Research Article Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

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1 Abstract and Applied Analysis Volume 212, Article ID 93518, 15 pages doi:1.1155/212/93518 Research Article Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup He Yang Department of Mathematics, Northwest Normal University, Lanzhou 737, China Correspondence should be addressed to He Yang, yanghe256@163.com Received 2 May 212; Revised 9 September 212; Accepted 27 September 212 Academic Editor: Dumitru Bǎleanu Copyright q 212 He Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results. 1. Introduction The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, economics, and science. Numerous applications can be found in electrochemistry, control, porous media, electromagnetic, see for example, 1 5 and references therein. Hence the study of such equations has become an object of extensive study during recent years, see 6 23 and references therein. In this paper, we consider the existence of the following fractional evolution equation: D q ut Aut ft, ut,gut, u x, t J, T, 1.1 where D q is the Caputo fractional derivative of order q, 1, A is the infinitesimal generator of a compact analytic semigroup S of uniformly bounded linear operators, f is the nonlinear term and will be specified later, and

2 2 Abstract and Applied Analysis Gut Kt, susds 1.2 is a Volterra integral operator with integral kernel K CΔ,R, Δ{t, s : s t T},R,. Throughout this paper, we denote by K : max t,s Δ Kt, s In some existing articles, the fractional differential equations were treated under the hypothesis that nonlinear term satisfies Lipschitz conditions or linear growth conditions. It is obvious that these conditions are not easy to be verified sometimes. To make the things more applicable, in this work, we will prove the existence of mild solutions for 1.1 under some new conditions. More precisely, the nonlinear term only satisfies some local growth conditions see conditions H 1 and H 2. These conditions are much weaker than Lipschitz conditions and linear growth conditions. The main techniques used here are fractional calculus, theory of analytic semigroup, and Schauder fixed point theorem. The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of a compact analytic semigroup and the definition of mild solutions of 1.1. InSection 3, we study the existence of mild solutions for 1.1. In Section 4, an example is given to illustrate the applicability of abstract results obtained in Section Preliminaries In this section, we introduce some basic facts about the fractional power of the generator of a compact analytic semigroup and the fractional calculus that are used throughout this paper. Let X be a Banach space with norm. Throughout this paper, we assume that A : DA X X is the infinitesimal generator of a compact analytic semigroup St t of uniformly bounded linear operator in X, that is, there exists M 1 such that St M for all t. Without loss of generality, let ρ A, where ρ A is the resolvent set of A. Then for any >, we can define A by A : 1 Γ t 1 Stdt. 2.1 It follows that each A is an injective continuous endomorphism of X. Hence we can define A by A :A 1, which is a closed bijective linear operator in X. It can be shown that each A has dense domain and that DA β DA for β. Moreover, A β x A A β x A β A x for every, β R and x DA μ with μ : max{, β, β}, where A I, I is the identity in X. For proofs of these facts, we refer to the literature We denote by X the Banach space of DA equipped with norm x A x for x DA, which is equivalent to the graph norm of A. Then we have X β X for β 1 with X X, and the embedding is continuous. Moreover, A has the following basic properties. Lemma 2.1 see 24. A has the following properties. i St : X X for each t> and. ii A Stx StA x for each x DA and t.

3 Abstract and Applied Analysis 3 iii For every t>,a St is bounded in X and there exists M > such that A St M t. 2.2 iv A is a bounded linear operator for 1 in X. In the following, we denote by CJ, X the Banach space of all continuous functions from J into X with supnorm given by u C sup t J ut for u CJ, X.From Lemma 2.1iv, sincea is a bounded linear operator for 1, there exists a constant C such that A C for 1. For any t, denote by S t the restriction of St to X.FromLemma 2.1i and ii, for any x X, we have Stx A Stx St A x St A x St x, Stx x A Stx A x St A x A x 2.3 as t. Therefore, St t is a strongly continuous semigroup in X,and S t St for all t. To prove our main results, the following lemma is also needed. Lemma 2.2 see 27. S t t is an immediately compact semigroup in X, and hence it is immediately norm-continuous. Let us recall the following known definitions in fractional calculus. For more details, see 16 2, 23. Definition 2.3. The fractional integral of order σ> with the lower limits zero for a function f is defined by I σ ft 1 Γσ t s σ 1 fsds, t >, 2.4 where Γ is the gamma function. The Riemann-Liouville fractional derivative of order n 1 < σ < n with the lower limits zero for a function f can be written as d n L D σ 1 ft Γn σ dt n t s n σ 1 fsds, t >, n N. 2.5 Also the Caputo fractional derivative of order n 1 <σ<nwith the lower limits zero for a function f C n, can be written as D σ ft 1 Γn σ t s n σ 1 f n sds, t >, n N. 2.6

