Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 209, 202᎐220Ž. 1997 ARTICLE NO. AY975358

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. ᮊ 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.¨ 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 ᮊ 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.ˆˆx, 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 ˆy. 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.ˆˆx, 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.ˆˆx, y and also to the pseudo-Lipschitzian property of Fy1 at Ž.Žˆˆy, 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.Žˆˆx, y where ˆx g Fy Ž.. ˆ , if there exist positive numbers ␣, ␤, ␥ and c ) 0 such that

1 dxŽ.,FyŽ.y FcdŽ. F Ž. x , y for all x g BxŽ.ˆˆ,␣, ygBy Ž.,␤ with dFxy ŽŽ..F␥. The notion of pseudo-Lipschitzianity is due to Aubinwx 1 : F is called pseudo-Lipschitzian around Ž.ˆˆx, y if there exist positive numbers ␣, ␤ and c ) 0 such that

dyŽ.,FxŽ.21Fcd Ž x , x2 . Ž. Ž. Ž.Ž for all x12, x g Bxˆˆ,␣ and y g By,␤ 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 ˆˆx and fxŽ.is surjective then the constant of surjection of F s Ä4f at ŽŽ..ˆˆx,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 prederi¨ati¨e of the set-valued function F at ˆx if, for all ␧ ) 0, there exists ␦ ) 0 such that

XYXYXY FxŽ.;Fx Ž .qA Žxyx .q␧5xyxB5Y

XY for x , x g BxŽ.ˆ,␦ 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 ˆx. 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 ˆx 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 ˆˆx. forms a strict prederivative for f at x. Conversely, if a bounded strict prederivative exists then f must be locally Lipschitz at ˆx. 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 ˆx is defined by

XX ѨfxŽ.ˆˆ[co½5A g L ŽX, Y .¬ ᭚ xnnnª x : ᭙n ᭚fxŽ.and lim fx Ž.sA nªϱ Žcf.wx 9.Ž . We note that Ѩ fx Ž.ˆ 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 ˆx 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´ differentiability 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 ␹ 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 ␹ Ž.A s 0, and conversely, if X and Y are Banach spaces and ␹ 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, ␹ Ž.A s ␹ Žco A .for all A. Obvi- ously, ␹ Ž.A is finite if and only if A is bounded, moreover, ␹ Ž.A F diamŽ.A . The second quantity describes the surjectivity properties of A. For an operator A g LŽ.X, Y , let

␴ Ž.A [ supÄ4c g R : cBYX; ABŽ.. Clearly, ␴ Ž.A is related to the constant of surjection of A:

␴ Ž.A s sur ŽA,0,0 . . If X and Y are Banach spaces, then, by Banach’s open mapping theorem, ␴Ž.Ais finite if and only if A is surjective. For an arbitrary family of operators A ; LŽ.X, Y , define ␴ Ž.A [ inf ␴ Ž.A . AgA 206 ZSOLT PALES´

In this case, however, ␴ Ž.A is not equal to the constant of surjection of the set-valued map x ¬ AŽ.x at Ž 0, 0 . . In this paper we consider single valued functions. In the central result we prove that a strict prederivative A provides a local surjection theorem for the given single valued function if we have

␹ Ž.A - ␴ Ž.A .1Ž.

More precisely, Theorem 3 below asserts the local surjection property for a parametrized family of functions assuming the existence of a convex strict prederivative A consisting of linear operators A such thatŽ. 1 is valid. Moreover, in this case the constant of surjection is nonsmaller than ␴Ž.Ay␹ Ž.A.If Aincreases, then ␴ Ž.A decreases and ␹ Ž.A increases, therefore the gap between these values decreases. This shows that the smaller A is, the bigger is the constant of surjection. At this point we mention three closely related results. Fabian and Preisswx 13 obtained a local surjection theoremŽ see Theorem 2 there. using operator generated strict prederivatives when the underlying Banach spaces are reflexive. The proof uses a slight generalization of Caristi’s fixed point principle. Another approach is due to Glover and Ralphwx 16 . They use the extended notion of Clarke’s Jacobian for a single valued locally Lipschitz mapping f when the range space Y is reflexiveŽ cf. Ralphwx 37 , Ioffe wx 20. : ѨfxŽ.ˆ consists of those operators A g L ŽX, Y .such that

U ² y , fxŽ.Ž.q␧hyfx: ²:yU,Ah lim sup ᭙h X, ᭙ yUUY . F ␧ g g ␧ª0, xªˆx A particular case of the main result inwx 16 is that if ␴ѨŽŽ..fxˆ )0 and the set-valued map x ª Ѩ fxŽ.satisfies certain continuity assumptions, then f is metrically regular at ˆx. 1 The result of Corollary 1.4 of Ioffewx 23 offers CyŽ.A y diam Ž.A as the lower estimate for the constant of surjection, where

CŽ.Assup infÄ455x : y g AŽ.x 55ys1 is the Banach constant of the family of operators A.If Aincreases, then 1 CŽ.Adecreases, therefore the behaviour of CyŽ.A y diam Ž.A cannot be described easily. The comparison of this estimate with our estimate for the constant of surjection is not obvious as well. In the proofs of our main results we shall use Banach’s open mapping theorem, Michael’s selection theorem, Ekeland’s variational principle, and Kakutani’s fixed point theorem. As an immediate consequence, we get INVERSE AND IMPLICIT FUNCTION THEOREMS 207

Grave’s and Lyusternik’s theorem, a generalization of Clarke’s inverse function theorem, Pourciau’s open mapping theorem, and the classical inverse and implicit function theorems. The organization of the paper is the following: In Section 2 we recall a generalization of Banach’s open mapping theorem to operators mapping from CŽ.T, X into C Ž.T, Y , where T is a compact Hausdorff topological space and X, Y are Banach spaces. The main results of the paper are contained in Section 3, where we prove theorems of local surjection type. In the last section, the prederivatives are assumed to consist of invertible linear operators, and the results obtained reduce to implicit and inverse function theorems.

