Research Article Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay

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1 International Journal of Differential Equations Volume 211, Article ID 79323, 2 pages doi:1.1155/211/79323 Researc Article Existence of te Mild Solutions for Impulsive Fractional Equations wit Infinite Delay Jaydev Dabas, 1 Arcana Cauan, 2 and Mukes Kumar 2 1 Department of Paper Tecnology, IIT Roorkee, Saaranpur Campus, Saaranpur 2471, India 2 Department of Matematics, Motilal Neru National Institute of Tecnology, Allaabad 211 4, India Correspondence sould be addressed to Jaydev Dabas, jay.dabas@gmail.com Received 25 May 211; Revised 22 July 211; Accepted 6 August 211 Academic Editor: D. D. Ganji Copyrigt q 211 Jaydev Dabas et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Tis paper is concerned wit te existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation wit te infinite delay and impulsive effects. Te existence and uniqueness of a mild solution is establised using a solution operator and te classical fixed-point teorems. 1. Introduction Tis paper is concerned wit te existence and uniqueness of a mild solution of an impulsive fractional-order functional differential equation wit te infinite delay of te form D t x t Ax t f t, x t,bx t, t J,T, t/ t k, ( ( Δx t k I k x t k)), k 1, 2,...,m, x t φ t, φ t B, 1.1 were T>, <<1, A: D A is te infinitesimal generator of an -resolvent family S t t, te solution operator T t t is defined on a complex Banac space, D is te Caputo fractional derivative, f : J B is a given function, and B is a pase space defined in Section 2. Here, t <t 1 < <t m <t m 1 T, I k C,, k 1, 2,...,m, are bounded functions, Δx t k x t k x t k,x t k lim x t k and x t k lim x t k represent te rigt and left limits of x t at t t k, respectively.

2 2 International Journal of Differential Equations We assume tat x t :,, x t s x t s, s, belongs to an abstract pase space B. Te term Bx t is given by Bx t K t, s x s, were K C D, R is te set of all positive continuous functions on D { t, s R 2 : s t T}. Differential equations wit impulsive conditions constitute an important field of researc due to teir numerous applications in ecology, medicine biology, electrical engineering, and oter areas of science. Many pysical penomena in evolution processes are modelled as impulsive differential equations and ave been studied extensively by several autors, for instance, see 1 3, for more information on tese topics. Impulsive integrodifferential equations wit delays represent matematical models for problems in te areas suc as population dynamics, biology, ecology, and epidemic and ave been studied by many autors 2 7. Te study of fractional differential equations as emerged as a new branc of applied matematics, wic as been used for construction and analysis of matematical models in science and engineering. In fact, te fractional differential equations are considered as models alternative to nonlinear differential equations. Many pysical systems can be represented more accurately troug fractional derivative formulation. For more detail, see, for instance, te papers 1, 3 5, 7 12 and references terein. Recently, in 4, te autor as establised sufficient conditions for te existence of a mild solution for a fractional integro-differential equation wit a state-dependent delay. Mopou and N Guérékata 7 ave investigated te existence and uniqueness of a mild solution for te fractional differential equation 1.1 witout impulsive conditions. Autors of 7 ave establised te results assuming tat A generates an -resolvent family S t t on a complex Banac space by means of classical fixed-point meto. In 5, Bencora et al. ave considered te following nonlinear functional differential equation wit infinite delay D q x t f t, x t, t,t, <q<1, x t φ t, t,, 1.2 were D q is Riemann-Liouville fractional derivative, φ B,witφ, and establised te existence of a mild solution for te considered problem using te Banac fixed-point and te nonlinear alternative of Leray-Scauder teorems. Motivated by te above-mentioned works, we consider te problem 1.1 to study te existence and uniqueness of a mild solution using te solution operator and fixed-point teorems. Te paper is organized as follows: in Section 2, we introduce some function spaces and notations and present some necessary definitions and preliminary results tat will be used to prove our main results. Te proof of our main results is given in Section 3.Intelast section one example is presented. 2. Preliminaries In tis section, we mention some definitions and properties required for establising our results. Let be a complex Banac space wit its norm denoted as,andl represents te Banac space of all bounded linear operators from into, and te corresponding norm is denoted by L.LetC J, denote te space of all continuous functions from J into wit supremum norm denoted by C J,. In addition, B r x, represents te closed ball in wit te center at x and te radius r. To describe a fractional-order functional differential equation wit te infinite delay, we need to discuss te abstract pase space B in a convenient way for details see 3. Let

