A New Hybrid-Extragradient Method for Generalized Mixed Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems

Abstract

In this paper, we introduce a new iterative scheme based on the hybrid method and the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz-continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend and unify some well-known strong convergence theorems in the literature.

Hybrid steepest-descent viscosity methods for triple hierarchical variational inequalities with constraints of mixed equilibria and bilevel variational inequalities

“…It is assumed as in [18] that Θ : C × C → R is a bifunction satisfying conditions (A1)-(A4) and ϕ : C → R is a lower semicontinuous and convex function with restrictions (B1) or (B2), where (A1) Θ(x, x) = 0 for all x ∈ C; (A2) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) 0 for any x, y ∈ C; (A3) Θ is upper-hemicontinuous, i.e., for each x, y, z ∈ C, lim sup t→0 + Θ(tz + (1 − t)x, y) Θ(x, y); (A4) Θ(x, ·) is convex and lower semicontinuous for each x ∈ C; (B1) for each x ∈ H and r > 0, there exists a bounded subset D x ⊂ C and y x ∈ C such that for any z ∈ C \ D x , Θ(z, y x ) + ϕ(y x ) − ϕ(z) + 1 r y x − z, z − x < 0;…”

“…In 2008, Peng and Yao [18] introduced the following generalized mixed equilibrium problem (GMEP) of finding x ∈ C such that Θ(x, y) + ϕ(y) − ϕ(x) + Ax, y − x 0, ∀y ∈ C.…”

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

“…(1.1) If k = 1, then SGMEP reduces to the generalized mixed equilibrium problem considered in [17]. More precisely, let ϕ : C → R be a real-valued function, A : H → H be a nonlinear mapping and Θ : C × C → R be a bifunction.…”

On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions and systems of generalized equilibria

“…The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see e.g., [8,9,10,11,12,14,18,20,23,24,25,28,29,30,34,35,36] and references therein, to name but a few.…”