Commentationes Mathematicae Universitatis Carolinae

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1 Commentationes Mathematicae Universitatis Carolinae John R. Giles; Scott Sciffer Fréchet directional differentiability and Fréchet differentiability Commentationes Mathematicae Universitatis Carolinae, Vol. 37 (1996), No. 3, Persistent URL: Terms of use: Charles University in Prague, Faculty of Mathematics and Physics, 1996 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Comment.Math.Univ.Carolin. 37,3(1996) Fréchet directional differentiability and Fréchet differentiability J.R. Giles, Scott Sciffer Abstract. Zajíček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any realvaluedfunctiononanormedlinearspace,thesetofpointswherethefunctionisfréchet directionally differentiable but not Fréchet differentiable is first category. Keywords: Gâteaux and Fréchet subdifferentiability, directional differentiability, strict and intermediate differentiability Classification: 58C20, 46G05 Areal-valuedfunction F onanopensubset Aofanormedlinearspace X issaidtobefréchetdifferentiableat x Aifthereexistsacontinuouslinear functional F (x)on Xwhere,given ǫ >0thereexistsaδ(ǫ, x) >0suchthat F(x+y) F(x) F (x)(y) < ǫ y forall y X, y < δ. In determining Fréchet differentiability properties, interest has focused on variants of Fréchet differentiability. The function F is said to be Fréchet subdifferentiable at x Aifthereexistsacontinuouslinearfunctional fon Xwhere,given ǫ >0 thereexistsaδ(ǫ, x) >0suchthat F(x+y) F(x) f(y) > ǫ y forall y X, y < δ. Inparticular,BorweinandPreissprovedthatwhen XisaBanachspacewith an equivalent norm Fréchet differentiable away from the origin, a lower semicontinuousfunction F onanopensubset Aof XisdenselyFréchetsubdifferentiable,[BP,p.521].RecentlyZajíčekprovedthatwhen XisanAsplundspace,a lowersemi-continuousfunction Fonanopensubset Aof Xhasthepropertythat the set of points where F is Fréchet subdifferentiable but not Fréchet differentiable isfirstcategoryin A,[Z2,p.485]. In this paper we study another variant of Fréchet differentiability. Given a real-valuedfunction F onanopensubset Aofanormedlinearspace X,wesay that Fhasaright-handderivativeat x Ainthedirection v Xif F + F(x+λv) F(x) (x)(v)= lim λ 0 + λ

3 490 J.R. Giles, Scott Sciffer exists.clearly F + (x)(v)ispositivelyhomogeneousin v.wesay Fisdirectionally differentiable at x Aif F + (x)(v) existsineverydirection v X andisa continuousfunctionin v. If F + (x)(v)isalsolinearin vthenwesaythat F is Gâteaux differentiable at x. We note that although F may have a right-hand derivativeat x Aineverydirection v X, F + (x)(v)neednotbecontinuousin v,evenif Fiscontinuousat x. Example. Consider FonR 2 definedinpolarcoordinatesby { cosθsin(r/cosθ) forcosθ 0and r 0 F(r, θ)= 0 forcosθ=0or r=0. Nowattheorigin { F 1 when cosθ 0 r = 0 when cosθ=0. However, if a locally Lipschitz function has a right-hand derivative at a point in every direction, then it is directionally differentiable at the point. This is the implication of the following well known result whose proof is included for the sake of completeness. Proposition 1. Consider a locally Lipschitz function ψ on an open subset A of anormedlinearspace X. Given x A,if ψ + (x)(v)existsforall v Xthen ψ + (x)(v)islipschitzin v. Proof:Since ψislocallylipschitzon A,thereexistsaK>0andaδ>0such that ψ(y) ψ(z) K y z forall y, z B(x; δ) A. Given u X, u 1and ǫ >0thereexistsa0 < δ 1 < δsuchthat ψ +(x)(u) ǫ < ψ(x+λu) ψ(x) for 0 < λ < δ 1 λ ψ(x+λv) ψ(x) + K u v λ for v X, v 1and 0 < λ < δ 1. Butthereexists0 < δ 2 < δ 1 suchthat Andso ψ(x+λv) ψ(x) λ < ψ + (x)(v)+ǫ for 0 < λ < δ 2. ψ +(x)(u) ǫ < ψ +(x)(v)+ ǫ+k u v forall u, v X, u, v 1.