3 International Journal of Differential Equations 3 :,, be a continuous function wit l t dt <. For any a>, we define B {ψ : a, suc tat ψ t is bounded and measurable} and equip te space B wit te norm ψ a, sup ψ s, ψ B. 2.1 s a, Let us define by { B ψ :,, suc tat for any c>, ψ c, B wit ψ and s ψ } s, <. 2.2 If B is endowed wit te norm ψb s ψ s,, ψ B, 2.3 ten it is known tat B, B is a Banac space. Now, we consider te space B {x :,T suc tat x Jk C J k, and tere exist x ( t ) ( k and x t k) wit x tk x ( } t k), x φ B,k 1,...,m, 2.4 were x Jk is te restriction of x to J k t k,t k 1,k, 1, 2,...,m. Te function B to be a seminorm in B, it is defined by x B sup{ x s : s,t } φ B, x B. 2.5 If x :,T, T >, is suc tat x B, ten for all t J, te following conditions old: 1 x t B, 2 x t B C 1 t sup <s<t x s C 2 t x B, 3 x t H x t B, were H>isaconstant and C 1 :,, is continuous, C 2 :,, is locally bounded, and C 1,C 2 are independent of x. For more details, see 6. A two parameter function of te Mittag-Lefller type is defined by te series expansion E,β z z k Γ ( k β ) 1 2πi k C μ β e μ μ dμ, z, β >, z C, 2.6

4 4 International Journal of Differential Equations were C is a contour wic starts and en at and encircles te disc μ z 1/2 counter clockwise. For sort, E z E,1 z. It is an entire function wic provides a simple generalization of te exponent function: E 1 z e z and te cosine function: E 2 z 2 cos z, E 2 z 2 cos z, and plays an important role in te teory of fractional differential equations. Te most interesting properties of te Mittag-Lefller functions are associated wit teir Laplace integral e λt t β1 E,β ωt dt λβ λ ω, Re λ>ω1/,ω>, 2.7 see 12 for more details. Definition 2.1. A closed and linear operator A is said to be sectorial if tere are constants ω R, θ π/2,π, M >, suc tat te following two conditions are satisfied: 1 ρ A θ,ω { λ C : λ / ω, M 2 R λ, A L λ ω, λ. arg λ ω <θ }, θ,ω 2.8 Sectorial operators are well studied in te literature. For details see 13. Definition 2.2 see Definition 2.3 in 1. LetA be a closed and linear operator wit te domain D A defined in a Banac space. Letρ A be te resolvent set of A. We say tat A is te generator of an -resolvent family if tere exist ω and a strongly continuous function S : R L suc tat {λ :Reλ>ω} ρ A and λ I A 1 x e λt S t xdt, Re λ>ω, x, 2.9 in tis case, S t is called te -resolvent family generated by A. Definition 2.3 see Definition 2.1 in 4. LetA be a closed linear operator wit te domain D A defined in a Banac space and >. We say tat A is te generator of a solution operator if tere exist ω and a strongly continuous function S : R L suc tat {λ :Reλ>ω} ρ A and λ 1 λ I A 1 x e λt S t xdt, Re λ>ω, x, 2.1 in tis case, S t is called te solution operator generated by A. Te concept of te solution operator is closely related to te concept of a resolvent family see 14 Capter 1. For more details on -resolvent family and solution operators, we refer to 14, 15 and te references terein.

