PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 9, Pages 1–8 (February 7, 2022) https://doi.org/10.1090/bproc/117 GRAPHICAL EKELAND’S PRINCIPLE FOR EQUILIBRIUM PROBLEMS MONTHER RASHED ALFURAIDAN AND MOHAMED AMINE KHAMSI (Communicated by Mourad Ismail) Abstract. In this paper, we give a graphical version of the Ekeland’s vari- ational principle (EVP) for equilibrium problems on weighted graphs. This version generalizes and includes other equilibrium types of EVP such as op- timization, saddle point, fixed point and variational inequality ones. We also weaken the conditions on the class of bifunctions for which the variational principle holds by replacing the strong triangle inequality property by a below approximation of the bifunctions. 1. Introduction Ekeland’s variational principle [10, 11, 16, 18] is a minimization theorem for a bounded from below proper lower semicontinuous function on complete metric spaces. This result provides one of the most powerful tools in nonlinear analysis, op- timization, geometry of Banach spaces, economics, control theory, sensitivity, fixed point theory, and game theory [3–5,9,12–15,19]. It is used to approximate the solu- tion through a simple minimization idea. Motivated by its wide applications, many authors have been interested in extending Ekeland’s variational principle to, for instance, weighted graphs [2] and equilibrium problems on complete metric spaces [8]. Inspired by these two papers, we aim to get a generalized form of the Ekeland’s variational principle for equilibrium problems on weighted graphs endowed with a metric distance. First we start by recalling the equilibrium problem. Definition 1.1 ([6, 17]). Let (X, d) be a metric space and M be a nonempty subset of X. Let F : M × M → R be a bifunction such that F (x, x) = 0 for all x ∈ M . The problem of finding x ∈ M such that F ( x, y) ≥ 0, for all y ∈ M, is called an equilibrium problem for F (·, ·). It is clear that the concept of an equilibrium problem as defined in the above definition is not dependent on the distance d(·, ·). Therefore, we may rephrase the above definition in a more abstract form to obtain the following: Received by the editors September 14, 2021, and, in revised form, November 6, 2021. 2020 Mathematics Subject Classification. Primary 49J40, 47H10; Secondary 54E50. Key words and phrases. Ekeland variational principle, fixed point, equilibrium problem, weighted graph. The authors were funded by the deanship of scientific research at King Fahd University of Petroleum & Minerals for this work through project No. IN171032. c 2022 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0) 1