On Differentiability and Surjectivity of a-Lipschitz Mappings.
JACoPo PEJS~C~OWlCZ - ALF0~S0 VIG:~0LI (Genova) (*) (**)
Summary. - In the ]irst part o] this paper the well known result that the Erdchet di]]erential and the asymptotic derivative o] a compact mapping is compact, is extended to ~-Lipschitz mappings (see Theorem 1.1, 1.2 and Corollaries). In the second part we introduce the new concept o/ s-quasibounded mappings de]ined on Banach spaces and obtain new s~r]ectivity theorems extending results previously obta- ined by other Authors (see Theorem 2.1, 2.2). In the third section, by restricting our atten- tion to the class o] asymptotically linear mappings, we give some extimates o] the s-quasinorm (see Theorem 3.1). As a byproduct we obtain a result of indipendent interest, regarding con- tinuous semigroups (see Corollary 3.4). 1~inally, Theorem 3.3 gives a measure o] sur]ectivity for the class o/strongly asymptotically linear mappings.
Introduction.
In the last few years the class of a-Lipsehitz mappings (see definition below) has been intensively studied. The survey paper of B. N. SADovsI¢IJ [14] furnishes a brief historical introduction to the subject matter and a comprehensive (though not complete) bibliography. Our aim here is to give some new results regarding surjectivity and differen- tiabflity of ~-Lipschitz mappings. In the first section of the present note we extend to the context of ~-Lipsehitz mappings some well known results regarding Fr6chet and asymptotic derivatives of completely continuous mappings. (See [1, 8, 11, 14]). In the second section we introduce the class of s-quasibounded mappings rep- resenting a natural extension of the concept of quasibounded ones. For densifying mappings satisfying a s-quasiboundedness condition some surjectivity results are given, extending results of [2~ 6, 10, 16, 17]. In the third section we specialize to the context of asymptotically linear mappings giving extimates of the s-quasinorm. We obtain also a result on continuous semi- groups (see Corollary 3.Q. Finally, for the class of strongly asymptotically linear mappings we are able to give a measure of s~rjeetivity (see Theorem 3.3).
1. - ~-Lipschitz mappings and differentiability.
We list first the definitions and notations to be used below.
(*) Entrain in Redazione il 10 matzo 1973. (**) Work performed un4er the auspices of the National Research Council of Italy (C.N.R.). 50 JACOPOPEJSACKOWICZ - ALFO~S0 VIGNOL].: On di~]erentiability, etc.
Let A c X be any bounded subset of a metric space X. By a(A) we define the measure o/non compactness of A as the infimnm of all positive numbers e such that A can be covered by a finite number of subsets of A with diameter less than e, (C. KUI~ATOWSt;I [9]). Let T: X -~ X be a continuous mapping'. T is said to be ~-Lipschitz with constant K, K>0, if fer any bounded subset AcX
If K< 1, then T is catled ~-contrac~ion. If for any bounded AaX such that ~(A) # 0 we have < then T is called deusi/ying. Let us note that a completely continuous mapping (i.e. continuous sending bounded sets into preeompaet ones) is =-Lipschitz with constant K= 0. For properties (with detailed proofs) of the measure of noncompactness and the theory of ~-Lipsehitz mappings see B. N. SADovs~;m [14] and 1% D. NUSSBAU~ [ll-a] and the references therein. The mapping T:DcE-->E defined on a open subset D of a Banach space E is said to be strongly Frdehet dif/erentiable at a point xo ~ D if it can be represen- ted in the following form
(1.1) T(xo + h)- T(xo) := T'(xo)h + w(xo, h), h ~ E, where T'(xo) is a linear mapping and the mapping w appearing on the right hand side of (1.1) satisfies lim IIw(xo, )ll _ o,
Let T: E --~ E be a mapping such that it can be represented in the form
T(x) = r( )x + w¢), where T'(~) is a linear mapping then the mapping Tis said to be asymptotically linear if
lira ]lw(x)]! _ 0, xeE, and the linear mapping T'(c~) is called the asymptotic derivative of T. The above mentioned concepts of differentiability may be found in the book of ~[. A. K~S~OSEL'S~:IJ [8]. The following results about Fr6chet and asymptotic derivatives due to M. A. K~ASNOSF_~'SICU may be found in [8]. JACoPo PEJSACgOW~Cz -ALFONSO VIONOLI: On differentiability, etc. 51
A) The strong Frdchet derivative o/ a completely continuous mapping is com- pletely continuous. B) The asymptotic derivative of a completely continuous mapping is completely continuous. Our first aim is to extend Kr~snosel'ski's results to the context of ~-Lipschitz mappings for strong Fr~chet derivatives and strong asymptotic derivatives respec- tively. In fact we will prove a slightly more general result by not requiring the linearity of the derivatives. Positive homogeneity will be sufficient for our purposes. Let T: B(O, R) -~E be a nonlinear mapping defined on the ball B(0, R) cE with center at the origin and radius R and let E be a Banach space. Suppose furthermore that T can be expressed as sum of two mappings S and R i.e.
