Electronic Journal of Differential Equations, Vol. 2010(2010), No. 31, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
STABILITY FOR A NON-LOCAL NON-AUTONOMOUS SYSTEM
OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS
WITH DELAYS
AHMED M. A. EL-SAYED, FATMA M. GAAFAR, EMAN M. A. HAMADALLA
Abstract. In this article, we establish sufficient conditions for the existence,
uniqueness and uniformly stability of solutions for a class of nonlocal nonautonomous system of fractional-order delay differential equations with several
delays.
1. Introduction
Let x(t) = (x1 (t), x2 (t), . . . , xn (t))0 , where 0 denoted the transpose of the matrix.
Let α ∈ (0, 1] and i = 1, 2, . . . , n. Consider the nonlocal problem
Dα xi (t) =
n
X
aij (t)xj (t) +
n
X
bij (t)xj (t − rj ) + hi (t),
t>0
(1.1)
j=1
j=1
x(t) = Φ(t)
for t < 0,
I β x(t)|t=0 = O,
and
lim Φ(t) = O
t→0−
β ∈ (0, 1]
(1.2)
(1.3)
where Dα denoted the Riemann-Liouville derivative of order α; A(t) = (aij (t))n×n ,
B(t) = (bij (t))n×n , H(t) = (hi (t))n×1 , Φ(t) = (φi (t))n×1 are given matrices; O is
the zero matrix; rj ≥ 0 are constants.
Fractional differential equations has been studied by various researchers because
they appear in various fields: physics, mechanics, engineering, electrochemistry,
economics; see for example [5]-[8], [11]-[14] and references therein.
In this work, we discuss the existence, uniqueness and stability of solution of
the non-autonomous time-varying delay system (1.1)-(1.3). Abd El-Salam and
El-Sayed [1] proved the existence of a unique uniformly stable solution for the nonautonomous system
c
Daα x(t) = A(t)x(t) + f (t) x(0) = x0 ,
t>0
where c Daα is the Caputo fractional derivatives (see [10]-[12]), A(t) and f (t) are continuous matrices. El-Sayed [3] proved the existence and uniqueness of the solution
2000 Mathematics Subject Classification. 34A12, 34A30, 34D20.
Key words and phrases. Riemann-Liouvile derivatives; nonlocal non-autonomous system;
time-delay system; stability analysis.
c
2010
Texas State University - San Marcos.
Submitted October 12, 2009. Published February 26, 2010.
1
2
A. M. A. EL-SAYED, F. M. GAAFAR, E. M. A. HAMADALLA
EJDE-2010/31
u(t) of the problem
c
Daα u(t) + CDaβ u(t − r) = Au(t) + Bu(t − r),
u(t) = g(t),
0≤β≤α≤1
t ∈ [a − r, a], r > 0
by the method of steps, where A, B, C are bounded operators defined on a Banach
space X. Zhang [15] established the existence of a unique solution for the delay
fractional differential equation
Dα x(t) = A0 x(t) + A1 x(t − r) + f (t),
t > 0,
x(t) = φ(t),
t ∈ [−r, 0]
by the method of steps, where A0 , A1 are constant matrices. a study of finite time
stability was shown there.
Here we prove the existence of a unique solution for (1.1)-(1.3), of the form


t<0
φi (t),
xi (t) = 0,
t=0

Pn
 α Pn
I { j=1 aij (t)xj (t) + j=1 bij (t)xj (t − rj ) + hi (t)}, t > 0.
This solution is in C((−∞, T ]), T < ∞, and is uniformly stable.
2. Preliminaries
In this section, we introduce notation, definitions, and preliminary facts which
are used thought this paper.
Definition 2.1. The fractional (arbitrary) order integral of a function f ∈ L1 [a, b]
of order α ∈ R+ is defined by
Z t
(t − s)α−1
f (s) ds.
Iaα f (t) =
Γ(α)
a
where Γ is the gamma function; see [9, 10, 11, 12].
Definition 2.2. The Riemann-liouville fractional (arbitrary) order derivatives of
order α ∈ (n − 1, n) of the function f is defined by
Z t
dn
1
d
Daα f (t) = n Ian−α f (t) =
( )n
(t − s)n−α−1 f (s)ds, t ∈ [a, b];
dt
Γ(n − α) dt
a
see [9, 10, 11, 12].
