RESEARCH Open Access Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions Ahmed El-Sayed 1* and Fatma Gaafar 2 * Correspondence: amasayed5@yahoo.com 1 Faculty of Science, Alexandria University, Alexandria, Egypt Full list of author information is available at the end of the article Abstract In this paper, we establish sufficient conditions for the existence of a unique solution for a class of nonlinear non-autonomous system of Riemann-Liouville fractional differential systems with different constant delays and non-local condition is. The stability of the solution will be proved. As an application, we also give some examples to demonstrate our results. Keywords: Riemann-Liouville derivatives, nonlocal non-autonomous system, time- delay system, stability analysis 1 Introduction Here we consider the nonlinear non-local problem of the form D α x i ( t ) = f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) , t ∈ ( 0, T ) , T < ∞ , (1) x(t)=(t)fort < 0 and lim t → 0 − (t)=0 , (2) I 1−α x ( t ) | t=0 =0 , (3) where D a denotes the Riemann-Liouville fractional derivative of order a Î (0, 1), x(t) =(x 1 (t), x 2 (t), , x n (t))’,where‘ denote the transpose of the matrix, and f i , g i :[0,T]× R n ® R are continuous functions, F(t)=(j i (t)) n ×1 are given matrix and O is the zero matrix, r j ≥ 0, j = 1, 2, , n, are constant delays. Recently, much attention has been paid to the existence of solution for fractional dif- ferential equations because they have appl ications in various fields of science and engi- neering. We can describe many physical and chemical processes, biological systems, etc., by fractional differential equations (see [1-9] and references therein). In this work, we discuss the existence, uniqueness and uniform of the solution of sta- bility non-local problem (1)-(3). Furthermore, as an application, we give some exam- ples to demonstrate our results. For the earlier work we mention: De la Sen [10] investigated the non-negative solu- tion and the stability and asymptotic properties of the solution of fractional differential dynamic systems involving delayed dynamics with point delays. El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 © 2011 El-Sayed and Gaafar; licensee Springer. This is an Open Access article distributed under the terms of the Creative Comm ons Attribution License (http://creativecom mons.org/licenses/by/2.0), which permits unrestricted use, distribu tion, and reproduction in any medium, provided the original work is properly cited. El-Sayed [11] proved the existence and uniqueness of the solution u(t)ofthepro- blem c D α a u(t )+C c D β a u(t − r)=Au(t)+Bu(t − r), 0 ≤ β ≤ α ≤ 1 , u ( t ) = g ( t ) , t ∈ [a − r, a], r > 0 by the method of steps, where A, B, C are bounded linear operators defined on a Banach space X. El-Sayed et al. [12] proved the existence of a unique uniformly stable solution of the non-local problem D α x i (t )= n j=1 a ij (t ) x j (t )+ n j=1 b ij (t ) x j (t − r j )+h i (t ), t > 0, x(t)=(t)fort < 0, lim t → 0 − (t)=O and I β x(t)| t=0 = O, β ∈ (0, 1) . Sabatier et al. [6] delt with Linear Matrix Inequality (LMI) stability conditions for fractional