Research Article Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

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1 Inernaional Differenial Equaions Volume 213, Aricle ID 49673, 9 pages hp://dx.doi.org/1.1155/213/49673 Research Aricle Nonlocal Problems for Fracional Differenial Equaions via Resolven Operaors Zhenbin Fan 1 and Gisèle Mophou 2 1 Deparmen of Mahemaics, Changshu Insiue of Technology, Suzhou, Jiangsu 2155, China 2 Laboraoire CEREGMIA, Universié des Anilles e de la Guyane, Campus Fouillole, Guadeloupe (FWI) Poine-à-Pire, France Correspondence should be addressed o Gisèle Mophou; gmophou@univ-ag.fr Received 7 April 213; Acceped 15 May 213 Academic Edior: Soiris Nouyas Copyrigh 213 Z. Fan and G. Mophou. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We discuss he coninuiy of analyic resolven in he uniform operaor opology and hen obain he compacness of Cauchy operaor by means of he analyic resolven mehod. Based on his resul, we derive he exisence of mild soluions for nonlocal fracional differenial equaions when he nonlocal iem is assumed o be Lipschiz coninuous and neiher Lipschiz nor compac, respecively. An example is also given o illusrae our heory. 1. Inroducion In his paper, we are concerned wih he exisence of mild soluions for a fracional differenial equaion wih nonlocal condiions of he form D α u () =Au() +J 1 α f (, u ()), < b, u () =u g(u), where D α is he Capuo fracional derivaive of order α wih <α<1, A:D(A) X Xis he infiniesimal generaor of a resolven S α (),, X is a real Banach space endowed wih he norm,andf and g are appropriae coninuous funcionsobespecifiedlaer. The heory of fracional differenial equaions has received much aenion over he pas weny years, since hey are imporan in describing he naural models such as diffusion processes, sochasic processes, finance, and hydrology. Many noions associaed wih resolven are developed such as inegral resolven, soluion operaors, α-resolven operaor funcions, (a, k)-regularized resolven, and α-order fracional semigroups. All of hese noions play a cenral role in he sudy of Volerra equaions, especially he fracional differenial equaions. Concerning he lieraure, we refer he (1) reader o he books [1, 2], he recen papers [3 2], and he references herein. On he oher hand, absrac differenial equaions wih nonlocal condiions have also been sudied exensively in he lieraure, since i is demonsraed ha he nonlocal problems have beer effecs in applicaions han he classical ones. I was Byszewski and Lakshmikanham [21] who firs sudied he exisence and uniqueness of mild soluions for nonlocal differenial equaions. And he main difficuly in dealing wih he nonlocal problem is how o ge he compacness of soluion operaor a zero, especially when he nonlocal iem is only assumed o be Lipschiz coninuous or coninuous. Many auhors developed differen echniques and mehods o solve his problem. For more deails on his opic, we refer o [1, 11, 22 33] and references herein. In his paper, we combine he above wo direcions and sudy he nonlocal fracional differenial equaion (1) governed by operaor A generaing an analyic resolven. A sandard approach in deriving he mild soluion of (1) is o define he soluion operaor Q. Then, condiions are given such ha some fixed poin heorems such as Browder s and Schauder s fixed poin heorems can be applied o ge a fixed poin for soluion operaor Q, whichgivesriseoa mild soluion of (1). The key sep of using his approach is

