansive mapping for each i = 1, 2, ... such that F := ∞ i=1 Fix(T i ) ∩ GSVI(C, A, B) ∅, and f ∈ Ω with contractive constant ∈ (0, 1). Let {V k } ∞ k=1 be defined by (7) and (8). Given sequences {ι k } ∞ k=1 , {τ k } ∞ k=1 in [0, 1] and { k } ∞ k=1 in (0, 1], the following conditions are satisfied: (C1) 0 < τ k ≤ 1 − , ∀k ≥ k 0 for some k 0 ≥ 1, and…”

“…( [5]) Let C be a nonempty closed convex subset of a Banach space X and let {T i } ∞ i=1 be an infinite family of nonexpansive self-mappings on C such that the set of common fixed points F := ∞ i=1 Fix(T i ) ∅. Let V k be a mapping, defined by (7), and let α i satisfy (8). Then, for each x ∈ C and i ≥ 1, lim k→∞ V i k x exists.…”

“…The system of variational inequalities plays a crucial tool in science, engineering and economics. Namely, the practical problem can be formulated in the form of a system of variational inequalities; see e.g., [1,2,[7][8][9]19] and the references therein. Note that problem (1) can be reduced to the following classical variational inequality (VI) of finding x * ∈ C such that…”

A modified iterative algorithm for finding a common element in Hilbert space

“…In 2014, Bnouhachem [2] proposed a modified projection method for computing a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in Hilbert space, and proved the strong convergence of the iteration sequences. Since the iterative algorithms played an important role for solving integral and differential equations, optimization problems, image reconstruction problems, game theory and other fields such as [5,16,19,20,22], the convergence and construction of the iteration algorithm for computing fixed points has attracted more and more attentions see, e.g., [3,6,12,18,21].…”