5 International Journal of Differential Equations 5 Definition 2.4. Te Riemann-Liouville fractional integral operator for order >, of a function f : R R and f L 1 R,, is defined by I f t f t, I f t 1 Γ t s 1 f s, >, t >, 2.11 were Γ is te Euler gamma function. Te Laplace transform of a function f L 1 R, is defined by f λ e λt f t dt, Re λ >ω, 2.12 provided te integral is absolutely convergent for Re λ >ω. Definition 2.5. Caputo s derivative of order for a function f :, R is defined as D t f t 1 Γ n t s n1 f n s I n f n t, 2.13 for n 1 <n, n N.If< 1, ten D t f t 1 Γ 1 t s f 1 s Obviously, Caputo s derivative of a constant is equal to zero. Te Laplace transform of te Caputo derivative of order >isgivenas L { Dt f t ; λ} n1 λ f λ λ k1 f k ; n 1 <n k Lemma 2.6. If f satisfies te uniform Holder condition wit te exponent β, 1 and A is a sectorial operator, ten te unique solution of te Caucy problem D t x t Ax t f t, x t,bx t, x t φ t, t t, t > t,t R, <<1, 2.16 is given by x t T t t ( x ( t )) t S t s f s, x s,bx s, t 2.17

6 6 International Journal of Differential Equations were T t E,1 At 1 e λt λ 1 2πi Br λ A dλ, S t t 1 E, At 1 2πi B r e λt 1 λ A dλ, 2.18 B r denotes te Bromwic pat. S t is called te -resolvent family, and T t is te solution operator, generated by A. Proof. Let t t u, ten we get D ux u t Ax u t f u t,x u t,bx u t, u > Taking te Laplace transform of 2.19, we ave λ L{x u t } λ 1 x ( t ) AL{x u t } L { f u t,x u t,bx u t }. 2.2 Since λ I A 1 exists, tat is, λ ρ A,from 2.2, weobtain L{x u t } λ 1 λ I A 1 x ( t ) λ I A 1 L { f u t,x u t,bx u t } By te inverse Laplace transform of 2.21, weget x u t E,1 Au x ( t ) u u s 1 ( E, A u s ) f s t,x s t,bx s t Set u t t,in 2.22, we ave ( x t E,1 A t t ) x ( t ) t t t s 1 E, ( A t t s ) f s t,x s t,bx s t On simplification, we obtain ( x t E,1 A t t ) x ( t ) t θ 1 ( E, A t θ ) f θ, x θ,bx θ dθ. t 2.24

7 International Journal of Differential Equations 7 Set T t E,1 At and S t t 1 E, At in We ave x t T t t x ( t ) t S t θ f θ, x θ,bx θ dθ. t 2.25 Tis completes te proof of te lemma. Now, we give te definition of a mild solution of te system 1.1 by investigating te classical solution of te system 1.1. Definition 2.7. A function x :,T is called a mild solution of 1.1 if te following ol: x φ B on, wit φ ; Δx t tk I k x t,k 1,...,m, te restriction k of x to te interval,t \{t 1,...,t m } is continuous and satisfies te following integral equation: φ t, t,, S t s f s, x s,bx s, t,t 1, x t T t t 1 ( x ( t ( ( ))) t 1) I1 x t 1 S t s f s, x s,bx s, t 1. T t t m x t m I m x t m t m S t s f s, x s,bx s, t t 1,t 2, t t m,t Now, we introduce te following assumptions: H1 tere exist μ 1,μ 2 > suc tat f t, ϕ, x f t, ψ, y μ 1 ϕ ψ B μ 2 x y, t I, ( ϕ, ψ ) B 2,x,y H2 for eac k 1,...,m, tere exists ρ k > suc tat Ik x I k y ρ k x y, x, y H3 max 1 i m { ( ) M S T M ( T 1 ρi μ1 C 1 μ 2B )} < 1, 2.29