T(x) = S(x) + R(x) , x e B(O, R) ,
where the mappings S and R satisfy the following conditions
i) S is positive homogeneous i.e. S(tx)-~tS(x), t>O~ xeB(O~R).
ii) R is such that ~lj~m[LR(x)l[/lIxlt = 0, ~ 0.
Then the following two lemmata hold.
LEM~k 1.1. - Let T, S and R be as above. Then if T is continuous so are S and R.
P~ooP.- Suppose the contrary an4 let S be discontinuous at some point xo e B(0, R) then for some s > 0 there exists a sequence {x,} converging to x0 such that lis(x.)-~(xo)tI > ~, n=l, 2, ....
Note that since S is homogeneous we may suppose tlxoll < 1. From property ii) we know that there exists t > 0 such that for I[xH~ 8 iIR(~)l] < 5 N. Therefore II T(tx~) -- T(txo)I] > ItS(tx,) - ~(txo)I! - IIR(tx.)11 - ]lR(txo)ll > ts - ~ ]Itx. 1I - ~ HtXo[I > 2 a contradiction since T is continuous and tx.-+ txo. 52 J2~CoPo PEJ'SAGg0WICZ - A~FONS0 VIGNOm: On di#erentiability, etc. LE~A 1.2. - Let R satisfy condition ii). Then /or any bounded subset AcE we have lira ~(R(tA)) _~ O, t>O. t-~o t P]~ooF. - Let A be any bounded subset of E and let M be a positive constant such that A cB(0, M) where B(O, M) is the ball centered at the origin with radius M. LeG s'== ~/2M, where s is any given positive constant. Then by ii) there exists a positive constant p=p(s') such that t*) iiR(x) ll< ']lx]l Jf Suppose t is such that 0< t tB(O, M) c B(O, p(s')) . Now from the inclusion A cB(O, M) and from (.) we get R(tA) c R(tB(O, M)) = R(B(O, tM)) c B(O, e'tM) -=: B(O, @/2)t) . Using the fact that a(B(O, R))=2R (see [4, 11-a]) we obtain ~(R(tA)) < st or a(R(tA)) < s, t and we are done since s is arbitrary. The following theorem is a consequence of the above lemmata. Tm~O~E~ 1.1. - Let T: B(O, R) --> E be a ~-Lipschitz mapping with constant K, such that it can be represented as a sum o/two mappings S and R satis/ying conditions i) and ii). Then S is also ~,.-Lipschitz wi$h the same constant K. P~ooF. - We have S(x) : T(x) -- R(x), x e.B(O, R). Hence for any A c B(0, R) and any t > 0 tS(A ) = S(tA ) c T(tA ) - R(tA ) . Therefore t~(S(A)) <~(T(tA)) + ~(R(tA)) or e(R(tA)) ~(S(A)) By taking the limit as t-~ 0 we get o~(S(A)) < k~(A), A c B(O, R). The continuity of S is ensured by Lemma 1.1. As a corollary to Theorem 1.1 we have COrOLLArY 1.1. - (B. N. SADOVSKIJ [1~:], R. D. NUSSBAU~ [ll-a], J. DA:NE~ [1]). Let T: D-)-E be a s¢-Lipschitz mapping with constant k defined on a open subset D of a Banaeh space E. Let T admits a (strong) Frgehet derivative T'(x,) at some point xo~ D. Then the mapping T'(xo) is also c<-Lipsehitz with constant k. RES~AI~K 1.1. - Let us note that Corollary 1.1 and therefore Theorem 1.1 represent direct extensions of analogous results obtained by ~[. A. K~ASNOSEL'SKXJ [8] for completely continuous mappings. :Now we consider a,symptotic derivatives. As above we will give first an abstract theorem and then obtain results about asymptotdc derivatives as immediate con- sequences of this theorem. First some notations. Let U={x~EtII~lt>R } and let T: U-~E be such that T(x) = S(x) + R(x), x e U, where S and R satisfy the following conditions i) S(tx)= tS(x), t> o, x e U, iii) lira ]iR(x)l]/llxl] = O. [lz[]--~oo Using tile same arguments of Theorem i.1 we get the following. THEO~]~ 1.2. - Let T: U-~E be a ~-Zipsehitz mapping with constant k~ such that it can be written as a sum o] two mappings S and R satis/ying conditions i) and iii). Then S is also a ~-Lipsehitz mapping with the same constant k. The proof of Theorem 1.2 is Mmost a word by word repetition, with only minor changes, of the proof of Theorem 1.1. As an immediate consequence of Theorem 1.2 we have the following COI~OLLAP~Y 1.2. -- Let T: U->E be a ~-Lipschitz mapping with constant k ad. mitting asymptotic derivative T'(oo). Then T~(oo) is also c~-Zipschitz with con- stant k. 54: JACOPOPEJSACI~0WICZ - AL~0~S0 VIGNOLI: On dif/erentiability, etc. R]~A~K 1.2. - Theorem 1.2 and Corollary 1.2 extend to the context of ~-Lipschitz mapping a result due to 5{. A. KI~ASNOSEL'SKIJ [8] obtained by him for completely continuous mappings. 2. - S-quasibounded mappings and surjectivity. We are going to introduce a new class of mappings and give some surjectivity results for these. Let us first recall a definition due to A. G~A~tS [6]. Let T: E-~ E be a continuous mapping. If the number ITI = lira sup ltT(x)lI is finite then the mapping T is said to be quasibounded and T is called the quasinorm of T. It is evident that any continuous linear mapping L: E-+E is quasibounded and [L]---= IlL]I, where ]ILII is the norm of L. As Granas pointed out in [6] the relation between quasibounded mappings and asymptotically differentiable mappings is given by the following evident fact: let T be asymptotically differentiable and let T~(oo) be its asymptotic derivative then tYt = lIT'(eo)l I. Surjeetivity results for qu~sibounded mappings w~re given in [16] and [17]. Further surjeetivity results for this class Of mappings are due to W. V. PEm~YS~¥~ [12]. Now we introduce the following class of mappings. In what follows~ unless otherwise stated~ E is a real or complex Banaeh space. Let T: E-+ E be continues. Given r > 0 consider the extended real number ~r(T) defined as follows ~r(~) = sup{lZl: ~x= T We need the following result. LE3G.< 2.1. - a) [T]~= [T-Fp]o; b) [~T]o----I)~I[T]o; e) [JlT]: I)~][T];d) 1] T is quasibounded then T is s-quasibounded and [T]< IT]. PI¢ooI~. - a) and b) are evident, e) follows from the relation Finally in order to prove d) observe that II~r(x)ii /I T(x)II ~,(T)< sup-- -< sup ,~,,o, llx]l ,~,,~>, llxll Hence for any p ~ E we have [T]. We are now in a position of proving the following theorem. TtIEOIt.E:~ 2.1. - .Let T: E-+ E be a densifying mapping~ a) If [T]o< 1, then T has a fined point. b) I] [T]~< 1, then p belongs to the image of I-- T. e) If [T] < 1~ then I--T is surjeetive. PzcooF. - Since a) ~ b) ~ e) it is enough to show a). By hypothesis there exists r>0 such that ~,(T)< 1. Let R: E-->B(O,r) be the radial retraction defined us follows x if Ilxl] Since R is ~-nonexpansive (i.e. ~(R(A))<~(A) for any bounded AcE), then R o T is a densifying mapping from B(0, r) into itself and hence it has a fixed point xaB(O,r). )row, if !tT(x)tldr, then x is also a fixed point of T. Suppose that il T(x)il > r, then r. T(x) Z~_.~_-II/(x) ll therefore :T(x) -- Ii _T(x)II x, where II T(x)]I > 1. r 56 JACOPOPEJSACI-IOWICZ - ALFONSO VIGNOLI: On di/]erentiability~ etc. But this contradicts the fact that 8~(T)< 1. Theorem 2.1 represents an extension of a surjectivity theorem for compact quasi- bounded mappings (see G~A~As [6]). This follows at once from the inequality [T] < tTI and from the fact that any compact opera¢or is a-Lipschitz with constant K--0. Let H be a ttilbert space and tet T: tt-+It. If ), is an eigenvalue of T with eigenvector x, then _ Therefore