- Open Access
Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions
Advances in Difference Equations volume 2011, Article number: 47 (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.
Here we consider the nonlinear non-local problem of the form
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.
El-Sayed  proved the existence and uniqueness of the solution u(t) of the problem
by the method of steps, where A, B, C are bounded linear operators defined on a Banach space X.
El-Sayed et al.  proved the existence of a unique uniformly stable solution of the non-local problem
Sabatier et al.  delt with Linear Matrix Inequality (LMI) stability conditions for fractional order systems, under commensurate order hypothesis.
Abd El-Salam and El-Sayed  proved the existence of a unique uniformly stable solution for the non-autonomous system
Bonnet et al.  analyzed several properties linked to the robust control of fractional differential systems with delays. They delt with the BIBO stability of both retarded and neutral fractional delay systems. Zhang  established the existence of a unique solution for the delay fractional differential equation
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.
Theorem 1 Let α, β ∈ R + . Then we have
, 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.
Theorem 2 Let f i , g i : [0, T] × R n → R be continuous functions and satisfy the Lipschitz conditions
and , .
Then there exists a unique solution × ∈ X of the problem (1)-(3).
Proof Let t ∈ (0, T). Then equation (1) can be written as
Integrating both sides, we obtain
From (3), we get
Operating by I α on both sides, we obtain
Differentiating both side is, we get
Now let F : X → X, defined by
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:
Since x ∈ C n and I 1-α x(t) ∈ C n (I), and f i , g i ∈ C(I) then I 1-α f i (t), I 1-α g i (t) ∈ C(I). Operating by I 1-α on both sides of (4), we get
Differentiating both sides, we obtain
which implies that
which completes the proof of the equivalence between (4) and (1).
Now we prove that . Since f 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
Similarly, we can prove
Then from (4),. Also from (2), we have .
Now for t ∈ (-∞, T], T < ∞, the continuous solution x(t) ∈ (-∞, T] of the problem (1)-(3) takes the form
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.
Proof Let x(t) and be two solutions of the system (1) under conditions (2)-(3) and , respectively. Then for t > 0, we have from (4)
Then we have,
Therefore, for δ > 0 s.t., we can find s.t. which proves that the solution x(t) is uniformly stable.
Example 1 Consider the problem
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 < ∞
Example 2 Consider the problem
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 < ∞
Example 3 Consider the problem (see )
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.
Gaul L, Kempfle S: Damping description involving fractional operators. Mech Syst Signal Process 1991, 5: 81-88. 10.1016/0888-3270(91)90016-X
Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Podlubny I: Fractional Differential Equation. Academic Press, San Diego; 1999.
Sabatier J, Moze M, Farges C: LMI stability conditions for fractional order systems. Comp Math Appl 2010, 59: 1594-1609. 10.1016/j.camwa.2009.08.003
Samko S, Marichev OL: Fractional Integral and Derivatives. Gordon and Breach Science Publisher; 1993.
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-X
Srivastava HM, Saxena RK: Operators of fractional integration and their applications. Appl Math Comput 2001, 118: 1-52. 10.1016/S0096-3003(99)00208-8
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 10.1186/1687-1812-2011-19
El-Sayed AMA: Fractional differential-difference equations. J Frac Calculus 1996, 10: 101-107.
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.
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.
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)
Bonnet C, Partington JR: Analysis of fractional delay systems of retarded and neutral type. Automatica 2002, 38: 1133-1138. 10.1016/S0005-1098(01)00306-5
Zhang X: Some results of linear fractional order time-delay system. Appl Math Comput 2008, 197: 407-411. 10.1016/j.amc.2007.07.069
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
6 Competing interests
The authors declare that they have no competing interests.
7 Authors' contributions section
All authors contributed equally to the manuscript and read and approved the final draft.
About this article
Cite this article
El-Sayed, A., Gaafar, F. Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions. Adv Differ Equ 2011, 47 (2011). https://doi.org/10.1186/1687-1847-2011-47
- Riemann-Liouville derivatives
- nonlocal non-autonomous system
- time-delay system
- stability analysis