order systems, under commensurate order hypothesis. Abd El-Salam and El-Sayed [13] proved the existence of a unique uniformly stable solution for the non-autonomous system c D α a x(t)=A(t)x(t)+f (t), x(0) = x 0 , t > 0 , where c D α a is the C aputo fractional derivatives (see [5-7,14]), A(t)andf(t) are contin- uous matrices. Bonnet et al. [15] analyzed several properties linked to the robust control of frac- tional differential systems with delays. They delt with the BIBO stability of both retarded and neutral fractional delay systems. Zhang [16] established the existence of a unique solution for the delay fractional differential equation D α x ( t ) = A 0 x ( t ) + A 1 x ( t − r ) + f ( t ) , t > 0, x ( t ) = φ ( t ) , t ∈ [−r,0] , by the method of steps, where A 0 , A 1 are constant matrices and studied the finite time stability for it. 2 Preliminaries Let L 1 [a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0 ≤ a <b < ∞ with the norm | |x|| L 1 = b a |x(t) |d t . Definition 1 The fractional (arbitrary) order integral of the function f(t) Î L 1 [a, b]of order a Î R + is defined by (see [5-7,14,17]) I α a f (t)= t a (t − s) α−1 ( α ) f (s)ds , where Γ (.) is the gamma function. Definition 2 The Caputo fractional (arbitrary) order derivatives of order a, n <a <n + 1, of the function f(t) is defined by (see [5-7,14]), c D α a f (t)=I n−α a D n f (t)= 1 ( n − α ) t a (t − s) n−α−1 f (s)ds, t ∈ [a, b] , El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 2 of 8 Definition 3 The Riemann-liouville fractional (arbitrary) order derivatives of order a, n <a <n + 1 of the function f (t) is defined by (see [5-7,14,17]) D α a f (t)= d n dt n I n−α a f (t)= 1 ( n − α ) d n dt n t a (t − s) n−α−1 f (s)ds, t ∈ [a, b] , The following theorem on the properties of fractional order integration and differen- tiation can be easily proved. Theorem 1 Let a, b Î R + . Then we have (i) I α a : L 1 → L 1 , and if f(t) Î L 1 then I α a I β a f (t)=I α+β a f (t ) . (ii) lim α→ n I α a = I n a , n = 1,2,3, uniformly. (iii) c D α f (t)=D α f (t) − (t − a) − α ( 1 − α ) f (a ) , a Î (0,1), f (t) is absolutely continuous. (iv) lim α→1 c D α a f (t)= df dt = lim α→1 D α f (t ) , a Î (0,1), f (t) is absolutely continuous. 3 Existence and uniqueness Let X =(C n (I), || . || 1 ), where C n (I) is the class of all continuous column n-vectors function. For x Î C n [0, T], the norm is defined by ||x|| 1 = n i=1 sup t∈ [ 0,T ] {e −Nt |x i (t ) | } , where N >0. Theorem 2 Let f i , g i :[0,T]×R n ® R be continuous functions and satisfy the Lipschitz conditions | f i (t , u 1 , , u n ) − f i (t , v 1 , , v n ) ≤ n j=1 h ij |u j − v j |, | g i (t , u 1 , , u n ) − g i (t , v 1 , , v n )|≤ n j =1 k ij |u j − v j | , and h = n i=1 |h i | = n i=1 max ∀ j |h i j | , k = n i=1 |k i | = n i=1 max ∀ j |k i j | . Then there exists a unique solution × Î X of the problem (1)-(3). Proof Let t Î (0, T). Then equation (1) can be written as d dt I 1−α x i (t )=f i (t , x 1 (t ), , x n (t )) + g i (t , x 1 (t − r 1 ), , x n (t − r n ) . Integrating both sides, we obtain I 1−α x i (t) −I 1−α x i (t)| t=0 = t 0 {f i (t, x 1 (t), , x n (t)) +g i (t, x 1 (t −r 