Strong convergence of composite general iterative methods for one-parameter nonexpansive semigroup and equilibrium problems

Abstract

In this paper, we introduce both explicit and implicit schemes for finding a common element in the common fixed point set of a one-parameter nonexpansive semigroup {T(s)|0 ≤ s <∞} and in the solution set of an equilibrium problems which is a solution of a certain optimization problem related to a strongly positive bounded linear operator. Strong convergence theorems are established in the framework of Hilbert spaces. As an application, we consider the optimization problem of a k-strict pseudocontraction mapping. The results presented improve and extend the corresponding results of many others. 2000 AMS Subject Classification: 47H09; 47J05; 47J20; 47J25.

Common Solution to Generalized Mixed Equilibrium Problem and Fixed Point Problem for a Nonexpansive Semigroup in Hilbert Space

“…This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results and method presented here improve and generalize the corresponding results and methods given in [5,9,10,15].…”

“…Finally, we have the following consequence of Theorem 4.1, which generalizes Theorem 3.1 due to Xiao [15].…”

“…Motivated by the work of Ceng et al [3], Xiao et al [15], Cianciaruso et al [5], Kazmi et al [8,9,10], and by the ongoing research in this direction, we suggest and analyze an explicit hybrid relaxed extragradient iterative method for approximating a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup.…”

“…Very recently, Xiao et al [15] introduced and studied the following implicit iterative scheme and obtained strong convergence theorem for EP(1.4) and FPP(1.1):…”