8 8 International Journal of Differential Equations were C 1 sup <τ<t C 1 τ and B t sup t,t K t, s < and M T sup T t L, t T M S sup Ce ωt( 1 t 1). t T 2.3 If, 1 and A A θ,ω, ten for any x and t >, we ave T t L Me ωt and S t L Ce ωt 1 t 1,t >, ω > ω. Hence, we ave T t L M T, S t L t 1 M S.See 1 for details. 3. Te Main Results Our first result is based on te Banac contraction principle. Teorem 3.1. Assume tat te assumptions (H1) (H3) are satisfied. If A A θ,ω, ten te system 1.1 as a unique mild solution. Proof. Consider te operator N : B B defined by φ t, Nx t. t,, S t s f s, x s,bx s, T t t 1 ( x ( t ( ( ))) 1) I1 x t t 1 S t s f s, x s,bx s, 1 t,t 1, t t 1,t 2, 3.1 T t t m x t m I m x t m t m S t s f s, x s,bx s, t t m,t. Let y :,T be te function defined by φ t, t, y t, t J, 3.2 ten y φ. For eac z C J, R wit z, we denote by z te function defined by, t, ; z t z t, t J. 3.3

9 International Journal of Differential Equations 9 If x satisfies 2.26, ten we can decompose x as x t y t z t for t J, wic implies x t y t z t for t J, and te function z satisfies z t. S t s f ( s, y s z s,b ( y s z s )), t,t 1, T t t 1 [ y ( t ( ) (( ( )) ( ))] 1) z t 1 I1 y t 1 z t 1 t 1 S t s f ( s, y s z s,b ( y s z s )), T t t m [ y t m z t m I m (( y t m ) z t m )] t m S t s f ( s, y s z s,b ( y s z s )), t t 1,t 2, t t m,t. 3.4 Set B {z B suc tat z } and let B be te seminorm in B defined by z B sup z t z B t J sup z t, z B, 3.5 t J tus B, B is a Banac space. We define te operator P : B B by Pz t. S t s f ( s, y s z s,b ( y s z s )), T t t 1 [ z ( t ( ( ))] 1) I1 z t 1 t 1 S t s f ( s, y s z s,b ( y s z s )), t,t 1, t t 1,t 2, 3.6 T t t m z t m I m z t m t m S t s f ( s, y s z s,b ( y s z s )), t t m,t.

10 1 International Journal of Differential Equations It is clear tat te operator N as a unique fixed-point if and only if P as a unique fixedpoint. To prove tat P as a unique fixed-point, let z, z B, ten for all t,t 1. We ave P z t P z t S t s f s, L ys z s,b y s z s f s, y s z s,b y s z s M S t s 1[ μ zs 1 z ( ) ( s B μ 2 B y s z s B y s z ) ] M S ( μ1 C 1 μ 2B ) T z z B. 3.7 For t t 1,t 2, we ave P z t P z t [ T t t 1 z t L 1 z t 1 I1 z t 1 I 1 z t 1 ] S t s f s, L ys z s,b y s z s f s, y s z s,b y s z s t 1 M T [ z t 1 z t 1 ρ 1 z t 1 z t 1 ] M S t s 1[ μ zs 1 z ( ) ( s B μ 2 B y s z s B y s z ) ] t 1 M T ( 1 ρ1 ) z z B M S ( μ1 C 1 μ 2B ) T z z B. 3.8 Similarly, wen t t i,t i 1,i 2,...,m,weget P z t P z t M T ( 1 ρi ) z z B Tus, for all t,t, we ave P z P z B max 1 i m { M S ( μ1 C 1 μ 2B ) T z z B. 3.9 } ( ) M S ( M T 1 ρi μ1 C 1 μ 2B ) T z z B. 3.1 Hence, P is a contraction map, and terefore it as an unique fixed-point z B, wic is a mild solution of 1.1 on,t. Tis completes te proof of te teorem. Te second result is establised using te following Krasnoselkii s fixed-point teorem.