4 Fréchet directional differentiability and Fréchet differentiability 491 Thisholdsforall ǫ >0andsoweconcludethat andso ψ + (x)(u) ψ + (x)(v)+k u v ψ + (x)(u) ψ + (x)(v) K u v forall u, v X, since ψ + (x)(v)ispositivelyhomogeneousin v. Forareal-valuedfunction Fonanopensubset Aofanormedlinearspace X wesaythat FisFréchetdirectionallydifferentiableat x Aif Fisdirectionally differentiableat xandgiven ǫ >0thereexists δ(ǫ, x) >0suchthat F(x+λv) F(x) F + λ (x)(v) < ǫ forall 0 < λ < δ andall v X, v 1. Ofcourse FisFréchetdifferentiableat x AifitisFréchetdirectionallydifferentiableat xand F + (x)(v)islinearin v. It is of interest to see how Fréchet directional differentiability and Fréchet subdifferentiability relate. A continuous convex function φ on an open convex subset Aofanormedlinearspace Xhasasubgradientateachpointofitsdomain; thatis,given x Athereexistsacontinuouslinearfunctional fon Xsuchthat φ(x+y) φ(x) f(y) forall y X, [Ph,p.7]. So φisfréchetsubdifferentiableateachpointinitsdomain. Butalso φ + (x)(y) existsandisacontinuoussublinearfunctionalin yateachpoint x A,[Ph, p.2],so φisdirectionallydifferentiableateachpointofitsdomain.however,the norm is always Fréchet directionally differentiable at the origin, but it need not befréchetdirectionallydifferentiableatanyotherpoint;thereexistson l 1 an equivalent norm which is Gâteaux differentiable away from the origin but which is nowhere Fréchet differentiable,[ph, p. 86]. Moregenerally,forareal-valuedfunction Fonanopensubset Aofanormed linearspace X,if F isfréchetdirectionallydifferentiableat x Athengiven ǫ >0thereexistsaδ>0suchthat F(x+y) F(x) > F + (x)(y) ǫ y forall y X, y < δ, andif F + (x)(y)isalsosublinearin ythenanysubgradient fof F + (x)at xsatisfies F(x+y) F(x) > f(y) ǫ y forall y X, y < δ, that is, F is Fréchet subdifferentiable at x. However, if F is Fréchet directionally differentiable at x then it is not necessarily Fréchet subdifferentiable at x; for the norm. on X, F =. isfréchetdirectionallydifferentiableat0butisnot Fréchet subdifferentiable at 0. We now establish for Fréchet directional differentiability a general result comparable to that of Zajíček for Fréchet subdifferentiability,[z2, p. 485].