2 2 Inernaional Differenial Equaions o prove he compacness of Cauchy operaor G associaed wih soluion operaor Q. When he operaor A generaes a compacsemigroup,iiswellknownhahecauchyoperaor G is also compac. However, o he bes of our knowledge, i is unknown when A generaes a compac resolven. The main difficuly of his problem lies in he fac ha here is no propery of semigroups for resolven. To his end, we will firs discuss he coninuiy of resolven in he uniform operaor opology in his paper. In fac, we prove ha he compac analyic resolven is coninuous in he uniform operaor opology. Based on his resul, we can prove he compacness of Cauchy operaor. As a consequence, we obain he exisence of mild soluions for (1) when he nonlocal iem is Lipschiz coninuous. A he same ime, we also derive he exisence of mild soluions for (1) wihou he Lipschiz or compac assumpion on he nonlocal iem g by using he echniques developed in [24, 3]. Acually, we only assume ha g is coninuous on C([, b], X) and g is compleely deermined on [δ, b] for some small δ>or g is coninuous on C([, b], X) wih L 1 ([, b], X) opology (see Corollaries 17 19). This paper has four secions. In Secion 2, we recall some definiions on Capuo fracional derivaives, analyic resolven,andmildsoluionso(1). In Secion 3, weprove he compacness of Cauchy operaor. Finally, in Secion 4 we esablish he exisence of mild soluions of (1) when he nonlocal iem saisfies differen condiions. An example is also given in his secion. 2. Preinaries Throughou his paper, le b > be fixed, and le N, R, and R + be he se of posiive inegers, real numbers, and nonnegaive real numbers, respecively. We denoe by X he Banach space wih he norm, C([, b], X) he space of all X-valued coninuous funcions on [, b] wih he norm u = sup{ u(), [, b]},andl p ([, b], X) he space of Xvalued Bochner inegrable funcions on [, b] wih he norm f L p =( b f() p d) 1/p,where1 p<. Also, we denoe by L(X) hespaceofboundedlinearoperaorsfromx ino X endowed wih he norm of operaors. Now, le us recall some basic definiions and resuls on fracional derivaive and fracional differenial equaions. Definiion 1 (see [1]). The fracional order inegral of he funcion f L 1 ([, b], X) of order α>is defined by J α 1 f () = Γ (α) ( s) α 1 f (s) ds, (2) where Γ is he Gamma funcion. Definiion 2 (see [1]). The Riemann-Liouville fracional order derivaive of order α of a funcion f L 1 ([, b], X) given on he inerval [, b] is defined by D α L f () = 1 d n Γ (n α) d n ( s) n α 1 f (s) ds, (3) where α (n 1, n], n N. Definiion 3 (see [1]). The Capuo fracional order derivaive of order α of a funcion f L 1 ([, b], X) given on he inerval [, b] is defined by D α 1 f () = Γ (n α) ( s) n α 1 f (n) (s) ds, (4) where α (n 1, n], n N. In he remainder of his paper, we always suppose ha < α<1and A is a closed and densely defined linear operaor on X. Definiion 4. Afamily{S α ()} L(X) of bounded linear operaors in X is called a resolven (or soluion operaor) generaing by A if he following condiions are saisfied: (S1) S α () is srong coninuous on R + and S α () = I; (S2) S α ()D(A) D(A) and AS α ()x = S α ()Ax for all x D(A) and ; (S3) he resolven equaion holds S α () x=x+ g α ( s) AS α (s) xds x D (A),. (5) For ω, θ R,le (ω, θ) := {λ C : arg (λ ω) <θ}. (6) Definiion 5. A resolven S α () is called analyic, if he funcion S α ( ) : R + L(X) admis analyic exension o a secor (, θ ) for some < θ π/2.ananalyic resolven S α () is said o be of analyiciy ype (ω,θ ) if for each θ<θ and ω>ω here is M 1 =M 1 (ω, θ) such ha S(z) M 1 e ω Re z for z (, θ), whererez denoes he real par of z. Definiion 6. AresolvenS α () is called compac for >if for every >, S α () is a compac operaor. According o Proposiion 1.2 in [2], we can give he following definiion of mild soluions for (1). Definiion 7. Afuncionu C([, b], X) is called a mild soluion of fracional evoluion equaion (1) if i saisfies u () =S α () [u g(u)] for every u X. + S α ( s) f (s,u(s)) ds, b Nex, we inroduce he Hausdorff measure of noncompacness β( ) defined on each bounded subse Ω of Banach space Y by (7) β (Ω) = inf {ε >; Ωhas a finie ε-ne in Y}. (8) Some basic properies of β( ) are given in he following lemma.