The concept of stability can be related to that of continuous dependence of
solution on their initial value. Consider the non-autonomous linear system
x0 (t) = A(t)x(t)
(2.1)
with the initial condition x(t0 ) = x0 .
Definition 2.3. The solution x = 0 of (2.1) is called stable if for any > 0, t0 ≥ 0,
there exist δ(, t0 ) > 0 such that kx(t, t0 , x0 )k < for t ≥ t0 as soon as kx0 k < δ.
And the solution x = 0 of (2.1) will be called uniformly stable if δ(, t0 ) can be
chosen independent of t0 : δ(, t0 ) ≡ δ(); see [2].
EJDE-2010/31
STABILITY FOR A NON-LOCAL PROBLEM
3
3. Existence and Uniqueness
Let X = (Cn (I), k · k1 ), where Cn (I) be the class of P
continuous column n-vectors
n
functions. For x ∈ Cn [0, T ], define the norm kxk = i=1 supt∈[0,T ] {e−N t |xi (t)|}.
Pn
Pn
For a matrix B define the norm kBk = i=1 |bi | = i=1 supt,j |bij |.
Theorem 3.1. Let aij , bij , hi , φi be in C(I). Then there exist a unique solution
x ∈ X of (1.1)-(1.3)
Proof. For t > 0, equation (1.1) can be written as
n
n
X
X
d 1−α
I
xi (t) =
aij (t)xj (t) +
bij (t)xj (t − rj ) + hi (t)
dt
j=1
j=1
integrating both sides of the above equation, we obtain
Z t X
n
n
X
I 1−α xi (t) − I 1−α xi (t)|t=0 =
{
aij (t)xj (t) +
bij (s)xj (s − rj ) + hi (s)} ds
0
j=1
j=1
then
I
1−α
t
Z
{
xi (t) =
0
n
X
aij (t)xj (t) +
j=1
n
X
bij (s)xj (s − rj ) + hi (s)} ds .
j=1
Applying the operator by I α , on both sides,
Ixi (t) = I α+1
n
X
aij (t)xj (t) +
n
X
bij (t)xj (t − rj ) + hi (t)
j=1
j=1
differentiating both side, we obtain
α
xi (t) = I {
n
X
aij (t)xj (t) +
j=1
n
X
bij (t)xj (t − rj ) + hi (t)},
i = 1, 2, . . . , n
(3.1)
j=1
Now let F : X → X, defined by
F xi = I α
n
X
aij (t)xj (t) +
j=1
n
X
bij (t)xj (t − rj ) + hi (t)
j=1
then
n
n
X
X
|F xi − F yi | = |I α {
aij (t){xj (t) − yj (t)} +
bij (t){xj (t − rj ) − yj (t − rj )}}|
j=1
=
t
Z
0
+
j=1
α−1
(t − s)
Γ(α)
n
X
n
X
{
aij (s){xj (s) − yj (s)}
j=1
bij (s){xj (s − rj ) − yj (s − rj )}}ds
j=1
n
(t − s)α−1 X
|aij (s)||xj (s) − yj (s)|ds
Γ(α)
j=1
t
Z
≤
0
Z
+
0
t
n
(t − s)α−1 X
|bij (s)||xj (s − rj ) − yj (s − rj )|ds
Γ(α)
j=1
4
A. M. A. EL-SAYED, F. M. GAAFAR, E. M. A. HAMADALLA
≤
n
X
0
Z
j=1 t,∀j
Z t
n
X
0
j=1
n
X
+
j=1
n
X
(t − s)α−1
|xj (s − rj ) − yj (s − rj )|ds
Γ(α)
α−1
(t − s)
Γ(α)
ai
≤ ai
t
sup |bij (t)|
