EXISTENCE OF MILD SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS.

Transcription

1 Joural of Fractioal Calculus ad Applicatios, Vol. 2. Ja. 22, No., pp.. ISSN: EXISTENCE OF MILD SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS. ZUFENG ZHANG, BIN LIU Abstract. I this article, we establish sufficiet coditios for the existece of mild solutios for fractioal evolutio differetial equatios by usig a ew fixed poit theorem. The results obtaied here improve ad geeralize may kow results. A example is also give to illustrate our results.. Itroductio Our aim i this paper is to study the olocal iitial value problem { D q xt) = Axt) + ft, xt)), t [, ], x) = gx), where D q is the Caputo fractioal derivative of order < q <, A is the ifiitesimal geerator of a strogly cotiuous semigroup of bouded liear operator i.e. C -semigroup) T t) i Baach space X, f : [, ] X X ad g : C[, ]; X) X are appropriate fuctios to be specified later. Fractioal differetial equatios have appeared i may braches of physics, ecoomics ad techical scieces [, 2]. There has bee a cosiderable developmet i fractioal differetial equatios i the last decades. Recetly, May authors are iterested i the existece of mild solutios for fractioal evolutio equatios. I [3], El-Borai discussed the followig equatio i Baach X, { D α ut) = Aut) + Bt)ut), u) = u, where A geerates a aalytic semigroup ad the solutio was give i terms of some probability desities. I [4], Zhou ad Jiao cocered the existece ad uiqueess of mild solutios for fractioal evolutio equatios by some fixed poit theorems. Cao et al. [5] studied the α-mild solutios for a class of fractioal evolutio equatios ad optimal cotrols i fractioal powder space. For more 2 Mathematics Subject Classificatio. 26A33, 47J35. Key words ad phrases. Existece, Fractioal evolutio equatios, Mild solutio, Measure of ocompactess. Correspodig author. Submitted Oct. 9, 2. Published March, 22. Research supported by NNSF of Chia 722) ad Uiversity Sciece Foudatio of Ahui Provice KJ22A265) ad KJ22B87). )

2 2 ZUFENG ZHANG, BIN LIU JFCA-22/2 iformatio o this subjects, the readers may refer to [6]-[] ad the refereces therei. Very recetly, Zhu [] used the measure of ocompactess to discuss problem ) whe q =. Motivated by this paper we cotiue to study the existece of mild solutios for problem ) with a fixed poit theorem related to the measure of ocompactess which is firstly used to deal with fractioal evolutio equatios. We obtai the existece results without the compactess o T t) which are differet from may existig papers such as [4, 6, 7]. The rest of the paper will be orgaized as follows. I sectio 2 we will recall some basic defiitios ad lemmas from the measure of ocompactess, fractioal derivatio ad itegratio. Sectio 3 is devoted to the existece results for problem ). We shall preset i Sectio 4 a example which illustrates our mai theorems. 2. Prelimiaries I this sectio, we itroduce otatios, defiitios, ad prelimiary results which are used i the rest of the paper. Throughout this paper, we deote by R + ad N the set of positive real umbers ad the set of positive itegers. Let X, ) be a real Baach space. We deote by C[, ]; X) the space of X-valued cotiuous fuctios o [, ] with the x = sup{xt) : t [, ]}. Let L p [, ]; X) be the space of X-valued Bocher fuctio o [,] with the orm x L p = xs)p ds) p, p <. Defiitio 2. [2]). The Riema-Liouville fractioal itegral of order q R + of a fuctio f : R + X is defied by I q ft) = Γq) t s) q fs)ds, t >, provided the right-had side is poitwise defied o R +, where Γ is the gamma fuctio. Defiitio 2.2 [2]). The Caputo fractioal derivative of order < q < of a fuctio f : C R + ; X) is defied by D q ft) = Γ q) t s) q f s)ds, t >. Let α defie the Hausdorff measure of ocompactess o both X ad C[, ]; X). To prove our results we eed the followig lemmas. Lemma 2.3 [2]). If W C[, ]; X) is bouded, the αw t)) αw ) for every t [, ], where W t) = {xt); x W }. Furthermore if W is equicotiuous o [, ], the αw t)) is cotiuous o [, ] ad αw ) = sup{αw t)); t [, ]}. Lemma 2.4 [3]). If {u } = L [, ]; X) is uiformly itegrable, the α{u } =) is measurable ad { } ) α u s)ds 2 α{u s)} =)ds. = Lemma 2.5 [4]). If W is bouded, the for each ϵ >, there is a sequece {u } = W such that αw ) 2α{u } =) + ϵ.

