Approximations of equilibrium problems (Q2903518)

The purpose of this paper is to deeply analyse, via a variational approach, the scalar equilibrium problem (EP): find \(\bar{x}\in C\) such that \(f(\bar{x},y)\geq 0\) for every \(y\in C,\) where \(C\subseteq \mathbb R^n\) is nonempty and closed, and \(f:C\times C\to \mathbb R\). In order to approximate the data, the constraint set, the bifunction of the problem, and some notions of convergence of \((C_k,f^k)\) to \((C,f)\) are given (namely, loopsided convergence in the maxinf framework, hypographical convergence, continuous convergence), and their properties are investigated. In Section 4, the assumptions of the existence results proved in [\textit{K. Fan}, Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 103--113 (1972; Zbl 0302.49019)] and [\textit{M. Bianchi, G. Kassay} and \textit{R. Pini}, J. Math. Anal. Appl. 305, No. 2, 502--512 (2005; Zbl 1061.49005)] are deeply inspected, and are related to variational convergence. Various applications of EPs are provided. In Section 6, existence and stability results are obtained via an asymptotic analysis: various coercive existence results already known are proved with a different approach, and some stability results are extended. A new notion of approximate solution for EPs is introduced, by encompassing notions given in [Zbl 1061.49005], [\textit{M. Bianchi, G. Kassay} and \textit{R. Pini}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, A, 460--468 (2010; Zbl 1180.49028)]; well-posedness is studied and characterized, and the results are employed in order to study the convergence of two numerical methods for pseudomonotone EPs.