for any p e H we have [T]~ This follows from the fact that lira sup KP, x>! ___ O. Hence we have the following extension of a fixed point theorem due to Krasnoscl'skij. CO~0LLAI~Y 2.1. - Let H be a Hilbert space and let T:H-+H be a densi/ying mapping. I] IJ lira sup l~otice that in any Banach space we have [T]o [T]o This upper boand for [T]o enables us to obtain as corollaries of Theorem 2.1 some well known fixed point theorems. Now we state a more general form of Theorem 2.1. T~o:~v,~ 2.2. - Let 2: E->E be a ~-~ipsehitz mapping with constant K. Then /or any A such that I~] > max(K, [T]} the mapping 2I--2 is sur~eetive. P~o0F. - Since > K the mapping A-~T is ~-contractive. By Lemma 2.1 e) [1-~T]--- [2]-~[T]< 1 and hence from Theorem 2.1 e) it follows that I--A-~T is surjective. Thus AI--T~-)~(1--A-~T) must be surjective. RE~AI~K 2.1. -- We shall give a simple example showing that the strict inequality IT]< IT t may indeed occur. Let T: R2-~R 2 be a rotation of an angle ~=~/2. Simple calculations show that [2]---- 0. On the other hand 1T] = I]TII= 1. 3. - Surjectivity theorems for asymptotically linear mappings. As it is transparent from the definition of the spectral quasinorm its effective computation is very hard to handle even in the case of simple nonlinear mappings. We shall see below that for asymptotically linear mappings the spectral radius of its asymptotic derivative gives a close upper bound for the spectral quasinorm. Given any bounded linear mapping A we denote by @(A)=limttA~ll~/" the spectral radius of A. The following theorem holds. THE0aE:~ 3.1. -- Let 2: E-~ E be an asymptotically linear mapping. Then [2] < d2'(~)). PlC0OF. - Set A = T'(c<)). Assume that [2] > Q(A). Then for some p e E we have [T]~ > y > @(A). Therefore for any r > 0 there exist 1~ and x~ such that IA~I > y and Ilx~1f = r for which t~x~ = T(x~)-~ p. Thus we get (*) Lx~ = 2(x,) + w(x~), where lim sup tlw(x)II _ O . Since Ar > @(A) for any r > 0, ~,I--A are invertible. Denoting by /t(~, A) the 58 J~ (**) r But since i~t > 7 we have that for any r > 0 liR(x, x)!I < ._Zo l~oreover the right hand-side series converges to a number M, because 7 > @(A). Thus from (**) it follows that I ~Vhich contradicts the fact that lim sup fl~,(x) II _ o. ri~:i-*oo flxll The following lemma will be used below in a remark to Theorem 3.1. L~M_~ 3.1. - Let X, Y, Z be Banach spaces. Let T: Y-+Z be a quasibounded mapping which is bounded ou bounded sets and let S: X --> Y be quasibounded. Then T oS: X-->Z is quasibounded and IT oSt<[T[[S[. PR.OOF. -- Given any e > 0 consider r> 0 such that for liyI1)r Since T is bounded on bounded sets there exists a number M such that IIT(y) ]l < M for any jtyll Hence []T°£~(~)II< M + (ITI+ ~S~)IIS(~)ii, for any xeX. Thus limsup llT oS(x)ll ( ~) =ITIISI+~. J~tcoPo PEZSACH0W~CZ - ALFONSO VIGNOLI: On differentiability, etc. 59 Since s is arbitrary the lemma si proved. t~I~K TO THEOI~,EM3.1. -- Let T be a quasibounde4 mapping which is bounded on bounded sets. Consider the sequence {IT~t~/~}n>]. By a standard argument (see YosDA [18]) and using Lemma 3.1 one can show that the above sequence is convergent. Furthermore if T is also asymptotically linear then lira tT~II'~ = lira tIT'~(c~)]l TM = ~(T'(c~)). n--> CO n-~¢:O This follows at once from the fact that tT--T'(c~)I= 0 and (T o S')(c~)= T'(c~) o S'(co) . Now we give seome consequences of Theorem 3.1. CO}~OL~¥ 