3 JFCA-22/2 MILD SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS 3 Lemma 2.6 [5]). Suppose that x, the x e )x 2πx + 2x ) < Γx + ) < x e )x 2πx + 2x.5 ). Lemma 2.7 [6] Fixed Poit Theorem). Let G be a closed ad covex subset of a real Baach space X, let A : G G be a cotiuous operator ad AG) be bouded. For each bouded subset B G, set A B) = AB), A B) = AcoA B))), = 2, 3,..., if there exist a costat k < ad a positive iteger such that for each bouded subset B G, αa B)) kαb), the A has a fixed poit i G. 3. Mai results I this sectio we will establish the existece results by usig the Hausdorff measure of ocompactess. Based o referece [6], we give the defiitio of the mild solutios of problem ) as follows. Defiitio 3.. By the mild solutio of problem ), we mea that the fuctio x C[, ]; X) which satisfies where xt) = St)gx) + St) = Ψ q θ) = π Remark 3.2 [6]). t s) q Tt s)fs, xs))ds, t [, ], ξ q θ)t t q θ)dθ, Tt) = q ξ q θ) = q θ q Ψq θ q ), θξ q θ)t t q θ)dθ, 2) ) q Γq + ) θ siπq), θ R +.! = ξ q θ) is the probability desity fuctio defied o R + ad θξ q θ)dθ = θ q Ψ qθ)dθ = Γ + q). To state ad prove our mai results for the existece of mild solutios of problem ), we eed the followig hypotheses: H) The C -semigroup {T t)} t geerated by A is equicotiuous ad M = sup{t t); t [, )} < +. H2) The fuctio g : C[, ]; X) X is completely cotiuous, moreover there exist positive costats c ad d such that gx) cx + d, for every x C[, ]; X). H3) The fuctio f : [, ] X X satisfies the Carathéodory type coditios, i.e. ft, ) : X X is cotiuous for a.e. t [, ] ad f, x) : [, ] X is strogly measurable for each x C[, ], X). H4) There exist a fuctio m L p [, ]; R + ), < p < q ad a odecreasig cotiuous fuctio Ω : R + R + such that ft, x) mt)ωx) for all x X

4 4 ZUFENG ZHANG, BIN LIU JFCA-22/2 ad a.e. t [, ]. H5) There exists L L [, ]; R + ) such that for each bouded D X, αft, D)) Lt)αD), for a.e. t [, ]. Remark 3.3. i) If A geerates a aalytic semigroup or a differetiable semigroup {T t)} t, the {T t)} t is a equicotiuous see [8]). ii) If ft, x) ft, y) Lt)x y, Lt) L [, ]; R + ), x, y X, the we ca get αft, D)) Lt)αD) for each bouded D X ad a.e. t [, ] see []). For each positive costat r, let B r = {x C[, ], X); x r}, the B r is clearly a bouded closed ad covex subset i C[, ], X). Lemma 3.4. Assume that hypotheses H)-H4) hold, the i) For ay fixed t, St) ad Tt) defied i 2) are liear ad bouded operators, i.e. for ay x X, St)x Mx, Tt)x M Γq) x. ii) St) ad Tt) are strogly cotiuous. iii) The set {t t s)q Tt s)fs, xs))ds; x B r } is equicotiuous o [, ]. Proof. i) ad ii) were give i [6], we oly check iii) as follows. For x B r, t < t 2, we have = = 2 t 2 s) q Tt 2 s)fs, xs))ds t s) q Tt s)fs, xs))ds 2 q θt 2 s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds q θt s) q ξ q θ)t t s) q θ)fs, xs))dθds 2 q θt 2 s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds t + q θt 2 s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds q θt s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds + q θt s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds q θt s) q ξ q θ)t t s) q θ)fs, xs))dθds 2 q θt 2 s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds t + q θ[t 2 s) q t s) q ]ξ q θ)t t 2 s) q θ)fs, xs))dθds + q θt s) q ξ q θ)[t t 2 s) q θ) T t s) q θ)]fs, xs))dθds

