(1.1) (f'(z),y-x)>f(y)-f(x).

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vlume 122, Number 4, December 1994 MEAN VALUE INEQUALITIES F. H. CLARKE AND YU. S. LEDYAEV (Cmmunicaed by Andrew Bruckner) Absrac. We prve a new ype f mean value herem, ne in which he funcinal differences are esimaed in muliple direcins simulaneusly. 1. Inrducin The Mean Value Therem f calculus in is inequaliy frm assers ha if / is a smh funcin n an inerval [x, y], hen here is a pin z in he inerval such ha f'(z)(y-x) >f(y)-f(x). The exensin f his resul mre han ne dimensin is sraighfrward; [x, y] is hen a line segmen and we have he inequaliy (1.1) (f'(z),y-x)>f(y)-f(x). Hwever, a resul such as his remains essenially ne-dimensinal, r unidirecinal. Our purpse is esimae he rae f grwh in muliple direcins simulaneusly. Le X and Y he cmpac subses f R" and f a smh funcin n R". Suppse fr simpliciy ha / equals 1 n y and 0 n X. Then fr any x X, y Y, i fllws frm he inequaliy (1.1) cied abve ha fr sme pin zxiy in he segmen [x, y] ne has (1.2) (f(zx,y),y-x)>l. Bu we will shw ha when X and Y are cnvex, here is a single pin z which fills he rle f zx>y in (1.2) fr all pairs (x, y); he pin z lies in he cnvex hull f XUY, which we dene [X, Y]. We remark ha he cnvexiy f X and Y cann simply be deleed in baining his cnclusin. Fr example, if in R" we ake X = {0} and Y = he uni sphere, and if / is a funcin which vanishes a 0 and is equal 1 n he sphere, he unifrm inequaliy cied abve becmes (f'(z), y) > 1 fr all uni vecrs y, which is clearly impssible. Received by he edirs March 10, Mahemaics Subjec Classificain. Primary 26B05. Key wrds and phrases. Mean value herem. The firs auhr was suppred in par by Naural Sciences and Engineering Research Cuncil (Canada) and le Fnds FCAR (Québec) American Mahemaical Sciey /94 $1.00+ $.25 per page

2 1076 F. H. CLARKE AND YU. S. LEDYAEV Therem 2.1 f he fllwing secin des n require ha / be cnsan n X and n Y. Fr example, when X and Y are cmpac i implies ha fr sme z in he "inerval" [X, Y], ne has (1.3) (f'(z),y-x) > min / - max / fr all x X, y Y. Furher, X and Y need n be disjin nr finie-dimensinal. Fr example, le us se X = Y = he clsed uni ball B in a Banach space. By aking y x be an apprpriae negaive muliple f f'(z) in an exended frm f (1.3), we bain frm Therem 2.1 belw he esimae (1.4) sup/-inf/>2inf /'(z) B B B fr any cninuusly differeniable funcin /. We d n knw wheher ( 1.4) has been ned previusly, bu i serves illusrae he deeper level f infrmain cnained in ur herem as cmpared he classical unidirecinal resuls. The herem is discussed and prved in he fllwing secins, geher wih sme varians. In a subsequen and mre echnical aricle we shall develp he applicains in nnsmh analysis and parial differenial equains ha were he riginal mivain fr his wrk. 2. The Therem Thrughu his aricle, X and Y are nnempy, clsed cnvex bunded subses f a Banach space E. We dene by [X, Y] he "inerval" {z E : z = Xx + (I -X)y fr sme X [0, I], x X, y Y}. (This cincides wih he cnvex hull f X U Y.) We are given a funcin / which is cninuusly Gâeaux differeniable n a neighbrhd f [X, Y]. Therem 2.1. Le e > 0 be given, and suppse ha a leas ne f X and Y is cmpac. Then here exiss a pin z in [X, Y] such ha (2.1) iaf f-sup f < (f'(z),y-x) + e Vx X,y Y Y X By swiching he rles f X and Y in he herem we bain Crllary 2.1. There exiss a pin w in [X, Y] such ha (2.2) sup f-iaff > (f'(w),y-x)-e \x X,y Y. Y X We nw derive a resul ha is clser he usual mean value herem. T sae i, we define w funcins n E* : 5(0 := max{(c, y -x) : x X, y Y}, s(i):=min{(ç, y-*):* *, y Y}. Ne ha when X, Y are singlens {x}, {y}, hen s(q = S(Ç) = (Ç, x - y). We nminae he fllwing as "mulidirecinal mean value herem".

