Mixed equilibrium problems and optimization problems

Abstract

In this paper, we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings. Furthermore, we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions. We also discuss some special cases. Results obtained in this paper improve the previously known results in this area.

Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems

“…If B = 0, then the problem (1.1) reduces the following mixed equilibrium problem (for short, MEP) of finding x ∈ C such that Θ(x, y) + ϕ(y) − ϕ(x) 0, ∀y ∈ C, (1. 3) which was studied by Ceng and Yao [5] (see also [39]). The set of solutions of the problem (1.3) is denoted by MEP(Θ, ϕ).…”

“…In 2008, Peng and Yao [22] studied an iterative method for the GMEP (1.1) related to an α-inverse-strongly monotone mapping B, the VIP (1.5) for a monotone and Lipschitz continuous mapping F and a nonexpansive mapping S, and proved strong convergence to a point z ∈ GMEP(Θ, ϕ, B) ∩ VI(C, F) ∩ Fix(S). In 2010, by using the method of Yao et al [39], Jaiboon and Kumam [12] also introduced an iterative method related to optimization problem for the MEP (1.3), the VIP (1.5) for an α-inverse-strongly monotone mapping F and a sequence {S n } of nonexpansive mappings, and showed strong convergence to a point z ∈ ∩ ∞ n=1 Fix(S n ) ∩ MEP(Θ, ϕ) ∩ VI(C, F).…”

The hybrid algorithm for the system of mixed equilibrium problems, the general system of finite variational inequalities and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings

“…The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problem as special cases (see [9][10][11][12][13][14][15][16][17][18][19]). In the last two decades, many articles have appeared in the literature on the existence of solutions of equilibrium problems; see, for example [13] and references therein.…”

“…In the last two decades, many articles have appeared in the literature on the existence of solutions of equilibrium problems; see, for example [13] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, (see [11][12][13][14][16][17][18][19][20][21][22][23][24]) and references therein.…”

A projected fixed point algorithm with Meir-Keeler contraction for pseudocontractive mappings

“…In the literature, there are a large number references associated with the fixed point algorithms for nonexpansive mappings and pseudocontractive mappings. See, for instance, [1][2][3][4][5][6][7]11] and [9,10,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. The first interesting result for finding the fixed points of the pseudocontractive mappings was presented by Ishikawa in 1974 as follows.…”