Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 24, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR SECOND-ORDER
IMPULSIVE BOUNDARY-VALUE PROBLEMS
ABDELKADER BOUCHERIF, ALI S. AL-QAHTANI, BILAL CHANANE
Abstract. In this article we discuss the existence of solutions of second-order
boundary-value problems subjected to impulsive effects. Our approach is based
on fixed point theorems.
1. Introduction
Differential equations involving impulse effects arise naturally in the description
of phenomena that are subjected to sudden changes in their states, such as population dynamics, biological systems, optimal control, chemotherapeutic treatment in
medicine, mechanical systems with impact, financial systems. For typical examples
see [9, 11]. For a general theory on impulsive differential equations the interested
reader can consult the monographs [2, 7, 14], and the papers [1, 5, 6, 8, 10, 12, 13, 15]
and the references therein. Our objective is to provide sufficient conditions on the
data in order to ensure the existence of at least one solution of the problem
(p(t)x0 (t))0 + q(t)x(t) = F (t, x(t), x0 (t)),
t 6= tk , t ∈ [0, 1],
0
∆x(tk ) = Uk (x(tk ), x (tk )),
0
∆x (tk ) = Vk (x(tk ), x0 (tk )),
k = 1, 2, . . . , m,
(1.1)
x(0) = x(1) = 0,
where x ∈ R is the state variable; F : R+ × R2 → R is a piecewise continuous
function; Uk and Vk represent the jump discontinuities of x and x0 , respectively, at
t = tk ∈ (0, 1), called impulse moments, with 0 < t1 < t2 < · · · < tm < 1.
2. Preliminaries
In this section we introduce some definitions and notations that will be used in
the remainder of the paper.
Let J denote the real interval [0, 1]. Let J 0 = J\{t1 , t2 , . . . , tm }. P C(J) denotes
the space of all functions x : J → R continuous on J 0 , and for i = 1, 2, . . . , m,
−
x(t+
i ) = lim→0+ x(ti + ) and x(ti ) = lim→0 x(ti − ) exist. We shall write
x(t−
i ) = x(ti ). This is a Banach space when equipped with the sup-norm; i.e.,
2000 Mathematics Subject Classification. 34B37, 34B15, 47N20.
Key words and phrases. Second order boundary value problems; impulse effects;
fixed point theorem.
c
2012
Texas State University - San Marcos.
Submitted September 12, 2011. Published February 7, 2012.
1
2
A. BOUCHERIF, A. S. AL-QAHTANI, B. CHANANE
EJDE-2012/24
kxk0 = supt∈J |x(t)|. Similarly, P C 1 (J) is the space of all functions x ∈ P C(J),
0 −
xis continuously differentiable on J 0 , and for i = 1, 2, . . . , m, x0 (t+
i ) and x (ti ) exist
0
0 −
1
and x (ti ) = x (ti ). For x ∈ P C (J) we define its norm by kxk1 = kxk0 + kx0 k0 .
Then (P C 1 (I), k · k1 ) is a Banach space.
The following linear problem plays an important role in our study.
(p(t)x0 (t))0 + q(t)x(t) = f (t),
t 6= tk , t ∈ [0, 1],
0
∆x(tk ) = Uk (x(tk ), x (tk )),
0
∆x (tk ) = Vk (x(tk ), x0 (tk )),
k = 1, 2, . . . , m,
(2.1)
x(0) = x(1) = 0,
To study (2.1) we first consider the problem without impulses
(p(t)x0 (t))0 + q(t)x(t) = f (t),
t ∈ [0, 1]
x(0) = x(1) = 0.
(2.2)
We shall assume, throughout the paper, that the following condition holds.
(H0)
(i) p ∈ C 1 (J : R), p(t) ≥ p0 > 0, for all t ∈ J.
(ii) q ∈ C(J : R), q(t) ≤ p0 π 2 , for all t ∈ J, and q(t) < p0 π 2 on a subset of
J of positive measure.
