Nonlinear Analysis: Theory, Methods & Applications
A localized version of Ky Fan’s minimax inequality
Introduction
Let X denote a real normed space. Then the minimax inequality discovered by Ky Fan in 1972 reads as follows (see [5]).
Theorem A. Let K⊂X be a compact convex set, . Assume that g fulfills the following properties:
- 1.
x→g(x,y) is quasiconcave on K for all fixed y∈K;
- 2.
y→g(x,y) is lower semicontinuous (lsc) on K for all fixed x∈K;
- 3.
g(x,x)≤0 for all x∈K.
This result has many interesting applications concerning fixed point theorems, variational inequalities, potential theory, mathematical economy, etc. It is also known that Ky Fan’s minimax inequality is one of the several statements which are equivalent to each other: Brouwer’s fixed point theorem, the lemma of Knaster–Kuratowski–Mazurkiewicz, the theorem of Browder concerning variational inequalities, the fixed point theorem of Kakutani, the fixed point theorem of Ky Fan–Glicksberg, the Nash equilibrium principle, the theorem of Gale–Nikaido–Debreu on Walras equilibrium in mathematical economics, etc. (see [6] Vol. IV, pp. 795–810]).
Consider now the special case of Theorem A, when x→g(x,y) is concave (instead of being quasiconcave only) for all y∈K and g(x,x)=0 for all x∈K. Then, if y0 satisfies the requirements of the statement,Therefore, taking the limit t→0+0, we getwhere g′ denotes the (existing by the concavity) directional derivative of g with respect to the first variable. By homogeneity of g′ in the h variable, we can extend this inequality to the conical hull of K−y0, that is, we getOn the other hand, if this inequality holds for some y0∈K and g is concave in the first variable, then (by known properties of concave functions)Thus g(x,y0)≤0 for all x∈K, therefore the conclusion of Theorem A is valid.
The aim of this note is to state a principle (similar to the above formulated one) for locally Lipschitz functions in terms of a generalized Clarke-type derivative (see Theorem 1 of Section 2). It will also be pointed out that Theorem 1 and a specialized version of Ky Fan’s minimax inequality are directly equivalent, i.e. they are deducible from each other.
In the subsequent two sections, we clarify how this local Ky Fan’s inequality relates to the results mentioned above. Using Theorem 1, we give new and simple proofs for extended versions of Schauder’s and Kakutani’s fixed point theorems for inward maps (see Corollaries 1 and 2 below). In the last section, we deduce a new variational inequality for upper demicontinuous set-valued maps which, in particular, contains that of Browder [2]. In this way, it will be clear that our localized version of Ky Fan’s minimax inequality is equally powerful as the above mentioned fixed point theorems and variational inequalities.
Section snippets
The generalized Clarke-type derivative – Main result
Let X be a real normed space, K⊂X be a nonempty compact convex set, and let be a given function, where D⊂X is an open set containing K. We say that f is locally Lipschitz on the diagonal of K×K with respect to the first variable if for each x∈K there exists Lx>0 and a neighbourhood Ux⊂D of x such thatfor all x′,x″∈Ux and y∈Ux∩K. The family of all functions with the above property will be denoted by .
Let . For each x∈K and h∈X, the generalized
Applications to fixed point theorems
First we show that, using Theorem 1, one can easily deduce an extended form of Schauder’s fixed point theorem (cf. [6] Vol. I, Theorem 2.6.2.A, p. 57]).
Corollary 1. Let K be a compact convex subset of a real normed space X and let ϕ:K→X be a continuous function such that ϕ(x)∈x+TK(x) for all boundary points x of K. Then ϕ admits a fixed point x0∈K.
Proof. Consider the function defined byClearly, x↦f(x,y) is a convex Lipschitz function with Lipschitz constant L=1. Thus
Applications to variational inequalities
In the sequel, let X be a real Banach space and be its dual space.
A set-valued mapping is said to be upper demicontinuous (udc) at x∈D if, for any h∈X, the real-valued functionis upper semicontinuous at x.
Clearly, if is an usc set-valued map, then it is also udc.
If , that is, if is a single valued map, then is udc at x∈D if and only if the operator is demicontinuous at x∈D, i.e., if for each sequence xn converging to x
References (6)
- J.P. Aubin, H. Frankowska, Set-Valued Analysis, Systems and Control: Foundations and Applications, vol. 2, Birkhäuser,...
- F. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Annalen 177 (1968)...
- F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Math. Soc. Series of Monographs and Advanced Texts, Wiley,...
Cited by (6)
Some applications of variational inequalities in nonsmooth analysis
2007, Nonlinear Analysis, Theory, Methods and ApplicationsEquilibria and fixed points of set-valued maps with nonconvex and noncompact domains and ranges
2006, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :The above three results are the source of many investigations and have been generalized in several directions with applications in various branches of mathematics: minimax theorems, minimax inequalities, variational inequalities, fixed point theory, coincidence theory, game theory, mathematical economics, optimization problems, etc. Especially during the last few decades, by applying various new methods (notably based on those introduced by Ky Fan), remarkable progress has been made in these areas of investigations, and also some fascinating new problems have been proposed (see, e.g. [1–14,17–21,23–31,33–41] and many references therein). Most of the work has centered around the equilibrium points, minimax inequalities and fixed points of maps on convex compact sets, but there are also a considerable number of papers devoted to maps on nonconvex or noncompact domains (see, e.g. [10,28,37,39,41]).
GABOR KASSAY - IN MEMORIAM
2022, Studia Universitatis Babes-Bolyai MathematicaFixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces
2003, Abstract and Applied AnalysisSome minimax type results of Clarke's generalized directional derivative and point-coderivative of multifunctions
2002, Acta Mathematica HungaricaOn Nash stationary points
1999, Publicationes Mathematicae Debrecen