2. EXTENSION OF BANACH’S OPEN MAPPING THEOREM

To formulate the main result of this section, introduce the following notation: If T is a Hausdorff topological space and X is a normed space, then CŽ.T, X denotes the space of all bounded continuous maps from T to X supplied with the supremum norm. Clearly, if X is a Banach space, then CŽ.T,Xis also a Banach space. If T is compact, then C Ž.T, X consists of all continuous functions mapping T into X. In the main result of this section we give a necessary and sufficient condition in order that a map A : CŽ.T, X ª C Ž.T, Y of the form Ž.Ž.Ax t sAtŽ.Ž.xt be open. We shall need the following three lemmas. LEMMA 1. Let X be a Banach space, Y be a normed space. Then, for all A,DgLŽ.X,Y,

␴Ž.Ay␴ Ž.DF55AyD.

Proof. It is enough to prove that

␴Ž.DF␴ Ž.Aq55AyD Ž.2 for all A, D g LXŽ.,Y. If ␴ Ž.D F 55A y D , then there is nothing to prove. In the remaining case, let 55A y D - c - ␴ Ž.D be arbitrary. Then

cBYX; DBŽ..3Ž.

We are going to show that, with q [ 55A y D rc, we have

Ž.1 y qcBYX;AB Ž..4Ž. 208 ZSOLT PALES´

Ž. Let y g 1 y qcBYnbe arbitrary. We show first that a sequence x g X can be constructed such that

ny1 ny1 xnXg Ž.1 y qq B and y s Dx niq ÝAx Ž.5 is1 hold for all n g N. Ž. Ž. By 3 and the choice of y, we can find x1 g 1 y qBX such that Ž. Dx1 s y, thus 5 holds true for n s 1. Assume that we have constructed x1,..., xn. By the second equation ofŽ. 5 , we get

n y y AxŽ.Ž.Ž.1qиии qxnns D y Ax F1yqqc. Ž. Ž . n Therefore, by 3 , xnq1 g 1 y qqBX can be chosen such that иии y y AxŽ.1q qxnns Dx q1. Thus, both relations ofŽ. 5 hold for n q 1 instead of n. Ýϱ Now observe that the sum is1 xiXconverges to an element x g B Žsince the norms of its elements can be majorized by terms of a convergent geometric series with sum equal to 1. . Therefore, taking the limit n ª ϱ in the second equation ofŽ. 5 , we arrive at y s Ax. This proves Ž. 4 . The relationŽ. 4 yields Ž 1 y qc .F␴ ŽA .. Therefore cy55AyDF␴Ž.A. Being c - ␴ Ž.D arbitrary, we obtainŽ. 2 . An obvious consequence of this lemma is that ␴ is a Lipschitz continuous function on LŽ.X, Y . Remark.IfYis also a Banach space, then the continuity of ␴ yields that the set of surjective operators in LŽ.X, Y is open. Indeed, if A : X ª Y is a surjective linear operator, then, by the open mapping theorem of Banach, ␴ Ž.A ) 0. By Lemma 1, then ␴ Ž.D ) 0 holds in a neighborhood of A. Thus the operators in this neighborhood are also surjective.

LEMMA 2. Let X be a Banach space, Y be a normed space, c ) 0 a fixed Ž. X constant, and define the set-¨alued mapping Fc : Y = L X, Y ª 2 by

FycŽ.,A[Ä4xgX¬Ax s y, 55x F cy 55.

Then Fc is a closed conëx älued set-älued mapping. Moreoër Fc is lower semicontinuous at a pointŽ.ˆ y, Aifcˆˆ)1r␴ Ž.A.

Proof. The closedness and convexity of values of Fc is obvious. To point out the lower semicontinuity at a point Ž.ˆy, Aˆˆwhen c ) 1r␴ Ž.A , Ž. Ž. let ˆˆx g Fycn,Aˆ be arbitrary and let y , Anbe a sequence with INVERSE AND IMPLICIT FUNCTION THEOREMS 209 lim Ž.Ž.y , A y, Aˆ . We have to show that there exists a sequence nªϱ nns ˆ xXsuch that x FyŽ.,A holds for large n and lim x x. nng g cnn nªϱns ˆ Ž. Ž. Let c ) c0 ) 1r␴ Aˆˆbe arbitrarily fixed. Then, by definition of ␴ A , there exists a vector x000such that Axˆ s ˆˆy and 55x F cy0 55. Define

55 5 5 5 52 5ˆ 555 ¡cyŽ.ˆˆqynnnyyyy qcAyAyˆ if 55ˆy / 0, ␭n[~ Ž.cycy055ˆ ¢0if55ˆys0, and

unn[ ␭ x0q Ž.1 y ␭nˆx.