5 JFCA-22/2 MILD SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS 5 = qi + I 2 + I 3 ), where I = I 2 = I 3 = 2 t From hypothesis H4), we have I MΩr) Γ + q) θt 2 s) q ξ q θ)t t 2 s) q θ)fs, xs))dθds, θ[t 2 s) q t s) q ]ξ q θ)t t 2 s) q θ)fs, xs))dθds, θt s) q ξ q θ)[t t 2 s) q θ) T t s) q θ)]fs, xs))dθds. 2 t t 2 s) q ms) ds MΩr) Γ + q) + η) p t 2 t ) +η) p) m L p, I 2 MΩr) Γ + q) MΩr)m L p Γ + q) t s) q t 2 s) q ) p ds ) p m L p ) p t s) η t 2 s) η )ds = MΩr)m L p t+η Γ + q) + η) p t +η 2 + t 2 t ) +η ) p MΩr)m L p Γ + q) + η) p t 2 t ) +η) p), where η = q p, ). Hece lim t 2 t I = ad lim t2 t I 2 =. O the other had, from H) ad the Lebesgue domiated covergece theorem, we get lim I 3 lim t 2 t t 2 t θt s) q ξ q θ)t t 2 s) q θ)fs, xs)) T t s) q θfs, xs))dθds =. θt s) q ξ q θ) lim t 2 t T t 2 s) q θ)fs, xs)) T t s) q θ)fs, xs))dθds Hece, 2 t 2 s) q Tt 2 s)fs, xs))ds t s) q Tt s)fs, xs))ds idepedetly of x B r as t 2 t. This completes the proof. Lemma 3.5. Suppose that < a <, b > are two fixed costats, let S = a +C a b a 2 b 2 ab b ) Γq + ) +C2 + +C Γ2q + ) Γ )q + ) +C. Γq + ) The, lim S =. Proof. Sice < a <, there exists a costat b > with a + b <.

6 6 ZUFENG ZHANG, BIN LIU JFCA-22/2 From < q <, we kow that there exists N such that, if > the q >. By Lemma 2.6 if >, the ) q ) q q q Γq + ) > 2πq >. e e Therefore, for >, we have Γq + ) < q e )q ). b O the other had, there exists 2 N such that < b for each > q 2. e )q Set 3 = max{, 2 }, for > 3, we divide S ito two parts where S = S + S, S = a + C a b Γq + ) + a 2 b 2 C2 Γ2q + ) + + a 3 b 3 C 3 Γ 3 q + ), S = C a 3+ 3 b 3+ Γ 3 + )q + ) + a C b 3+2 Γ 3 + 2)q + ) + + b C Γq + ). For > 3, we have = C a 3+ 3 b 3+ Γ 3 + )q + ) + a C b 3+2 Γ 3 + 2)q + ) + + b C Γq + ) C a 3+ 3 b 3+ a 3 2 b 3+2 b S 3+)q e ) q ) + 3+ C )q e ) q ) C C 3+ a 3 b 3+ + C 3+2 a 3 2 b Cb a + b). q e )q ) I view of a + b <, we have lim + S =. Sice lim + S = is obvious, we obtai lim + S =. The proof is completed. Theorem 3.6. If hypotheses H)-H5) are satisfied, the there is at least oe mild solutio for problem ) provided that there exists a costat r such that where η = q p Mcr + d) + MΩr) + η) p Γq) m L p is defied i the proof of Lemma 3.4. Proof. Defie operator F : C[; ], X) C[, ]; X) by F x)t) = St)gx) + r, 3) t s) q Tt s)fs, xs))ds, t [, ]. We ca easily show that F is cotiuous by the usual techiques see [4]). For ay x B r, we have F x)t) St)gx) + t s) q Tt s)fs, xs))ds = ξ q θ)t t q θ)gx)dθ + q t s) q θξ q θ)t t s) q θ)dθfs, xs))ds