+
+
(t − s)α−1
|xj (s) − yj (s)|ds
Γ(α)
sup |aij (t)|
j=1 t,∀j
n
X
≤
t
Z
EJDE-2010/31
0
rj
Z
bi
0
Z
t
bi
rj
j=1
Z
n
t
X
|xj (s) − yj (s)|ds
(t − s)α−1
|xj (s − rj ) − yj (s − rj )|ds
Γ(α)
(t − s)α−1
|xj (s − rj ) − yj (s − rj )|ds
Γ(α)
(t − s)α−1
|xj (s) − yj (s)|ds
Γ(α)
j=1 0
n Z t
X
+ bi
j=1
rj
(t − s)α−1
|xj (s − rj ) − yj (s − rj )|ds
Γ(α)
and
e−N t |F xi − F yi |
n Z t
X
(t − s)α−1 −N (t−s) −N s
≤ ai
e
e
|xj (s) − yj (s)|ds
Γ(α)
j=1 0
n Z t
X
(t − s)α−1 −N (t−s+rj ) −N (s−rj )
+ bi
e
e
|xj (s − rj ) − yj (s − rj )|ds
Γ(α)
j=1 rj
Z t
n
X
(t − s)α−1 −N (t−s)
−N t
≤ ai
sup{e
|xj (t) − yj (t)|}
e
ds
Γ(α)
t
0
j=1
n Z t−rj
X
(t − θ − rj )α−1 −N (t−θ) −N θ
e
e
|xj (θ) − yj (θ)|dθ
+ bi
Γ(α)
j=1 0
Z N t α−1 −u
n
X
1
u
e
≤ ai
sup{e−N t |xj (t) − yj (t)|} α
du
N
Γ(α)
t
0
j=1
Z t−rj
n
X
(t − θ − rj )α−1 −N (t−θ)
−N t
+ bi
sup{e
|xj (t) − yj (t)|}
e
dθ
Γ(α)
t
0
j=1
Z t−rj α−1
n
X
ai
u
≤ α kx − yk + bi
sup{e−N t |xj (t) − yj (t)|}
e−N u e−N rj du
N
Γ(α)
t
0
j=1
Z
n
X
ai
e−N rj N (t−rj ) uα−1 e−u
−N t
≤ α kx − yk + bi
sup{e
|xj (t) − yj (t)|}
du
N
Nα 0
Γ(α)
t
j=1
≤
n
X
ai
e−N rj
kx
−
yk
+
b
sup{e−N t |xj (t) − yj (t)|}
i
α
N
Nα
t
j=1
EJDE-2010/31
STABILITY FOR A NON-LOCAL PROBLEM
≤
n
ai
bi X
kx
−
yk
+
sup e−N t |xj (t) − yj (t)|
Nα
N α j=1 t
≤
ai + bi
kx − yk .
Nα
5
Then
kF x − F yk =
≤
n
X
i=1
n
X
i=1
≤
sup e−N t |F xi − F yi |
t
ai + bi
kx − yk
Nα
kAk + kBk
kx − yk.
Nα
Now choose N large enough such that kAk+kBk
< 1, so the map F : X → X is a
Nα
contraction and it has a fixed point x = F x and hence, there exist a unique column
vector x ∈ X which is the solution of the integral equation (3.1).
We now prove the equivalence between the integral equation (3.1) and the nonlocal problem (1.1)-(1.3). Indeed, since x ∈ Cn (I) and I 1−α x(t) ∈ Cn (I) applying
the operator I 1−α on both sides of (3.1), we obtain
n
n
X
X
I 1−α xi (t) = I 1−α I α {
aij (t)xj (t) +
bij (t)xj (t − rj ) + hi (t)},
j=1
i = 1, 2, . . . , n
j=1
n
n
X
X
= I{
aij (t)xj (t) +
bij (t)xj (t − rj ) + hi (t)} .
j=1
j=1
Differentiating both sides,
n
n
X
X
DI 1−α xi (t) = DI{
aij (t)xj (t) +
bij (t)xj (t − rj ) + hi (t)}, .
j=1
Then
Dα xi (t) =
n
X
aij (t)xj (t) +
j=1
j=1
n
X
bij (t)xj (t − rj ) + hi (t),
t>0
j=1
which proves the equivalence of (3.1) and (1.1).