1 ), , x n (t −r n ))}ds . From (3), we get I 1−α x i (t )= t 0 {f i (t , x 1 (t ), , x n (t )) + g i (t , x 1 (t − r 1 ), , x n (t − r n ))}ds . Operating by I a on both sides, we obtain Ix i ( t ) = I α+1 {f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) } . El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 3 of 8 Differentiating both side is, we get x i ( t ) = I α {f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) }, i =1,2, , n . (4) Now let F : X ® X, defined by Fx i = I α {f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) } . then | Fx i − Fy i | = |I α {f i (t, x 1 (t), , x n (t)) − f i (t, y 1 (t), , y n (t)) + g i (t, x 1 (t − r 1 ), , x n (t − r n )) − g i (t, y 1 (t − r 1 ), , y n (t − r n ))}| ≤ t 0 (t − s) α−1 (α) |f i (s, x 1 (s), , x n (s)) − f i (s, y 1 (s), , y n (s))|ds + t 0 (t − s) α−1 (α) |g i (s, x 1 (s − r 1 ), , x n (s − r n )) − g i (s, y 1 (s − r 1 ), , y n (s − r n ))|d s ≤ t 0 (t − s) α−1 (α) n j=1 h ij |x j (s) − y j (s)|ds + t 0 (t − s) α−1 (α) n j =1 k ij |x j (s − r j ) − y j (s − r j )|ds and e −Nt |Fx i − Fy i |≤h i n j=1 t 0 (t − s) α− 1 (α) e −N(t−s) e −Ns |x j (s) − y j (s)|ds + k i n j=1 t r j (t − s) α−1 (α) e −N(t−s+r j ) e −N(s−r j ) |x j (s − r j ) − y j (s − r j )|d s ≤ h i n j=1 sup t {e −Nt |x j (t) − y j (t)|} t 0 (t − s) α−1 (α) e −N(t−s) ds + k i n j=1 sup t {e −Nt |x j (t) − y j (t)|}e −Nr j t r j (t − s) α−1 (α) e −N(t−s) ds ≤ h i n j=1 sup t {e −Nt |x j (t) − y j (t)|} 1 N α Nt 0 u α−1 e −u (α) du + k i n j=1 sup t {e −Nt |x j (t) − y j (t)|} e −Nr j N α N(t−r j ) 0 u α−1 e −u (α) du ≤ h i N α ||x − y|| 1 + k i N α n j=1 sup t {e −Nt |x j (t) − y j (t)|} ≤ h i + k i N α ||x − y|| 1 and | |Fx − Fy|| 1 = n i=1 sup t e −Nt |Fx i − Fy i |≤ n i=1 h i + k i N α ||x − y|| 1 ≤ h + k N α ||x − y|| 1 . Now choose N largeenoughsuchthat h+k N α < 1 ,sothemapF : X ® X is a contrac- tion and hence, there exists a unique column vector x Î X which is the solution of the integral equation (4). El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 4 of 8 Now we complete the proof b y proving the equivalence between the integral equa- tion (4) and the non-local problem (1)-(3). Indeed: Since x Î C n and I 1-a x(t) Î C n (I), and f i , g i Î C(I)thenI 1-a f i (t), I 1-a g i (t) Î C(I). Operating by I 1-a on both sides of (4), we get I 1−α x i (t )=I 1−α I α {f i (t , x 1 (t ), , x n (t )) + g i (t , x 1 (t − r 1 ), , x n (t − r n )) } = I {f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) }. Differentiating both sides, we obtain DI 1−α x i ( t ) = DI{f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) } , which implies that D α x i ( t ) = f i ( t, x 1 ( t ) , , x n ( t )) + g i ( t, x 1 ( t − r 1 ) , , x n ( t − r n )) , t > 0 , which completes the proof of the equivalence between (4) and (1). Now we prove that lim t → 0 + x i =0 .Sincef i (t, x 1 (t), , x n (t)), g i (t, x 1 (t-r 1 ), , x n (t- r n )) are continuous on [0, T] then there exist constants l i , L i , m i , M i such that l i ≤ f i (t, x 1 (t), , x n (t)) ≤ L i and m i ≤ g i (t, x 1 (t - r 1 ) ), , x n (t-r n )) ≤ M i , and we have I α f i (t , x 1 (t ), , x n (t )) = t 0 (t − s) α− 