Báo cáo hóa học: " Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems" doc

RESEARC H Open Access Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems Amar Debbouche Correspondence: amar_debbouche@yahoo.fr Department of Mathematics, Faculty of Science, Guelma University Guelma, Algeria Abstract In this article, sufficient conditions for the existence result of quasilinear multi-delay integro-differential equations of fractional orders with nonlocal impulsive conditions in Banach spaces have been presented using fractional calculus, resolvent operators, and Banach fixed point theorem. As an application that illustrates the abstract results, a nonlocal impulsive quasilinear multi-delay integro-partial differential system of fractional order is given. AMS Subject Classifications. 34K05, 34G20, 26A33, 35A05. Keywords: Fractional integrodifferential systems, resolvent operators, nonlocal and impulsive conditions, fixed point theorem Introduction Many fractional models can be represented by the following system d α u(t ) dt α + A(t , u(t ))u(t)=f (t, u(t), u(β(t ))) + t  0 g(t, s , u(s), u(γ (s))) ds , (1:1) u ( 0 ) + h ( u ) = u 0 , (1:2) u ( t i ) = I i ( u ( t i )), (1:3) in a Banach space X, where 0 <a ≤ 1, t Î [0, a], u 0 Î X, i = 1, 2, , m and 0 <t 1 <t 2 < ··· <t m <a. We assume that -A(t,.) is a closed linear operator defined on a dense domain D(A)inX into X such that D(A) is independent of t. It is assumed also that -A(t,.) gen- erates an evolution operator in the Bana ch space X. The functions f : JX r+1 ® X, g : Λ ×X k+1 ® X, h : PC(J, X) ® X, u(b)=(u(b 1 ), , u(b r )), u(g)=(u(g 1 ), , u(g k )), and b p , g q : J ® J are given, where p = 1, 2, , r and q = 1, 2, , k. Here J =[0,a]andΛ ={(t, s). 0 ≤ s ≤ t ≤ a}. Let PC (J, X) consist of functions u from J into X, such that u(t)iscon- tinuous at t ≠ t i and left continuous at t = t i andtherightlimit u (t + i ) exists for i =1, 2, , m.ClearlyPC(J, X) is a Banach space with the norm ||u|| PC =sup tÎJ ||u(t)||, and let u(t i )=u(t + i ) − u(t − i ) constitutes an impulsive condition. Fractional differential equations have proved to be valuable tools in the modelling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [1-5]). They involve a wide area of applications by bringing into a broader paradigm concepts Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 © 2011 D ebbouche; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attrib ution License (http://creativ ecom mons.org/license s/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. of physics and mathematics [6-8]. There has been a significant development in frac- tional differential and partial differential equations in recent years, see Kilbas et al. [9,10], also in fractional nonlinear systems with delay and fractional variational princi- ples with delay, see Baleanu et al. [11,12]. The existence results to evolution equations with nonlocal conditions in Banach space was studied first by By szewski [13,14], subsequently, many author s were pointed in the same field, see reference therein. Deng [15] indicated that, using the nonlocal condition u(0) + h(u)=u 0 to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result t han using the usual local Cauchy problem u(0) = u 0 . Let us observe also that since Deng’s papers, the func- tion h is considered h(u)= p  k =1 c k u(t k ) , where c k , k = 1, 2, , p are given constants and 0 ≤ t 1 <···<t p ≤ a.However,among the previous research on nonlocal cauchy