3.1. - Zet T: E--~E be an asymptotically linear mapping which is ~-Lipschitz with constant k. Then for any A such that 1).] >max{k; ljm IT~l 1~} : = max {k; (T'(c~))} the mapping AI--T is surjeetive. RElgARK 3.1. -- It can be shown that AI--T i salso proper on closed bounded sets. Co]¢o~¥ 3.2. - If T: E->E is a compact quasinilpotent (i.e. lim IT~II'": 0) asymptotically linear mapping then for any ~ v~ 0, ~I--T is surjective. We shall say that a mapping T is finite if IT] = 0. Then we have the following. Co~o~I~¥ 3.3. - Zet G(A; T)~ 21--T, where ~ and T satisfy the conditions of Corollary 3.1. Then for any finite ~-I~ipsehitz mapping C with constant k'< %- -max{k, @(T'(c~))} the perturbed mapping G(A; T)- C is surjeetive. In particular this holds for any /inite compact perturbation. P~ooF. - First, note that the mapping T-~C is a-Lipschitz with constant k+k'. Second, T+C is asymptotically linear having the same asymptotic derivative of T. Indeed, we have ]T ~- C--T'(co)I I~E~.~RK 3.2. -- Corollary 3.3 is a generalized version of a pert~trbation theorem due to :NAsm~D and Wo~G [10]. The following corollary is of independent interest. COI¢OLLA]~Y 3.4. -- f~et E be a Banach space and let S: R+×E-+ E be a continuous semigroup o] densifying asymptotically linear mappings. If lira I#,lu'< 1, t,--~co then the semigroup S has either a fixed point or the set o] ]ixed points o] %1 is unbounded. 60 JAGOPOPEJSACtt0W'£CZ - ALFONSO VI~OLI: On differentiabitity, etc. PI~oo~. - V~e denote by Y~ the set of fixed points of Ss. For any t e R + we have (,) n Itn llu lim IS, [ = lim IS.,l""'= ( ~.->oolim Then by Theorem 2.1 and 3.1 I- ~S~ is surjective and hence F~ is nonempty for all t>O. Assume that /~ is bounded, then E~ must be compact since & is 4ensifying. Now, the family {F~:,}~>, is a family of closed subsets of F~ having the finite intersection property. In fact, given any finite subfamily {tP'I/~ 1 ~ ..,~ -~llln~} set d= n~'n2 ...~. Then F~/d is a nonempty set contained in f-] /~:~. Thus the t=1 intersection of the family is nonempty and hence there exists an x belonging to /~:~ for any natural number n. On the other hand we have that ~q~:, == g~, gild therefore we get that x a Ft for any rational t. Finally, by the continuity of g we obtain that x is g fixed point of St for any t>0. In order to obtain more precise informazion about the range of surjeetivity of a mapping (see Theorem 3.3 below) we have to restrict our attention to a subclass of the class of asymptotically linear operators. Namely, a mapping T: E--->E from a Banach space into itself id said to be strongly asymptotically linear if there exists a bounded linear mapping A: E-+ E such that lim ItT@)--A(,s)I t = O . I1~i[--~¢¢ Clearly any strongly asymptotically linear mapping f is asympto-~ieatly linear and T'(oo) = A. The following lemma will be useful in the sequel. LEPTA 3.2. - .Get T: E-->E be a surjective strongly asymptotically linear mappi~g which is bounded on bounded sets. Then the image of T:(oo) is dense in E. PROOF. - Given any p ~E and s > 0 take r > 0 such that (*) HT(x) - T'(~o) Since T is bounded on bounded sets, there exists M such that ~[T(x)i[ < M if Ilxi[ < r. Let s¢>l be such that ~I!Pll > M. From the surjectivity of T it follows that, there exists x such that T(x)= ~p. Certainly ]lxl] >r and hence by (,) II~p--T'(oo)x][< e. Taking x'=ct-~x, where ~>1 we get I[p-T'(oo)vgIl< s which