11 International Journal of Differential Equations 11 Teorem 3.2. Let B be a closed-convex and nonempty subset of a Banac space. LetP and Q be two operators suc tat i Px Qy B wenever x, y B, ii P is compact and continuous; iii Q is a contraction mapping, ten tere exists z B suc tat z Pz Qz. Now, we make te following assumptions: H4 f : J B is continuous, and tere exist two continuous functions μ 1,μ 2 : J, suc tat f t, ψ, x μ 1 t ψ B μ 2 t x, ( t, ψ, x ) J B H5 te function I k : is continuous, and tere exists Ω > suc tat Ω max 1 k m,x B r { I k x } Before going furter, we need te following lemma. Lemma 3.3 see Lemma 3.2 in 7. Let C 1 sup C 1 τ, <τ<t C 2 sup C 2 τ, <τ<t μ 1 sup μ 1 τ, <τ<t μ 2 sup μ 2 τ <τ<t 3.13 ten for any s J, μ 1 s y s z s B μ 2 s B y s z s μ 1 [ μ 2 C 2 φ ] B C 1 sup z τ τ s s K s, τ z τ dτ If z <r, r>, ten μ 1 s ys z s B μ 2 s B y s z s μ 1 [ C 2 ] φb C 1 r μ 2 rb λ Teorem 3.4. Suppose tat te assumptions (H1), (H4), (H5) are satisfied wit [ ] M S ( μ1 C 1 μ 2B ) T < 1, 3.16 ten te impulsive problem 1.1 as at least one mild solution on,t.

12 12 International Journal of Differential Equations Proof. Coose r M T r Ω M S T λ/ and consider B r {z B : z B r}, ten B r is a bounded, closed-convex subset in B. Let Γ 1 : B r B r and Γ 2 : B r B r be defined as, t,t 1, T t t 1 [ z ( t ( ( 1) I1 z t 1))], t t1,t 2, Γ 1 z t. T t t m z t m I m z t m, t t m,t, 3.17 S t s f ( s, y s z s,b ( y s z s )), Γ 2 z t t 1. t S t s f ( s, y s z s,b ( y s z s )), t m S t s f ( s, y s z s,b ( y s z s )), t,t 1, t t 1,t 2, t t m,t Step 1. Let z, z B r, ten sow tat Γ 1 z Γ 2 z B r,fort,t 1, we ave Γ 1 z t Γ 2 z t S t s f s, L ys z s,b y s z s M S t s 1[ μ 1 s ys z s B μ 2 s B y s z s ], 3.19 and by using Lemma 3.3, we conclude tat Γ 1 z Γ 2 z B M S λt <r. 3.2 Similarly, wen t t i,t i 1, i 1,...,m, we ave te estimate Γ 1 z t Γ 2 z t T t t i z t i I i z t i S t s L f s, y s z s,b y s z s t i

13 International Journal of Differential Equations 13 M T ( z B ) Ii z t i [ S t s L μ 1 s y s z s B μ 2 s B y s z s ] t i M T r Ω M S T λ <r, 3.21 wic implies tat Γ 1 z Γ 2 z B r. Step 2. We will sow tat te mapping Γ 1 z t is continuous on B r. For tis purpose, let {z n } n 1 be a sequence in B r wit lim z n z B r, ten for t t i,t i 1, i, 1,...,m, we ave Γ 1 z n t Γ 1 z t T t t i L [ z n t i z t i I i z n t i I i z t i ] Since te functions I i,i 1, 2,...,m are continuous, ence lim n Γ 1 z n Γ 1 z in B r wic implies tat te mapping Γ 1 is continuous on B r. Step 3. Uniform boundedness of te map Γ 1 z t is an implication of te following inequality: for t t i,t i 1, i, 1,...,m, we ave Γ 1 z t T t t i L [ z t i Ii z t i ] M T r Ω Step 4. To sow tat te map 3.17 is equicontinuous, we proceed as follows. Let u, v t i,t i 1, t i u<v t i 1, i, 1,...,m, z B r, ten we obtain Γ 1 z v Γ 1 z u T v t i T u t i L z t i I i z t i r Ω T v t i T u t i L Since T is strongly continuous, te continuity of te function t T t allows us to conclude tat lim u v T v t i T u t i L, wic implies tat Γ 1 B r is equicontinuous. Finally, combining Step 1 to Step 4 togeter wit Ascoli s teorem, we conclude tat te operator Γ 1 is compact.