problems, few are concerned with mild solu- tions of fractional semilinear differential equations, see Mophou and N’Guérékata [16], and others with fractional nonlocal boundary value problems, for instance, Ahmad et al. [17,18]. The theory of impulsive differential equations has been emerging as an important area of investigation in recent years, because all the structures of its emergence have deep physical background a nd realistic mathematica l model. The theory of impulsive differential equations appears as a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. It has seen co nsiderable development in the last decade, see the mono- graphs of Bainov and Simeonov [19], Lakshmikantham et al. [20], and Samoilenko and Perestyuk [21] where numerous properties of their solutions are studied, and detailed bibliographies are given. Recently, the existence of solutions of fractional abstract differential equations with nonlocal initial condition was investigated by N’ Guérékata [22] and Li [23]. Much attention has been paid to existence results for the impulsive differential and integro- differential equations of fractional order in abstract spaces, see Benchohra et al. [2,24]. Several authors have studied the existence of solutions of abstract quasilinear evolution equations in Banach space [25-27]. Regarding this article, it generalizes previous results concerned the existence of solu- tions to nonlocal and impulsiv e integrodifferential equati ons of quasilinear type with delays of arbitrary orders. Section “Preliminaries” is devoted to a review of some essen- tial results. In next section, we state and prove our main results, the last section deals to giving an example to illustrate the abstract results. 1 Preliminaries Let X and Y be two Banach spaces such that Y is dense ly and continuously embedded in X. For any Banach space Z,thenormofZ is denoted by ||·|| Z .Thespaceofall bounded linear operators from X to Y is denoted by B(X, Y)andB(X, X) is written as B(X). We recall some definitions in fractional calculus from Gelfand-Shilov [28] and Podlubny [29], then some known facts of the theory of semigroups from Pazy [30]. Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 2 of 10 Definition 2.1 The fractional integral of order with the lower limit zero for a func- tion f Î C([0, ∞)) is defined as I α f (t)= 1 (α) t  0 f (s) (t − s) 1−α ds, t > 0, 0 <α<1 , provided the right side is pointwise defined on [0, ∞), where Γ is the gamma func- tion. Riemann-Liouville derivative of order a with the lower limit zero for a functio n f Î C([0, ∞)) can be written as L D α f (t)= 1 (1 − α) d dt t  0 f (s) (t − s) α ds, t > 0, 0 <α<1 . The Caputo derivative of order for a function f Î C([0, ∞)) can be written as C D α f ( t ) = L D α ( f ( t ) − f ( 0 )) , t > 0, 0 <α <1 . Remark 2.1 (1) If f Î C 1 ([0, ∞)), then C D α f (t)= 1 (1 − α) t  0 f  (s) (t − s) α ds = I 1−α f  (t ), t > 0, 0 <α <1 . (2) The Caputo derivative of a constant is equal to zero. (3) If f is an abstract function with values in X, then integrals which appear in Defini- tion 2.1 are taken in Bochner’s sense. Definition 2.2 A two parameter family of bounded linear operators U(t, s), 0 ≤ s ≤ t ≤ a,onX is called an evolution system if the following two conditions are satisfied (i) U(t, t)=I, U(t, r)U(r, s)=U(t, s) for 0 ≤ s ≤ r ≤ t ≤ a, (ii) (t, s) ® U(t, s) is strongly continuous for 0 ≤ s ≤ t ≤ a. More detail about evolution system and quasilinear equation of evolution