4 4 Abstract and Applied Analysis Remark The Caputo derivative of a constant is equal to zero. 2 If f is an abstract function with values in X, then integrals which appear in Definition 2.3 are taken in Bochner s sense. Lemma 2.5 see 12. A measurable function h : J X is Bochner integrable if h is Lebesgue integrable. For x X, we define two families {Ut} t and {V t} t of operators by Utx η q θst q θxdθ, V tx q θη q θst q θxdθ, <q<1, 2.7 where η q θ 1 q θ 1 1/q ρ q ( θ 1/q), ρ q θ 1 π 1 n 1 θ qn 1 Γ( nq 1 ) sin ( nπq ), n! n1 θ,, 2.8 η q is a probability density function defined on,, which has properties η q θ for all θ, and η q θdθ 1, θη q θdθ 1 Γ ( q 1 ). 2.9 The following lemma follows from the results in 7, Lemma 2.6. The operators U and V have the following properties. i For fixed t and any x X, we have Utx M x, V tx qm Γ ( 1 q ) x M Γ ( q ) x. 2.1 ii The operators Ut and V t are strongly continuous for all t. iii Ut and V t are norm-continuous in X for t>. iv Ut and V t are compact operators in X for t>. v For every t>, the restriction of Ut to X and the restriction of V t to X are normcontinuous. vi For every t>, the restriction of Ut to X and the restriction of V t to X are compact operators in X. Based on an overall observation of the previous related literature, in this paper, we adopt the following definition of mild solution of 1.1.

5 Abstract and Applied Analysis 5 Definition 2.7. By a mild solution of 1.1, we mean a function u CJ, X satisfying ut Utx t s q 1 V t sfs, us,gusds 2.11 for all t J. 3. Existence of Mild Solutions In this section, we give the existence theorems of mild solutions of 1.1. The discussions are based on fractional calculus and Schauder fixed point theorem. Our main results are as follows. Theorem 3.1. Assume that the following condition on f is satisfied. H 1 There exists a constant β, 1 such that f : J X X X β satisfies: i for each x, y X X, the function f,x,y : J X β is measurable; ii for each t J, the function ft,, : X X X β is continuous; iii for any r>, there exists a function g r L J, R such that sup x r, y K Tr f ( t, x, y ) β g r t, t J, 3.1 and there is a constant γ> such that 1 t lim inf r r g r s ds γ<. 1 q t s 3.2 If x X and MC β γ<γq,then1.1 has at least one mild solution. Proof. Define an operator Q by Qut Utx t s q 1 V t sfs, us,gusds, t J. 3.3 It is not difficult to verify that Q : CJ, X CJ, X. We will use Schauder fixed point theorem to prove that Q has fixed points in CJ, X. For any r>, let B r : {u CJ, X : ut r, t J}. We first show that there is a positive number r such that QB r B r. If this were not the case, then for each r>,

6 6 Abstract and Applied Analysis there would exist u r B r and t r J such that Qu r t r >r.thus,fromlemma 2.6i and H 1 iii, weseethat r r< Qu r t r Ut r x t r s q 1 V tr sfs, u r s,gu r s ds r M x t r s q 1 A β V t r s A β fs, u r s,gu r sds 3.4 M x MC β Γ ( q ) r t r s q 1 g r sds. Dividing on both sides by r and taking the lower limit as r, we have MC β γ Γ ( q ), 3.5 which is a contradiction. Hence QB r B r for some r>. To complete the proof, we separate the rest of proof into the following three steps. Step 1. Q : B r B r is continuous. Let {u n } B r with u n u B r as n. From the assumption H 1 ii, for each s J, we have fs, u n s,gu n s fs, us,gus 3.6 as n. Since fs, u n s,gu n s fs, us,gus β 2g r s, by the Lebesgue dominated convergence theorem, for each t J, we have Qu n t Qut t s q 1 V t s [ fs, u n s,gu n s fs, us,gus ] ds t s q 1 A β V t s MC β Γ ( q ) A β[ fs, u n s,gu n s fs, us,gus ] ds t s q 1 fs, un s,gu n s fs, us,gus β ds 3.7 as n, which implies that Q : B r B r is continuous.