14 14 International Journal of Differential Equations Now, it only remains to sow tat te map Γ 2 is a contraction mapping. Let z, z B r and t t i,t i 1, i, 1,...,m, ten we ave Γ 2 z t Γ 2 z t t i S t s L f ( s, ys z s,b ( y s z s )) f ( s, y s z s,b ( y s z s )) M S t s 1[ μ 1 z s z s B μ 2 B ( y s z s ) B ( y s z s ) ] t i M S ( μ1 C 1 μ 2B ) T z z B, 3.25 since M S / μ 1 C 1 μ 2B T < 1, wic implies tat Γ 2 is a contraction mapping. Hence, by te Krasnoselkii fixed-point teorem, we can conclude tat te problem 1.1 as at least one solution on,t. Tis completes te proof of te teorem. Our last result is based on te following Scaefer s fixed-point teorem. Teorem 3.5. Let P be a continuous and compact mapping on a Banac space into itself, suc tat te set {x : x νpx for some ν 1} is bounded, ten P as a fixed-point. Lemma 3.6 see 5. Let v :,T, be a real function, w is nonnegative and locally integrable function on,t, and tere are constants a> and <<1 suc tat v s v t w t a t s Ten tere exists a constant K suc tat w s v t w t ak, for every t,t t s Teorem 3.7. Assume tat te assumptions (H4)-(H5) are satisfied, and if A A θ,ω and M T < 1, ten te impulsive problem 1.1 as at least one mild solution on,t. Proof. We define te operator P : B B as in Teorem 3.3. Note tat P is well defined in. We complete te proof in te following steps. B Step 1. For te continuity of te map P,let{z n } be a sequence in B suc tat zn z in B. Since te function f is continuous on J B, Tis implies tat f ( s, y s z n s,b ( y s z n s )) f ( s, y s z s,b ( y s z s )) as n. 3.28

15 International Journal of Differential Equations 15 Now, for every t,t 1,weget Pz n t Pz t S t s f s, L ys z n s,b y s z n s f s, y s z s,b y s z s T M S ε, 3.29 were ε>, ε asn. Moreover, we ave Pz n t Pz t M T [ z n t i z t i Ii z n t i I i z t i ] S t s f s, L ys z n s,b y s z n s f s, y s z s,b y s z s t i [ M T z n t i z t i I i z n t i I i z t i ] T M S ε, 3.3 were ε>, ε asn, for all t t i,t i 1,i 1,...,m. Te impulsive functions I k,k 1,...,mare continuous, ten we get lim n Pzn Pz B Tis implies tat P is continuous. Step 2. P maps bounded sets into bounded sets in B. To prove tat for any r>, tere exists a γ>suc tat for eac z B r {z B : z B r}, ten we ave Pz B γ, ten for any z B r,t,t 1, we ave Pz t S t s f s, L ys z s,b y s z s M S t s 1[ μ 1 s ys z B s μ 2 s B y s z s ] Using Lemma 3.3, weobtain Pz t M S T / λ. Similarly, we ave Pz t M T r Ω M S T λ, t t i,t i 1, i 1,...,m. 3.33