3 Inernaional Differenial Equaions 3 Lemma 8 (see [34]). Le Y be a real Banach space and le B, C Y be bounded; hen he following properies are saisfied: (1) B is precompac if and only if β(b) = ; (2) β(b) = β(b) = β(conv B),whereB and conv B mean heclosureandconvexhullofb,respecively; (3) β(b) β(c) when B C; (4) β(b + C) β(b) + β(c), whereb+c={x+y;x B, y C}; (5) β(b C) max{β(b), β(c)}; (6) β(λb) = λ β(b) for any λ R; (7) if he map Q:D(Q) Y Zis Lipschiz coninuous wih consan k,henβ(qb) kβ(b) for any bounded subse B D(Q),whereZ is a Banach space. The map Q:W Y Yis said o be a β-conracion if here exiss a posiive consan k<1such ha β(qc) kβ(c) for any bounded closed subse C W,whereY is a Banach space. Lemma 9 ([34], Darbo-Sadovskii). If W Y is bounded closed and convex, he coninuous map Q : W W is a β-conracion, hen he map Q has a leas one fixed poin in W. 3. Compacness of Cauchy Operaors Le Cauchy operaor G : C([, b], X) C([, b], X) be defined by (Gf) () = S α ( s) f (s) ds, [, b]. (9) If S α () is a compac C -semigroup, i is well known ha G is compac. However, i is unknown in case of compac resolven. The main difficuly is ha he resolven does no have he propery of semigroups. Thus, i seems o be more complicaed o prove he compacness of Cauchy operaor. Here, we will firs discuss he coninuiy of resolven in he uniform operaor opology. Then, we can give he posiive answer o he above problem. In he remainder of his paper, we always assume ha M=sup [,b] S α () < +. Lemma 1. Suppose S α () is a compac analyic resolven of analyiciy ype (ω,θ ). Then he following are hold: (i) h S α ( + h) S α () = for >; (ii) h + S α ( + h) S α (h)s α () = for >; (iii) h + S α () S α (h)s α ( h) = for >. Proof. (i) Le S α () be an analyic resolven of analyiciy ype (ω,θ ), and le >be given. Then, by means of Cauchy inegral formula, we have S α () = 1 2πi z = sin θ S α (z) dz. 2 (1) (z ) Thus, for any ω > ω, < θ < θ, here exiss a consan M 1 =M 1 (ω, θ) such ha S α () M 1(2π) 1 π e ω(+ sin θ cos γ) ( sin θ) 1 dγ π M 1 e 2ω ( sin θ) 1. (11) Now, le x 1 and h <. Ifollowsfrom(11) hahere exiss a consan M such ha S α (+h) x S α () x = +h M x S α (s) x ds +h s 1 ds M ln (+h) ln as h, (12) which implies ha S α () is coninuous in he uniform operaor opology for >,hais, h S α (+h) S α () = for >. (13) (ii) Le x 1, >,andε > be given. Since S α () is compac, he se W := {S α ()x : x 1} is also compac. Thus, here exiss a finie family {S α ()x 1, S α ()x 2,...,S α ()x m } W such ha for any x wih x 1, here exiss x i (1 i m)suchha S α () x S α () x i ε 3 (M+1). (14) From he srong coninuiy of S α (),, here exiss < h 1 < min{, b} such ha S α () x i S α (h) S α () x i ε 3, h h 1,1 i m. (15) On he oher hand, from (i), here exiss <h 2 < min{, b} such ha S α (+h) x S α () x ε 3, h h 2, x 1. (16)

4 4 Inernaional Differenial Equaions Thus, for h min{h 1,h 2 } and x 1, ifollowsfrom (14) (16)ha S α (+h) x S α (h) S α () x S α (+h) x S α () x + S α () x S α () x i Firs, we claim ha he se GD l is equiconinuous on C([, b], X). In fac, le 1 2 band f D l ;hen we have I:= (Gf) ( 2) (Gf)( 1 ) = 2 1 S α ( 2 s)f(s) ds S α ( 1 s)f(s) ds + S α () x i S α (h) S α () x i + S α (h) S α () x i S α (h) S α () x S α (+h) x S α () x (17) 1 S α ( 2 s) S α ( 1 s) f (s) ds 2 + S α ( 2 s)f(s) ds 1 (2) + (M+1) S α () x S α () x i + S α () x i S α (h) S α () x i 1 l S α ( 2 s) S α ( 1 s) ds+ml( 2 1 ). ε 3 + ε 3 + ε 3 If 1 =,iiseasyoseeha ε, 2 I=, uniformly for f D l. (21) which implies ha h + S α ( + h) S α (h)s α () = for all >. (iii) Le >and <h<min{, b}. Then, here exiss M>such ha If < 1 <b,for<δ< 