3 MEAN VALUE INEQUALITIES 1077 Crllary 2.2. Le bh X and Y be cmpac, and le g be any cninuus funcin n E* saisfying s(c)<g(0 < S(C) fr all C E\ Then here exiss a pin u [X, Y] such ha (2.3) g(f'(u)) f(y)-f(x). Prf. By an eviden limiing prcedure in which he e f he herem ges zer, here is a pin z such ha min(/(7)-/w) = mm/-max/ < s(f'(z)) < g(f'(z)). Similarly, we bain a pin w saisfying g(f'(w)) < S(f'(w)) < maxf-miaf=max(f(y)-f(x)). Y X. I fllws frm he inermediae value herem ha sme pin u beween z and w saisfies (2.3). D Example 1. As an illusrain f he "mulidirecinal equaliy" embdied in he crllary, cnsider a C1 funcin / n he fllwing subse Í2 f R2 : Q := {(x, y) : 0 < x < 1, 0 < y < 1}, and suppse ha f(0, ) = 0, f(l, ) = 1. Then here is a pin w = (u, v) in Q such ha fx(u,v) = i + \f;(u,v)\. This fllws frm Crllary 2.2 by aking g(u, v)=s(u,v) = u- \v\, X = {(u,v) il:u = 0}, Y = {(u,v) ïl:u=l}. When X and Y are singlens {x}, {y}, i is eviden ha (2.3) reduces he classical equaliy f(y)-f(x) = (f'(z),y-x). In cnras his classical unidirecinal case, i is n always pssible, hwever, affirm (2.3) fr sme z lying in he relaive inerir f [X, Y]. We illusrae by a cunerexample. Example 2. In w dimensins, we define ( u + v2(l-v)2/2-(u-v)2/2 ifu<v, J(u,v)-^ u + v2^_vyi2-v2(u-v)2/2 if u>v, AT ={(0,0)}, r = {(l,u):0<;<l}. Then / is cninuusly differeniable, equal 0 n X, and 1 n Y. Therem 2.1 implies ha fr sme (z, w) in he (clsed) riangle frmed by aking he cnvex hull f XöY, ne has (2.4) (f'(z,w), (l,v))>l fr all >e [0,1].

