Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Abstract

Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the setFix (S i ) of fixed points of a finite family of nonexpansivein a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element ofFix (S i ) ∩ GMEP , which is the unique solution of a variational inequality.

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

“…In this case, the variational inequality defined on such feasible set is also called a hierarchical variational inequality (HVI). Yamada's method is subsequently extended and modified to solve more complex problems, containing finite or infinite nonexpansive mappings (see, e.g., [3,6,40] and references therein). In [40], based on the Yamada result, Zeng and Yao introduced an implicit method that converges weakly to a solution of a variational inequality, involving a Lipschitz continuous and strongly monotone mapping in a Hilbert space H, where the feasible set is that of common fixed points of a finite family of nonexpansive mappings on H. In [7], Ceng et al extended this result from nonexpansive mappings to Lipschitz pseudocontractive mappings and strictly pseudocontractive mappings on H. Recently, in [4], Buong and Anh proposed a strongly convergent implicit method, which is a modification of Yamada's result.…”

Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces

“…This problem is a fundamental problem in the variational analysis, optimization theory, and mechanics; see e.g., [8,11,17,24,[29][30][31] and the references therein. A large number of algorithms for solving this problem are essentially projection algorithms that employ projections onto the feasible set C of the VI, or onto some related sets, so as to iteratively reach a solution.…”

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

“…The GMEP (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see e.g., [20,25,26,29,36].…”