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R E S E A R C H Open AccessStability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions Ahmed El-Sayed1* and Fatma Gaafar2 * Correspond

R E S E A R C H Open Access

Stability of a nonlinear non-autonomous

fractional order systems with different delays and non-local conditions

Ahmed El-Sayed1* and Fatma Gaafar2

* 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, x1(t), , x n (t)) + g i (t, x1(t − r1 ), , x n (t − r n )), t ∈ (0, T), T < ∞, (1)

where Dadenotes the Riemann-Liouville fractional derivative of ordera Î (0, 1), x(t)

= (x1(t), x2(t), , xn(t))’, where ‘ denote the transpose of the matrix, and fi, gi: [0, T] ×

Rn® R are continuous functions, F(t) = (ji(t))n × 1are given matrix and O is the zero matrix, rj≥ 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 applications 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 solusolu-tion of fracsolu-tional differential dynamic systems involving delayed dynamics with point delays

© 2011 El-Sayed and Gaafar; licensee Springer 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

El-Sayed [11] proved the existence and uniqueness of the solution u(t) of the pro-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 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) for t < 0, lim

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) = x0, t > 0,

wherec D α a is the Caputo fractional derivatives (see [5-7,14]), A(t) and f(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) = A0x(t) + A1x(t − r) + f (t), t > 0, x(t) = φ(t), t ∈ [−r, 0],

by the method of steps, where A0, A1 are constant matrices and studied the finite time stability for it

2 Preliminaries

Let L1[a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0≤ a

<b < ∞ with the norm||x|| L1 =b

a |x(t)|dt Definition 1 The fractional (arbitrary) order integral of the function f(t) Î L1[a, b] of ordera Î R+

is defined by (see [5-7,14,17])

I α a f (t) =

 t a

(t − s) α−1

whereΓ (.) 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 a n −α D n f (t) = 1

(n − α)

 t (t − s) n −α−1 f (s)ds, t ∈ [a, b],

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 a −α 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 : L1→ L1, and if f(t) Î L1thenI α a I a β f (t) = I α+β a f (t) (ii)α→nlimI 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 f (t),a Î (0,1), f (t) is absolutely continuous

3 Existence and uniqueness

Let X = (Cn(I), || ||1), where Cn(I) is the class of all continuous column n-vectors

function For x Î Cn[0, T], the norm is defined by||x||1=n

i=1supt ∈[0,T] {e −Nt |x i (t)|}, where N > 0

Theorem 2 Let fi, gi : [0, T] × Rn ® R be continuous functions and satisfy the Lipschitz conditions

|f i (t, u1, , u n)− f i (t, v1, , v n)≤

n



j=1

h ij |u j − v j|,

|g i (t, u1, , u n)− g i (t, v1, , v n)| ≤

n



j=1

k ij |u j − v j|,

andh =n

i=1 |h i| =n

i=1 max ∀j |h ij|,k =n

i=1 |k i| =n

i=1max∀j |k ij| Then there exists a unique solution ×Î X of the problem (1)-(3)

Proof Let tÎ (0, T) Then equation (1) can be written as d

dt I

1−α x

i (t) = f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n)

Integrating both sides, we obtain

I1−αx i (t)−I1−αx i (t)| t=0=

 t

0

{f i (t, x1(t), , x n (t)) + g i (t, x1(t −r1), , x n (t −r n))}ds

From (3), we get

I1−α x i (t) =

 t

0 {f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}ds.

Operating by Iaon both sides, we obtain

Ix i (t) = I α+1 {f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}

Differentiating both side is, we get

x i (t) = I α {f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}, i = 1, 2, , n (4) Now let F : X® X, defined by

Fx i = I α {f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}

then

|Fx i − Fy i | = |I α {f i (t, x1(t), , x n (t)) − f i (t, y1(t), , y n (t))

+ g i (t, x1(t − r1), , x n (t − r n))− g i (t, y1(t − r1), , y n (t − r n))}|

≤

t 0

(t − s) α−1

(α) |f i (s, x1(s), , x n (s)) − f i (s, y1(s), , y n (s))|ds

+

 t 0

(t − s) α−1

(α) |g i (s, x1(s − r1), , x n (s − r n))− g i (s, y1(s − r1), , y n (s − r n))|ds

≤

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)|ds

≤ 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

+ 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

≤ 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

N α ||x − y||1 Now choose N large enough such that h+k N α < 1, so the map F : X ® X is a contrac-tion and hence, there exists a unique column vector xÎ X which is the solution of the

integral equation (4)

Now we complete the proof by proving the equivalence between the integral equa-tion (4) and the non-local problem (1)-(3) Indeed:

Since x Î Cnand I1-ax(t)Î Cn(I), and fi, giÎ C(I) then I1-a fi(t), I1-agi(t)Î C(I)

Operating by I1-aon both sides of (4), we get

I1−αx i (t) = I1−αI α {f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}

= I{f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}

Differentiating both sides, we obtain

DI1−αx i (t) = DI{f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n))}, which implies that

D α i (t) = f i (t, x1(t), , x n (t)) + g i (t, x1(t − r1), , x n (t − r n)), t > 0,

which completes the proof of the equivalence between (4) and (1)

Now we prove thatlimt→0 +x i= 0 Since fi(t, x1(t), , xn(t)), gi(t, x1(t - r1), , xn(t

-rn)) are continuous on [0, T] then there exist constants li, Li, mi, Misuch that li≤ fi(t,

x1(t), , xn(t))≤ Liand mi≤ gi(t, x1(t - r1) ), , xn(t - rn)) ≤ Mi, and we have

I α f i (t, x1(t), , x n (t)) =

 t

0

(t − s) α−1

(α) f i (s, x1(s), , x n (s))ds,

which implies

l i

 t

0

(t − s) α−1

(α) ds ≤ I α f i (t, x1(t), , x n (t)) ≤ L i

 t

0

(t − s) α−1

l i t α

(α + 1) ≤ I α f i (t, x1(t), , x n (t))≤ L i t α

(α + 1)

and lim

t→0 +I α f i (t, x1(t), , x n (t)) = 0.

Similarly, we can prove lim

t→0 +I α i (t, x1(t − r1), , x n (t − r n)) = 0

Then from (4),limt→0 +x i (t) = 0 Also from (2), we havelimt→0 −(t) = O Now for tÎ (-∞, T], T < ∞, the continuous solution x(t) Î (-∞, T] of the problem (1)-(3) takes the form

x i (t) =

⎧

⎪

⎪

t

0

(t −s) α−1

(α) {f i (s, x1(s), , x n (s)) + g i (s, x1(s − r1), , 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-autonomous linear system (1) is stable if for any ε > 0, there exists δ > 0 such that for any two solutions x(t) = (x1(t), x2(t), , xn(t))’

and ˜x(t) = (˜x (t), ˜x (t), , ˜x (t)) with the initial conditions (2)-(3) and

||x(t) − ˜x(t)||1< εrespectively, 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, x1(t), , x n (t)) − f i (t, ˜x1(t), , ˜x n (t))

+ g i (t, x1(t − r1 ), , x n (t − r n))− g i (t, ˜x1(t − r1 ), , ˜x n (t − r n))}|

≤t

0

(t − s) α−1

(α) | f i (s, x1(s), , x n (s)) − f i (s, y1(s), , y n (s))|ds

+

 t

0

(t − s) α−1

(α) |g i (s, x1(s − r1 ), , x n (s − r n))− g i (s, ˜x1(s − r1 ), , ˜x n (s − r n))|ds

≤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)|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)− ˜x j (s − r j)|ds

≤ h i

N α ||x j (t) − ˜x j (t)||1

 Nt

0

u α−1 e −u

+ 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

+ 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

≤ h i

N α ||x j (t) − ˜x j (t)||1+ k i

N α

n



j=1

e −Nr jsup

t {e − Nt |x j (t) − ˜x j (t)|}

+ k i

N α

n



j=1

e −Nr jsup

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

N α ||x − ˜x||1+ k

N α || − ˜||1

i.e 1 −h + k

N α

||x − ˜x||1≤ k

N α || − ˜||1and ||x − ˜x||1≤ k

N α 1−h + k

N α

−1

|| − ˜||1

Therefore, for δ > 0 s.t.|| − ˜||1< δ, we can find ε = k

N α

1− h+k

N α

−1

δ s.t

||x − ˜x||1≤ εwhich proves that the solution x(t) is uniformly stable

5 Applications

Example 1Consider the problem

D α 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

I1−α x(t)|t=0 = O,

where A(t) = (aij(t))n×nand(g i (t, x1(t − r1), , 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 stable solution

xÎ X on (-∞, T], T < ∞

Example 2Consider the problem

D α 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

I1−α x(t)|t=0 = O,

where B(t) = (bij(t))n×n, and(f i (t, x1(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 3Consider the problem (see [12])

D α 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) for t < 0and lim

t→0 −(t) = O

I1−α x(t)|t=0 = O,

where A(t) = (aij(t))n×nB(t) = (bij(t))n×n, and H(t) = (hi(t))n×1 are given continuous matrices, then the problem has a unique uniformly stable solution xÎ 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

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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.

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