Abstract
In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.
Highlights
Given a nonempty set C and a bifunction f : C × C → R, the equilibrium problem is defined as follows: find x ∈ C such that f (x, y) ≥ 0, ∀y ∈ C. It seems that the earliest mathematical formulation of the problem above belongs to Nikaido and Isoda [20]: they used it as an auxiliary problem to study the existence of solutions of Nash equilibrium problem
1 Università Cattolica del Sacro Cuore, Milan, Italy 2 Babes-Bolyai University, Cluj, Romania 3 Università degli Studi di Milano-Bicocca, Milan, Italy Journal of Global Optimization rium problem was coined by Muu and Oettli [19], while Blum and Oettli [5] adopted this terminology perhaps because it is equivalent to find the equilibrium point of several problems, namely, minimization problems, saddle point problems, Nash equilibrium problems, variational inequality problems, fixed point problems, and so forth
Related to (EP) in literature has been considered the so-called quasi equilibrium problem (QEP, for short), that is an equilibrium problem with a constraint set depending on the current point
Summary
Given a nonempty set C and a bifunction f : C × C → R, the equilibrium problem (in the sequel EP, for short) is defined as follows: find x ∈ C such that f (x, y) ≥ 0, ∀y ∈ C. While there is an extensive literature on existence results, stability of the solutions, and solution methods concerning equilibrium problems (for a recent survey see [4]), the investigation of quasi equilibrium problems, initially introduced by Mosco (see [18]), received some attention only quite recently These problems arise from quasi variational inequalities, which are well-known tools to model equilibria in several frameworks (see, for instance, [15]). Konnov ([17]) proposed to apply in a finitedimensional setting a regularized version of the penalty method to establish existence results for the general quasi equilibrium problem under weak coercivity conditions by replacing the quasi equilibrium problem with a sequence of usual equilibrium problems We apply these techniques to (QEP) in real Banach spaces, under the assumption of topological, or Brezis pseudomonotonicity of f and weak lower semicontinuity of the constraint map.
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