Parametric Borwein-Preiss variational principle and applications

Abstract

Abstract. A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.The tool for proving this parametric smooth variational principle is a useful lemma about continuous ε-minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

Approximate solutions of quasiequilibrium problems in Banach spaces

“…⊓ ⊔ Remark 2.1 Theorem 2.3 recovers Lemma 2.1 in [8] where the author assumes Y a Banach space, C X = X, C Y closed and ε = ε ′ . In regards to Assertion (b), the assumption of openness of the images of F assumed in [8] does not guarantee the openness of the lower sections of F and also the existence of a continuous selection as showed in [ Returning back to Theorem 2.2, the mistake in its proof resides in the fact that the authors apply Lemma 2.3 choosing…”

“…In the present note a counterexample which contradicts the existence of approximate solutions is exhibited. The mistake in the proof stems from an incorrect application of a result [8] about continuous ε-minimizers of quasiconvex functions depending on a parameter which seems worthy in itself. We furnish a slightly improved version of this result which allows to establish a new existence result of approximate solutions of quasiequilibrium problems.…”

Variational principles for monotone variational inequalities: The single-valued case

“…when the solution depends in a continuous way on a parameter. Such results are presented in [14], [15], [24], [7].…”

“…✷ Remark 8. The parametric Borwein-Preiss variational principle in [15] requires equi-lower semi-continuity of the functions {f (x, y) : y ∈ Y 0 } for every bounded set Y 0 ⊂ Y , and the convex functions f (x, .) are not required to be Gâteaux differentiable.…”

Borwein–Preiss vector variational principle

“…Because of that, the Borwein-Preiss variational principle is often referred to as the smooth variational principle. It has found numerous applications and paved the way for a number of smooth principles [3][4][5][6][7][8][9][10][11]. Among the known extensions of the Borwein-Preiss variational principle, we mention the work by Li and Shi [12, Theorem 1], where the principle was extended to metric spaces (of course at the expense of losing the smoothness), covering also the conventional Ekeland variational principle.…”