A localized version of Ky Fan’s minimax inequality

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, $\text{g:K×K→}\text{R}$. 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,$\text{g(y}{}_0\text{+t(x−y}{}_0\text{),y}{}_0\text{)−g(y}{}_0\text{,y}{}_0\text{)}\text{t}\text{≤0}\text{for}\text{all}\text{t∈[0,1],}\text{x∈K.}$Therefore, taking the limit t→0+0, we get$\text{g′(y}{}_0\text{,y}{}_0\text{)(h)≤0}\text{for}\text{all}\text{h∈K−y}{}_0\text{,}$where 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 get$\text{g′(y}{}_0\text{,y}{}_0\text{)(h)≤0}\text{for}\text{all}\text{h∈}\text{cone}\text{(K−y}{}_0\text{).}$On 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)$\text{g(y}{}_0\text{+h,y}{}_0\text{)−g(y}{}_0\text{,y}{}_0\text{)≤g′(y}{}_0\text{,y}{}_0\text{)(h)≤0}\text{fo}\text{h∈K−y}{}_0\text{.}$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.