We want to prove that limt→0+ xi = 0. Since xj (s), aij (s), hi (s) are continuous
on [0, T ], there exist constants lj , Lj , mi , Mi such that lj ≤ aij (s)xj (s) ≤ Lj and
mi ≤ hi (s) ≤ Mi . We have
Z t
n
n
X
(t − s)α−1 X
α
I {
aij (t)xj (t) + hi (t)} =
aij (s)xj (s) + hi (s) ds
Γ(α)
0
j=1
j=1
which implies
n
X
j=1
lj + mi
Z
0
t
n
X
(t − s)α−1
ds ≤ I α {
aij (t)xj (t) + hi (t)}
Γ(α)
j=1
Z t
n
X
(t − s)α−1
≤{
Lj + Mi }
ds
Γ(α)
0
j=1
6
A. M. A. EL-SAYED, F. M. GAAFAR, E. M. A. HAMADALLA
EJDE-2010/31
which in turn implies
n
X
n
n
X
X
tα
tα
α
{
lj + mi }
≤I {
aij (t)xj (t) + hi (t)} ≤ {
Lj + Mi }
Γ(α + 1)
Γ(α + 1)
j=1
j=1
j=1
and
n
X
lim I α {
aij (t)xj (t) + hi (t)} = 0.
t→0+
j=1
Since bij (s), φj (s − rj ) are continuous on [0, rj ], there exist constants kj , Kj such
that kj ≤ bij (s)φj (s − rj ) ≤ Kj . Also bij (s), xj (s − rj ) are continuous on [rj , T ],
then there exist a constants kj∗ , KJ∗ such that kj∗ ≤ bij (s)xj (s − rj ) ≤ Kj∗ . Let
k = min∀j {kj , kj∗ } and K = max∀j {Kj , Kj∗ }, we have
Iα
n
X
bij (t)xj (t − rj )
j=1
Z
n
(t − s)α−1 X
bij (s)xj (s − rj )ds
Γ(α)
j=1
t
=
0
Z
rj
=
0
Z t
n
n
(t − s)α−1 X
(t − s)α−1 X
bij (s)φj (s − rj )ds +
bij (s)xj (s − rj )ds
Γ(α)
Γ(α)
rj
j=1
j=1
which implies
n
X
Z
Z t
n
X
(t − s)α−1
(t − s)α−1
∗
ds +
kj
ds
Γ(α)
Γ(α)
rj
j=1
rj
kj
0
j=1
≤ Iα
n
X
bij (t)xj (t − rj )
j=1
≤
n
X
j=1
rj
Z
Kj
0
Z t
n
X
(t − s)α−1
(t − s)α−1
ds +
Kj∗
ds
Γ(α)
Γ(α)
rj
j=1
which implies
n
X
kj
j=1
≤
n
X
j=1
n
n
X
tα
(t − rj )α X ∗ (t − rj )α
+
kj
bij (t)xj (t − rj )
−
≤ Iα
Γ(α + 1) Γ(α + 1)
Γ(α + 1)
j=1
j=1
Kj
n
tα
(t − rj )α X ∗ (t − rj )α
−
+
Kj
.
Γ(α + 1) Γ(α + 1)
Γ(α + 1)
j=1
Then
k
n
X
tα
tα
≤ Iα
bij (t)xj (t − rj ) ≤ K
Γ(α + 1)
Γ(α + 1)
j=1
and
lim+ I α
t→0
Then from (3.1) limt→0+ xi = 0.
n
X
bij (t)xj (t − rj ) = 0.
j=1
EJDE-2010/31
STABILITY FOR A NON-LOCAL PROBLEM
7
Now for t ∈ (−∞, T ], T < ∞, the solution of (1.1)-(1.3) takes the form


t<0
φi (t),
xi (t) = 0,
t=0

Pn
 α Pn
I { j=1 aij (t)xj (t) + j=1 bij (t)xj (t − rj ) + bi (t)}, t > 0
4. Stability
In this section we study the stability of the solution of the nonlocal problem
(1.1)-(1.3).
Definition 4.1. The solution of the non-autonomous linear system (1.1) is stable if for any > 0, there exist δ > 0 such that for any two solutions x(t) =
(x1 (t), x2 (t), . . . , xn (t))0 and x
e(t) = (e
x1 (t), x
e2 (t), . . . , x
en (t))0 with the initial conβ
e
ditions (1.2)-(1.3) and {I x
e(t)|t=0 = 0, β ∈ (0, 1], x
e(t) = Φ(t)
for t < 0 and
e
e
limt→0 Φ(t) = O}, respectively, one has kΦ(t) − Φ(t)k ≤ δ, then kx(t) − x
e(t)k < for all t ≥ 0.
