New extragradient methods for solving equilibrium problems in Banach spaces

Abstract

In this paper, three new algorithms are proposed for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a 2-uniformly convex and uniformly smooth Banach space. The algorithms are constructed around the $$\phi $$ -proximal mapping associated with cost bifunction. The first algorithm is designed with the prior knowledge of the Lipschitz-type constant of bifunction. This means that the Lipschitz-type constant is an input parameter of the algorithm while the next two algorithms are modified such that they can work without any information of the Lipschitz-type constant, and then they can be implemented more easily. Some convergence theorems are proved under mild conditions. Our results extend and enrich existing algorithms for solving equilibrium problem in Banach spaces. The numerical behavior of the new algorithms is also illustrated via several experiments.