A Novel Method for Solving Nonmonotone Equilibrium Problems

In this paper, we propose a novel projection method for finding a solution of equilibrium problems EP(C, f) in Euclidean spaces in which the bifunction does not require to be satisfied any monotonicity. Unlike methods for solving such a problem used in the literature, shrinking projection methods which may raise cost of computation, we suggest using an adaptive projection method incorporate with linesearch procedures to solve these problems when the bifunction does not satisfy the Lipschitz-type condition. These linesearches are unnecessary when f satisfies the Lipschitz-type condition with constants L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document}. In case these constants are unknown, we propose to use the adaptive step sizes. All algorithms are proven to converge to a solution of the considered problem, and some numerical examples are also reported to illustrate the efficiency of the proposed algorithms.