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Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces

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Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 209, 202]220Ž. 1997 ARTICLE NO. AY975358 Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces Zsolt Pales U Institute of Mathematics and Informatics, Lajos Kossuth Uni¨ersity, H-4010, Debrecen, View metadata, citation and similar papers at core.ac.ukPf. 12, Hungary brought to you by CORE Submitted by Jean-Pierre Aubin provided by Elsevier - Publisher Connector Received May 6, 1993 We extend the classical inverse and implicit function theorems, the implicit function theorems of Lyusternik and Graves, and the results of Clarke and Pourciau to the situation when the given function is not smooth, but it has a convex strict prederivative whose measure of noncompactness is smaller than its measure of surjectivity. The proof of the main results requires Banach's open mapping theorem, Michael's selection theorem, Ekeland's variational principle, and Kaku- tani's fixed point theorem. Q 1997 Academic Press 1. INTRODUCTION The aim of the present note is to extend some classical inverse and implicit function theorems to the nonsmooth situation. Let X and Y be normed spaces and F be a set-valued function acting between them. If a point Ž.x, y belongs to the graph of F, then the modulus of surjection of F at Ž.x, y introduced by Ioffewx 20, 23 is defined by SurŽ.Ž.F, x, yt[supÄ4r G 0:By Ž.Ž.,r;FBxŽ.,t . ŽŽ.Here Bz,s denotes the closed ball of radius s around z in the Ž. corresponding space; for B 0, 1 ; Z we simply write BZ. The constant of * The first version of this paper was prepared when the author received a Research Fellowship of the Humboldt Foundation in Saarbrucken.È The support of the Humboldt Foundation is gratefully acknowledged. The preparation of the second revised version was supported by Hungarian National Foundation for Scientific ResearchŽ. OTKA , Grant T- 016846. 202 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. INVERSE AND IMPLICIT FUNCTION THEOREMS 203 surjection is then defined by SurŽ.Ž.F, x, yt surŽ.F, x, y s lim inf . tª0 t Afirst order local surjection theorem is a statement offering a positive lower estimate for the constant of surjection at a given point Ž.ÃÃx, y . As it has been pointed out by Ioffe, a first order local surjection theorem always implies solvability of the inclusion y g FxŽ.under small perturbations y of Ãy. It also turns out that a necessary condition in extremal problems can always be obtained in the form that the constant of surjection is zero for a certain set-valued function obtained from the data of the mathematical program. In the terminology by Penot inwx 35 , a set-valued function F with positive constant of surjection in a neighborhood of Ž.ÃÃx, y is called open around F at a linear rate. It is one of the main results of Penotwx 35 that this property is equivalent to the metric regularity of F at Ž.ÃÃx, y and also to the pseudo-Lipschitzian property of Fy1 at Ž.ŽÃÃy, x . For special cases of this result, see Borweinwx 6 .. The notion of metric regularity was stressed first by Robinsonwx 38 : The set-valued function F is called metrically regular around Ž.ŽÃÃx, y where Ãx g Fy Ž.. à , if there exist positive numbers a, b, g and c ) 0 such that 1 dxŽ.,FyŽ.y FcdŽ. F Ž. x , y for all x g BxŽ.ÃÃ,a, ygBy Ž.,b with dFxy ŽŽ..Fg. The notion of pseudo-Lipschitzianity is due to Aubinwx 1 : F is called pseudo-Lipschitzian around Ž.ÃÃx, y if there exist positive numbers a, b and c ) 0 such that dyŽ.,FxŽ.21Fcd Ž x , x2 . Ž. Ž. Ž.Ž for all x12, x g BxÃÃ,a and y g By,b lFx1. For other important Lipschitzian properties of multifunctions, see Rockafellarwx 39 .. The classical surjection theorem of Graveswx 17 and Lyusternik wx 29 states that if F s Ä4f where f is a continuously differentiable function X near ÃÃx and fxŽ.is surjective then the constant of surjection of F s Ä4f at ŽŽ..ÃÃx,fx is positive. The sequential procedure of the proof by Lyusternik has been extended in the work of Dmitruk, Milyutin, and Osmolovskiwx 11 . Another approach to surjection theorems is due to Ioffewx 19 , who initiated the application of the Ekeland variational principle in this field. The first application of this principle to set-valued functions is likely due to Aubin wx1 .