Then ␭nnG 0 for all n g N, further ␭ ª 0 and therefore unª ˆx as nªϱ. Moreover, we have

Auˆ nns ˆˆy and 5u 5F ␭nc0q Ž.1 y ␭ncy55Fcy 55ˆ, hence

555555555555ynnnnyAu F yyˆˆˆyq Auˆˆnnnny Au F yyyqcAyAyn. Ž. Ž Choose n0 so that we have ␭nn- 1 and c ) 1r␴ A for n G n0. This is . possible by Lemma 1. Then, for n G n0 , there exists ¨n g X such that ynnnnny Au sA¨ and

2 55¨nnnnnFcy 5yAu 5Fcy 5yˆˆy 5qcA 5ˆyAyn 555.

Define now xnnn[ u q ¨ . Then, by the previous relations and the definition of ␭nn, we have Axnsynand

55xnnnF 55uq 55¨

2 F␭nc0qŽ.1y␭nncy55ˆˆqcy 5yy 5qcA 5ˆyAy nn 555 ˆscy 5 5, Ž. that is, xncnng Fy,A. Thus the proof is complete. Remark. The proof of this lemma could have been also deduced from Propositions 1.2.6 or 1.5.1 ofwx 2 ; however, a direct proof was more convenient here.

LEMMA 3Ž cf. Bartle and Graveswx 5, Theorem 4. . Let X be a Banach space, Y be a normed space, T be a paracompact space, and let A : T ª LŽ.X,Y be a bounded continuous operator ¨alued function. Then ŽA xt .Ž.[At Ž. xt Ž..6Ž. 210 ZSOLT PALES´ defines a bounded linear map A : CŽ.T, X ª C Ž.T, Y , moreo¨er

␴ ŽA .s ␴ Ž.Ž.At Ž. tgTsinf ␴ Ž.AtŽ..7Ž. tgT Proof. The boundedness and linearity of A is obvious. ␥ ␴ ŽŽ..␴ Ž.A Define 0 [ inf t g T At G0. We are going to show first that G ␥ 00.If␥ s0, then there is nothing to show. Otherwise let 0 - ␥ - ␥ 0 be fixed. We have to prove that

␥ BCŽT,Y.; A BCŽT,X..8Ž.

Fix y g ␥ BCŽT , Y . arbitrarily. Making use of Lemma 2, we get that the set-valued map y y ␥55 5y 5 ftŽ.[F1r␥Ž. Ž.t,At Ž. sÄ4xgX¬AtŽ. xs Ž.t, xF Ž.t is nonempty closed convex valued and lower semicontinuous on T. Now we are in the position to apply Michael’s theoremŽ cf.w 30; 2, Theorem 9.1.2, p. 355; 43, Theorem 9.G, p. 466x. on the existence of continuous selections of lower semicontinuous maps. Therefore, we can find a continuous function x : T ª X such that 1 AtŽ.Ž.xt y Ž.t and xŽ.t y Ž.t s F ␥ are valid for all t g T. In other words, we have 1 Ax y and 55x 55y 1. s F ␥ F

ThusŽ. 8 is proved. It follows fromŽ. 8 and the arbitrariness of ␥ that Ž. ␴AG␥0 is also satisfied. To complete the proof ofŽ. 7 , we have to show that here equality holds. Ž. If this were not so, for some t0 g T and for some ␥, we have ␴ A ) ␥ ) ␴ŽŽ..At0 . Then the ball ␥ BY is not contained in AtŽ.0 BX , i.e., there exists Ž. Ž. yg␥BY such that y f At0 BX . On the other hand, since ␥ - ␴ A , Ž. Ž. hence the constant function y t ' y is in A BCŽT , Y ., i.e., y g At BX for Ž. all t g T. This means a contradiction when t s t0. Thus 7 is proved. THEOREM 1. Let X and Y be Banach spaces, let T be a compact Hausdorff space, and let A : T ª LŽ.X, Y be a continuous operator ¨alued function. Then the bounded linear map A : CŽ.T, X ª C Ž.T, Y defined by Ž.6 is open if and only if AŽ. t is surjecti¨e for all t g T. Remark. The result of this theorem specializes to Banach’s open mapping theorem if the space T consists of one element only. INVERSE AND IMPLICIT FUNCTION THEOREMS 211

Proof. If A is open, then ␴ Ž.A ) 0, whence, by Lemma 3, we get ␴ŽŽ..At )0 for all t g T. Thus all the maps At Ž.are surjective. Conversely, if all the operators AtŽ.are surjective, then they are also open by Banach’s open mapping theorem. Therefore ␴ ŽŽ..At )0 for all ␴ ␴ ŽŽ.. tgT. Then continuity of now yields that inf t g T At )0. Then, by Lemma 3, we have ␴ Ž.A ) 0. Thus A is open. Remark. The results stated in Lemma 3 and Theorem 1 were discov- ered and formulated in a somewhat different from by Bartle and Graves wx5, Theorem 4 . Since at that time Michael’s selection theorem was not known, they used a direct and more complicated approachŽ only requiring the notion of paracompactness.Ž. to obtain 7 .

