Approachability and fixed points for non-convex set-valued maps

Fixed point theory for compact upper semi-continuous or lower semi-continuous set valued maps

“…In section 2 we use a recent Leray-Schauder type alternative due to Ben-El-Mechaiekh and Idzik [3] for u.s.c., compact approachable maps with nonempty, compact values to establish a new fixed point result for such maps. Our theory generalizes some fixed point results in Ben-El-Mechaiekh and Deguire [2] and Granas [9]. In section 3 we establish via Michael's selection theorem [11] a nonlinear alternative of Leray-Schauder type for l.s.c., compact multifunctions with closed, convex values.…”

Minimax Inequality and Equilibria with a Generalized Coercivity

“…Let X be a subset of a topological vector space and F : X → Y be a correspondence. A family {(C i , K i )} i∈I of pair of sets is coercing for F if and only if it satisfies conditions (i), (ii) of Definition 1 and the following one: Note that in case where the family is reduced to one element, condition (C) appeared first in this generality (with two sets K and C) in [3] and generalizes condition of Karamardian [8] and Allen [1]. Condition (C) is also an extension of the coercivity condition given by Fan [6].…”

The von Neumann minimax theorem revisited

“…Theorem 3 (A Nonlinear Alternative [1]). Let X and Y be two convex subsets of topological vector spaces and letf , f, g,g : X × Y −→ R be four functions satisfying:…”

“…The aim of this note is to present a simple and elegant approach to the von Neumann Theorem in relation to contributions by Professors J. Dugundji and A. Granas [9,10]. The M. Sion [25] generalization of the minimax theorem to quasiconcave/convex functions can be formulated as a Nonlinear Alternative [1], which turns out to be equivalent to the Dugundji-Granas version of the KKM Principle, the Browder-Ky Fan Fixed Point Theorem, and a Coincidence Principle for dual Ky Fan type set-valued maps. We include what we believe is the most elementary proof of Maurice Sion's version of the minimax theorem based on a theorem of C. Berge [4] equivalent to a Helly type result of V. Klee [15] on the intersection of a family of convex sets.…”