1 (α) f i (s, x 1 (s), , x n (s))ds , which implies l i t 0 (t − s) α−1 (α) ds ≤ I α f i (t , x 1 (t ), , x n (t )) ≤ L i t 0 (t − s) α−1 (α) ds ⇒ l i t α ( α +1 ) ≤ I α f i (t , x 1 (t ), , x n (t )) ≤ L i t α ( α +1 ) and lim t → 0 + I α f i (t , x 1 (t ), , x n (t )) = 0 . Similarly, we can prove lim t → 0 + I α g i (t , x 1 (t − r 1 ), , x n (t − r n )) = 0 . Then from (4), lim t→0 + x i ( t ) = 0 . Also from (2), we have lim t→0 − ( t ) = O . Now for t Î (-∞, T], T < ∞, the continuous solution x(t) Î (-∞, T] of the problem (1)-(3) takes the form x i (t)= ⎧ ⎪ ⎨ ⎪ ⎩ φ i (t), t < 0 0, t =0 t 0 (t−s) α−1 (α) {f i (s, x 1 (s), , x n (s)) + g i (s, x 1 (s − r 1 ), , x n (s − r n ))}ds, t > 0 . 4 Stability In this section we study the stability of the solution of the non-local problem (1)-(3) Definition 5 The solution of the non-auto nomous linear system (1) is stable if for any ε > 0, there exists δ > 0 such that for any two solutions x(t)=(x 1 (t), x 2 (t), , x n (t))’ and ˜ x ( t ) = ( ˜ x 1 ( t ) , ˜ x 2 ( t ) , , ˜ x n ( t )) with the initial conditions (2)-(3) and El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 5 of 8 ||x ( t ) − ˜ x ( t ) || 1 < ε respectively, one has || ( t ) − ˜ ( t ) || 1 ≤ δ ,then ||x ( t ) − ˜ x ( t ) || 1 < ε for all t ≥ 0. Theorem 3 The solution of the problem (1)-(3) is uniformly stable. Proof Let x(t)and ˜ x ( t ) be two solutions of the system (1) under conditions (2)-(3) and {I β ˜ x ( t ) | t=0 =0, ˜ x ( t ) = ˜ ( t ) , t < 0 and lim t→0 ˜ ( t ) = O } , respectively. Then for t >0, we have from (4) | x i − ˜ x i | = |I α {f i (t, x 1 (t), , x n (t)) − f i (t, ˜ x 1 (t), , ˜ x n (t)) + g i (t, x 1 (t − r 1 ), , x n (t − r n )) − g i (t, ˜ x 1 (t − r 1 ), , ˜ x n (t − r n ))}| ≤ t 0 (t − s) α−1 (α) | f i (s, x 1 (s), , x n (s)) − f i (s, y 1 (s), , y n (s))|ds + t 0 (t − s) α−1 (α) |g i (s, x 1 (s − r 1 ), , x n (s − r n )) − g i (s, ˜ x 1 (s − r 1 ), , ˜ x n (s − r n ))|d s ≤ t 0 (t − s) α−1 (α) n j=1 h ij |x j (s) − ˜ x j (s)|ds + t 0 (t − s) α−1 (α) n j =1 k ij |x j (s − r j ) − ˜ x j (s − r j )|ds and e −Nt |x i − ˜ x i |≤h i n j=1 t 0 (t − s) α− 1 (α) e −N(t−s) e −Ns |x j (s) − ˜ x j (s)|ds + k i n j=1 r j 0 (t − s) α−1 (α) e −N(t−s+r j ) e −N(s−r j ) |φ j (s − r j ) − ˜ φ j (s − r j )|d s + k i n j=1 t r j (t − s) α−1 (α) e −N(t−s+r j ) e − N(s−r j ) |x j (s − r j ) − ˜ x j (s − r j )|ds ≤ h i N α ||x j (t) − ˜ x j (t) || 1 Nt 0 u α−1 e −u (α) du + k i n j=1 sup t {e −Nt |φ j (t) − ˜ φ j (t) |} e −Nr j N α Nt N(t−r j ) u α−1 e −u (α) du + k i n j=1 sup t {e −Nt |x j (t) − ˜ x j (t) |} e −Nr j N α N(t−r j ) 0 u α−1 e −u (α) du ≤ h i N α ||x j (t) − ˜ x j (t) || 1 + k i N α n j=1 e −Nr j sup t {e − Nt |x j (t) − ˜ x j (t) |} + k i N α n j=1 e −Nr j sup t {e −Nt |ϕ j (t) − ˜ φ j (t) |} ≤ h i + k i N α ||x − ˜ x|| 1 + k i N α || − ˜ || 1 . Then we have, | |x − ˜ x|| 1 ≤ n i=1 h i + k i N α ||x − ˜ x|| 1 + n i=1 k i N α || − ˜ || 1 ≤ h + k N α ||x − ˜ x|| 1 + k N α || − ˜ || 1 i.e. 1 − h + k N α ||x − ˜ x|| 1 ≤ k N α || − ˜ || 1 and ||x − ˜ x|| 1 ≤ k N α 1 − h + k N α − 1 || − ˜ || 1 El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 6 of 8 Therefore, for δ >0s.t. || − ˜ || 1 < δ , we can find ε = k N α 1 − h+k N α −1 δ s.t. || x − ˜ x || 1 ≤ ε which proves that the solution x(t) is uniformly stable. 