Theorem 4.2. The solution of the nonlocal delay system (1.1)-(1.3) is uniformly
stable.
Proof. Let x(t) and x
e(t) be two solutions of the system (1.1) under the conditions
β
e
e = O, respectively.
(1.2)-(1.3) and I x
e(t)|t=0 = 0, x
e(t) = Φ(t),
t < 0 and limt→0 Φ(t)
Then for t > 0, from (3.1), we have
|xi − x
ei | = |I α {
n
X
aij (t)(xj (t) − x
ej (t)) +
j=1
t
Z
≤
0
+
≤
n
(t − s)α−1 X
{
|aij (s)||xj (s) − x
ej (s)|
Γ(α)
j=1
n
X
|bij (s)||xj (s − rj ) − x
ej (s − rj )|}ds
j=1
n
X
t
Z
(t − s)α−1
|xj (s) − x
ej (s)|ds
Γ(α)
sup |aij (t)|
0
Z
sup |bij (t)|
+
j=1 t,∀j
n
X Z t
0
j=1
n
X
+
j=1
n
X
j=1
and
e−N t |xi − x
ei |
t
(t − s)α−1
|xj (s − rj ) − x
ej (s − rj )|ds
Γ(α)
(t − s)α−1
|φj (s) − φej (s)| ds
Γ(α)
ai
+
bij (t)(xj (t − rj ) − x
ej (t − rj )}|
j=1
j=1 t,∀j
n
X
≤
n
X
0
Z
rj
bi
0
Z
t
bi
rj
(t − s)α−1
|φj (s − rj ) − φej (s − rj )|ds
Γ(α)
(t − s)α−1
|xj (s − rj ) − x
ej (s − rj )|ds
Γ(α)
8
A. M. A. EL-SAYED, F. M. GAAFAR, E. M. A. HAMADALLA
≤ ai
n Z
X
t
EJDE-2010/31
(t − s)α−1 −N (t−s) −N s
e
e
|xj (s) − x
ej (s)|ds
Γ(α)
j=1 0
n Z rj
X
+ bi
j=1 0
n Z t
X
(t − s)α−1 −N (t−s+rj ) −N (s−rj )
e
e
|φj (s − rj ) − φej (s − rj )|ds
Γ(α)
(t − s)α−1 −N (t−s+rj ) −N (s−rj )
e
e
|xj (s − rj ) − x
ej (s − rj )|ds
Γ(α)
j=1 rj
Z t
n
X
(t − s)α−1 −N (t−s)
−N t
≤ ai
e
sup{e
|xj (t) − x
ej (t)|}
ds
Γ(α)
t
0
j=1
n Z 0
X
(t − θ − rj )α−1 −N (t−θ) −N θ
e
e
|φj (θ) − φej (θ)|dθ
+ bi
Γ(α)
j=1 −rj
n Z t−rj
X
(t − θ − rj )α−1 −N (t−θ) −N θ
+ bi
e
e
|xj (θ) − x
ej (θ)|dθ
Γ(α)
j=1 0
Z N t α−1 −u
ai
u
e
ej (t)k
du
≤ α kxj (t) − x
N
Γ(α)
0
Z 0
n
X
(t − θ − rj )α−1 −N (t−θ)
−N t
e
+ bi
sup{e
|φj (t) − φj (t)|}
e
dθ
Γ(α)
t
−rj
j=1
Z t−rj
n
X
(t − θ − rj )α−1 −N (t−θ)
+ bi
sup{e−N t |xj (t) − x
ej (t)|}
e
dθ
Γ(α)
t
0
j=1
+ bi
≤
≤
ai
kxj (t) − x
ej (t)k
Nα
Z
n
X
e−N rj N t
uα−1 e−N u
+ bi
sup{e−N t |φj (t) − φej (t)|}
du
α
N
Γ(α)
t
N (t−rj )
j=1
Z
n
X
e−N rj N (t−rj ) uα−1 e−u
+ bi
du
sup{e−N t |xj (t) − x
ej (t)|}
Nα 0
Γ(α)
t
j=1
n
ai
bi X −N rj
kx
(t)
−
x
e
(t)k
+
e
sup{e−N t |xj (t) − x
ej (t)|}
j
j
Nα
N α j=1
t
+
≤
n
ai
bi X
kx
(t)
−
x
e
(t)k
+
sup{e−N t |xj (t) − x
ej (t)|}
j
j
Nα
N α j=1 t
+
≤
n
bi X −N rj
e
sup{e−N t |φj (t) − φej (t)|}
N α j=1
t
n
bi X
sup{e−N t |φj (t) − φej (t)|}
N α j=1 t
ai + bi
bi
e .