shows that the image of T'(oo) is dense. JACoI'o PEJ'SACgow~cz - A L~O~SO V]~oI~: On di)~erentiability, etc. 61 I~E~tCK 3.3. -Let T:R-~R be the mapping defined by T(x)--=signx.V'X. Then T is surjective asymptotically linear and T'(c~) ----- 0. This example shows that the assumption of strong asymptotic linearity of T cannot be weakened to asym- ptotic linearity. THEO~E~ 3.2. - Let T: E-~ E, where E is a complex Banach space, be a strongly asymptotically linear ~-Lipschitz mapping with constant k. Then max {k, [r]} = ma {k, PROOF. - From Theorem 3.1 it follows that max(k, IT]} On the other hand if ]A1 >max{k, [T]} then by Theorem 2.2 AI--T is snr- jective. But )d[--T is ~ strongly asymptotically linear mapping which is bounded on bounded sets and hence by Lemma 3.2 (~I--T)'(c~)~--2I--T'(c~) has dense image. Furthermore Corollary 1.2 tells us that T'(c~) is also ~-Lipschitz with the same constant k. Since t)~] > k from a result of 1~. D. NuSS]~A~ [ll-b] it follows that AI--Tt(c~) is a Fredholm mapping of index 0. This together with the dense image implies that 2I~T'(co)is invertible for any ~ such that 121 >max(k, [T]}. Hence max{k, IT]} <~(T'(co)) and the theoremis proved. We have the following. C0~OI~LA~¥ 3.5. - Let T: E--~ E, where E is a complex Banach space, be a strongly asymptotically liq~ear a-Lipschitz mapping with constant k< lina ]T-I TM. Then IT] = ITol-o. I~E~At~K 3.4. -- Corollary 3.5 is false in the case when E is a real Banach space (see l~emark 2.1). Let us denote by r(T) any of the three quantities appearing in Corollary 3.5. If T is compact and E complex then r(T) is in some sense a measure of surjectivity for mappings of the form 2I--T. ~Iore precisely the following theorem holds. TIIEOICE~ 3.3. - Let T: E-* E, where E is a complex Banach space, be a strongly asymptotically linear ~-Lipsvhitz mapping with constant k< r(T). Then a) For any 2 such that ]21 > r(T) the mapping ~I ~ T (and any/inite compact perturbation o] it) is surjevtive (and proper on closed bounded sets). b) For any tt< r(T) there exists a number ), such that ttdI~]dr(T) /or which ~I--T jails to be sur~eetive. 5 - Annali eli Matematica 62 JACOPO PEJSACKOWICZ - ALFONSO VIGNOLI: On differentiability, etc. PROOF. -- a) follows from Corollary 3.3. The argument ~lsed at the end of the proof of Theorem 3.2 shows that if 21- 2" is sllrjec£ive for IiI> k then ). belongs to the resolvent set of T'(c~). Therefore in order to prove b) it suffices to take belonging to the spectmlm of T'(c~) with 1~] >ma.x{#,/~}. Such ~ i always exists. RanSACK 3.5. -- Part a) of Theorem 3.3 holds also if E is a real Banach space. I~EM~K 3.6. - In the second part of this paper we have used the following evident facts: a) T'(c<,) is unique; b) (T + %)'(oo)= T'(oo) + %'(00); c) (1~ o%)'(c~)= T'(~) 0%'(00). I~E)~AI~K 3.7. -- It should be pointed out that there exists another measure of noneompae~ness. Let A c E be any bounded subset. By JE(A) define the ball measure of noncompactness (see L. S. GOLDSmEIN, I. Z. G0Cn-BF~G and A. S. MARKUS [5]) as the infimum of those s > 0 such that A can be covered by u finite number of balls with radii less than e. The results of the present note are valid also for the ball Ineasure of noncompactness. For a detailed discussion of the different abstract measures of noncompactness we refer to B. 57. SADOVSXIJ [14]. 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