4 1078 F. H. CLARKE AND YU. S. LEDYAEV We shall see ha (z, w) cann lie in he relaive inerir f he riangle. Fr if (2.4) hlds, hen necessarily f'u(z, w) > 1 (ake v = 0). Bu when w < z, we have f'u(z, w) = 1 -w2(z-w), which is sricly less han 1 in he relaive inerir. (We can g n shw ha he pins (z, w) saisfying (2.4) are hse n he segmen (0,0) (1,0), and he pin (1, 1).) 3. Prf f he Therem Le e' belng (0, e), and le M dene he quaniy sup inf v (/'(z). V-x). ze[x,y] xex.yey We shall esablish he exisence f a pin (x, y) in X x Y such ha AY)-Ax) <M + e', which clearly implies (2.1). Ne ha M belngs (-, ] ; we may limi urselves he case in which M is finie, fr herwise he exisence f z saisfying (2.1) is rivial. We define a mulifuncin Y a [X, Y] wih values in X x Y as fllws: where f':e^e* Y(z):={(x,y) XxY:(f'(z),y-x) denes he Gâeaux derivaive. <M + e'}, Lemma 3.1. There exiss a lcally Lipschiz funcin y : E -> E such ha (3.1) y(z) Y(z)frallz [X,Y], (3.2) y(z) XxY fr all z E. Prf. We begin by cnsrucing an pen cvering f E. Le z belng [X, Y]. By definiin f M, here exiss a pin (xz,yz) in X x Y such ha (f'(z),yz-xz) <M + e'/2. Since /' is cninuus a z, here is an pen neighbrhd U(z) f z such ha (3.3) (f'(z'), yz - xz) < M + e' Vz' U(z). Fix any pin (x, y) X x Y. Fr z ^ [X, Y] (which is a clsed se), le U(z) be an pen neighbrhd f z such ha U(z) n [X, Y] = 0, and define (xz,yz) = (x, y). We have an pen cvering {U(z)}zeE f E. Since a Banach space is paracmpac, here is a lcally finie refinemen {Va}aeA f his cvering (see fr example [3]). Thus fr each a here exiss a pin za suchha Va is cnained in U(za). Als, fr each z, he se f a fr which z Va is finie. We nw use a classical cnsrucin define a cerain pariin f uniy. Le Pa(z) dene he disance frm z E\Va ; hen pa(z) = 0 iff z Va. We define <Pa(z) = Pa(z)I[ Y, P<*-(2)) Xa'eA J

5 MEAN VALUE INEQUALITIES 1079 (ne ha fr each z, nly finiely many erms in he sum are nnzer and ha he sum is psiive). The funcins z pa(z) are glbally Lipschiz wih Lipschiz cnsan 1 ; i fllws ha each pa is lcally Lipschiz. We have ha (3.4) J pa(z) = l \/z E. aea Thus {cpa(')}a is a lcally Lipschiz pariin f uniy. We prceed cnsruc a funcin y as fllws: y(z) ' = Yl (*z '?J ^«(z) aea where he funcin z (xz, yz) was defined earlier. Again, nly finiely many erms in his sum are nnzer, and i fllws ha y is lcally Lipschiz. Frm (3.4), because X and Y are cnvex, i fllws ha y has values in X x Y, which esablishes (3.2). We need nly esablish (3.1). Le z e [X, Y]. Le a be any index in A fr which z Va. Since Va is a subse f U(za), frm (3.3) we deduce which gives (f'(z),yza-xza) <M + e', (xza,yza) Y(z). Bu y(z) is a finie cnvex cmbinain f such pins (xza,yza), and Y(z) is cnvex. This esablishes (3.1) and cmplees he prf f he lemma. D We nw suppse ha X is cmpac; he case in which Y is cmpac is handled by simply swiching he rles f X and Y in wha fllws. We pause prve an apparenly nvel "gemeric cnrllabiliy" resul ha may be f independen ineres. Lemma 3.2. Le u and v be lcally Lipschiz mappings frm E X and Y respecively, where X is cmpac. Then here exiss a sluin z n [0, 1] f he differenial equain (3.5) z() = v(z())-u(z()) such ha fr each [0, I], z() Y + (l-)x. Prf. Since u and v are bunded, i is well knwn [2] ha fr any x, here is a unique sluin z defined n [0, ) f (3.5) wih iniial cndiin z(0) = x, in he sense ha here is a Bchner inegrable funcin z fr which (3.5) hlds, and such ha fr all > 0, z() = x + z(s)ds. Furher, he sluin z(, x) depends cninuusly upn [0, 1] and x.