1,wehave 1 δ I l S α ( 2 s) S α ( 1 s) ds S α () S α (h) S α ( h) S α () S α (+h) + S α (+h) S α (h) S α () + S α (h) S α () S α (h) S α ( h) (18) bl 1 +l S α ( 2 s) S α ( 1 s) 1 δ ds+ml( 2 1 ) sup s [, 1 δ] S α ( 2 s) S α ( 1 s) S α () S α (+h) + S α (+h) S α (h) S α () +M S α () S α ( h). I follows from (i) and (ii) ha h + S α () S α (h) S α ( h) = for >. (19) This complees he proof. Lemma 11. Suppose S α () is a compac analyic resolven of analyiciy ype (ω,θ ). Then Cauchy operaor G defined by (9) is a compac operaor. Proof. We will show G : C([, b], X) C([, b], X) is compac by using he Arzela-Ascoli heorem. Le D l = {f C([, b], X) : f l} be any bounded subse of C([, b], X). + 2Mlδ + Ml ( 2 1 ). (22) Noe ha from Lemma 1(i), we know S α () is operaor norm coninuous uniformly for [δ,b].combining his and he arbirariness of δ wih he above esimaion on I,wecan conclude ha 1 2 I=, uniformly for f D l. (23) Thus, GD l is equiconinuous on C([, b], X). Nex, we will show ha he se {(Gf)() : f D l } is precompac in X for every [,b].iiseasyoseehahe se {(Gf)() : f D l } is precompac in X.Now,le< b be given and <ε<.then {S α (ε) S α ( s ε) f (s) ds :f D l } (24)

5 Inernaional Differenial Equaions 5 is precompac since S α (ε) is compac. Moreover, for arbirary ε<δ<,wehave S α (ε) S α ( s ε) f (s) ds S α ( s) f (s) ds l S α (ε) S α ( s ε) S α ( s) ds δ l S α (ε) S α ( s ε) S α ( s) ds +l S α (ε) S α ( s ε) S α ( s) δ ds δ l S α (ε) S α ( s ε) S α ( s) ds+lδ(m2 +M). (25) From Lemma 1(iii), we know S α (ε) S α ( s ε) S α ( s), (26) as ε for s [, δ]. Then, i follows from he Lebesgue dominaed convergence heorem and he arbirariness of δ ha ε S α (ε) On he oher hand, S α (ε) Thus, S α ( s ε) f (s) ds S α ( s) f (s) ds =. S α ( s ε) f (s) ds S α ( s) f (s) ds S α (ε) S α ( s ε) f (s) ds + S α ( s) f (s) ds S α ( s) f (s) ds S α ( s) f (s) ds S α (ε) S α ( s ε) f (s) ds (27) S α ( s) f (s) ds +lεm. (28) ε S α (ε) S α ( s ε) f (s) ds S α ( s) f (s) ds =, (29) which implies ha {(Gf)() : f D l } is precompac in X by using he oal boundedness, and herefore G is compac in view of Arzela-Ascoli heorem. 4. Nonlocal Problems In his secion, we always assume ha u Xand ha he operaor A generaes a compac analyic resolven S α () of analyiciy ype (ω,θ ), and we will prove he exisence of mild soluions of (1) when he nonlocal iem g is assumed o be Lipschiz coninuous and neiher Lipschiz nor compac, respecively. Le r be a fixed posiive real number and W r := {u C([, b],x) : u r}. (3) Clearly, W r isaboundedclosedandconvexse.wemakehe following assumpions. (H1) f:[,b] X Xis coninuous. (H2) g : C([, b], X) X is Lipschiz coninuous wih Lipschiz consan k such ha Mk < 1. Under hese assumpions, we can prove he firs main resul in his paper. Theorem 12. Assume ha condiions (H1) and (H2) are saisfied. Then he nonlocal problem (1) has a leas one mild soluion provided ha M[ u + sup g (u) +b sup f (s,u(s)) ] r. u W r s [,b],u W r (31) Proof. We consider he soluion operaor Q : C([, b], X) C([, b], X) defined by (Qu)() =S α () [u g(u)]+ S α ( s) f (s,u(s)) ds := (Q 1 u) () +(Q 2 u) (), [, b]. (32) I is easy o see ha he fixed poin of Q is he mild soluion of nonlocal Cauchy problem (1). Subsequenly, we will prove ha Q has a fixed poin by using Lemma 9 (Darbo-Sadovskii s fixed poin heorem). Firsly, we prove ha he mapping Q is coninuous on C([, b], X). For his purpose, le {u n } n 1 be a sequence in C([, b], X) wih n u n =uin C([, b], X).Then Qu n Qu M[ g(u n) g(u) +bsup f(s,u n (s)) f(s,u(s)) ]. s [,b] (33) By he coninuiy of f and g,wededucehaq is coninuous on C([, b], X).