3. SURJECTIVITY OF NONSMOOTH MAPPINGS

We formulate one of the main results of the paper:

THEOREM 2. Let X and Y be Banach spaces, D ; X be an open set, f : D ª Y, andˆ x g D. Assume that f has a conëx strict prederiätië A;LŽ.X,Y atˆ x such that ␹ Ž.A - ␴ Ž.A . Then there exists a neighborhood U;D=YofŽŽ..ˆˆ x,fx,a positië constant L, and a function ␾ : U ª D such that

f Ž.␾ Ž.x, y s y and x y ␾ Ž.x, y F Lfx Ž.yy Ž.9 hold for allŽ. x, y g U. Moreo¨er, L can be arbitrarily close to 1rŽŽ.␴ A y ␹ A.. The proof of this theorem is postponed after that of Theorem 4 below. The last statement of the theorem expresses that the constant of surjection of f at Žˆˆx, fx Ž..is nonsmaller than ␴ Ž.A y ␹ Ž.A . We immediately obtain two corollaries:

COROLLARY 1Ž Lyusternik’s and Graves’ Inverse Function Theorems, cf. wx29; 17; 2, Theorem 3.4.2, p. 95; 3, Theorem 2.3.1, p. 161; 20; 24, p. 30. . Assume that f in Theorem 2 is strictly Frechet´ˆˆ differentiable at x and fXŽ. x maps X onto Y. Then the conclusion of Theorem 2 holds. X Proof. Take A s Ä fxŽ.ˆ 4in Theorem 2. Then ␹ Ž.A s 0 and ␴ Ž.A ) 0 by Banach’s open mapping theorem. Thus, the conditions of Theorem 2 are satisfied.

COROLLARY 2Ž cf. Pourciau’s Open Mapping Theoremwx 36. . Let X and Y be finite dimensional normed spaces and assume that f : D ª Y is a locally Lipschitz function atˆˆ x. Assume that the elements of Ѩ fŽ. x are matrices of full rank. Then the conclusion of Theorem 2 holds. 212 ZSOLT PALES´

Proof. The Clarke generalized Jacobian A s Ѩ fxŽ.ˆ is a compact convex strict prederivative of f at ˆx as we have mentioned it in the Introduc- tion. Therefore, ␹ Ž.A s 0. On the other hand, ␴ Ž.A ) 0 for all A g A. Hence, by compactness of A and the continuity of ␴ , ␴ Ž.A ) 0. Thus the statement follows directly. Actually Theorem 2 is the consequence of the following more general result:

THEOREM 3. Let X, Y be a Banach spaces and P be a topological space. Let further D ; X be an open set, f : P = D ª Y, ˆˆx g D, and p g P. Assume that the following conditions hold:

Ž.i The mapping p ¬ fp Ž,ˆˆx . is continuous at p. Ž.ii There exist a conëx family of operators A ; LŽ.X, Y such that ␹Ž.A-␴ Ž.A. Ž.iii A is a uniform strict prederiätië for x ª fp Ž,x . nearˆ p, that is, Ž. for all ␧ ) 0, there exists a neighborhood W␧; P = Dofˆˆ p,x such that XYXYXY fpŽ.Ž.Ž,x yfp,x gAxyx .q␧5xyxB5Y, X Y forŽ.Ž. p, x , p, x g W␧. Then, for all ␹ Ž.A - ␤ - ␥ - ␴ Ž.A , there exist a neighborhood U of ŽŽ..ˆˆp,x,fp ˆˆ,x and a function ␾ : U ª D such that fpŽ.,␾Ž.p,x,y sy and 1 Ž.10 xy␾Ž.p,x,y F fp Ž.,xyy ␥y␤ are satisfied for allŽ. p, x, y g U. Proof. Let ␹ Ž.A - r - ␤ - ␥ - ␴ Ž.A be arbitrary. Then, by defini- Ž. Ä4 tion of ␹ A , there exists a finite family of operators A1,..., An ;A such that n A BAŽ.,r. A ; Di is1 Ä4 Denote by T the convex hull of A1,..., An . Then T ; A is compact and Ž. convex, and A ; T q rBLŽ X, Y .. Taking ␧ 0 s ␤ y r in condition iii , we have a neighborhood W P = D such that ␧ 0 ; XYXYXY fpŽ.Ž.Ž,x yfp,x gAxyx .q␧05xyxB5Y XY XY ;TŽ.xyxq␤5xyxB5Y,11Ž. whenever Žp, xX.Ž, p, xY. W . g ␧ 0 INVERSE AND IMPLICIT FUNCTION THEOREMS 213