can be found in [30, Chap. 5 and Sect. 6.4, respectively]. Let E be the Banach space formed from D(A) with the graph norm. Since - A(t)isa closed operator, it follows that - A(t) is in the set of bounded operators from E to X. Definit ion 2.3 [31-33] A resolvent operators for problem (1.1)-(1.3) is a bounded operators valued function R u (t, s) Î B(X), 0 ≤ s ≤ t ≤ a, the space of bounded linear operators on X, having the following properties: (i) R u (t, s) is strongly continuous in s and t, R u (s, s)=I,0≤ s ≤ a,||R u (t, s)|| ≤ Me N (t, s) for some constants M and N. (ii) R u (t, s)E ⊂ E, R u (t, s) is strongly continuous in s and t on E. (iii) For x Î X, R u (t, s)x is continuously differentiable in s Î [0, a] and ∂R u ∂s (t , s)x = R u (t , s)A(s, u(s))x . (iv) For x Î X and s Î [0, a], R u (t, s)x is continuously differentiable in t Î [s, a] and ∂R u ∂ t (t , s)x = −A(t, u(t))R u (t , s)x , Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 3 of 10 with ∂R u ∂ s (t , s) x and ∂R u ∂ t (t , s) x are strongly continuous on 0 ≤ s ≤ t ≤ a. Here R u (t, s) can be extracted from the evolution operato r of the generator - A(t, u). The resolvent operator is similar to the evolution operator for nonaut onomous differential equations in a Banach space. Let Ω be a subset of X. Definition 2.4 (Compare [31] with [7,22,34]) By a mild solution of (1.1)-(1.3) we mean a function u Î PC(J : X) with values in Ω satisfying the integral equation u (t)=R u (t,0)u 0 − R u (t,0)h(u) + 1 (α) t  0 (t − s) α−1 R u (t, s)[f (s, u(s), u(β( s))) + s  0 g(s, η, u(η), u(γ (η)))dη]d s +  0<t i <t R u (t, t i )I i (u(t i )), t ∈ J (2:1) for all u 0 Î X. Definition 2.5 (Compare [35,36] with [2]) By a classical solution of (1.1)-(1.3) on J, we mean a function u with values in X such that: (1) u is continuous function on J \{t 1 , t 2 , , t m } and u(t) Î D(A), (2) d α u dt α exists and continuous on J 0 ,0<a <1, (3) u satisfies (1.1) on J 0 , the nonlocal condition (1.2) and the impulsive condition (1.3), where J 0 = (0, a]\{t 1 , t 2 , , t m }. We assume the following conditions (H 1 ) h : PC(J : Ω) ® Y is Lipschitz continuous in X and bounded in Y , i.e., there exist constants k 1 > 0 and k 2 > 0 such that ||h(u)|| Y ≤ k 1 , | |h(u) − h(v)|| Y ≤ k 2 max t∈ J ||u − v|| PC , u, v ∈ PC(J : X) . For the conditions (H 2 ) and (H 3 ) let Z be taken as both × and Y. (H 2 ) g : Λ × Z k+1 ® Z is continuous and there exist constants k 3 > 0 and k 4 > 0 such that t  0 ||g(t, s, u 1 , , u k+1 ) − g(t, s, v 1 , , v k+1 )|| Z ds ≤ k 3 k+1  q=1 ||u q − v q || Z , u q , v q ∈ X, q =1, , k +1 , k 4 =max  t  0 ||g(t, s,0, ,0)|| Z ds :(t, s) ∈   . (H 3 ) f : J ×Z r+1 ® Z is continuous and the re exist constants k 5 > 0 an d k 6 > 0 such that ||f (t, u 1 , , u r+1 ) − f (t, v 1 , , v r+1 )|| Z ≤ k 5 r+1  p=1 ||u p − v p || Z , u p , v p ∈ X, p =1, , r +1 , k 6 =max t∈ J ||f (t,0, ,0)|| Z . (H 4 ) b p , g q : J ® J are bijective absolutely continuous and there exist constants c p >0 and b q > 0 such that β  p (t ) ≥ c p and γ  q (t ) ≥ b q , respectively, for t Î J, p = 1, , r and q = 1, , k. (H 5 ) I i : X ® X are continuous and there exist constants l i >0,i = 1, 2, , m such that ||I i ( u ) − I i ( v ) || ≤ l i ||u − v||, u, v ∈ X . Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 4 of 10 Let us take M 0 = max ||R u (t, s)|| B(Z) ,0≤ s ≤ t ≤ a, u Î Ω. (H 6 ) There exist positive constants δ 1 , δ 2 , δ 3 Î (0, δ /3] and l 1 , l 2 , l 3 Î [0, 1 3 )such that δ 1 = M 0 ||u 0 || Y + M 0 k 1 , δ 2 = M 0 θ, δ 3 = M 0 ξ, and λ 1 = Ka||u 0 || Y + k 1 Ka + M 0 k 2 , λ 2 = Kaθ + M 0 σ [k 5 (1+1/c 1 + ···+1/c r )+k 3 (1+1/b 1 + ···+1/b k )] , λ 3 = Kaξ + M 0 m  i =1 l i , where r = s [k 5 (1/c 1 + ··· +1/c r )+ k 3 (1/b 1 + ··· +1/b k )], θ = sδ (k 3 + k 5 )+ rδ + s (k 4 + k 6 ), σ = a α  ( 1+α ) and ξ = m  i =1 (l i δ + ||I i (0)|| ) . Main results Lemma 3.1 Let R u (t, s) the resolvent operators for the fractional problem (1.1)-(1.3). There exists a constant K > 0 such that | |R u (t , s)ω − R v (t , s)ω|| ≤ K||ω|| Y t  s ||u(τ ) − v(τ )||dτ, for every u, v Î PC(J : X) with values in Ω and every ω Î Y , see [30, lemma 4.4, p. 202]. Let S δ ={u : u Î PC(J : X), u(0) + h(u)=u 0 , Δu(t i )=I i (u(t i )), ||u|| ≤ δ}, for t Î J, δ > 0, u 0 Î X and i = 1, , m. Lemma 3.2 | |ϕ ( t ) || Y ≤ θ , where ϕ(t)= 1 (α) t  0 (t − s) α−1 ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, τ , u(τ ), u(γ (τ )))dτ ⎤ ⎦ ds . Proof We have | |ϕ(t)|| Y ≤ 1 (α) t  0 (t − s) α−1 [||f (s, u(s), u(β 1 (s)), , u(β r (s)) − f (s,0, ,0)|| + || f (s,0, ,0)|| + s  0 ||g(s, τ , u(τ ), u(γ 1 (τ )), , u(γ k (τ )) − g(s, τ,0, ,0)||dτ + s  0 ||g(s, τ ,0, ,0)||dτ ⎤ ⎦ ds . Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 5 of 10 Using H 2 ,H 3 , and H 4 , we get ||ϕ(t ) || Y ≤ 1 (α) t  0 (t − s) α−1 [k 5 (||u(s)|| + ||u(β 1 (s))|| + ···+ ||u(β r (s))||)+k 6 + k 3 (||u(s)|| + ||u(γ 1 (s))|| + ···+ ||u(γ k (s))||)+k 4 ]ds ≤ 1 (α) t  0 (t − s) α−1 [k 5 {δ + ||u(β 1 (s))||(β  1 (s)/c 1 )+···+ ||u(β r (s))||(β  r (s)/c r )} + k 6 + k 3 {δ + ||u(γ 1 (s))||(γ  1 (s)/b 1 )+···+ ||u(γ k (s))||(γ  k (s)/b k )} + k 4 ]ds ≤ σδ(k 3 + k 5 )+σ (k 4 + k 6 ) + k 5 c 1 (α) β 1 (t)  β 1 (0) (t − β −1 1 (τ )) α−1 ||u(τ )||dτ + ···+ k 5 c r (α) β r (t)  β r (0) (t − β −1 r (τ )) α−1 ||u(τ )||dτ + k 3 b 1 (α) γ 1 (t)  γ 1 ( 0 ) (t − γ −1 1 (η)) α−1 ||u(η)||d η + ···+ k 3 b k (α) γ k (t)  γ k ( 0 ) (t − γ −1 k (η)) α−1 ||u(η)||d η . Hence the required result. Theorem 3.3 Suppose that the operator -A(t, u) generates the resol vent operator R u (t, s) with ||R u (t, s)||≤ Me N(t-s) . If the hypotheses (H 1 )-(H 6 ) are satisfied, then the frac- tional integro-differential equation (1.1) with nonlocal condition (1.2) and impulsive condition (1.3) has a unique mild solution on J for all u 0 Î X. Proof Consider a mapping P on S δ defined by (Pu)(t)=R u (t,0)u 0 − R u (t,0)h(u) + 1 (α) t  0 (t − s) α−1 R u (t, s) ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, η, u(η), u(γ (η)))dη ⎤ ⎦ d s +  0<t i <t R u (t, t i )I i (u(t i )). We shall show that P : S δ ® S δ . For u Î S δ , we have ||Pu(t)|| Y ≤||R u (t,0)u 0 || + ||R u (t,0)h(u)|| +       1 (α) t  0 (t − s) α−1 R u (t, s) ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, η, u(η), u(γ (η)))dη ⎤ ⎦ ds       +  0<t i <t ||R u (t, t i )||(||I i (u(t i )) − I i (0)|| + ||I i (0)||). Using H 1 , Lemma 3.2 and H 5 , we get ||Pu(t) Y || ≤ M 0  ||u 0 || + k 1 + θ + m  i=1 (l i δ + ||I i (0)||)  . From assumption H 6 , one gets ||(Pu μ )(t)|| Y ≤ δ. Thus, P maps S δ into itself. Now for u, v Î S δ , we have ||Pu ( t ) − Pv ( t ) || ≤ I 1 + I 2 + I 3 , where I 1 = ||R u (t,0)u 0 − R v (t,0)u 0 || + ||R u (t,0)h(u) − R v (t,0)h(v)||, I 2 = 1 (α) t  0 (t − s) α−1 ||R u (t, s) ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, η, u(η), u(γ (η)))dη ⎤ ⎦ − R v (t, s)[f (s, v(s), v(β(s))) + s  0 g(s, η, v(η), v(γ (η)))dη]||ds Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 6 of 10 and I 3 = m  i =1 ||R u (t , t i )I i (u(t i )) − R v (t , t i )I i (v(t i ))|| . Applying Lemma 3.1 and H 1 , we get I 1 ≤||R u (t ,0)u 0 − R v (t ,0)u 0 || + ||R u (t ,0)h(u) − R v (t ,0)h(u)| | + ||R v (t ,0)h(u) − R v (t ,0)h(v)|| ≤{Ka||u 0 || Y + k 1 Ka + M 0 k 2 } max τ ∈ J ||u(τ ) − v(τ )||. Also, we apply Lemmas 3.1,3.2, H 2 ,H 3 ,H 4 , and H 6 , we obtain I 2 ≤ 1 (α) t  0 (t − s) α−1 ⎧ ⎨ ⎩       R u (t, s) ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, η, u(η), u(γ (η)))dη ⎤ ⎦ − R v (t, s) ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, η, u(η), u(γ (η)))dη ⎤ ⎦       +       R v (t, s) ⎡ ⎣ f (s, u(s), u(β(s))) + s  0 g(s, η, u(η), u(γ (η)))dη ⎤ ⎦ − R v (t, s) ⎡ ⎣ f (s, v(s), v(β(s))) + s  0 g(s, η, v(η), v(γ (η)))dη ⎤ ⎦       ⎫ ⎬ ⎭ ds ≤ Kaθmax τ ∈J ||u(τ ) − v(τ )|| + M 0 1 (α) t  0 (t − s) α−1 ⎧ ⎨ ⎩ k 5 ⎡ ⎣ ||u(s) − v(s)|| + r  p=1 ||u(β p (s)) − v(β p (s))||(β  p (s)/c p ) ⎤ ⎦ + k 3 ⎡ ⎣ ||u(s) − v(s)|| + k  q=1 ||u(γ q (s)) − v(γ q (s))||(γ  q (s)/b q ) ⎤ ⎦ ⎫ ⎬ ⎭ ds ≤ Kaθmax τ ∈J ||u(τ ) − v(τ )|| + M 0 σ [k 5 (1+1/c 1 + ···+1/c r )+k 3 (1+1/b 1 + ···+1/b k )]max τ ∈ J ||u(τ ) − v(τ )||. Again, Lemma 3.1, H 5 and H 6 , we have I 3 ≤ m  i=1 {||R u (t, t i )I i (u(t i )) − R v (t, t i )I i (u(t i ))|| + ||R v (t, t i )I i (u(t i )) − R v (t, t i )I i (v(t i ))|| } ≤  K m  i=1 (l i δ + ||I i (0)||) a + M 0 m  i=1 l i  max τ ∈J ||u(τ ) − v(τ )||. It follows from these estimations that | |Pu(t) − Pv(t) || ≤ λ max τ ∈ J ||u(τ ) − v(τ )||, where 0 ≤ l < 1. Thus P is a contraction on S δ . From the contraction mapping theo- rem, P has a unique fixed point u Î S δ which is the mild solution of (1.1)-(1.3) on J. Theorem 3.4 Assume that (i) Conditions (H 1 )-(H 6 ) hold, (ii) Y is a reflexive Banach space with norm ||·||, (iii) The functions f and g are uniformly Hölder continuous in t Î J. Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 7 of 10 Then the problem (1.1)-(1.3) has a unique classical solution on J. Proof From (i), applying Theorem 3.3, the problem (1.1)-(1.3) has a unique mild solu- tion u Î S δ Set ω(t)=f (t, u(t), u(β(t))) + t  0 g(t, s , u(s), u(γ (s))) ds . In order to prove the regularity of the mild solution, we use the further assumptions, it is easy to conclude that the function ω(t) is also uniformly Hölder continuous in t Î J. Consider the following fractional differential equation d α v(t) dt α + A(t, u)u(t)=ω(t) , (3:1) with the nonlocal condition (1.2) and impulsive condition (1.3). According to Pazy [30], the late problem has a unique solution v on J intoX given by v(t)=R u (t ,0)u 0 − R u (t ,0)h(u)+ 1 (α) t  0 (t − s) α−1 R u (t , s)ω(s)d s +  0<t i <t R u (t , t i )I i (u(t i )). Noting that, each term on the right-hand side belongs to D(A), using the uniqueness of v(t), we have that u(t) Î D(A). It follows that u is a unique classical solution of (1.1)-(1.3) on J. Application Consider the nonlinear integro-partial differential equation of fractional order ∂ α u(x, t) ∂t α +  |q|≤2m a q (x, t)u(x, t)D q x u(x, t)=F(x, t, u, w 1 )+ t  0 G(x, t, s, u(x, s), w 2 (s))ds , (4:1) u (x,0)+ p  k =1 c k u(x, t k )=g( x ) , (4:2) u(x, t k )=  R n ρk(y, x)u(y, t k )dy , (4:3) where 0 <a ≤ 1, 0 ≤ t 1 < ··· <t p ≤ a, x Î R n , D q x = D q 1 x 1 D q n x n , D x i = ∂ ∂x i , q=(q 1 , ,q n )is an n-dimensional multi-index, |q|=q 1 + ··· + q n , and w i , i = 1, 2, is given by w i (x, t)=  |q|≤2m−1 b qi (x, t)D q x u(x,sint)+    |q|≤2m−1 c q i (x, t)D q y u(y,sint)dy . Let L 2 (R n ) be the set of