Lemma 2.1. If (H0) is satisfied, then for any nonzero x ∈ C 2 (J : R) with x(0) =
x(1) = 0,
Z 1
{p(t)(x0 (t))2 − q(t)x2 (t)}dt > 0.
0
Proof. The proof of this lemma is presented in [3]. We shall reproduce it here for
the sake of completeness. Since q(t) ≤ p0 π 2 on a subset of J of positive measure,
we have
p(t)(x0 (t))2 − q(t)x2 (t) > p0 ((x0 (t))2 − π 2 x2 (t)).
This inequality yields
Z 1
Z 1
{p(t)(x0 (t))2 − q(t)x2 (t)}dt > p0
{(x0 (t))2 − π 2 x2 (t)}dt.
0
0
We show that
Z
J (x) =
1
{(x0 (t))2 − π 2 x2 (t)}dt ≥ 0
0
for all functions x ∈ C 2 (J : R) with x(0) = x(1) = 0. The function u that minimizes
J (x) satisfies the Euler-Lagrange equation (see [4])
u00 + π 2 u = 0,
and the boundary conditions u(0) = u(1) = 0. Then u(t) = sin πt or u(t) = 0, and
J (u) = 0. Since J (x) ≥ J (u) it follows that J (x) ≥ 0, and so
Z 1
{p(t)(x0 (t))2 − q(t)x2 (t)}dt > 0.
0
This completes the proof of the lemma.
EJDE-2012/24
EXISTENCE OF SOLUTIONS
3
Lemma 2.2. If (H0) is satisfied, then the linear problem
(p(t)x0 (t))0 + q(t)x(t) = 0
(2.3)
x(0) = x(1) = 0.
has only the trivial solution.
Proof. Assume on the contrary that (2.3) has a nontrivial solution x0 . Then (2.3)
implies [(p(t)x00 (t))0 + q(t)x0 (t)]x0 (t) = 0 which yields
Z 1
0=
[(p(t)x00 (t))0 + q(t)x0 (t)]x0 (t) dt
0
1
Z
[(p(t)x00 (t))0 ]x0 (t) dt +
=
1
Z
0
q(t)x20 (t) dt
0
Z
=−
1
2
[p(t)x02
0 (t) − q(t)x0 (t)] dt < 0.
0
This is a contradiction. See Lemma 2.1. Therefore x0 ≡ 0 is the only solution of
(2.3).
It is well known that the unique solution of (2.2) is given by
Z 1
x(t) =
G(t, s)f (s)ds,
0
where G(·, ·) : J × J → R is the Green’s function corresponding to (2.3).
Lemma 2.3. The solution to (2.1) is
1
Z
G(t, s)f (s)ds −
x(t) =
0
+
m
X
∂G(t, tk )
k=1
m
X
∂s
p(tk )Uk (x(tk ), x0 (tk ))
(2.4)
0
G(t, tk )p(tk )Vk (x(tk ), x (tk )).
k=1
Proof. We shall use of superposition principle and write x(t) = y(t) + z(t) + w(t),
where y(t) solves the problem
(p(t)y 0 (t))0 + q(t)y(t) = f (t),
t ∈ J,
∆y(tk ) = 0,
0
∆y (tk ) = 0,
k = 1, 2, . . . , m,
(2.5)
y(0) = y(1) = 0,
while z(t) solves the problem
(p(t)z 0 (t))0 + q(t)z(t) = 0,
t 6= tk , t ∈ J,
0
∆z(tk ) = Uk (x(tk ), x (tk )),
∆z 0 (tk ) = 0,
k = 1, 2, . . . , m,
z(0) = z(1) = 0,
(2.6)
4
A. BOUCHERIF, A. S. AL-QAHTANI, B. CHANANE
EJDE-2012/24
and w(t) solves the problem
(p(t)w0 (t))0 + q(t)w(t) = 0,
t 6= tk , t ∈ J,
∆w(tk ) = 0,
0
∆w (tk ) = Vk (x(tk ), x0 (tk )),
k = 1, 2, . . . , m,
(2.7)
w(0) = w(1) = 0.