6 1080 F. H. CLARKE AND YU. S. LEDYAEV Cnsider nw he funcin F n X defined by i F(x) = / u(z(s, x))ds. Then F is a cninuus map frm X X, s by he Schauder fixed pin herem [4, Therem 2.3.7], here is a pin x in X such ha F(x) = x. Le z be he sluin f (3.5) saisfying z(0) =x. Then ~z() = x + j v{z(s))ds- cmpleing he prf f he lemma. i I u(~z(s))ds = / v(z(s))ds+ / u(z(s))ds Y + (l-)x, D We prceed nw apply Lemma 3.2 he funcin y = (u, v) prduced by Lemma 3.1. The funcin z() has values in [X, F],s y(z()) belngs T(z(r)), which gives (f'(z()),z()) = (f'(z()),v(z())-u(z()))<m + e'. The lef side f his inequaliy is he derivaive f he funcin > f(z()), which, as a lcally Lipschiz funcin, is absluely cninuus. Inegraing he inequaliy ver [0, 1] herefre yields f(y)-f(x) <M + e', where y := z(l), x := z(0). This implies he exisence f a pin z saisfying (2.1). D Remark 3.1. An analysis f he prf shws ha he nly requiremen n / is he fllwing: n a neighbrhd f [X, Y], f is defined and Gâeaux differeniable; he Gâeaux derivaive /' is cninuus a each pin f [X, Y]. Remark 3.2. The prf echnique applies vecr-valued funcins. As an illusrain, suppse ha / has values in Rm, and le < dene he usual cmpnen-wise parial rdering in ha space. Le r be any vecr in Rw saisfying r<f(y)-f(x) (i.e., r < f(y) - f(x) fr all (x, y) X x Y ), and le A be a vecr such ha fr any z [X, Y], fr sme (x, y) X x Y (depending n z ), ne has (f'(z),y-x) <A (i.e., sric inequaliy in each cmpnen). We deduce r < A, by an bvius adapain f he prf. This raher indirec saemen is equivalen ha f he herem when m = 1.

7 MEAN VALUE INEQUALITIES TWO VARIANTS OF THE THEOREM We nw presen w exensins f he herem which weaken in urn he smhness and cnvexiy requiremens f Therem 2.1. Fr he firs, we recall w nins frm nnsmh analysis [1]. Le / : E > R be a lcally Lipschiz funcin defined n a Banach space E. The generalized direcinal derivaive a x in direcin v is given by A), ^, f(x'+xv)-f(x') / (jc;î;) := hm sup -A- ^-^, x'-> x * A 0 and he generalized gradien (r subdifferenial) f / a x is he nnempy se defined by df(x) := [UE*: f(x ; v) > (v, Q Vv e). Then df(x) reduces {f'(x)} if / is sricly differeniable (r cninuusly differeniable) a x [1, Prpsiin 2.2.4]. We assume nw ha X and Y are as in Therem 2.1, bu we weaken he regulariy f / he fllwing: / is lcally Lipschiz n [X, Y]; i.e., here crrespnds each z in [X, Y] an pen neighbrhd f z in E in which / saisfies a Lipschiz cndiin. Therem 4.1. Under hese hypheses, fr any e > 0, here exis a pin z in [X, Y] andan elemen Ç f df(z) such ha iaf f-sup f <(Ç,y-x) + e Vx X, y Y. y x Prf. We merely indicae hw adap he prf f Therem 2.1 he presen seing. The mulifuncin Y is nw defined as fllws: r(z) := {(x,y) XxY:(C,y-x) < M + e' VÇeO/(z)} = {(x,y) XxY:f(z;y-x) < M + e'}, where M is defined as befre bu wih df fr /'. Because f (z; v) is upper semicninuus as a funcin f z [ 1, Prpsiin 2.1.1], he prf f Lemma 3.1 requires nly rivial mdificain; Lemma 3.2 requires nne a all. When he final sep f he prf is reached, we have a funcin z() saisfying (C,z()) <M + e' VC Ö/(z(r)). The funcin» f(z()) is Lipschiz (and hence absluely cninuus), and is derivaive a /, when i exiss, lies in he se {(C, (0) : C G df(z())}, as fllws readily frm a direc argumen, r frm a chain rule f nnsmh calculus [1, Therem ]. The preceding inequaliy herefre yields /(z(l))-/(z(0)) which, as befre, cmplees he prf. D <M + e', The analgus crllaries hse f Secin 2 can be derived in he nnsmh seing abve; we limi urselves here saing jus ne.