6 6 Inernaional Differenial Equaions Secondly,weclaimhaQW r W r,wherew r is defined by (3). In fac, for any u W r,by(31), we have b Qu M( u + g (u) + f (s,u(s)) ds) M( u + sup g (u) +b sup f (s,u(s)) ) u W r s [,b],u W r r, (34) which implies ha Q maps W r ino iself. Now, according o Lemma 9,iremainsoprovehaQ is a β-conracion in W r. From condiion (H2), we ge ha Q 1 : W r C([, b], X) is Lipschiz coninuous wih consan Mk. In fac, for u, V W r,by(h2),wehave Q 1u Q 1 V M g (u) g(v) Mk u V. (35) Thus, i follows from Lemma 8-(3) ha β(q 1 W r ) Mkβ(W r ). For operaor Q 2 :W r C([, b], X), wehaveq 2 u= Gf u,wheref u ( ) = f(, u( )) is coninuous on [, b] and G is he Cauchy operaor defined by (9). Thus, in view of Lemma 11, weknowhaq 2 is compac on C([, b], X), and hence β(q 2 W r )=.Consequenly, β(qw r ) β(q 1 W r )+β(q 2 W r ) Mkβ(W r ). (36) Since Mk<1,hemappingQ is a β-conracion on W r.by Darbo-Sadovskii s fixed poin heorem, he operaor Q has a fixed poin in W r,whichisjushemildsoluionofnonlocal Cauchy problem (1). Now, we give he following echnical condiion on funcion g. (H) g : C([, b], X) X is coninuous, and he se g(convqw r ) is precompac, where convb denoes he convex closed hull of se B C([, T], X), andq is given by (32). Remark 13. I is easy o see ha condiion (H) is weaker han he compacness and convexiy of g. The same hypohesis canbeseenfrom[24, 3], where he auhors considered he exisence of mild soluions for semilinear nonlocal problems of ineger order when A is a linear, densely defined operaor on X which generaes a C -semigroup. Afer he proof of our main resuls, we will give some special ypes of nonlocal iem g which is neiher Lipschiz nor compac, bu saisfies he condiion (H) in he nex Corollaries. Theorem 14. Assume ha condiions (H1) and (H) are saisfied. Then he nonlocal problem (1) hasaleasonemild soluion provided ha (31) holds. Proof. We will prove ha Q has a fixed poin by using Schauder s fixed poin heorem. According o he proof of Theorem 12, wehaveprovenhaq : W r W r is coninuous. Nex, we will prove ha here exiss a se W W r such ha Q:W Wis compac. For his purpose, le < band <δ<. I is easy o see ha he se {S α (δ) S α ( δ) [u g(u)] :u W r } (37) is precompac since S α (δ) is compac. On he oher hand, by Lemma 1(iii), we obain δ S α () [u g(u)] S α (δ) S α ( δ) [u g(u)] =, u W r, (38) which implies ha Q 1 W r () is precompac in X by using he oal boundedness. Nex, we claim Q 1 W r is equiconinuous on [η, b] for any small posiive number η. In fac, for u W r and η 1 2 b,wehave (Q 1u) ( 2 ) (Q 1 u) ( 1 ) S α ( 2 ) S α ( 1 ) ( u + g (u) ). (39) By Lemma 1(i), S α () is operaor norm coninuous for >. ThusS α () is operaor norm coninuous uniformly for [η, b], and hence 2 1 (Q 1u) ( 2 ) (Q 1 u) ( 1 ) =, uniformly for u W r, (4) ha is, Q 1 W r is equiconinuous on [η, b]. NoehaQ 2 is compac by Lemma 11. Therefore, we have proven ha QW r () is precompac for every (,b]and QW r is equiconinuous on [η, b] for any small posiive number η. Now, le W = convqw r,wegehaw is a bounded closed and convex subse of C([, b], X) and QW W. Iis easy o see ha QW() is precompac in X for every (, b] and QW is