It is clear that
1
Z
y(t) =
G(t, s)f (s)ds,
t ∈ I.
0
For k = 1, 2, . . . , m, set
∂G(t, tk )
p(tk )Uk (x(tk ), x0 (tk )), t ∈ J,
∂s
wk (t) = G(t, tk )p(tk )Vk (x(tk ), x0 (tk )), t ∈ J.
zk (t) = −
Using the properties of Green’s function and its derivatives we can prove that the
functions zk and wk , k = 1, 2, . . . , m, P
are the solutions
Pm of problems (2.6) and (2.7),
m
respectively. Consequently, x = y + k=1 zk + k=1 wk is a solution of problem
(2.1).
3. Nonlinear Problem
In this section we present our main results on the existence of solutions for
nonlinear boundary-value problems for the second-order impulsive control system.
Consider the problem
(p(t)x0 (t))0 + q(t)x(t) = F (t, x(t), x0 (t)),
t 6= tk , t ∈ J,
0
∆x(tk ) = Uk (x(tk ), x (tk )),
0
∆x (tk ) = Vk (x(tk ), x0 (tk )),
k = 1, 2, . . . , m,
(3.1)
x(0) = x(1) = 0,
where x ∈ R is the state variable; F : R+ × R2 → R is a piecewise continuous
function; Uk and Vk are impulsive functions representing the jump discontinuities
of x and x0 at t ∈ {t1 , t2 , . . . , tm }.
The nonlinear system
(p(t)x́(t)´) + q(t)x(t) = F (t, x(t), x0 (t))
x(0) = x(1) = 0,
is equivalent to the nonlinear integral equation
Z 1
x(t) =
G(t, s)F (s, x(s), x0 (s))ds,
(3.2)
for all t ∈ J
0
It follows from Lemma 2.3 that any solution of (3.1) satisfies
Z 1
m
X
W (t, tk )p(tk )Uk (x(tk ), x0 (tk ))
x(t) =
G(t, s)F (s, x(s), x0 (s))ds −
0
+
m
X
k=1
k=1
0
G(t, tk )p(tk )Vk (x(tk ), x (tk )).
(3.3)
EJDE-2012/24
EXISTENCE OF SOLUTIONS
where W (t, tk ) =
∂G(t,tk )
.
∂s
Let
K = max{|G(t, s)| : (t, s) ∈ J × J},
M = sup{|
5
L = max{|W (t, s)| : (t, s) ∈ J × J},
∂W (t, s)
∂G(t, s)
| : (t, s) ∈ J × J}, N = sup{|
| : (t, s) ∈ J × J},
∂t
∂t
P = max{K, L, M, N }.
For the next theorem we use the following assumptions:
(H1) F (·, ·, ·) is continuous on J 0 and satisfies the Lipschitz condition
|F (t, x1 , y1 ) − F (t, x2 , y2 )| ≤ β(|x1 − y1 | + |x2 − y2 |).
(H2) Uk and Vk are continuous and satisfy the Lipschitz conditions
|Uk (x1 , y1 ) − Uk (x2 , y2 )| ≤ ck (|x1 − y1 | + |x2 − y2 |),
|Vk (x1 , y1 ) − Vk (x2 , y2 )| ≤ dk (|x1 − y1 | + |x2 − y2 |),
Pm
Pm
(H3) 2P (β + R k=1 ck + R k=1 dk ) < 1 .
Theorem 3.1. Under assumptions (H0)–(H3), problem (3.1) has a unique solution.