16 16 International Journal of Differential Equations Tis implies tat Pz B T M T r Ω M S λ γ, t,t Step 3. We will prove tat P B r is equicontinuous. Let u, v,t 1,witu<v, we ave Pz v Pz u u v S v s S u s L f s, y s z s,b y s z s u S v s L f s, ys z s,b y s z s Q 1 Q 2, 3.35 were Q 1 u λ u S v s S u s L f s, y s z s,b y s z s S v s S u s L Since S v s S u s L 2 M S t 1 s 1 L 1 I,R for s,t 1 and S v s S u s asu v, S is strongly continuous. Tis implies tat lim u v Q 1, Q 2 v u S v s f s, L ys z s,b y s z s λ M S v u Hence, lim u v Q 2. Similarly, for u, v t i,t i 1,witu<v, i 1,...,m, we ave Pz v Pz u T v t i T u t i L [ z ( t i ) Ii ( z ( t i )) ] Q1 Q Since T is also strongly continuous, so T v t i T u t i asu v. Tus,from te above inequalities, we ave lim u v Pz v Pz u. So, P B r is equicontinuous. Finally, combining Step 1 to Step 3 wit Ascoli s teorem, we conclude tat te operator P is compact. Step 4. We sow tat te set E { z B suc tat z νpz for some <ν<1} 3.39

17 International Journal of Differential Equations 17 is bounded. Let z E, ten z t νpz t for some <ν<1. Ten for eac t,t 1, we ave z t ν S t s f s, L ys z s,b y s z s ν M S t s 1 f s, ys z s,b y s z s, 3.4 for t t i,t i 1,i 1,...,m,weget [ ( z t ν T t t i L z t i I i z t i ) t S t s L t i f s, ys z s,b y s z s ] [ t M T z t i M T Ω M S t s 1 f s, ys z s,b y s z s ]. t i 3.41 ten for all t,t, we ave z t M T Ω 1 M T M S 1 M T M T Ω 1 M T t s 1[ μ 1 s ys z s B μ 2 s B y s z s ] M S μ 1 C 2 φb T ) (1 M T 3.42 M S 1 M T ( μ ω 1 ω 2 1 C 1 μ 2 B ) t s 1 sup z τ, τ s t s 1 sup z τ τ s were ω 1 M T Ω/ 1 M T M S μ 1 C 2 φ B T / 1 M T and ω 2 M S /1 M T μ 1 C 1 μ 2 B.Letτ,s be suc tat sup τ s z τ z τ, s t. If τ,t, ten 3.43 can be written as z t ω 1 ω 2 t s 1 z s. 3.43

18 18 International Journal of Differential Equations Using Lemma 3.6, tere exists a constant K, and 3.43 becomes z t ω 1 [1 ] [ K t s 1 ω 1 1 K T ] As a consequence of Scaefer s fixed-point teorem, we deduce tat P as a fixed-point on,t. Tis completes te proof of te teorem. 4. Applications To illustrate te application of te teory, we consider te following partial integrodifferential equation wit fractional derivative of te form D q t 2 u t, x u t, x x2 H t, x, s t Q u s, x k s, t e u s,x, x,π, t,b, t/ t k, u t, u t, π, t, 4.1 u t, x φ t, x, t,, x,π, Δu t i x i q i t i s u s, x, x,π, were D q t is Caputo s fractional derivative of order <q<1, <t 1 <t 2 < <t n <bare prefixed numbers, and φ B.Let L 2,π, and define te operator A : D A by Aw w wit te domain D A : {w : w, w are absolutely continuous, w, w w π }, ten Aw n 2 w, w n w n, w D A, 4.2 n 1 were w n x 2/π sin nx, n N is te ortogonal set of eigenvectors of A. It is well known tat A is te infinitesimal generator of an analytic semigroup T t t in and is given by T t w e n2t w, w n w n, w, and every t>. 4.3 n 1

