arXiv:1701.00152v1 [math.OC] 31 Dec 2016 Existence results for equilibrium problem John Cotrina * Yboon Garc´ ıa * January 3, 2017 Abstract In this work, we introduce the notion of regularization of bifunctions in a similar way as the well- known convex, quasiconvex and lower semicontinuous regularizations due to Crouzeix. We show that the Equilibrium Problems associated to bifunctions and their regularizations are equivalent in the sense of having the same solution set. Also, we present new existence results of solutions for Equilibrium Problems. Keywords: Equilibrium Problems, Convex Feasibility problems, Monotonicity generalized, Convexity gen- eralized, Coercivity conditions, Upper sign property. MSC (2000): 47J20, 49J35, 54C60, 90C37 1 Introduction Given a real Banach space X , a nonempty subset K of X and a bifunction f : K × K → R. The Equilibrium Problem, (EP) for short, is defined as follows: Find x ∈ K such that f (x, y) ≥ 0 for all y ∈ K. (EP) Equilibrium Problems have been extensively studied in recent years (e.g., [4–6, 8, 9, 16, 17, 19–21]). Par- ticularly, It is well known that many problems such as variational inequality problems, fixed-point problems, Nash equilibrium problems and optimization problems, among others, can be reformulated as equilibrium problems. (see for instance [6, 15, 21, 22]). A recurrent subject in the analysis of this problem is the connection between the solution sets of (EP) and the solution set of the following problem: Find x ∈ K such that f (y,x) ≤ 0 for all y ∈ K. (CFP) This can be seen as a dual formulation of (EP) and it corresponds to a particular case of the convex feasibility problem (cfr. [12, 13]). It was proved in [21] that if f is upper semicontinuous in the first argument, convex and lower semicon- tinuous in the second one and it vanishes on the diagonal K × K, then every solution of (CFP) is a solution of (EP), and moreover both solution sets trivially coincide under pseudomonotonicity of f . In order to establish the nonemptiness of the solution set of (CFP) and the inclusion of this set in solution set of (EP) in [5], Bianchi and Pini introduced the concept of local convex feasibility problem and the upper sign continuity for bifunctions as an adaptation of the set-valued map introduced in [18], by Hadjisavvas. They adaptated the existence result for variational inequalities developed by Aussel and Hadjisavvas in [2]. Basically, they proved that every solution of (CFP) is a local solution of (CFP) and all local solution of (5.1.1) is a solution of (EP). Following the same way, in [9], Castellani and Giuli introduced the concept of upper sign property for bifunction as a local property which is weaker than the upper sign continuity and they extend the result obtained by Bianchi and Pini. Our aim in this paper is to provide sufficient conditions for the existence of solutions under weak as- sumptions on the bifunction and some coercivity conditions. We introduce, in Section 3, the regularization * Universidad del Pac´ ıfico. Av. Salaverry 2020, Jes´ us Mar´ ıa, Lima, Per´ u. Email: {cotrina je,garcia yv}@up.edu.pe 1