The function T ¬ ␴ Ž.T is continuous on T, therefore, by the compactness of T, we can find a constant c such that ␥ - c - ␴ Ž.T for all T g T. Making use of Lemma 3, we obtain that the map A : CŽ.T, X ª C Ž.T, Y defined by ŽAx .Ž.T s T x Ž.T is surjective and c - ␴ Ž.A , i.e.,

cBCŽT,Y.; A BCŽT,X. Ž.12 is valid. Let ␧ [ ␥ y ␤. Choose ␦ ) 0 and the neighborhood U of ŽŽ..ˆˆp, x, fp ˆˆ,x so that ␦␧␦ X Ž.p,tgW␧,55xyˆxF,fpŽ.,xyyF , 0 33 Ž. whenever p, x, y g U and t g ˆx q ␦BX . This is possible, since f is obviously continuous at Ž.p, x and W is a neighborhood of Ž.p, x . ˆˆ ␧0 ˆˆ Let Ž.p, x, y g U be fixed. To show the existence of ␾ s ␾Ž.p, x, y satisfyingŽ. 10 , we apply Ekeland’s variational principle Ž cf.w 12; 2, Theorem 3.3.1, p. 91; 9, Theorem 7.5.1, p. 266x. to the complete metric space M [ ˆˆx q ␦BXand function t ¬ fpŽ.,tyy. Thus there exists ␾ g x q ␦BX satisfying the following two relations:

fpŽ.,␾yyq␧55xy␾FfpŽ.,xyy Ž.13 and

fpŽ.,␾yyFfp Ž.,tyyq␧55ty␾ for t g ˆx q ␦BX.14Ž. The first inequality gives the second relation inŽ. 10 . To complete the proof, we have to show that fpŽ.,␾syholds also true. It follows fromŽ. 13 that 1 ␦ 2␦ 55xy␾FfpŽ.,xyyF 55␾yˆxF . ␧ 33« On the other hand, it follows fromŽ. 12 that the constant function y Ž.T ' y yfpŽ.,␾ is the image of a function x g C ŽT, X .with norm not greater than y y fpŽ.,␾ rc, that is, there exists, an element xTfor all T g T Ž. such that the map T ¬ xT is continuous cf. Lemma 2 and 1 ␧␦ ␦ yyfpŽ.,␾sTxTTand 55x F y y fpŽ.,␾ F - c3c3 Ž.15 214 ZSOLT PALES´ holds for all T g T. Then t s ␾ q xTXg ˆx q ␦B , therefore we conclude fromŽ. 14 that

fpŽ.,␾yyFfp Ž,␾qxTT .Ž.yfp,␾yTx q ␧ 55xT for T g T.16Ž. T Define the set-valued map ⌽ : T ª 2 by

⌽Ž.T[Ä4SgT¬fpŽ.Ž.,␾qxTTyfp,␾ySx F ␤ 55xT. Then one can check that ⌽Ž.T is convex, compact and further, byŽ. 11 and the choice of ␦, it is also nonempty for all T g T. We point out now that ⌽ is upper semicontinuous.

Let T0 g T be an arbitrary fixed point, Tnn, S two sequences with Ž. Snng⌽T, and limit points T00and S , respectively. Then we have fp,␾ x fpŽ.,␾ Sx ␤55x . Ž.qTnnny y TF Tn

Taking the limit n ª ϱ and using the continuity of the mapping T ¬ xT , Ž. we obtain this inequality for n s 0, whence we get S00g ⌽ T . This completes the proof of the upper semicontinuity. Now we are in the position to apply Kakutani’s fixed point theoremŽ cf. wx2, Theorem 3.2.3, p. 87; 43, Theorem 9.B, p. 452. . Thus there exists a fixed point Tˆˆˆof ⌽, i.e., there exists T g T satisfying T g ⌽Ž.Tˆ. In other words, Tˆ satisfies

fpŽ.,␾qxTTˆˆyfpŽ.,␾yTxˆ F ␤ 55xTˆ. Putting T s TˆintoŽ. 16 and using the second inequality ofŽ. 15 , we arrive at ␥ fpŽ.,␾yyF Ž␤q␧ .55xTTˆˆs␥ 55x F fpŽ.,␾yy. c Since ␥ - c, hence this inequality yields 5fpŽ.,␾yy5s0, which was to be proved. Remark. In the very special case, when A consists of one element only, Theorem 3 reduces to the general inverse function theorems contained in wx3, Theorem 2.3.1, p. 161; 24, p. 34 . Observe that in that case we do not need Kakutani’s fixed point theorem andŽ. implicitly Michael’s selection theorem, therefore our approach offers then a very short proof for both of these results requiring only Banach’s open mapping theorem and Ekeland’s variational principle. Since ␥ y ␤ can be arbitrarily close to ␴ Ž.A y ␹ Ž.A , hence the constant of surjection of the function fxppŽ.[fp Ž,x .at Žˆˆx, fx Ž..is close to ␴Ž.Ay␹ Ž.Awhen p is close to ˆp. INVERSE AND IMPLICIT FUNCTION THEOREMS 215

It is easy to check that the conditionŽ. iii of Theorem 3 is fulfilled if we have Ž. Ž . iv For all ␧ ) 0, there exists a neighborhood W␧; P = D of ˆˆp, x such that XXYY fpŽ.Ž.Ž.Ž.,x yfpˆˆ,x yfp,x qfp,x XY XY F␧5xyx5for Ž.Ž.p, x , p, x g W␧ Ž.17 Ž.v A;L ŽX,Y .is a strict prederivative of the mapping x ¬ fpŽ.ˆ,x at ˆx. One can see that assumptionŽ. iv is always satisfied if f is continuously Ž.partially differentiable with respect to the second variable, which is a usual assumption in the classical implicit function theorems. Proof of Theorem 2. Taking the topological space P [ Ä40 and fŽ.0, x sfxŽ., Theorem 3 yields Theorem 2 at once. Remark. Taking the parameter space P [ Ä40, 1, 1r2, 1r3, . . . , Theo- rem 4 reduces to a sequential version of Lyusternik’s theorem.