5 492 J.R. Giles, Scott Sciffer Theorem2. Givenareal-valuedfunction F onanopensubset Aofanormed linearspace X,thesetofpoints W Awhere F isfréchetdirectionallydifferentiable, but not Fréchet differentiable, is first category in A. Proof:Foreach n, p Nconsidertheset W n,p consistingofthosepointsin A where F is Fréchet directionally differentiable and F(x+λv) F(x) F λ +(x)(v) <1 p (i) forall 0 < λ 2 n andall v X, v 1, (ii) thereexist u, v X, u, v 1 and F + (x)(u)+f + (x)(v) F + (x)(u+v) >16 p. Now W= n,p 2 W n,p andweshowthatforeach n, p 2, W n,p isnowheredense in A. Supposeonthecontrarythatforsome n, p 2, W n,p isdenseinsomeopen subset Uof A. Wemayassumethat0 W n,p Uand F(0)=0. Weexamine estimatesof Fnear0.For m > n,by(i),wehave F( 1 n u) 1 n F + (0)(u) < 1 np, F(1 m v) 1 m F + (0)(v) < 1 mp and F( 1 m (u+v)) 1 m F + (0)(u+v) < 1 mp. Wenowchoose msufficientlylargethat B(0;2/m) Uand,tosatisfythecontinuityof F + (0),suchthat F + (0)(1 n u) F + (0)(x+1 n u) < 1 np forall x B(0;2/m). Since W n,p isdensein U, bythe continuityof F + (0), thereexistsan x m B(0;2/m) W n,p suchthatboth F + (0)(1 m v) F + (0)(x m) < 1 mp and F + (0)(1 m (u+v)) F + (0)(x m+ 1 m u) < 1 mp. However,byproperty(i), F(x m ) F + (0)(x m) <(1/p) x m andso F(x m ) F + (0)(1 m v) F(x m) F + (0)(x m) + F + (0)(x m) F + (0)(1 m v) < 1 p x m + 1 mp < 3 mp....(a)

6 Fréchet directional differentiability and Fréchet differentiability 493 Similarly,when m 2nwehave F(x m + 1 m u) F + (0)(1 m (u+v)) F(x m + 1 m u) F + (0)(x m+ 1 m u) + F + (0)(x m+ 1 m u) F + (0)(1 m (u+v)) < 1 p x m+ 1 m u + 1 mp < 4 mp since x m+ 1 m u 3 m 2 n. Also F(x m + 1 n u) F +(0)( 1 n u)...(b) F(x m + 1 n u) F + (0)(x m+ 1 n u) + F + (0)(x m+ 1 n u) F + (0)(1 n u) < 1 p x m+ 1 n u + 1 np < 3 np. Since x m W n,p, F(x m +(1/m)u) F(x m ) F(x m+(1/n)u) F(x m ) 1/m 1/n F(x m +(1/m)u) F(x m ) F + 1/m (x m)(u) + F(x m +(1/n)u) F(x m ) F 1/n +(x m )(u)...(c) < 2 p. But F(x m +(1/m)u) F(x m ) 1/m F + (0)((1/m)(u+v)) F + (0)((1/m)v) 1/m F(x m+(1/n)u) F(x m ) 1/n F + (0)((1/n)u) F + (0)((1/m)v) 1/n m F +(0)((1/m)(u+v)) F(x m +(1/m)u) m F +(0)((1/m)v) F(x m ) n F + (0)((1/n)u) F(x m+(1/n)u) n F + (0)((1/m)v) F(x m) F +(0)(u+v) F +(0)(v) F +(0)(u) n m F +(0)(v) 4 p 3 p 3 p 3n mp from(a),(b)and(c) 16 p n m F + (0)(v) 12 p 3 p foranychoiceof m np F + (0)(v). Thisisourcontradiction,soweconclude that W is first category.