7 Abstract and Applied Analysis 7 Step 2. QB r t : {Qut : u B r } is relatively compact in X for all t J. It follows from 2.9 and 3.3 that QB r {Qu : u B r } {x } is compact in X. Hence it is only necessary to consider the case of t>. For each t,t,ɛ,t, and any δ>, we define a set Q ɛ,δ B r t by Q ɛ,δ B r t : {Q ɛ,δ ut : u B r }, 3.8 where Q ɛ,δ ut δ η q θst q θdθx ɛ q t s q 1 θη q θs ( t s q θ ) dθ fs, us,gusds δ [ Sɛ q δ η q θst q θ ɛ q δdθx q δ ɛ t s q 1 δ θη q θs ( t s q θ ɛ q δ ) ] dθ fs, us,gusds. 3.9 Then the set Q ɛ,δ B r t is relatively compact in X since by Lemma 2.2, the operator S ɛ q δ is compact in X. For any u B r and t,t, from the following inequality: δ Qut Q ɛ,δ ut η q θst q θdθx δ q t s q 1 θη q θs ( t s q θ ) dθ fs, us,gusds q t s q 1 θη q θs ( t s q θ ) dθ fs, us,gusds δ ɛ q t s q 1 θη q θs ( t s q θ ) dθ δ fs, us,gusds δ M x η q θdθ qmc grl β qmc β gr L t ɛ t s q 1 ds t s q 1 ds θη q θdθ δ θη q θdθ

8 8 Abstract and Applied Analysis δ M x η q θdθ MC β T q g L r δ θη q θdθ MC β grl Γ ( q 1 ) ɛq. 3.1 One can obtain that the set QB r t is relatively compact in X for all t,t. And since it is compact at t, we have the relatively compactness of QB r t in X for all t J. Step 3. QB r : {Qu CJ, X : u B r } is equicontinuous. For τ,t,by3.3, we have Quτ Qu Uτx x τ τ s q 1 V τ sfs, us,gusds Uτ I x MC β grl Γ ( q 1 ) τ q Hence it is only necessary to consider the case of t>. For <t 1 <t 2 T, bylemma 2.1 and Lemma 2.6i, we have Qut 2 Qut 1 Ut 2 x Ut 1 x 2 t 2 s q 1 V t 2 sfs, us,gusds 1 t 1 s q 1 V t 1 sfs, us,gusds Ut 2 Ut 1 x 2 t 2 s q 1 V t 2 sfs, us,gusds t 1 1 [ t 2 s q 1 t 1 s q 1] V t 2 sfs, us,gusds 1 t 1 s q 1 fs, us,gusv t 2 s V t 1 sds Ut 2 Ut 1 x MC β g L r Γ ( q 1 ) t 2 t 1 q MC β grl [ Γ ( q 1 ) t q 2 tq 1 t 2 t 1 q]

9 Abstract and Applied Analysis 9 1 t 1 s q 1 V t 2 s V t 1 s fs, us,gusds I 1 I 2 I 3 I From Lemma 2.6v, weseethati 1 ast 2 t 1 independently of u B r.fromthe expressions of I 2 and I 3, it is clear that I 2 andi 3 as t 2 t 1 independently of u B r. For any ɛ,t 1, we have I 4 1 ɛ 1 t 1 s q 1 V t 2 s V t 1 s fs, us,gusds t 1 ɛ t 1 s q 1 V t 2 s V t 1 s fs, us,gusds 1 q C β g L r T q ɛ q sup V t 2 s V t 1 s 2MC β g L r s t 1 ɛ Γ ( q 1 ) ɛq It follows from Lemma 2.6v that I 4 ast 2 t 1 and ɛ independently of u B r. Therefore, we prove that QB r is equicontinuous. Thus, the Arzela-Ascoli theorem guarantees that Q is a compact operator. By the Schauder fixed point theorem, the operator Q has at least one fixed point u in B r, which is a mild solution of 1.1. This completes the proof. Remark 3.2. In assumption H 1 iii, if the function g r t is independent of t, then we can easily obtain a constant γ> satisfying 3.2. For example, if there is a constant a f > such that f ( t, x, y ) β a f ( 1 x y ) 3.14 for all x, y X and t J, then for any r >, x, y X with x r, y K Tr, we have ft, x, y β a f a f 1 K Tr g r t, where g r t is independent of t. Thus, γ : 1/qa f T q 1 K T > is the constant in 3.2. More generally, if f satisfies the following condition: H2 there is a constant β, 1 such that f : J X X X β satisfies: i for each x, y X X, the function f,x,y : J X β is measurable, ii for any r>, there exists a function l L J, R such that f ( t, x 1,y 1 ) f ( t, x2,y 2 ) β lt ( x 1 x 2 y 1 y 2 ) 3.15 for any x i,y i X with x i r, y i K Tr i 1, 2 and t J, then we have the following existence and uniqueness theorem.

