Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions
© El-Sayed and Gaafar; licensee Springer. 2011
Received: 1 March 2011
Accepted: 27 October 2011
Published: 27 October 2011
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.
where D α denotes the Riemann-Liouville fractional derivative of order α ∈ (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, Φ(t) = (ϕ i (t)) n × 1are 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 differential equations because they have applications in various fields of science and engineering. 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 stability non-local problem (1)-(3). Furthermore, as an application, we give some examples to demonstrate our results.
For the earlier work we mention: De la Sen  investigated the non-negative solution and the stability and asymptotic properties of the solution of fractional differential dynamic systems involving delayed dynamics with point delays.
by the method of steps, where A, B, C are bounded linear operators defined on a Banach space X.
Sabatier et al.  delt with Linear Matrix Inequality (LMI) stability conditions for fractional order systems, under commensurate order hypothesis.
by the method of steps, where A 0, A 1 are constant matrices and studied the finite time stability for it.
Let L 1[a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0 ≤ a < b < ∞ with the norm .
where Γ (.) is the gamma function.
The following theorem on the properties of fractional order integration and differentiation can be easily proved.
, and if f(t) ∈ L 1 then .
, n = 1,2,3,... uniformly.
, α ∈ (0,1), f (t) is absolutely continuous.
, α ∈ (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 ∈ C n [0, T], the norm is defined by , where N > 0.
and , .
Then there exists a unique solution × ∈ X of the problem (1)-(3).
Now choose N large enough such that , so the map F : X → X is a contraction and hence, there exists a unique column vector x ∈ X which is the solution of the integral equation (4).
Now we complete the proof by proving the equivalence between the integral equation (4) and the non-local problem (1)-(3). Indeed:
which completes the proof of the equivalence between (4) and (1).
Then from (4),. Also from (2), we have .
In this section we study the stability of the solution of the non-local problem (1)-(3)
Definition 5 The solution of the non-autonomous 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 with the initial conditions (2)-(3) and respectively, one has , then for all t ≥ 0.
Theorem 3 The solution of the problem (1)-(3) is uniformly stable.
Therefore, for δ > 0 s.t., we can find s.t. which proves that the solution x(t) is uniformly stable.
where A(t) = (a ij (t)) n×n and are given continuous matrix, then the problem has a unique uniformly stable solution x ∈ X on (-∞, T], T < ∞
where B(t) = (b ij (t)) n×n , and are given continuous matrices, then the problem has a unique uniformly stable solution x ∈ X on (-∞, T], T < ∞
where A(t) = (a ij (t)) n×n B(t) = (b ij (t)) n×n , and H(t) = (h i (t)) n×1are given continuous matrices, then the problem has a unique uniformly stable solution x ∈ X on (-∞, T], T < ∞.
- Garh M, Rao A, Kalla SL: Fractional generalization of temperature fields problems in oil strata. Mat Bilten 2006, 30: 71-84.Google Scholar
- Gaul L, Kempfle S: Damping description involving fractional operators. Mech Syst Signal Process 1991, 5: 81-88. 10.1016/0888-3270(91)90016-XView ArticleGoogle Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I: Fractional Differential Equation. Academic Press, San Diego; 1999.Google Scholar
- Sabatier J, Moze M, Farges C: LMI stability conditions for fractional order systems. Comp Math Appl 2010, 59: 1594-1609. 10.1016/j.camwa.2009.08.003MathSciNetView ArticleGoogle Scholar
- Samko S, Marichev OL: Fractional Integral and Derivatives. Gordon and Breach Science Publisher; 1993.Google Scholar
- Saxena RK, Kalla SL: On a fractional generalization of free electron laser equation. Appl Math Comput 2003, 143: 89-97. 10.1016/S0096-3003(02)00348-XMathSciNetView ArticleGoogle Scholar
- Srivastava HM, Saxena RK: Operators of fractional integration and their applications. Appl Math Comput 2001, 118: 1-52. 10.1016/S0096-3003(99)00208-8MathSciNetView ArticleGoogle Scholar
- De La Sen M: About robust of Caputo linear fractional dynamic system with time delays through fixed point theory. J Fixed Point Theory Appl 2011, 2011: 19. Article ID 867932 10.1186/1687-1812-2011-19View ArticleGoogle Scholar
- El-Sayed AMA: Fractional differential-difference equations. J Frac Calculus 1996, 10: 101-107.MathSciNetGoogle Scholar
- El-Sayed AMA, Gaafar FM, Hamadalla EMA: Stability for a non-local non-autonomous system of fractional order differential equations with delays. Elec J Diff Equ 2010, 31: 1-10.MathSciNetGoogle Scholar
- Abd-Salam SA, El-Sayed AMA: On the stability of some fractional-order non-autonomous systems. Elec J Qual Theory Diff Equ 2007, 6: 1-14.View ArticleGoogle Scholar
- Podlubny I, El-Sayed AMA: On two definitions of fractional calculus. Preprint UEF (ISBN 80-7099-252-2), Slovak Academy of Science-Institute of Experimental Phys. UEF-03-96 ISBN 80-7099-252-2(1996) Google Scholar
- Bonnet C, Partington JR: Analysis of fractional delay systems of retarded and neutral type. Automatica 2002, 38: 1133-1138. 10.1016/S0005-1098(01)00306-5MathSciNetView ArticleGoogle Scholar
- Zhang X: Some results of linear fractional order time-delay system. Appl Math Comput 2008, 197: 407-411. 10.1016/j.amc.2007.07.069MathSciNetView ArticleGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.