Proof. Define an operator ω : P C 1 (J) → P C 1 (J) by
Z 1
m
X
W (t, tk )p(tk )Uk (x(tk ), x0 (tk ))
ω(x)(t) =
G(t, s)F (s, x(s), x0 (s))ds −
0
+
k=1
m
X
G(t, tk )p(tk )Vk (x(tk ), x0 (tk )).
k=1
It is clear that any solution of (3.1) is a fixed point of ω and conversely any fixed
point of ω is a solution of (3.1).
We shall show that ω is a contraction. Let x, y ∈ P C(J), then
Z
1
|G(t, s)||F (s, x(s), x0 (s)) − F (s, y(s), y 0 (s))|ds
kω(x) − ω(y)k0 ≤ sup
t∈J
0
m
X
|W (t, tk )|p(tk )|Uk (x(tk ), x0 (tk )) − Uk (y(tk ), y 0 (tk ))|
+
+
k=1
m
X
|G(t, tk )|p(tk )|Vk (x(tk ), x0 (tk )) − Vk (y(tk ), y 0 (tk ))|
k=1
≤ sup
t∈J
+R
+R
1
nZ
|G(t, s)|(β(kx − yk0 + kx0 − y 0 k0 ))ds
0
m
X
k=1
m
X
|W (t, tk )|ck (kx − yk0 + kx0 − y 0 k0 )
o
|G(t, tk )|dk (kx − yk0 + kx0 − y 0 k0 ) .
k=1
Now, by using (H1) and (H2), we have
kω(x) − ω(y)k0 ≤ βKkx − yk1 + RL
m
X
k=1
ck kx − yk1 + RK
m
X
k=1
dk kx − yk1 . (3.4)
6
A. BOUCHERIF, A. S. AL-QAHTANI, B. CHANANE
EJDE-2012/24
We have
d
ω(x)(t) =
dt
1
Z
0
+
m
X ∂W (t, tk )
∂G(t, s)
F (s, x(s), x0 (s))ds −
Uk (x(tk ), x0 (tk ))
∂t
∂t
k=1
m
X
∂G(t, tk )
k=1
∂t
Vk (x(tk ), x0 (tk )).
Let x, y ∈ P C(J), then
n Z 1 ∂G(t, s)
d
d
ω(x) − ω(y)k0 ≤ sup
||F (s, x(s), x0 (s)) − F (s, y(s), y 0 (s))|ds
|
dt
dt
∂t
t∈J
0
m
X
∂W (t, tk )
||Uk (x(tk ), x0 (tk )) − Uk (y(tk ), y 0 (tk ))|
+
|
∂t
k
+
k=1
m
X
k=1
|
o
∂G(t, tk )
||Vk (x(tk ), x0 (tk )) − Vk (y(tk ), y 0 (tk ))| .
∂t
Conditions (H1) and (H2) imply
k
m
m
k=1
k=1
X
X
d
d
ω(x)− ω(y)k0 ≤ βM kx−yk1 +RN
ck kx−yk1 +RM
dk kx−yk1 . (3.5)
dt
dt
From (3.4) and (3.5) we obtain
d
d
ω(x) − ω(y)k0
dt
dt
m
m
X
X
ck + RK
dk kx − yk1
≤ βK + RL
kω(x) − ω(y)k1 = kω(x) − ω(y)k0 + k
k=1
m
X
+ (βM + RN
k=1
ck + RM
m
X
dk )kx − yk1
k=1
k=1
m
m
X
X
≤ 2P β + R
ck + R
dk kx − yk1
k=1
k=1
Condition (H3) implies that ω is a contraction. By the Banach fixed point theorem
ω has a unique fixed point x, which is the unique solution of (3.1).
For the next Theorem, we use the following assumptions:
(H4) F : [0, 1] × R2 → R is continuous on J 0 and there exists h : J × R+ → R+ a
Caratheodory function, nondecreasing with respect to its second argument
such that
|F (t, x, y)| ≤ h(t, |x| + |y|),
a.e. t ∈ [0, 1].
