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Electronic Journal of Qualitative Theory of Differential Equations
2008, No. 16, 1-9; http://www.math.u-szeged.hu/ejqtde/

Stability of a monotonic solution of a non-autonomous
multidimensional delay differential equation of arbitrary
(fractional) order
A. M. A. El-Sayed
Faculty of Science, Alexandria University, Alexandria, Egypt
e.mail: amasayed@hotmail.com
E. M. El-Maghrabi
Faculty of Science, Benha University, Benha 13518, Egypt
e.mail: esam mh@yahoo.com

Abstract
We are concerned here with the existence of monotonic and uniformly asymptotically stable solution of an initial-value problem of non-autonomous delay differential
equations of arbitrary (fractional) orders.

Key words: Fractional calculus, delay differential equations, monotonic solutions, asymptotically stable solution.

1

Introduction

In a number of papers [1, 2, 3, 7] stability and existence of solution of some equations of
fractional order has been investigated.
In [2] the authors proved the stability (and some other properties concerning the existence
and uniqueness) of solution of the problem of non-autonomous system
Dtα0 X(t) = A(t) X(t) + f (t),

α ∈ (0, 1],

X(t0 ) = X 0 ,

EJQTDE, 2008 No. 16, p. 1

where the coefficients of the matrix functions A(t) and f (t) are absolutely continuous functions. And the problem
d α
d
X(t) = A(t)
I X(t) + f (t),
dt

dt t0

α ∈ (0, 1],

X(t0 ) = X 0

where the coefficients of the matrix A(t) are bounded and measurable and the function f (t)
is integrable.
Let αi ∈ (0, 1], βj ∈ [0, 1), i = 1, 2, . . . , n, j = 1, 2, . . . , m, and ri , σj are positive
constants. Consider the initial-value problem
d
x(t) − f (t, x(t),Drα11 x(t − r1 ), . . . , Drαnn x(t − rn ))
dt

(1.1)

m
x(t − σm )),
= g(t, x(t), Dσβ11 x(t − σ1 ), . . . , Dσβm

x(t) = x0 ,

t > 0,

t ≤ 0,

(1.2)

We prove here the existence of uniformly asymptotically stable and monotonic increasing
solution x ∈ C[0, T ] of (1.1) with

dx
dt

∈ C[0, T ].

As an application the special (linear) case
n

m

i=1

j=1

X
X
d
β
ai (t)Drαii x(t − ri ) +
x(t) − a(t) x(t) =
bj (t)Dσjj x(t − σj )
dt
x(t) = x0

(1.3)

t≤0

and the initial value problem of the multidimensional neutral delay differential equation

!
m
n
X
X
d
bj (t) x(t − σj )
ai (t) x(t − ri ) =
x(t) −
dt
j=1
i=0
(1.4)
x(t) = 0

t≤0

will be studied

2

Preliminaries

Let L1 [a, b] denote the space of all Lebesgue integrable functions on the interval [a, b],
0 ≤ a < b < ∞.

EJQTDE, 2008 No. 16, p. 2

Definition 2.1. The fractional (arbitrary) order integral of the function f ∈ L1 [a, b] of
order β ∈ R+ is defined by (see [4] - [6])
Iaβ

f (t) =

Z

a

t

(t − s)β
Γ(β)

− 1

f (s) ds,

where Γ(.) is the gamma function.
Definition 2.2. The (Caputo) fractional-order derivative D α of order α ∈ (0, 1] of the
function g(t) is defined as (see [4] - [6])
Daα g(t) = Ia1

− α

d
g(t),
dt

t ∈ [a, b].

The following properties are some of the main ones of the fractional derivatives and
integrals. Let β, γ ∈ R+ and α ∈ (0, 1). Then we have
(i) Iaβ : L1 → L1 , and if f (x) ∈ L1 , then Iaγ Iaβ f (x) = Iaγ+β f (x).
limβ→n Iaβ f (x) = Ian f (x) uniformly on [a,b], n = 1, 2, 3, · · · ,
Rx
where
Ia1 f (x) = a f (s) ds.

(ii)

(iii) limβ→0 Iaβ f (x) = f (x) weakly.

(iv) If f (x) is absolutely continuous on [a, b], then limα→1 Daα f (x) =

df (x)
dx .