4. INVERSE AND IMPLICIT FUNCTION THEOREMS

In this section we restate the results of the previous section with the additional assumption that all the operators in A are injective. We shall need the following lemma:

LEMMA 4. Let X and Y be normed spaces and A : X ª Y be an injecti¨e bounded linear operator. Then 55Ax G ␴ Ž.A 55 x for all x g X.18Ž. Proof. If ␴ Ž.A s 0, then there is nothing to prove. When ␴ Ž.A ) 0 andŽ. 18 is not valid, then there exist c - ␴ Ž.A and x g X such that 55Ax - cx 55. However, we have c Ax g cBYX; AB , 55Ax therefore there exists ˆx g BX such that c AxsAxŽ.ˆ. ž/55Ax The injectivity of A then yields c x s ˆx. 55Ax 216 ZSOLT PALES´

Taking the norms of both sides of this equation, we arrive at

cx55 1- s55ˆxF1. 55Ax

This contradiction verifiesŽ. 18 .

Remark. If the linear operator A : X ª Y admits a bounded inverse, 1 then it follows from the above lemma that 5 Ay 5 F 1r␴ Ž.A . Now we are able to reformulate Theorem 3.

THEOREM 4. Let X, Y be Banach spaces and P be a topological space. Let further D ; X be an open set, f : P = D ª Y, ˆˆx g D, and p g P. Assume that conditions Ž.i , Ž ii . , and Žiii . of Theorem 3 are satisfied and, in addition, all the operators A g A are injecti¨e. Then there exist a neighborhood V of ŽŽˆˆˆp,fp,x ..Ž and a uniquely determined . function ␺ : V ª D such that

fpŽ.,␺Ž.p,y sy for Ž. p, y g V,19Ž. p¬␺Ž.p,fpŽ.ˆˆ,x Ž.20 is continuous at p s ˆp, moreo¨er, for all ␧ ) 0, there exists a neighborhood ŽŽ.. V␧;Vofˆˆˆ p,fp,x such that

XYy1XY XY ␺Ž.Žp,yy␺p,y .gA Žyyy .q␧5yyyB5X X Y forŽ.Ž. p, y , p, y g V␧.21 Ž. Ž. Ž. Proof. Let ␹ A - r - ␤ - ␥ - ␴ A be fixed arbitrarily. Let ␧ 0 s ␤ Ž. yr-␴Aand W s W␧ be the corresponding neighborhood in condi- 0 X Y tionŽ. iii of Theorem 3. First we show that the equality fpŽ,x.Žsfp,x. X Y XY X ŽŽwhenever p, x .Žand p, x ..are in W always implies x s x .If fpŽ,x. Ž Y.Ž. sfp,x, then, by assumption iii , there exists A g A and y g BY so that XY XY AxŽ.yx q␧05xyxy5s0. ŽXY. XY Thus 5Axyx 5555s␧0 yxyx5, whence, using Lemma 4,

XY XY XY XY ␧00555555xyxG␧yxyxsAxŽ.Ž.yx G␴ Ax55yx. XY Ž. If x / x , then we arrive at the obvious contradiction ␧ 0 G ␴ A . Thus X Y necessarily x s x . The assumptions of Theorem 3 are satisfied, therefore there exist a neighborhood U and ␾ satisfying the requirements of Theorem 3. As we have seen in the proof of Theorem 3, U was constructed so that INVERSE AND IMPLICIT FUNCTION THEOREMS 217

ŽŽp,␾p,x,y ..W, whenever Žp, x, y .U. Define V as the projection g␧ 0 g of U on the space P = Y and define ␺ : V ª D by ␺ Ž.Ž.Ž.p, y [ ␾ p, x, y for p, x, y g U. The definition of ␺ is correct, since when Žp, xX, y.Žand p, xY, y.are in U, then ŽŽp, ␾ p, xX, y..and Žp, ␾ Ž p, xY, y.. are in W ; further, by the first ␧ 0 equation inŽ. 10 ,

X Y fpŽ.Ž.,␾Ž.p,x,y sysfp,␾ Žp,x,y .. X Y Therefore then ␾Žp, x , y.Žs ␾ p, x , y.Ž.. Clearly, 19 is satisfied. Substituting x [ ˆˆˆx and y [ fpŽ.,xinto the inequalityŽ. 10 , we get 1 ˆˆˆˆˆˆxy␺Ž.p,fpŽ.,x F fpŽ.,xyfpŽ.,x . ␥y␤ ByŽ. i , the right hand side tends to zero as p tends to ˆp. Therefore ␺ŽŽ..p,fpˆˆ,x ª ˆxs␺ ŽŽ.. ˆp,fp ˆˆ,x. Thus the continuity ofŽ. 20 is proved. Putting y [ yY and x [ ␺ Žp, yX.Žinto the second relation in 10. , we get

XY1 XY ␺Ž.Žp,yy␺p,y .F55yyy,22Ž. ␥y␤ X Y whenever Žp, y .Ž, p, y .g V, i.e., ␺ is a locally Lipschitz function of its second variable. Thus, by the continuity of the functionŽ. 20 , ␺ is continuous at Ž.ˆˆp, y . To prove the third assertion of the theorem, for ␧ ) 0, choose V␧ ; V so Ž. Ž␺ Ž .. Ž X .Ž Y. that p, y g V␧ implies p, p, y g W␧␥ Ž␥y␤ .. Let p, y , p, y g V␧ . Then, substituting xX [ ␺ Ž p, yX. and xY [ ␺ Ž p, yY.Žin condition iii. of Theorem 3 and usingŽ. 22 , we obtain