(H5) Uk and Vk are continuous and there exist ak > 0 and bk > 0 such that
|Uk (x(tk ), y(tk ))| ≤ ak , |Vk (x(tk ), y(tk ))| ≤ bk , k = 1, 2, . . . , m.
R1
Pm
(H6) lim%→+∞ sup %1 0 h(t, %)dt + k=1 R(ak + bk ) < 1/(2P ).
Theorem 3.2. Under assumptions (H0), (H4)–(H6), problem (3.1) has at least
one solution.
EJDE-2012/24
EXISTENCE OF SOLUTIONS
7
Proof. The proof is given in two steps.
Step 1. A priori bound on solutions. Let x ∈ P C 1 (J) be a solution of (3.1).
Z 1
m
X
x(t) =
G(t, s)F (s, x(s), x0 (s))ds −
W (t, tk )p(tk )Uk (x(tk ), x0 (tk ))
0
+
k=1
m
X
G(t, tk )p(tk )Vk (x(tk ), x0 (tk )),
k=1
and
0
1
Z
x (t) =
0
+
m
X ∂W (t, tk )
∂G(t, s)
F (s, x(s), x0 (s))ds −
p(tk )Uk (x(tk ), x0 (tk ))
∂t
∂t
k=1
m
X
∂G(t, tk )
∂t
k=1
p(tk )Vk (x(tk ), x0 (tk )).
It is easy to see that
1
Z
|F (s, x(s), x0 (s))|ds + RL
|x(t)| ≤ K
0
m
X
|Uk (x(tk ), x0 (tk ))|
k=1
+ RK
m
X
|Vk (x(tk ), x0 (tk ))|,
k=1
and
|x0 (t)| ≤ M
1
Z
|F (s, x(s), x0 (s))|ds + RN
0
+ RM
m
X
|Uk (x(tk ), x0 (tk ))|
k=1
m
X
|Vk (x(tk ), x0 (tk ))|.
k=1
Conditions (H4), (H5) and (H6) lead to
Z 1
m
X
kxk0 + kx0 k0 ≤ (K + M )
h(s, kxk0 + kx0 k0 )ds +
R((L + N )lk + (K + M )pk ).
0
k=1
0
Since kxk1 = kxk0 + kx k0 and h is nondecreasing, then
Z 1
m
X
kxk1 ≤ 2P
h(s, kxk1 )ds +
R(2P ak + 2P bk ),
0
k=1
or
Z
kxk1 ≤ 2P (
1
h(s, kxk1 )ds +
0
m
X
R(ak + bk )).
k=1
Let β0 = kxk1 . Then the above inequality gives
Z
m
X
1
1 1
R(ak + bk ) .
≤
h(s, β0 )ds +
2P
β0 0
(3.6)
k=1
Condition (H6) implies that there exists r > 0 such that for all β > r, we have
Z
m
X
1 1
1
h(s, β)ds +
R(ak + bk ) <
.
(3.7)
β 0
2P
k=1
8
A. BOUCHERIF, A. S. AL-QAHTANI, B. CHANANE
EJDE-2012/24
Comparing (3.6) and (3.7) we see that β0 ≤ r . Hence we have kxk1 ≤ r.
Step 2. Existence of solutions. Let Ω = {x ∈ P C 1 (J) : kxk1 < r + 1}. Then Ω
is an open convex subset of P C 1 (J). Define an operator H by
Z 1
m
X
H(λ, x)(t) = λ
G(t, s)F (s, x(s), x0 (s))ds + λ
W (t, tk )Uk (x(tk ), x0 (tk ))
0
+λ
k=1
m
X
G(t, tk )Vk (x(tk ), x0 (tk )),
0 ≤ λ ≤ 1.
k=1
Then H(λ, ·) : Ω̄ → P C 1 (J) is compact and has no fixed point on ∂Ω (see [6]). It
is an admissible homotopy between the constant map H(0, ·) ≡ 0 and H(1, ·) ≡ ω.
