The extragradient-Armijo method for pseudomonotone equilibrium problems and strict pseudocontractions

Abstract

In this article, we present a new iteration method for finding a common element of the set of fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions without Lipschitz-type continuous conditions. The iterative process is based on the extragradient method and Armijo-type linesearch techniques. We obtain weak convergence theorems for the sequences generated by this process in a real Hilbert space.

Fixed Point Solution Methods for Solving Equilibrium Problems

“…These methods can be considered as an improvement of the viscosity approximation method in [21] and Armijo-type linesearch techniques in [5]. Then, the algorithms are modified by projecting on a suitable convex set to obtain a new variant which possesses a better convergence property.…”

“…where I is the identity mapping on C. In [5], Anh and Hien introduced an Armijo-type iteration method for finding a common element of the set of fixed points of strict pseudocontractions S i (i = 1, 2, . .…”

“…Then, convergent properties of the method are established, among them the global convergence is proved under pseudomonotone assumptions of the bifunction and without Lipschitz-type continuous conditions. This technique also has extensively developed by many authors for solving equilibrium problems and others (see [2,5,7,11,19,20]). …”

“…Then, the convergence is established under continuously differentiable and strongly monotone assumptions of the bifunction. Recently, Anh et al in [5] introduced an Armijo-type method for pseudomonotone equilibrium problems and fixed point problems. The method is based on the auxiliary problem principle and the Armijo-type linesearch techniques.…”

“…Motivated by fixed point techniques of Takahashi and Takahashi in [21] and an improvement set of extragradient-type iteration methods in [5], we introduce new iteration algorithms for finding a common of the solution set of an equilibrium problem with a monotone bifunction and the solution set of a pseudomonotone equilibrium problem. In the first one, the second equilibrium problem is not required to satisfy any monotonicity property, but it must satisfy a certain Lipschitz-type continuous condition.…”