(v) If f (x) = k 6= 0, k is a constant, then Daα k = 0.

3

Existence of solution

Let C[0, T ] be the class of continuous functions. For x ∈ C[0, T ] we use the norm kxk1 =
supt e−N t |x(t)| , N > 0.
Let f : [0, T ]×Rn+1 → R+ , g : [0, T ]×Rm+1 → R+ and f, g satisfy the Lipschitz condition
|f (t, x0 , x1 , . . . , xn ) − f (t, y0 , y1 , . . . , yn )| ≤ k1
|g(t, x0 , x1 , . . . , xm ) − g(t, y0 , y1 , . . . , ym )| ≤ k2

n
X

i=0
m
X

|xi − yi | , k1 > 0
(3.1)
|xj − yj | , k2 > 0

j=0

Theorem 3.1.

If f and g, satisfy the Lipschitz condition (3.1), then the Cauchy problem

(1.1) - (1.2) has a monotonic increasing solution x ∈ C[0, T ] and

dx
dt

∈ C[0, T ].

EJQTDE, 2008 No. 16, p. 3

Proof. Let y(t) =

d
dt

x(t) then

x(t) = x0 + I y(t), which implies

x(t − ri ) = x0 +

Z

t

y(θ − ri )dθ,

i = 0, 1, 2, . . . , n

ri

i.e.
x(t − ri ) = x0 + Iri y(t − ri ),
from which we obtain
i
Ir1−α
i

d
i
x(t − ri ) = Ir1−α
y(t − ri ),
i
dt

i = 0, 1, . . . , n .

d
1−β
x(t − σj ) = Iσj j y(t − σj ),
dt

j =, 0, 1, . . . , m .

Similarly we obtain
1−βj

Iσj

Now equation (1.1) can be written as
n
1
y(t) = f (t, x0 + Iy(t), Ir1−α
y(t − r1 ), . . . , Ir1−α
y(t − rn ))
n
1
n
1
+ g(t, x0 + I y(t), Iσ1−β
y(t − σ1 ), . . . , Iσ1−β
y(t − σn )),
n
1

(3.2)

Define the operator F : C[0, T ] → C[0, T ] by
n
1
y(t − rn ))
F y(t) = f (t, x0 + Iy(t), Ir1−α
y(t − r1 ), . . . , Ir1−α
n
1
n
1
+ g(t, x0 + I y(t), Iσ1−β
y(t − σ1 ), . . . , Iσ1−β
y(t − σn )),
n
1

then


1−αn
1
y(t
−
r
)
F y(t) − F z(t) = f t, x0 + Iy(t), Ir1−α
y(t
−
r
),
.
.
.
,
I
n
1
r
n
1



n
1
z(t − rn )
− f t, x0 + Iz(t), Ir1−α
z(t − r1 ), . . . , Ir1−α
n
1


1−βn
1
+ g t, x0 + I y(t), Iσ1−β
y(t
−
σ
),
.
.
.
,
I
y(t
−
σ
)
1
n
σn
1


1−βn
1
− g t, x0 + I z(t), Iσ1−β
z(t
−
σ
),
.
.
.
,
I
z(t
−
σ
)
,
1
n
σn
1

EJQTDE, 2008 No. 16, p. 4

and



n
1
y(t − rn )
e−N t |F y(t) − F z(t)| = e−N t f t, x0 + Iy(t), Ir1−α
y(t − r1 ), . . . , Ir1−α
n
1


n
1
z(t − rn ) 
− f t, x0 + Iz(t), Ir1−α
z(t − r1 ), . . . , Ir1−α
n
1
 


1−βn
1
+ e−N t g t, x0 + I y(t), Iσ1−β
y(t
−
σ
),
.
.
.
,
I
y(t
−
σ
)
1
n
σn
1



n
1
− g t, x0 + I z(t), Iσ1−β
z(t − σ1 ), . . . , Iσ1−β
z(t − σn )  ,
n
1
Z t
≤ (k1 + k2 )
e−N (t) |y(s) − z(s)|ds
0

+ k1
+ k2

n Z
X

t

i=1 ri
m Z t
X

e−N (t)

(t − s)−αi
|y(s − ri ) − z(s − ri )|ds
Γ(1 − αi )

e−N (t)