equiconinuous on [η, b] for any small posiive number η. Moreover, we have ha g(w) = g(convqw r ) is precompac due o condiion (H). Thus, we can now claim ha Q:W Wis compac. In fac,iiseasyoseehaq 1 W() = u g(w) is precompac since g(w) = g(convqw r ) is precompac. I remains o prove ha Q 1 W is equiconinuous on [, b].tohisend,leu W and 1 < 2 b;henwehave (Q 1u)( 2 ) (Q 1 u)( 1 ) S α ( 2 ) S α ( 1 ) u g(u). (41) In view of he compacness of g(w) and he srong coninuiy of S α () on [, b], we obain he equiconinuous of Q 1 W on [, b]. Thus,Q 1 : W C([, b], X) is compac by Arzela- Ascoli heorem, and hence Q:W Wis also compac. Now, Schauder s fixed poin heorem implies ha Q has a fixed poin on W, which gives rise o a mild soluion of nonlocal problem (1). The following heorem is a direc consequence of Theorem 14.

7 Inernaional Differenial Equaions 7 Theorem 15. Assume ha condiions (H1) and (H) are saisfied for each r>.if g (u) u f (, x) x, u,, x (42) uniformly for [,b], hen he nonlocal problem (1) has a leas one mild soluion. Remark 16. I is easy o see ha if here exis consans L 1, L 2 >and γ 1, γ 2 [, 1) such ha g (u) L 1(1+ u ) γ 1, f (, x) L 2(1+ x ) γ 2 for [,b], hen condiions (42) are saisfied. (43) Nex, we will give special ypes of nonlocal iem g which is neiher Lipschiz nor compac, bu saisfies condiion (H). We give he following assumpions. (H3) g : C([, b], X) X is a coninuous mapping which maps W r ino a bounded se, and here is a δ=δ(r) (, b) such ha g(u) = g(v) for any u, V W r wih u(s) = V(s), s [δ,b]. (H4) g : (C([, b], X), L 1) Xis coninuous. Corollary 17. Assume ha condiions (H1) and (H3) are saisfied. Then he nonlocal problem (1) has a leas one mild soluion on [, b] provided ha (31) holds. Proof. Le (QW r ) δ = {u C([, b],x) ;u() = V () for [δ, b], u () =u(δ) for [, δ), where V QW r }. (44) From he proof of Theorem 14, weknowha(qw r ) δ is precompac in C([, b], X). Moreover,bycondiion(H3), g(convqw r ) = g(conv(qw r ) δ ) is also precompac in C([, b], X). Thus, all he hypoheses in Theorem 14 are saisfied. Therefore, here is a leas one mild soluion of nonlocal problem (1). Corollary 18. Le condiion (H1) be saisfied. Suppose ha g(u) = p j=1 c ju( j ),wherec j are given posiive consans, and < 1 < 2 < < p b. Then he nonlocal problem (1) has a leas one mild soluion on [, b] provided ha p M [ u + c j r+b sup f (s,u(s)) ] r. (45) j=1 s [,b],u W [ r ] Proof. I is easy o see ha he mapping g wih g(u) = p j=1 c ju( j ) saisfies condiion (H3). And all he condiions in Corollary 17 aresaisfied.soheconclusionholds. Corollary 19. Assume ha condiions (H1) and (H4) are saisfied. Then he nonlocal problem (1) has a leas one mild soluion on [, b] provided ha (31) holds. Proof. According o Theorem 14, iissufficienoproveha he hypohesis (H) is saisfied. For arbirary ε>, here exiss <δ<bsuch ha δ u(s) ds <εfor all u QW r.le (QW r ) δ = {u C([, b],x) ;u() = V () for [δ, b], u () =u(δ) for [, δ), where V QW r }. (46) From he proof of Theorem 14, weknowha(qw r ) δ is precompac in C([, b], X), which implies ha (QW r ) δ is precompac in L 1 ([, b], X). Thus,QW r is precompac in L 1 ([, b], X) as i has an ε-ne (QW r ) δ. By condiion g : (C([, b], X), L 1) X is coninuous and convqw r (L)convQW r, i follows ha condiion (H) is saisfied, where (L)convB denoes he convex and closed hull of B in L 1 ([, b], X). Therefore, he nonlocal problem (1) has a leas one mild soluion on [, b]. Finally, we give a simple example o illusrae our heory. Example 2. Consider he following fracional parial differenial hea equaion in R: D α w (, x) = 2 w (, x) x 2 +J 1 α F (, w (, x)), <x<1,<<1, w (, ) =w(, 1), w x (, ) =w x (, 1), 1, w (, x) =w (x) p j=1 c j w( j,x), <x<1,< j 1, (47) where < α < 1 and D α denoes he Capuo fracional derivaive. We inroduce he absrac frame as follows. Le X={u C([, 1], R) : u() = u(1)}. LeA be he linear operaor in X defined by D(A) = {u X : u,u X} and for u D(A), Au = u.then,iiswellknownhaa generaes a compac C semigroup T() for >on X.From he subordinaion principle [6, Theorems 3.1 and 3.3], A also generaes a compac resolven S α () for >. Assume ha f : [,1] X X is a coninuous funcion defined by f(, z)(x) = F(, z(x)), x 1and g : C([, 1], X) X is also a coninuous funcion defined by g(u)(x) = w (x) p j=1 c ju( j )(x),, x 1,where u()(x) := w(, x).

8 8 Inernaional Differenial Equaions Under hese assumpions, he fracional parial differenial hea equaion (47) can be reformulaed as he absrac problem (1). If heinequaliy p M [ w + c j r+ sup f (s,u(s)) ] r (48) j=1 s [,1],u W [ r ] holds for some consan r>, here exiss a leas one mild soluion for fracional equaion (47)in view of Corollary 18. Acknowledgmens This work was compleed when he firs auhor visied he Pennsylvania Sae Universiy. The auhor is graeful o Professor Albero Bressan and he Deparmen of Mahemaics for heir hospialiy and providing good working condiions. The work was suppored by he NSF of China (11134) and Jiangsu Overseas Research & Training Program for Universiy Prominen Young & Middle-aged Teachers and Presidens. References [1] I. Podlubny, Fracional Differenial Equaions,vol.198ofMahemaics in Science and Engineering, Academic Press, San Diego, Calif, USA, [2] J. 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9 Inernaional Differenial Equaions 9 [29] J. Liang and Z. Fan, Nonlocal impulsive Cauchy problems for evoluion equaions, Advances in Difference Equaions,vol. 211, Aricle ID , 17 pages, 211. [3] X. M. Xue, Exisence of semilinear differenial equaions wih nonlocal iniial condiions, Aca Mahemaica Sinica, vol.23, no. 6, pp , 27. [31] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fracional evoluion equaions, Nonlinear Analysis,vol.11,no.5,pp , 21. [32] L. Zhu and G. Li, Exisence resuls of semilinear differenial equaions wih nonlocal iniial condiions in Banach spaces, Nonlinear Analysis,vol.74,no.15,pp ,211. [33] L. Zhu, Q. Huang, and G. Li, Exisence and asympoic properies of soluions of nonlinear mulivalued differenial inclusions wih nonlocal condiions, Mahemaical Analysis and Applicaions,vol.39,no.2,pp ,212. [34] J. BanaśandK. Goebel, Measures of Noncompacness in Banach Spaces,vol.6ofLecure Noes in Pure and Applied Mahemaics, Marcel Dekker, New York, NY, USA, 198.

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