5 Applications Example 1 Consider the problem D α x i (t )= n j=1 a ij (t ) x j (t )+ n j=1 g ij (t , x j (t − r j ), t > 0 x(t)=(t)fort < 0and lim t→0 − (t)=O I 1−α x ( t ) | t=0 = O, where A(t)=(a ij (t)) n×n and (g i (t , x 1 (t − r 1 ), , x n (t − r n ))) =( n j =1 g ij (t , x j (t − r j )) are given continuous matrix, then the problem has a unique uniformly stab le solution x Î X on (-∞, T], T < ∞ Example 2 Consider the problem D α x i (t )= n j=1 f ij (t , x j (t )) + n j=1 b ij (t ) x j (t − r j ), t > 0 x(t)=(t)for t < 0and lim t→0 − (t)=O I 1−α x ( t ) | t=0 = O, where B(t)=(b ij (t)) n×n , and (f i (t , x 1 (t ), , x n (t ))) =( n j =1 f ij (t , x j (t ))) are given con - tinuous matrices, then the problem has a unique uniformly stable solution x Î X on (-∞, T], T < ∞ Example 3 Consider the problem (see [12]) D α x i (t )= n j=1 a ij (t ) x j (t )+ n j=1 b ij (t ) x j (t − r j )+h i (t ), t > 0 x(t)=(t)fort < 0and lim t→0 − (t)=O I 1−α x ( t ) | t=0 = O, where A(t)=(a ij (t)) n×n B(t)=(b ij (t)) n×n ,andH(t)=(h i (t)) n×1 are given continuous matrices, then the problem has a unique uniformly stable solution x Î X on (-∞, T], T < ∞. Author details 1 Faculty of Science, Alexandria University, Alexandria, Egypt 2 Faculty of Science, Damanhour University, Damanhour, Egypt Authors’ contributions section All authors contributed equally to the manuscript and read and approved the final draft. Competing interests The authors declare that they have no competing interests. Received: 1 March 2011 Accepted: 27 October 2011 Published: 27 October 2011 References 1. Garh, M, Rao, A, Kalla, SL: Fractional generalization of temperature fields problems in oil strata. Mat Bilten. 30,71–84 (2006) El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 7 of 8 2. Gaul, L, Kempfle, S: Damping description involving fractional operators. Mech Syst Signal Process. 5,81–88 (1991). doi:10.1016/0888-3270(91)90016-X 3. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) 4. 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Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) doi:10.1186/1687-1847-2011-47 Cite this article as: El-Sayed and Gaafar: Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions. Advances in Difference Equations 2011 2011:47. 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 El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 http://www.advancesindifferenceequations.com/content/2011/1/47 Page 8 of 8 . RESEARCH Open Access Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions Ahmed El-Sayed 1* and Fatma Gaafar 2 * Correspondence: amasayed5@yahoo.com 1 Faculty. AMA, Gaafar, FM, Hamadalla, EMA: Stability for a non-local non-autonomous system of fractional order differential equations with delays. Elec J Diff Equ. 31,1–10 (2010) 13. Abd-Salam, SA, El-Sayed,. non-negative solu- tion and the stability and asymptotic properties of the solution of fractional differential dynamic systems involving delayed dynamics with point delays. El-Sayed and Gaafar Advances