7 JFCA-22/2 MILD SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS 7 Mcr + d) + MΩr) Γq) Mcr + d) + MΩr) Mcr + d) + t s) q ms)ds Γq) MΩr) + η) p Γq) m L p. t s) q p ds ) p m L p The from 3) we get F x r which meas that F : B r B r is a bouded operator. Let B = cof B r. By Lemma 2.5 ad the coditio gx) is compact, we get for ay B B ad ϵ >, there is a sequece {x } = B such that αf Bt)) = αf Bt)) ) 2α t s) q Tt s)fs, {x } =)ds +ϵ 4 t s) q αtt s)fs, {x } =))ds + ϵ 4M t s) q Ls)α{x } Γq) =)ds + ϵ 4M Γq) αb) t s) q Ls)ds + ϵ. From the fact that there is a cotiuous fuctio ϕ : [, ] R + such that for ay γ >, t s) q Ls) ϕs) ds < γ. We choose γ < Γq) 4M ad let M = max{ ϕt) : t [, ]}, the αf Bt)) 4M [ ] Γq) αb) t s) q Ls) ϕs) ds + t s) q ϕs) ds +ϵ 4M ) γ Γq) αb) + Mtq +ϵ. q From ϵ > is arbitrary, it follows that αf Bt)) a + b Γq + ) tq )αb), where a = 4Mγ Γq), b = 4MM. From Lemma 2.5, we kow for ay ϵ >, there exists a sequece {y } = cof B) such that αf 2 Bt)) = αf cof Bt))) ) 2α t s) q Tt s)fs, {y } =)ds +ϵ 4 t s) q αtt s)fs, {y } =))ds + ϵ 4M t s) q Ls)αF Bs))ds + ϵ Γq) 4M Γq) αb) [t s) q b Ls) ϕs) + ϕs) ]a + Γq + ) sq )ds + ϵ

8 8 ZUFENG ZHANG, BIN LIU JFCA-22/2 4M [a Γq) αb) + btq Γq + ) ) t s) q Ls) ϕs) ds t ] +M t s) q b a + Γq + ) sq )ds +ϵ a 2 bt q + 2a Γq + ) + b2 t 2q ) αb) + ϵ. Γ2q + ) From ϵ > is arbitrary, it follows that αf 2 Bt)) a 2 bt q + 2a Γq + ) + b2 t 2q ) αb). Γ2q + ) By the method of mathematical iductio, for ay positive iteger ad t [, ], we obtai αf Bt)) a + C a bt q Γq + ) + C2 a 2 b 2 t 2q Γ2q + ) + ) αb). +C a b t )q Γ )q + ) + C +C b t q Γq + ) Therefore, by Lemma 3.4 ad Lemma 2.3, we get αf B) a + Ca b Γq + ) + C2 a 2 b 2 Γ2q + ) + b b ) αb). a Γ )q + ) + C Γq + ) The from Lemma 3.4, there exists a positive iteger such that a + C a b Γq + ) + a 2 b 2 C2 Γ2q + ) + +C ab ) Γ )q + ) + b C = k <. Γ q + ) The αf B) kαb). From Lemma 2.7 we coclude that F has at least oe fixed poit i B, i.e. the olocal value problem ) has at least oe mild solutio i B. The proof is completed. Corollary 3.7. If the hypotheses H)-H5) are satisfied, the there is at least oe mild solutio for ) provided that Proof. m L p < lim if T + [T McT + d)] + η) p Γq). 4) MΩT ) 4) implies that there exists a costat r > such that Mcr + d) + MΩr) + η) p Γq) m L p r. The by Theorem 3.6 we kow the corollary is true. 4. A example Let X = L 2 R ). Cosider the followig fractioal parabolic olocal Cauchy problem. { D q ut, z) = Lu)t, z) + ft, ut, z)), t [, ], z R, u, z) = m i= Kz, y)ut R i, y)dy, z R, 5)