10 1 Abstract and Applied Analysis Theorem 3.3. Assume that the condition H 2 is satisfied. If x X and MC β T q 1 K T l L < Γq 1,then1.1 has a unique mild solution. Proof. For any r>, if x, y X with x r, y K Tr, then from H 2 ii, we have f ( t, x, y ) β lt1 K Tr bt g r t, 3.16 where bt ft,, β. Therefore, the condition H 1 iii is satisfied with γ 1 K TT q l L /q.bytheorem 3.1, 1.1 has at least one mild solution u B r. Let u 1,u 2 B r be the solutions of 1.1. We show that u 1 u 2. Since u 1 t Qu 1 t and u 2 t Qu 2 t for all t J, we have u 1 t u 2 t Qu 1 t Qu 2 t MC β Γ ( q ) MC β Γ ( q ) t s q 1 V t s [ fs, u1 s,gu 1 s fs, u 2 s,gu 2 s ] ds t s q 1 A β V t s A β[ fs, u 1 s,gu 1 s fs, u 2 s,gu 2 s ] ds t s q 1 fs, u 1 s,gu 1 s fs, u 2 s,gu 2 s β ds t s q 1 ls u 1 s u 2 s Gu 1 s Gu 2 s ds MC β l L Γ ( q ) 1 K T t s q 1 u 1 s u 2 s ds By using the Gronwall-Bellman inequality see 14, Theorem 1, we can deduce that u 1 t u 2 t for all t J, which implies that u 1 u 2. Hence 1.1 has a unique mild solution u B r. This completes the proof. Remark 3.4. In Theorem 3.3, we only assume that f satisfies a local Lioschitz condition see condition H 2, and an existence and uniqueness result is obtained. If ft, u, v ft, u : J X X, then the assumption H 2 deletes the linear growth condition 3 of assumption Hf in 12. Therefore, the Theorem 3.3 extends and improves the main result in An Example Assume that X L 2,π equipped with its natural norm and inner product defined, respectively, for all u, v L 2,π, by ( π 1/2 u X ux dx) 2, u, v π uxvxdx. 4.1

11 Abstract and Applied Analysis 11 Consider the following fractional partial differential equation: 1/2 ( 2 ux, t t1/2 x ux, t g x, t, ux, t, ), 2 Kt, sux, sds t,t, x,π u,t uπ, t, t,t, ux, u x, x,π, 4.2 where T> is a constant. Let the operator A : DA X X be defined by DA : { v X : v X, v vπ }, Au 2 u x It is well known that A has a discrete spectrum with eigenvalues of the form n 2,n N, and corresponding normalized eigenfunctions given by z n 2/π sinnx. Moreover, A generates a compact analytic semigroup St t in X, and Stu e n2t u, z n z n. n1 4.4 It is not difficult to verify that St LX e t for all t. Hence, we take M 1. The following results are also well known. I The operator A can be written as Au n 2 u, z n z n n1 4.5 for every u DA. II The operator A 1/2 is given by A 1/2 u n u, z n z n n1 4.6 for each u DA 1/2 : {v X : n<v,z n >z n X} and A 1/2 LX 1. n1 Lemma 4.1 see 28. If m DA 1/2,thenm is absolutely continuous, m X and m X A 1/2 m X. Let X 1/2 DA 1/2, 1/2, where x 1/2 : A 1/2 x X for all x DA 1/2. Assume that g :,π,t R R R satisfies the following conditions. i For each x, t,π,t, the function gx, t,, is continuous. ii For each ξ, η R 2, the function g,,ξ,η is measurable.

