A generalized KKMF principle

Abstract

International audienceWe present in this paper a generalized version of the celebrated Knaster–Kuratowski–Mazurkiewicz–Fan's principle on the intersection of a family of closed sets subject to a classical geometric condition and a weakened compactness condition. The fixed point formulation of this generalized principle extends the Browder–Fan fixed point theorem to set-valued maps of non-compact convex subsets of topological vector spaces

Minimax Inequality and Equilibria with a Generalized Coercivity

“…Since P is KF -majorized and X is paracompact, it follows from Remark 2 that there exists a KF correspondence Ψ such that P (x) ⊆ Ψ(x), ∀x ∈ X. By Theorem 3.2 in [2], the correspondence co Ψ admits a maximal element, which is also a maximal element for P .…”

“…For each x ∈ X, let F (x) = {y ∈ X : f (x, y) ≤ 0}. We have to show that F satisfies all conditions of Theorem 3.1 in [2]. By (a), F (x) is compactly closed in X for each x ∈ X.…”

“…Theorem 3.2 in [2] can be rephrased in terms of the existence of maximal elements as follows: Proposition 1. Let X be a non-empty convex and paracompact subset of a topological vector space.…”

“…Condition (C) is also an extension of the coercivity condition given by Fan [6]. Fore more examples about correspondences admitting a coercing family (when I is a singleton), see [2]. By the following example, we can see that the notion of coercing family is very general: Example 1.…”

“…We firstly recall the notion of coercing family for set-valued maps (also called correspondences) defined by Ben-El-Mechaiekh, Chebbi and Florenzano in [2]. As an example, we give the very general coercivity condition obtained by Ding and Tan in [5].…”