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Lipschitz Mappings 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~<t we have 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<p(e')/M we have 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)) <kt~(A) + ~(R(tA)) , JACoPo PEJSACHOWICZ - ALFo~*so VmNOLI: On di]Terentiability~ etc. 53 or e(R(tA)) ~(S(A))<ke(A) -~ t 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.
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