kx − x
ek + α kΦ − Φk
Nα
N
EJDE-2010/31
STABILITY FOR A NON-LOCAL PROBLEM
9
Then
kx − x
ek =
≤
n
X
i=1
n
X
i=1
≤
sup e−N t |xi − x
ei
t
n
X
ai + bi
bi
e
kx
−
x
e
k
+
kΦ − Φk
α
Nα
N
i=1
kAk + kBk
kBk
e
kx − x
ek + α kΦ − Φk;
α
N
N
i.e.,
1−
kAk + kBk kAk
e
kx − x
ek ≤ α kΦ − Φk
α
N
N
and
kAk + kBk −1
e
kx − x
ek ≤ 1 −
kΦ − Φk;
Nα
−1
e < δ, we can find = 1 − kAk+kBk
therefore, for δ > 0 such that kΦ − Φk
δ such
Nα
that kx − x
ek ≤ which proves that the solution x(t) is uniformly stable.
References
[1] S. A. Abd-Salam, A. M. A. El-Sayed; On the stability of some fractional-order nonautonomous systems, E. J. Qualitative Theory of Diff. Equ., No. 6 (2007), 1-14.
[2] C. Corduneanu; Principle of differential and integral equations, Allyn and Bacon, Inc.,
Boston, Massachusetts (1971).
[3] A. M. A. El-Sayed; Fractional differential-difference equations, J. of frac. Calculus vol. 10
(1996), 101-107.
[4] A. M. A. El-Sayed; Nonlinear functional differential equations of arbitrary orders, Nonlinear
Analysis: Theory, Methods and Applications, Vol. 33, No. 2 (1998), 181-186.
[5] M. Garh, A. Rao, S. L. Kalla; emphFractional generalization of temperature fields problems
in oil strata, Mat. Bilten 30 (2006) 71-84.
[6] L. Gaul, P. Klein, S. Kempfle; Damping description involving fractional operators, Mech.
Syst. Signal Process. 5 (1991) 81-88.
[7] R. Hilfer; Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
[9] K. S. Miller, B. Ross; An introduction to the fractional calculus and fractional differential
equations, John Wiley, New York (1993).
[10] I. Podlubny, A. M. A. El-Sayed; On two definitions of fractional calculus, preprint UEF
(ISBN 80-7099-252-2), Solvak Academy of Science-Institute of Experimental phys, (1996),
03-06.
[11] I. Podlubny; Fractional differential equations, Acad. press, San Diego-New York-London
(1999).
[12] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional integral and derivatives, Gordon and
Breach science Publisher, (1993).
[13] R. K. Saxena, S. L. Kalla; On a fractional generalization of free electron laser equation, Appl.
Math. Comput. 143 (2003), 89-97.
[14] H. M. Srivastava, R. K. Saxena; Operators of fractional integration and their applications,
Appl. Math. Comput. 118 (2001), 1-52.
[15] X. Zhang; Some results of linear fractional order time-delay system, Appl. Math. Comput.
197 (2008), 407-411.
Ahmed M. A. El-Sayed
Faculty of Science, Alexandria University, Alexandria, Egypt
E-mail address: amasayed@hotmail.com
10
A. M. A. EL-SAYED, F. M. GAAFAR, E. M. A. HAMADALLA
EJDE-2010/31
Fatma M. Gaafar
Faculty of Science, Damanhour, Alexandria University, Alexandria, Egypt
E-mail address: gaafarfatma@yahoo.com
Eman M. A. Hamadalla
Faculty of Science, Alexandria University, Alexandria, Egypt
E-mail address: emanhamdalla@hotmail.com