all square integrable functions on R n .WedenotebyC m (R n ) the set of all continuous real-valued functions defined on R n which have continuous part ial derivatives of order less than or equal to m.By C m 0 (R n ) we denote the set of all Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 8 of 10 functions f Î C m (R n ) with compact supports. Let H m (R n ) be t he completio n of C m 0 (R n ) with respect to the norm ||f || 2 m =  |q|≤m  R n |D q x f (x)| 2 dx . It is supposed that (i) The operator A(t , u)=−  | q |≤2m a q (x, t)u(x, t)D q x is uniformly elliptic on R n .In other words, all the coefficients a q ,|q|=2m, are continuous and bounded on R n and there is a positive number c such that (−1) m+1  | q |=2m a q (x, t)u(x, t)ξ q ≥ c|ξ | 2m , for all x Î R n and all ξ ≠ 0, ξ Î R n , ξ q = ξ q 1 1 ξ q n n and | ξ| 2 = ξ 2 1 + + ξ 2 n . (ii) All the coefficients a q ,|q|=2m, satisfy a uniform Hölder condition on R n . Under these conditions the operator A with domain of definition D(A)=H 2m (R n ) generates an evolution operator defined on L 2 (R n ), and i t is well known that H 2m (R n ) is dense in X = L 2 (R n ) and the initial function g(x) is an element in Hilbert space H 2m ( R n ), see [14,15,35]. Applying Theorem 3.3, this achieves the proof of the existence of mild solu- tions of the system (4.1)-(4.3). In addition, (iii) If the coefficients b q , c q ,|q| ≤ 2m - 1 satisfy a uniform Hölder condition on R n and the operators F and G satisfy There are numbers L 1 , L 2 ≥ 0 and l 1 , l 2 Î (0, 1) such that  |q|≤2m−1  R n |F(x, t, u, D q x w 1 ) − F(x, s, u, D q x w ∗ 1 )| 2 dx ≤ L 1 (|t − s| λ 1 + |w 1 − w ∗ 1 | 2 dx) . and  |q|≤2m−1  R n | G(x, t, η, u, D q x w 2 ) − G(x, s, η, u, D q x w 2 )| 2 dx ≤ L 2 |t − s| λ 2 . for all t, s Î I,(t, h), (s, h) Î Δ,andallx Î R n . Applying Theorem 3.4, we deduce that (4.1)-(4.3) has a unique strong solution. Competing interests The author declare that he has no competing interests. Received: 15 December 2010 Accepted: 24 May 2011 Published: 24 May 2011 References 1. Agrawal OP, Defterli O, Baleanu D: Fractional optimal control problems with several state and control variables. J Vibr Control 2010, 16(13):1967-1976. 2. Benchohra M, Slimani BA: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron J Diff Eqns 2009, 10:1-11. 3. Hilfer R: Applications of fractional calculus in physics. World Scientific, Singapore 2000. 4. Li F, N’Guerekata GM: Existence and uniqueness of mild solution for fractional integrodifferential equations. Adv Differ Equ 2010, 10, Article ID 158789. 5. Oldham KB, Spanier J: The fractional calculus. Academic Press, New York, London; 1974. 6. 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Advances in Difference Equations 2011 2011:5. Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Debbouche Advances in Difference Equations 2011, 2011:5 http://www.advancesindifferenceequations.com/content/2011/1/5 Page 10 of 10 . existence result of quasilinear multi-delay integro-differential equations of fractional orders with nonlocal impulsive conditions in Banach spaces have been presented using fractional calculus,. RESEARC H Open Access Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems Amar Debbouche Correspondence: amar_debbouche@yahoo.fr Department. results, a nonlocal impulsive quasilinear multi-delay integro-partial differential system of fractional order is given. AMS Subject Classifications. 34K05, 34G20, 26A33, 35A05. Keywords: Fractional