7 494 J.R. Giles, Scott Sciffer Areal-valuedfunction F onanopensubset Aofanormedlinearspace Xis saidtobestrictlydifferentiableat x Aif F(z+ λv) F(z) lim existsforall v X z x λ λ 0 + andthislimitis F (x)(v),where F (x)isacontinuouslinearfunctionalon X. Further, F issaidtobeuniformlystrictlydifferentiableat x Aifthislimit existsandisapproacheduniformlyforall v X, v 1. Forareal-valuedfunctiononanopensubsetofanormedlinearspacetheset of points where the function is Fréchet differentiable but not uniformly strictly differentiableisfirstcategoryinitsdomain,[z1,p.158].sousingthisfactwecan make the following deduction. Corollary2.1. Givenareal-valuedfunction Fonanopensubset Aofanormed linear space X, the set of points where F is Fréchet directionally differentiable but not uniformly strictly differentiable is first category in A. Forareal-valuedfunctiononanopensubsetofanormedlinearspacethe set of points where the function is both Fréchet subdifferentiable and Fréchet directionally differentiable contains the set of points where the function is Fréchet differentiable. So from Theorem 2 and Zajíček s result for Fréchet subdifferentiability,[z2, p. 485], we can produce the following relation between points of Fréchet subdifferentiability and Fréchet directional differentiability. Corollary 2.2. Considerareal-valuedfunction F onanopensubset Aofa normed linear space X. (i) The set of points where F is Fréchet directionally differentiable but not Fréchet subdifferentiable is first category in A. (ii) If F islowersemi-continuouson Aand X isanasplundspacethen the set of points where F is Fréchet subdifferentiable but not Fréchet directionally differentiable is first category in A. WepointedoutbeforeTheorem2thatthereexistson l 1 anequivalentnorm which is Gâteaux differentiable away from the origin but is nowhere Fréchet differentiable. Now this norm is everywhere Fréchet subdifferentiable but Fréchet directionally differentiable only at the origin, so there is no obvious improvement tobemadetocorollary2.2(ii). InfactforanyBanachspace Xwhichisnot an Asplund space there exists a continuous convex function φ on an open convex subset A of X where the set of points of Fréchet differentiability is first category in A. Sothesetofpointswhere φisfréchetsubdifferentiablebutnotfréchet differentiableisresidualin AandfromTheorem2,thesetofpointswhere φis Fréchet subdifferentiable but not Fréchet directionally differentiable is residual in A.WeshouldpointoutthatforLipschitzfunctionsonthereallineitmaybe that the set of points of Fréchet subdifferentiability is first category; there is a Lipschitz function ψ where the set of points of differentiability is first category,[gs2,

8 Fréchet directional differentiability and Fréchet differentiability 495 p.210],soitisnotpossiblethatboth ψand ψarefréchetsubdifferentiableon a residual set. For convex functions there is a useful continuity characterization of Fréchet directional differentiability which has had interesting geometrical consequences. Givenacontinuousconvexfunction φonanopenconvexsubset Aofanormed linearspace X,thesubdifferentialof φat xistheset φ(x)={f X : f(y) φ +(x)(y) forall y X}. We say that the subdifferential mapping x φ(x) is restricted norm upper semi-continuousat x Aifgiven ǫ >0thereexistsaδ>0suchthat φ(y) φ(x)+ǫb(x) forall y B(x; δ). Thefunction φisfréchetdirectionallydifferentiableat x Aifandonlyifthe subdifferential mapping x φ(x) is restricted norm upper semi-continuous at x,[gm,theorem3.2].ithasbeenshownthatabanachspace XisanAsplund space if it has an equivalent norm which is Fréchet directionally differentiable on theunitsphere,[cp,p.453]. It is instructive to see that there is no comparable continuity characterization of Fréchet directional differentiability for locally Lipschitz functions. Given a locally Lipschitzfunction ψonanopensubset Aofanormedlinearspace X,theClarke subdifferentialof ψat x Aistheset ψ(x)={f X ψ(z+ λy) ψ(z) : f(y) limsup forall y X}. z x λ λ 0 + Now ψisstrictlydifferentiableat x Aifandonlyif ψ(x)issingletonand is uniformly strictly differentiable at x A if and only if the subdifferential mapping x ψ(x) is single-valued and norm upper semi-continuous at x,[gs1, p. 374]. There exists a locally Lipschitz function which is Fréchet differentiable and strictly differentiable at a point but not uniformly strictly differentiable at that point,[gs1, p. 373]; so the function is Fréchet directionally differentiable at the point but its subdifferential is not restricted norm upper semi-continuous there. But also there exists a Lipschitz function on the real line whose subdifferential mapping is everywhere restricted norm upper semi-continuous but nowhere singlevalued,[gs2,p.210],sobycorollary2.1thesetofpointswherethefunctionis not Fréchet directionally differentiable is residual. We might well ask how our Theorem 2 generalizes for Gâteaux differentiability. For locally Lipschitz functions on certain Banach spaces we have sufficient information about generic sets of points of differentiability to achieve a generalization. Givenareal-valuedfunction Fonanopensubset Aofanormedlinearspace X wesaythat Fisintermediatelydifferentiableat x Aifthereexistsacontinuous linearfunctional fon Xsuchthat F(x+λv) F(x) F(x+λv) F(x) lim inf f(v) limsup λ 0 + λ λ 0 + λ forall v X.