12 12 Abstract and Applied Analysis iii For each t,t and ξ, η R, g,t,ξ,η is differentiable, and / xgx, t, ξ, η X. iv g,,, gπ,,,. v There exist the functions l 1,l L,T, R such that x g( x, t, ξ, η ) l 1t ( ξ η ) l t 4.7 for all x, t, ξ, η,π,t R R. Define ft, ut, Gutx gx, t, ux, t, Kt, sux, sds. Then, for each φ X 1/2, from assumptions iii and iv, we have ( ) π ( f t, φ, Gφ,zn g x, t, φx, t, 1 n π ( ( x g x, t, φx, t, ) Kt, sφx, sds 2 π sinnxdx )) Kt, sφx, sds 2 π cosnxdx. 4.8 This implies from II that f :,T X 1/2 X 1/2 X 1/2. Moreover, for any r>, by Minkowski inequality, assumption v and Lemma 4.1, we have sup f ( t, φ, Gφ ) ( t 1/2 sup φ 1/2 r φ x g x, t, φx, t, Kt, sφx, sds) 1/2 r X π ( sup t 2 φ 1/2 r x g x, t, φx, t, Kt, sφx, sds) dx ( π [ sup l 1 t( φx, t ) Kt, sφx, sds φ 1/2 r l t] 2 dx ) 1/2 sup φ 1/2 r [ l1 t ( φ X K T φ ) X l t ] [ ] sup 1 K Tl 1 ta A 1/2 φ l t X φ 1/2 r 1/2 1 K Trl 1 t l t g r t. 4.9 Therefore, f satisfies the condition H 1 with γ 2T 1/2 1 K T l 1 L.Thus,4.2 has at least one mild solution provided that γ< π due to Theorem 3.1.

13 Abstract and Applied Analysis 13 Assume furthermore that the function g satisfies the following: vi for any r>, there exists a function l 2 L,T, R such that x g( ) x, t, ξ 1,η 1 x g( ) x, t, ξ 2,η 2 l 2t ( ξ 1 ξ 2 ) η 1 η for x, t, ξ 1,η 1, x, t, ξ 2,η 2,π,T R R with ξ i r and η i K Tr, i 1, 2. Then for each φ 1,φ 2 X 1/2,byLemma 4.1, we have ( ) ( ) f t, φ1,gφ 1 f t, φ2,gφ 1/2 2 A 1/2[ f ( ) ( )] t, φ 1,Gφ 1 f t, φ2,gφ 2 X ( x g x, t, φ 1 x, t, ) Kt, sφ 1 x, sds ( x g x, t, φ 2 x, t, Kt, sφ 2 x, sds) X ( π ( t ) x g x, t, φ 1 x, t, Kt, sφ 1 x, sds ( x g x, t, φ 2 x, t, Kt, sφ 2 x, sds) ( π ( l 2t φ 1 x, t φ 2 x, t Kt, sφ 1 x, sds ) 2 Kt, sφ 2 x, sds dx ( π l 2 t φ1 x, t φ 2 x, t ) 1/2 2 dx ( π Kt, sφ 1 x, sds 2 Kt, sφ 2 x, sds dx 1/2 2 dx 1/2 1/2

14 14 Abstract and Applied Analysis l 2 t ( φ1 φ X 2 ) Gφ1 Gφ X 2 ( A 1/2 l 2 t A 1/2( ) φ 1 φ 2 X A 1/2 A 1/2( ) ) Gφ 1 Gφ 2 X A 1/2 l 2 t( ( ) φ 1 φ 2 X A 1/2( ) ) Gφ 1 Gφ 2 X l 2 t( φ 1 φ 1/2 2 ) Gφ 1 Gφ 1/ This shows that f satisfies the condition H 2. Hence by Theorem 3.3, the mild solution of 4.2 is unique. Acknowledgments This research was supported by the National Natural Science Foundation of China Grant no , the Fundamental Research Funds for the Gansu Universities and The Project of NWNU-LKQN References 1 L. Gaul, P. Klein, and S. Kemple, Damping description involving fractional operators, Mechanical Systems and Signal Processing, vol. 5, no. 2, pp , W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophysical Journal, vol. 68, no. 1, pp , R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, Relaxation in filled polymers: a fractional calculus approach, Journal of Chemical Physics, vol. 13, no. 16, pp , F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, pp , Springer, Vienna, Austria, R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2. 6 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, vol. 14, no. 3, pp , M. M. El-Borai, Semigroups and some nonlinear fractional differential equations, Applied Mathematics and Computation, vol. 149, no. 3, pp , V. Lakshmikantham, S. Leela J, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp , Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications, vol. 59, no. 3, pp , J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp , R.-N. Wang, T.-J. Xiao, and J. Liang, A note on the fractional Cauchy problems with nonlocal initial conditions, Applied Mathematics Letters, vol. 24, no. 8, pp , H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp , 27.

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