9 JFCA-22/2 MILD SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS 9 where D q is the Caputo fractioal partial derivative of order < q <, f is a give fuctio, m is a positive iteger, < t < t 2 < < t m <, Kz, y) L 2 R R ; R + ). Moreover, u Lu)t, z) = a ij z) t, z) + b i z) u t, z) + cz)ut, z), z i z j z i i,j= where give coefficiets a ij, b i, c, i, j =, 2,..., satisfy the usual uiformly ellipticity coditios. We defie a operator A by A = L with the domai i= DA) = {v ) X : H 2 R )}. From [9], we kow that A geerates a aalytic, ocompact semigroup {T t)} t o L 2 R ). I additio, there exists a costat M > such that M = sup{t t); t [, )} < +. The the system 5) ca be reformulated as follows i X, { D q xt) = Axt) + ft, xt)), t [, ], x) = gx), where xt) = ut, ), that is xt)z = ut, z), z R. The fuctio g : C[, ], X) X is give by m gx)z = K g xt i )z), i= where K g vz) = Kz, y)vy)dy for v X, z R. R Let s take q = 2, ft, xt)) = t 4 si xt). Firstly, we have H) ad H3) are satisfied. The from ft, xt)) t 4, we get H4) holds with Ωx) =. From ft, xt)) ft, yt)) t 4 x y ad Remark 3.3 we get that H5) is satisfied. Furthermore, ote that K g : X X is completely cotiuous ad assume that c = m K R R 2 z, y)dydz) 2, we get H2) is satisfied. If M c <, the there exists a costat r which satisfies 3). Accordig to Theorem 3.6, problem 5) has at least oe mild solutio provided that Mc <. Refereces [] K.B. Oldham, J. Spaier, The Fractioal Calculus, Academic Press, New York, Lodo, 974. [2] I. Podluby, Fractioal Differetial Equatio, Academic Press, Sa Diego, 999. [3] M.M. El-Borai, Some probability desities ad fudametal solutios of fractioal evolutio equatios, Chaos Solitos ad Fractals 4 22) [4] Y. Zhou, F. Jiao, Nolocal Cauchy problem for fractioal evolutio equatios, No. Aal. 2) [5] J. Cao, Q. Yag, Z. Huag, Optimal mild solutios ad weighted pseudo-almost periodic classical solutios of fractioal itegro-differetial equatios, No. Aal. 74 2) [6] J. Wag, Y. Zhou, A class of fractioal evolutio equatios ad optimal cotrols, No. Aal. 2 2) [7] X. Shu, Y. Lai, Y. Che, The existece of mild solutios for impulsive fractioal partial differetial equatios, No. Aal. 74 2) [8] C. Cuevas, C. Lizama, Almost automorphic solutios to a class of semiliear fractioal differetial equatios, Appl. Math. Lett. 2 28) [9] E. Herádez, D. O Rega, K. Balachadra, O recet developmets i the theory of abstract differetial equatios with fractioal derivatives, No. Aal. 73 2)

10 ZUFENG ZHANG, BIN LIU JFCA-22/2 [] C. Lizama, A operator theoretical approach to a class of fractioal order differetial equatios, Appl. Math. Lett. 24 2) [] T. Zhu, C. Sog, G. Li, Existece of mild solutios for abstract semiliear evolutio equatios i Baach spaces, No. Aal ) [2] J.Baas, K. Goebel, Measure of Nocompactess i Baach Spaces, i: Lecture Notes i Pure ad Applied Math, vol.6, Marcle Dekker, New York, 98. [3] H.Möch, Boudary value problems for oliear ordiary differetial equatios of secod order i Baach spaces, No. Aal. 4 98) [4] D. Bothe, Multivalued perturbatio of m-accretive differetial iclusios, Israel J. Math ) [5] F. Wag, Y. Zhao, A two-sided iequality of gamma fuctio, J. Math. Res. Expo ) [6] L. Liu, F. Guo, C. Wu, Y. Wu, Existece theorems of global solutios for oliear Volterra type itegral equatios i Baach spaces, J. Math. Aal. Appl ) [7] F. Maiardi, P. Paradisi, R. Goreflo, Probability distributios geerated by fractioal diffusio equatios, i: J. Kertesz, I. KodorEds), Ecoophysics: A Emergig Sciece, Kluwer, Dordrecht, 2. [8] A. Pazy, Semigroups of Liear Operators ad Applicatios to Partial Differetial Equatios, Spriger-Verlag, New York, 983. [9] B. Maslowski, D. Nualart, Evolutio equatios drive by a fractioal Browia motio, J. Fu. Aal ) Zufeg Zhag School of Mathematics ad Statistics, Huazhog Uiversity of Sciece ad Techology, Wuha, 4374, Chia address: jshyzzf@sohu.com Bi Liu School of Mathematics ad Statistics, Huazhog Uiversity of Sciece ad Techology, Wuha, 4374, Chia address: biliu@mail.hust.edu.c