XY XY yyygA␺Ž.Žp,yy␺p,y . XY q␧␥Ž.Ž.Ž. ␥ y ␤␺p,yy␺p,yBY XYXY ;A␺Ž.Žp,yy␺p,y .q␧␥ 55y y yBY. X Y In other words, for all such p, y , y , there exists an operator A g A such that XY XYX yyygA␺Ž.Žp,yy␺p,y .q␧␥ 55y y yЉ BY. Taking the inverse of A, we get

XYy1XY XYy1 ␺Ž.Žp,yy␺p,y .gAy Žyy .q␧␥ 5y y yAB5Y.23Ž. Ž. y1 Ž. Ž. However, ␥ BYX; AB , therefore ABYX;1r␥B. Thus 21 follows fromŽ. 23 at once. 218 ZSOLT PALES´

Taking the value y s 0 in the previous result, it reduces to an implicit function theorem. The following result will be an inverse function theorem. It is the reformulation of Theorem 2 with injective operators.

THEOREM 5. Let X and Y be Banach spaces, D ; X be an open set, f : D ª Y, andˆ x g D. Assume that f has a conëx strict prederiätië A;LŽ.X,Y atˆ x such that ␹ Ž.A - ␴ Ž.A and, in addition, all A g A are injectië. Then there exist a neighborhood V ; Y of fŽ.ˆ x and a function ␺ : V ª D such that

f Ž.␺ Ž.y s y for y g V Ž.24

1 1 and Ays ÄAy: A g A4is a strict prederi¨ati¨eof␺ at fŽ.ˆ x . Proof. Taking the topological space P [ Ä40 and writing fŽ.0, x [ fx Ž. and ␺ Ž.x [ ␺ Ž0, x ., the statement follows from Theorem 4 directly. One has only to observe thatŽ. 21 specialized to this situation means that Ay1 is a strict prederivative for ␺ at fxŽ.ˆ. Now we give two immediate corollaries of Theorem 5:

COROLLARY 3. Assume that f in Theorem 5 is strictly Frechet´ differen- X tiable atˆˆ x and fŽ. x is a bijecti¨e operator. Then there exist a neighborhood V of fŽ.ˆ´ x and ␺ : V ª D such that Ž.24 holds and ␺ is strictly Frechet differentiable at fŽ.ˆˆˆ x with ␺ XŽfx Ž ..[ Ž fxXŽ..y1. COROLLARY 4Ž Clarke’s Inverse Function Theorem, cf.w 8; 9, Theorem 7.1.1, p. 253x. . Let X and Y be finite dimensional normed spaces and assume that f : D ª Y is a locally Lipschitz function atˆ x. Assume that the elements of ѨfŽ.ˆ x are in¨ertible matrices. Then the conclusion of Theorem 5 holds with AsÄѨfxŽ.ˆ4. The proofs of these corollaries are analogous to that of Corollary 1 and Corollary 2. We note that Corollary 4 is still more general than the original result of Clarke, since it states not only the existence of an inverse function, but also describes its strict prederivative. We note that this result of Clarke has recently been generalized by Kummerwx 27, 28 . The results of Kummer offer necessary and sufficient conditions for the Lipschitzian invertibility of functions.

ACKNOWLEDGMENT

The author is grateful to Professor Ernst Albrecht, who called his attention to the work of Bartle and Graveswx 5 . INVERSE AND IMPLICIT FUNCTION THEOREMS 219