19 International Journal of Differential Equations 19 From tese expressions, it follows tat T t t is a uniformly bounded compact semigroup, so tat R λ, A λa 1 is a compact operator for all λ ρ A, tat is, A A θ,ω.let s e 2s,s<, ten l s 1/2, and define φ s sup φ θ L 2. θ s, 4.4 Hence, for t, φ,b B, were φ θ x φ θ, x, θ, x,,π.setu t x u t, x, f ( t, φ, Bu t ) x H t, x, θ Q ( φ θ x ) dθ Bu t x, 4.5 were Bu t x k s, t eu s,x. Ten wit tese settings, te above equation 4.1 can be written in te abstract form of te equations 1.1. Te functions H, k, andq are satisfying some conditions, and q i : R R are continuous and d i s q2 i s < for i 1, 2,...,n. Suppose furter tat 1 te function H t, x, θ is continuous in,b,π, and H t, x, θ, H t, x, θ dθ p 1 t, x <, 2 te function Q is continuous and for eac θ, y,,π, Q u θ x e2s u s, L 2. Now, we can see tat f t, φ, Bu t L 2 π π ( t, x, θ Q ( φ θ x ) ) 2 1/2 dθ Bu t x dx ( ( ) 2 t, x, θ e 2s φ s L 2 dθ) dx ( 2 1/2 k s, t e ) u s,x dx π π ( ( ) 2 t, x, θ e 2s sup φ s L 2 dθ) dx s θ, π ( 2 1/2 k s, t e ) u s,x dx π ( 2 φ t, x, θ dθ) dx 1/2 B Bu t L 2 1/2 1/2

20 2 International Journal of Differential Equations [ π ] ( ) 1/2 2dx p t, x φb Bu t L 2 p t φb Bu t L If we take μ 1 t p t and μ 2 t 1, ence f satisfies H4, and similarly we can sow tat I k satisfy H5. All conditions of Teorem 3.7 are now fulfilled, so we deduce tat te system 4.1 as a mild solution on,t. Acknowledgment Te autors express teir deep gratitude to te referees for teir valuable suggestions and comments for improvement of te paper. References 1.-B. Su, Y. Lai, and Y. Cen, Te existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Analysis, Teory, Meto & Applications, vol. 74, no. 5, pp , J. Y. Park, K. Balacandran, and N. Annapoorani, Existence results for impulsive neutral functional integrodifferential equations wit infinite delay, Nonlinear Analysis. Teory, Meto & Applications, vol. 71, no. 7-8, pp , Zang,. Huang, and Z. Liu, Te existence and uniqueness of mild solutions for impulsive fractional equations wit nonlocal conditions and infinite delay, Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp , R. P. Agarwal, B. De Andrade, and G. Siracusa, On fractional integro-difierential equations wit state-dependent delay, Computers & Matematics wit Applications, vol. 62, no. 3, pp , M. Bencora, J. Henderson, S. K. Ntouyas, and A. Ouaab, Existence results for fractional order functional differential equations wit infinite delay, Journal of Matematical Analysis and Applications, vol. 338, no. 2, pp , J. K. Hale and J. Kato, Pase space for retarded equations wit infinite delay, Funkcialaj Ekvacioj, vol. 21, no. 1, pp , G. M. Mopou and G. M. N Guérékata, Existence of mild solutions of some semilinear neutral fractional functional evolution equations wit infinite delay, Applied Matematics and Computation, vol. 216, no. 1, pp , M. Sateri and D. D. Ganji, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical tecnique, International Journal of Differential Equations, vol. 21, Article ID , 1 pages, Z. Z. Ganji, D. D. Ganji, A. D. Ganji, and M. Rostamian, Analytical solution of time-fractional Navier- Stokes equation in polar coordinate by omotopy perturbation metod, Numerical Meto for Partial Differential Equations, vol. 26, no. 1, pp , D. Araya and C. Lizama, Almost automorpic mild solutions to fractional differential equations, Nonlinear Analysis. Teory, Meto & Applications, vol. 69, no. 11, pp , J. T. Macado, V. Kiryakova, and F. Mainardi, Recent istory of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp , I. Podlubny, Fractional Differential Equations, Acadmic press, New York, NY, USA, M. Haase, Te Functional Calculus for Sectorial Operators, vol. 169 of Operator Teory: Advances and Applications, Birkäuser, Basel, Switzerland, J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monograps in Matematics, Birkäuser, Basel, Switzerland, C. Lizama, Regularized solutions for abstract Volterra equations, Journal of Matematical Analysis and Applications, vol. 243, no. 2, pp , 2.

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