(t − s)−βj
|y(s − σj ) − z(s − σj )|ds
Γ(1 − βj )

σj

j=1

which implies that
kF y − F zk1 ≤ (k1 + k2 ) ky − zk1
+ k1 ky − zk1
+ k2 ky − zk1

Z

t

e−N (t−θ) dθ

0

n
X

i=1
m
X

t−ri

e

Z

e−N σj

Z

t−σj

−N ri

0

j=1

≤

k1 + k2
ky − zk1
N
n Z N (t−ri )
X
+ k1 ky − zk1
+ k2 ky − zk1

i=1 0
m Z N (t−σj )
X
0

j=1



where

n

X 1
k1 + k2
+ k1
≤ 
N
N 1−αi
i=1



e−N (t−ri −θ)

˜ =  k1 + k2 + k1
K
N

n
X
i=1

(t − ri − θ)−αi
dθ
Γ(1 − αi )

e−N (t−σj −θ)

0

(t − σj − θ)−βj
dθ
Γ(1 − βj )

du
e−u u−αi
−α
i
N
Γ(1 − αi ) N
e−u u−βj
du
−β
j
N
Γ(1 − βj ) N

m
X
1 
˜ ky − zk ,
ky − zk1 < K
+ k2
1
1−βj
N
j=1

1
+ k2
N 1−αi

n
X
j=1

˜ < 1, then
Now we choose N large enough such that K



1 
N 1−βj

kF y − F zk1 < ky − zk1 ,
EJQTDE, 2008 No. 16, p. 5

and the map F is contraction and has a unique positive fixed point y ∈ C[0, T ] which proves
the existence of a unique solution x ∈ C[0, T ] of the Cauchy problem (1.1) - (1.2). Now the
solution of (1.1) - (1.2) can be written as x(t) = x0 + I y(t), this implies that
which prove that

4

dx
dt

dx
dt

=y>0

∈ C[0, T ] and x is monotonic increasing.

Stability

In this section we study the stability of the solution of the initial value problem (1.1) (1.2).
Theorem 4.1. If the solution of the initial value problem (1.1) - (1.2) exists, then this
solution is uniformly asymptotically stable.
Proof. Let y(t) be a solution of problem (1.1) - (1.2) and y ∗ (t) be a solution of problem
(1.1) with the initial value x∗ (t) = x∗0

t ≤ 0 then


n
1
e−N t |y(t) − y ∗ (t)| ≤ e−N t f (t, x0 + Iy(t), Ir1−α
y(t − r1 ), . . . , Ir1−α
y(t − rn ))
n
1

n
1
y ∗ (t − rn ))
− f (t, x∗0 + Iy ∗ (t), Ir1−α
y ∗ (t − r1 ), . . . , Ir1−α
n
1


n
1
y(t − σn )) ,
+ e−N t g(t, x0 + I y(t), Iσ1−β
y(t − σ1 ), . . . , Iσ1−β
n
1


∗
1−βn ∗
1
− g(t, x∗0 + I y ∗ (t), Iσ1−β
y
(t
−
σ
),
.
.
.
,
I
y
(t
−
σ
))
,
1
n
σ
n
1


Z t
−N (t−s) −N s
∗
−N t
∗
e
e
|y(s) − y (s)|ds
≤ (k1 + k2 ) e
|x0 − x0 | +
0

+k1 e−N t

n Z
X

t

i=1 ri
n Z t
X

(t −
|y(s − ri ) − y ∗ (s − ri )| ds
Γ(1 − αi )
s)−αi

(t − s)−βj
|y(s − σi ) − y ∗ (s − σi )| ds
Γ(1
−
β
)
j
σ
j
j=1


Z t
−N t
∗
−N (t−s) −N s
∗
≤ (k1 + k2 ) e
|x0 − x0 | +
e
e
|y(s) − y (s)|ds
+k2 e−N t

0

+ k1
+ k2

n Z t−ri
X

i=1 0
n Z t−σj
X
j=1

0

e−N (t−θ)

(t − ri − θ)−αi −N θ
e
|y(θ) − y ∗ (θ)|dθ
Γ(1 − αi )

e−N (t−θ)