Ž See also the monograph of Aubin and Frankowskawx 2 .. For further works, where Ekeland's principle is used andror the connec- tion between local surjection, metric regularity, and pseudo-Lipschitzianity 204 ZSOLT PALES is studied, we refer to Borwein and Zhuangwx 7 , Cominetti w 10 x , Frankowska wx14 , Ioffe w 20]22 x , Jourani and Thibault w 25, 26 x , Mordukhovich w 31, 32 x , Penotwx 34, 35 , and Thibault wx 40 . We mention that, in the finite dimensional case, the complete descrip- tion of local openness, metric regularity, and Lipschitzian properties has recently been found by Mordukhovichwx 31 . The characterization is given in terms of the nonconvex subdifferential invented by Mordukhovich. The necessary and sufficient conditions for the Lipschitzian invertibility of single-valued functions in finite dimensional setting has been also found by Kummerwx 27, 28 . Here, Thibault's directional derivative is used to express the result. In the infinite dimensional case, in order to handle the nonsmoothness of a given single valued or set-valued function F we need other substitutes of differentiability of functions. We shall use strict prederivatives as first order approximants to nonsmooth functions. The use of this notion was initiated by Ioffewx 20, 21 . Y A homogeneous set-valued function A : X ª 2 is called a strict pre- deri¨ati¨e of the set-valued function F at Ãx if, for all « ) 0, there exists d ) 0 such that XYXYXY FxŽ.;Fx Ž .qA Žxyx .q«5xyxB5Y XY for x , x g BxŽ.Ã,d in the domain of F.If Fis a single valued function, that is, F s Ä4f , then we also say that A is a strict prederivative for f at Ãx. Then the above inclusion reduces to XY XY XY fxŽ.gfx Ž .qA Žxyx .q«5xyxB5Y. The principle of obtaining a surjection theorem for the function F is that we assume the local first order approximability of F by a strict prederivative A, and then we try to show that the properties of A can be transferred to F. In many cases, the strict prederivative A can be generated by a family of linear operators in the following sense: There exists a family A ; LŽ.X, Y such that AŽ.x s Ä4Ax : A g A .IfAis a family of operators, then we say that it is a strict prederivative of F at Ãx if the set-valued map A defined above is a strict prederivative. From now on, we shall not use distinct notation for A and A. The question whether a homogeneous set-valued function A with closed convex values can be represented by a convex family of linear operators can be tested using the results of Pales wx 33 . If F s Ä4f , where f is a locally Lipschitz function, then the set of operators A g LŽ.X, Y such that 55A F L Žwhere L is the local Lipschitz constant of f at ÃÃx. forms a strict prederivative for f at x. Conversely, if a bounded strict prederivative exists then f must be locally Lipschitz at Ãx. INVERSE AND IMPLICIT FUNCTION THEOREMS 205 If X and Y are finite dimensional spaces, then Clarke's generalized Jacobian of a locally Lipschitz function f at Ãx is defined by XX ­fxŽ.ÃÃ[co½5A g L ŽX, Y .¬ ' xnnnª x : ;n 'fxŽ.and lim fx Ž.sA nª` Žcf.wx 9.Ž . We note that ­ fx Ž.à is never empty, since f is nondifferentiable only on a set of measure zero by Rademacher's differentiability theorem.. It is knownŽ see Ioffewx 20, Corollary 9.11, Proposition 10.9 that Clarke's generalized Jacobian is a strict prederivative for locally Lipschitz functions. We shall refer to this fact later, when we specialize our results to the finite dimensional setting. A function f admits a one element strict prederivative at Ãx if and only if it is strictly Frechet differentiable, therefore the strict prederivative is the extension of strict Frechet derivative to the case when the latter does not exist. We note that continuous GateauxŽ. Hadamard, Frechet differentia- bility in a neighborhood of a point always implies the strict Frechet differentiability at the point in question. We shall need two numbers associated to any family of linear operators. The first quantity, the measure of noncompactness, measures the size, the largeness of A ; LŽ.X, Y : n x Ž.A inf r 'n N, 'A ,..., A A: A BAŽ.,r . A [ ½5¬ g 1 nigA A;D is1 Clearly, if A is compact, then x Ž.A s 0, and conversely, if X and Y are Banach spaces and x Ž.A s 0, then the closure of A is compact. One can see that this function slightly differs from the noncompactness measure of A introduced by Kuratowski, but is satisfies the usual properties of noncompactness measures, for instance, x Ž.A s x Žco A .for all A. Obvi- ously, x Ž.A is finite if and only if A is bounded, moreover, x Ž.A F diamŽ.A .
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