Báo cáo hóa học: " Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions" pot

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:

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

/>© 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
/>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
/>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
/>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
/>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
/>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
/>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. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam
(2006)
5. Podlubny, I: Fractional Differential Equation. Academic Press, San Diego (1999)
6. Sabatier, J, Moze, M, Farges, C: LMI stability conditions for fractional order systems. Comp Math Appl. 59, 1594–1609
(2010). doi:10.1016/j.camwa.2009.08.003

7. Samko, S, Marichev, OL: Fractional Integral and Derivatives. Gordon and Breach Science Publisher (1993)
8. Saxena, RK, Kalla, SL: On a fractional generalization of free electron laser equation. Appl Math Comput. 143,89–97
(2003). doi:10.1016/S0096-3003(02)00348-X
9. Srivastava, HM, Saxena, RK: Operators of fractional integration and their applications. Appl Math Comput. 118,1–52
(2001). doi:10.1016/S0096-3003(99)00208-8
10. 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, 19 (2011). Article ID 867932. doi:10.1186/1687-1812-2011-19
11. El-Sayed, AMA: Fractional differential-difference equations. J Frac Calculus. 10, 101–107 (1996)
12. 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. 31,1–10 (2010)
13. Abd-Salam, SA, El-Sayed, AMA: On the stability of some fractional-order non-autonomous systems. Elec J Qual Theory
Diff Equ. 6,1–14 (2007)
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)
15. Bonnet, C, Partington, JR: Analysis of fractional delay systems of retarded and neutral type. Automatica. 38, 1133–1138
(2002). doi:10.1016/S0005-1098(01)00306-5
16. Zhang, X: Some results of linear fractional order time-delay system. Appl Math Comput. 197, 407–411 (2008).
doi:10.1016/j.amc.2007.07.069
17. 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
/>Page 8 of 8