REFERENCES

1. J.-P. Aubin, Lipschitz behaviour of solutions to convex minimization problems, Math. Oper. Res. 9 Ž.1984 , 87᎐111. 2. J.-P. Aubin and H. Frankowska, ‘‘Set-Valued Analysis, Systems and Control: Foundations and Applications,’’ Vol. 2, Birkhauser¨ Verlag, BostonrBaselrBerlin, 1990. 3. V. M. Alekseev, V. M. Tihomirov, and S. V. Fomin, ‘‘Optimal Control,’’ Nauka, Moscow, 1979. 4. D. Aze, An inversion theorem for set valued maps, Bull. Austral. Math. Soc. 37 Ž.1988 , 411᎐414. 5. R. G. Bartle and L. M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 Ž.1952 , 400᎐413. 6. J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 Ž.1986 , 9᎐52. 7. J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for opennes and regularity of set valued and single valued maps, J. Math. Anal. Appl. 134 Ž.1988 , 441᎐459. 8. F. H. Clarke, On the inverse function theorem, Pacific J. Math. 62 Ž.1976 , 97᎐102. 9. F. H. Clarke, ‘‘Optimization and Nonsmooth Analysis,’’ Canad. Math. Soc. Ser. Mono- graphs Adv. Texts, Wiley᎐Interscience, New York, 1983. 10. R. Cominetti, Metric regularity, tangent sets and second order optimality conditions, Appl. Math. Optim. 21 Ž.1990 , 265᎐287. 11. A. V. Dmitruk, A. A. Milyutin, and N. P. Osmolovski, Lyusternik’s theorem and the theory of extremum, Russian Math. Sur¨eys 35 Ž.1980 , 11᎐51. 12. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 Ž.1974 , 324᎐353. 13. M. Fabian and D. Preiss, A generalization of the interior mapping theorem of Clarke and Pourciau, Comment. Math. Uni¨. Carolin. 28 Ž.1987 , 311᎐324. 14. H. Frankowska, An open mapping principle for set valued maps, J. Math. Anal. Appl. 127 Ž.1987 , 172᎐180. 15. H. Frankowska, Higher order inverse function theorems, in ‘‘Analyse Non Lineaire’’´ Ž.Attouch, Aubin, Clarke, and Ekeland, Eds. , pp. 283᎐304, Gauthier᎐Villars, Paris, and C.R.M., Universite´´ de Montreal, 1989. 16. B. M. Glover and D. Ralph, First order approximations to nonsmooth mappings with application to metric regularity, Numer. Funct. Anal. Optim. 15 Ž.1994 , 599᎐620. 17. L. M. Graves, Topics in functional calculus, Bull. Amer. Math. Soc. 41 Ž.1935 , 641᎐662. 18. H. Halkin, Mathematical programming without differentiability, in ‘‘Calculus of Varia- tions and Optimal Control Theory’’Ž. D. L. Russel, Ed. , pp. 279᎐287, Academic Press, New York, 1976. 19. A. D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 Ž.1979 , 61᎐69. 20. A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 Ž.1981 , 1᎐56. 21. A. D. Ioffe, Nonsmooth analysis and theory of fans, in ‘‘Convex Analysis and Optimiza- tion’’Ž. J.-P. Aubin and R. Vinter, Eds. , pp. 93᎐118, Pitman, London, 1982. 22. A. D. Ioffe, On the theory of subdifferential, in ‘‘Mathematics for OptimizationrMath- ematical Studies Series,’’ North-Holland, Amsterdam, 1986. 23. A. D. Ioffe, On the local surjection property, Nonlinear Anal. 11 Ž.1987 , 565᎐592. 24. A. D. Ioffe and V. M. Tihomirov, ‘‘Theory of Extremal Problems,’’ North-Holland, Amsterdam, 1979. 25. A. Jourani and L. Thibault, Approximate subdifferential and metric regularity, the finite dimensional case, Math. Programming 47 Ž.1990 , 203᎐218. 220 ZSOLT PALES´

26. A. Jourani and L. Thibault, Approximations and metric regularity in math programming in Banach spaces, Math. Oper. Res. 18 Ž.1993 , 390᎐401. 27. B. Kummer, An implicit function theorem for C 0, 1 equations and parametric C1, 1-optimization, J. Math. Anal. Appl. 158 Ž.1991 , 35᎐46. 28. B. Kummer, Lipschitzian inverse functions, directional derivatives, and applications in C1,1-optimization, J. Optim. Theory Appl. 70 Ž.1991 , 561᎐582. 29. L. A. Lyusternik, On extremum conditions for functionals, Mat. Sb. 41, No. 3Ž. 1934 , 390᎐401. 30. E. Michael, Continuous selections I, Ann. of Math. 63 Ž.1956 , 361᎐381; II, 64 Ž.1956 , 562᎐580; III 65 Ž.1957 , 375᎐390. 31. B. Mordukhovich, Complete characterization of openness, metric regularity, and Lips- chitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 Ž.1993 , 1᎐36. 32. B. Mordukhovich, Lipschitzian stability of constraint systems and generalized equations, Nonlinear Anal. 22 Ž.1994 , 173᎐206. 33. Zs. Pales,´ Linear selections for set-valued functions and extensions of bilinear forms, Arch. Math. 62 Ž.1994 , 427᎐432. 34. J.-P. Penot, On regularity conditions in mathematical programming, Math. Programming Study 19 Ž.1982 , 167᎐199. 35. J.-P. Penot, Metric regularity, openness and Lipschitzian behaviour of multifunctions, Nonlinear Anal. 13 Ž.1989 , 629᎐643. 36. B. H. Pourciau, Analysis and optimization of Lipschitz continuous mappings, J. Optim. Theory Appl. 22 Ž.1977 , 311᎐351. 37. D. Ralph, A chain rule for nonsmooth composite functions via minimization, Bull. Austral. Math. Soc. 49 Ž.1994 , 129᎐138. 38. S. M. Robinson, Stability theory for systems of inequalities. Part II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 Ž.1976 , 497᎐513. 39. R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Anal. 9 Ž.1985 , 867᎐865. 40. L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal. 6 Ž.1982 , 1037᎐1053. 41. T. Szilagyi,´ Generalization of the implicit function theorem and of Banach’s open mapping theorem, Acta Sci. Math. 38 Ž.1977 , 391᎐396. 42. J. Warga, Derivative containers, inverse functions and controllability, in ‘‘Calculus of Variations and Optimal Control Theory’’Ž. D. L. Russel, Ed. , pp. 13᎐45, Academic Press, New York, 1976. 43. E. Zeidler, ‘‘Nonlinear Functional Analysis and Its Applications,’’ Vol. I, Springer-Verlag, BerlinrHeidelbergrNew York, 1984.