9 496 J.R. Giles, Scott Sciffer Clearlyif F + (x)(v)existsat x Aforall v Xand F isintermediatelydifferentiable at x then F is Gâteaux differentiable at x. A GSGspaceisaBanachspacewhichcontainsadensecontinuouslinear imageofanasplundspace. EveryclosedlinearsubspaceofaGSGspaceisa weakasplundspace. IthasbeenshownthatalocallyLipschitzfunction ψon anopensubset Aofaclosedlinearsubspace XofaGSGspaceisintermediately differentiableonaresidualsubsetof A,[FP,p.733].Sowecanmakeanimmediate deduction. Theorem3. ForalocallyLipschitzfunction ψonanopensubset Aofaclosed linearsubspace XofaGSGspace,theset {x A:ψ +(x)(v) existsforall v X and isfirstcategoryin A. ψ is not Gâteaux differentiable at x} For locally Lipschitz functions on separable Banach spaces rather more can be stated. ForalocallyLipschitzfunction ψonanopensubset Aofaseparable Banachspace Xtheset {x A:ψ isgâteauxdifferentiableat x and ψ is not strictly differentiable at x} isfirstcategoryin A,[GS2,p.210].SofromTheorem3wededucetheextended result. Theorem4. ForalocallyLipschitzfunction ψonanopensubset Aofaseparable Banach space, the set {x A:ψ + (x)(v) existsforall v X and ψ is not strictly differentiable at x} isfirstcategoryin A. We should make the following remarks about these results. Theorem 3 does not holdgenerallyforallbanachspaces;on l thecontinuoussemi-norm pdefined for x=(x 1, x 2,..., x n,...)by p(x) =limsup n x n has p + (x)(v)existing ateach x l forall v l,but pisnowheregâteauxdifferentiable,[ph, p.13]. EvenforalocallyLipschitzfunctiononanopenintervaloftherealline, Theorem 4 has no obvious improvement; there exists a locally Lipschitz function ψ everywhere differentiable on(a, b) and which is strictly differentiable only on a setoflessthanfullmeasure,[m,p.975].

10 Fréchet directional differentiability and Fréchet differentiability 497 References [BP] Borwein J.M., Preiss D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303(1987), [CP] Contreras M.D., Paya R., On upper semi-continuity of duality mappings, Proc. Amer. Math. Soc. 121(1994), [FP] Fabian M., Preiss D., On intermediate differentiability of Lipschitz functions on certain spaces, Proc. Amer. Math. Soc. 113(1991), [GM] Giles J.R., Moors W.B., Generic continuity of restricted weak upper semi-continuous set-valued mappings, Set-valued Analysis, to appear. [GS1] Giles J.R., Scott Sciffer, Continuity characterizations of differentiability of locally Lipschitz functions, Bull. Austral. Math. Soc. 41(1990), [GS2] Giles J.R., Scott Sciffer, Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces, Bull. Austral. Math. Soc. 47(1993), [M] Marcus S., Sur les fonctions dérivées, intégrables, au sens de Riemann et sur les dérivées partialles mixtes, Proc. Amer. Math. Soc. 9(1958), [Ph] Phelps R.R., Convex Functions, Monotone Operators and Differentiability, Springer- Verlag, Lecture Notes in Math. 1364, 2nd ed., [Z1] Zajíček L., Strict differentiability via differentiability, Act. U. Carol. 28(1987), [Z2] Zajíček L., Fréchet differentiability, strict differentiability and subdifferentiability, Czech. Math. J. 41(1991), The University of Newcastle, NSW 2308, Australia (Received May 5, 1995)