(t − σj − θ)−βj −N θ
e
|y(θ) − y ∗ (θ)|dθ
Γ(1 − βj )

EJQTDE, 2008 No. 16, p. 6

and
(k1 + k2 )
ky − y ∗ k1
ky − y ∗ k1 ≤ (k1 + k2 ) e−N t kx0 − x∗0 k1 +
N
n Z N (t−ri )
X
e−u u−αi
du
∗
+ k1 ky − y k1
−α
i
N
Γ(1 − αi ) N
i=1 0
Z
n
N (t−σj )
X
e−u u−βj
du
+k2 ky − y ∗ k1
−β
j
N
Γ(1 − βj ) N
0
j=1

−N t

≤ (k1 + k2 ) e
kx0 − x∗0 k1


n
n
X
X
1
1 
k1 + k2
+ k1
+ k2
+
ky − y ∗ k1
1−α
1−β
i
j
N
N
N
i=1

this implies that

j=1


−1
˜
ky − y ∗ k1 ≤ 1 − K
(k1 + k2 ) e−N t kx0 − x∗0 k1
where
˜ =
K
Now

n

n

i=1

i=1

X 1
X 1
k1 + k2
+ k1
+
k
2
N
N 1−αi
N 1−βi

x(t) − x (t) = x0 −
∗

x∗0

+

Z

t

Z

t

!

< 1.

(y(s) − y ∗ (s))ds

0

this implies that
e−N t |x(t) − x∗ (t)| ≤ e−N t |x0 − x∗0 | +

e−N (t−s) e−N s |(y(s) − y ∗ (s))|ds

0

then
1
ky − y ∗ k1
N

−1 k + k
1
2
˜
) e−N t kx0 − x∗0 k1
≤ e−N t kx0 − x∗0 k1 + 1 − K
(
N


k1 + k2
e−N t kx0 − x∗0 k1 ,
≤
1+
˜
N (1 − K)

kx − x∗ k1 ≤

e−N t kx0 − x∗0 k1 +

therefore limt→∞ kx − x∗ k1 = 0, then the solution of the system (1.1) is uniformly asymptotically stable.

EJQTDE, 2008 No. 16, p. 7

5

Application

(1) Consider now the initial value problem (1.3). Letting
f (t, x(t), Drα11 x(t − r1 ), . . . , Drαnn x(t − rn )) = a(t) x(t) +

n
X

ai (t)Drαii x(t − ri )

i=1

and
g(t, x(t), Dσβ11 x(t

−

m
σ1 ), . . . , Dσβm
x(t

− σm )) =

m
X

bj (t)Dσβj x(t − σj )

j=1

where ai and bj , i, j = 0, 1, 2 · · · are positive continuous functions on [0, T ] with |ai (t)| ≤ a
and |bj (t)| ≤ b. Then our results here can be applied to the initial value problem (1.3) and
prove the existence of positive monotonic and uniformly asymptotically stable solution for
the initial value problem (1.3).
(2) Consider now the initial value problem (1.4). Letting αi → 1 and β → 0 in (1.3)
and use the proberties of the fractional derivative ([1]) we obtain (1.4).

References
[1] El-Sayed A. M. A. , fractional differential-difference equation , J.Frac. Calsulus, Vol.
10,(1996)100–107.
[2] El-Sayed A. M. A. and Abd El-Salam Sh. A. , On the stability of some fractional-order
non-autonomous systems, EJQTDE, No. 6,(2007)1–14.
[3] Delbosco D. and Rodino L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996) 609–625.
[4] Podlubny, I. and EL-Sayed, A. M. A. On two definitions of fractional calculus, Preprint
UEF 03-96 (ISBN 80-7099-252-2), Slovak Academy of Science-Institute of Experimental phys. (1996).
[5] Podlubny, I. Fractional differential equations, Acad. press, San Diego-New YorkLondon 1999.
[6] Samko, S., Kilbas, A. and Marichev, O. L. Fractional integrals and derivatives, Gordon
and Breach Science Publisher, 1993.
EJQTDE, 2008 No. 16, p. 8

[7] Bai C., Positive solutions for nonlinear fractional differential equations with coefficient
that changes sign, Nonlinear Analysis 64 (2006) 677–685

(Received January 3, 2008)

EJQTDE, 2008 No. 16, p. 9