8 1082 F. H. CLARKE AND YU. S. LEDYAEV Crllary 4.1. In addiin he hypheses f he herem, le bh X and Y be cmpac. Then here exis z in [X, Y] and Ç e df(z) such ha min / - max f < (Ç,y-x) Vx X, y Y. The secnd exensin f he herem invlves a mapping p : E -» E which admis, a each pin z in a neighbrhd f [X, Y], a Fréche derivaive Dp(z) (hus, a cninuus linear funcinal frm E iself). We assume ha a each such z, he map z' > Dcp(z') is cninuus fr he naural induced nrm plgies (i.e., 7)p(z) := supm<, \\Dcp(z)(u)\\ ). The hypheses n X, Y are hse f Therem 2.1 and we assume ha / is cninuusly differeniable n a neighbrhd f cp([x, Y]). ( * denes he adjin.) Therem 4.2. Fr any e > 0, here exiss a pin z in [X, Y] such ha iaf f-supf < (Dp(z)*f'(p(z)),y-x) + Vx X,y Y. 9(Y) <p(x) Prf. We adap he prf f Therem 2.1 given in Secin 3 his exended seing. The definiin f he mulifuncin Y is mdified as fllws: T(z) := {(x, y) X x Y : (Dp(z)* f'(cp(z)),y-x) <M + e'}, where M nw signifies he quaniy sup iaf (Dp(z)*f'(p(z)),y-x), ze[x,y] xex,yey which as befre can be suppsed finie. The prf hen prceeds alng he same lines as befre. In he final phase, we arrive a a smh funcin z() such ha (Dp(z()Yf'(cp(z())),z()) <M + e'. Inegrain and he chain rule give f(p(z(l)))-f(cp(z(0))) <M + e', which yields he cnclusin f Therem 4.2. D Remark. The prf echnique adaps easily he case in which cp(, z) depends n he real parameer. In his nnaunmus seing, he cnclusin f he herem becmes inf /- sup f< (f'(p(,z)),p'(,z)) P(l> Y) <p(0,x) + (Dcp(,zYf'(cp(,z)),y-x)+e Vx X,y Y, where cp' denes he i-derivaive. We mi he deails. I is pssible cmbine simulaneusly he w exensins f his secin; here is an example whse prf is mied. Crllary 4.2. Le X and Y be cmpac, le p be as abve, and suppse f is lcally Lipschiz in a neighbrhd f <p([x, Y]). Then here exiss z in [X, Y] and Ç df(p(z)) such ha min / - max / < (Dcp(z)* Ç, y - x) Vx X, y Y. 9(Y) 9(X) We bain he nex resul by applying Crllary 4.1 he (nnsmh, glbally Lipschiz) funcin f(z) := d9(x)(z) (disance he se p(x)) and upn ning ha any elemen f df has nrm a ms 1.

9 MEAN VALUE INEQUALITIES 1083 Crllary 4.3. Le X and Y be cmpac, le p be as abve, and suppse he Hausdrff disance beween p(x) and cp(y) equals A > 0. Then here exis a pin z in [X, Y] and an elemen f E* f nrm a ms 1 such ha A < (Dp(z)* Ç, y - x) Vx X,y Y. The abve can be viewed as a "linearized separain herem" fr he generally nncnvex ses p(x), cp(y). References 1. F. H. Clarke, Opimizain and nnsmh analysis, Classics Appl. Mah., vl. 5, SIAM, Philadelphia, 1990 (riginally published by Wiley, New Yrk, 1983). 2. K. Deimling, Ordinary differenial equains in Banach spaces, Lecure Nes in Mah., vl. 596, Springer-Verlag, New Yrk, J. Dugundji, Tplgy, Allyn and Bacn, Bsn, D. R. Smar, Fixed pin herems, Cambridge Univ. Press, Cambridge, Cenre de recherches mahémaiques, Universié de Mnréal, C.P A, Mnréal, Québec, Canada H3C 3J7 Seklv Mahemaical Insiue, Vavilv sr. 42, Mscw , Russia