By Pellacci B.
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Extra info for Multiple critical points for nondifferentiable functionals involving Hardy potentials
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7 THE KKM PRINCIPLE In this Section we state a fundamental result of Knaster-KuratowskiMazurkiewicz [227] expressing through the statement that a given intersection is nonempty. Let X be a real vector space and A C X a subset. mapping G : A ~ 2x is called a KKM mapping if A set-valued n UG(Xi) conv{xl," . 50) i=l for each finite subset {Xl," . , xn} of A. 19 Let X be a real Banach space, C C X a nonempty convex subset and F : C ~ X a mapping. Let us define G : C ~ 2x by G(X) = {v E C: (F(v),v - x) :s O}, X E C.
9) is valid if F is strictly differentiable. If in addition either 9 is regular at F(u) or F'(u) is surjective, then (goF)O(u;v) =go(F(u);F'(u)v), Vu E U, v E X and 8(g 0 F)(u) = 8g(F(u» 0 F'(u), VuE U. 10) (ii) Let f : U -+ R be a locally Lipschitz function on an open set U of a Banach space X and let h : R -+ R be a locally Lipschitz function. Then one has 8(h 0 f)(u) c co(8h(J(u» . 11) where the notation co stands for weak* -closed convex hull. 11) the equality holds and the symbol co is superfluous.
21 is the analytic form of the famous Ky-Fan principle [129]. 22 Let X be a reflexive Banach space and C a nonempty closed convex and bounded subset of X. c. on C, for each fixed x E C. Then there exists Yo E C such that f(x, Yo) ::; 0, V x E C. ° Proof. 21 with f = g, = and A = o. 21 cannot occur. 8 MINTY'S PRINCIPLE Let X be a real Banach space and C a nonempty closed convex subset of X. : 0, V v E C, 26 VARIATIONAL AND HEMIVARIATIONAL INEQUALITIES is equivalent to the following one (Av, v - u) ~ 0, V v E C.