Since H(0, ·) is essential then H(1, ·) is essential which implies that ω ≡ H(1, ·) has
a fixed point in Ω. This fixed point is a solution of our problem.
The following assumptions are used in the next theorem.
(H7) there exists g ∈ L1 (J) such that
|F (t, x, y)| ≤ g(t) for almost t ∈ J,
x, y ∈ R.
2
(H8) Uk : R → R is continuous and there exists αk > 0 such that
|Uk (x(tk ), y(tk ))| ≤ αk (kxk0 + kyk0 ), k = 1, 2, . . . , m.
2
(H9) Vk : R → R is continuous and there exists βk > 0 such that
|Vk (x(tk ), y(tk ))| ≤ βk (kxk0 + kyk0 ), k = 1, 2, . . . , m.
Pm
(H10) 2P R k=1 (αk + βk ) < 1.
Theorem 3.3. Under assumptions (H0), (H7)–(H10), equation (2.4) has at least
one solution.
Proof. The proof is given in two steps.
Step1. A priori bound on solutions. We have
Z 1
m
X
W (t, tk )p(tk )Uk (x(tk ), x0 (tk ))
x(t) =
G(t, s)F (s, x(s), x0 (s))ds −
0
+
k=1
m
X
G(t, tk )p(tk )Vk (x(tk ), x0 (tk )).
k=1
and
0
1
Z
x (t) =
0
+
m
X ∂W (t, tk )
∂G(t, s)
F (s, x(s), x0 (s), (Sx)(s))ds +
p(tk )Uk (x(tk ), x0 (tk ))
∂t
∂t
k=1
m
X
∂G(t, tk )
k=1
p(tk )Vk (x(tk ), x0 (tk )).
∂t
It is easy to see that
Z
1
|x(t)| ≤ K
|F (s, x(s), x0 (s))|ds + RL
0
+ RK
m
X
k=1
m
X
k=1
|Vk (x(tk ), x0 (tk ))|,
|Uk (x(tk ), x0 (tk ))|
EJDE-2012/24
EXISTENCE OF SOLUTIONS
9
and
Z
0
1
0
|x (t)| ≤ M
|F (s, x(s), x (s))|ds + RN
0
m
X
|Uk (x(tk ), x0 (tk ))|
k=1
+ RM
m
X
|Vk (x(tk ), x0 (tk ))|.
k=1
From (H7), (H8) and (H9), we obtain
kxk0 + kx0 k0 ≤ (K + M )kgkL1 +
m
X
R(L + N )αk (kxk0 + kx0 k0 )
k=1
+
m
X
R(K + M )βk (kxk0 + kx0 k0 ).
k=1
Setting µ = 2P R
Pm
k=1 (αk + βk ), we obtain
kxk1 ≤ 2P kgkL1 + µkxk1 .
Then (1 − µ)kxk1 ≤ 2P kgkL1 . Using condition (H10) we obtain
2P
)kgkL1 := r1 .
kxk1 ≤ (
1−µ
Step 2. Existence of solutions. Let Ω1 = {x ∈ P C 1 (J) : kxk1 < r1 + 1}. The
rest of the proof is similar to that of Theorem 3.2, and it is omitted.
Acknowledgements. The authors are grateful to the King Fahd University of
Petroleum and Minerals for its support.
References
[1] B. Ahmad; Existence of solutions for second-order nonlinear impulsive boundary-value problems, Elect. J. Diff. Equ. 2009 (2009) no. 68, 1–7.
[2] D. d. Bainov, P. S. Simeonov; Impulsive Differential Equations: Periodic Solutions and
Applications, Longman Scientific and Technical, Essex, England, 1993.
[3] A. Boucherif, J Henderson; Topological methods in nonlinear boundary value problems, Nonlinear Times and Digest 1 (1994), 149-168.
[4] L. Elsgolts; Differential Equations and Calculus of Variations, MIR Publishers, Moscow,
1970.
[5] L. H. Erbe, Xinzhi Liu; Existence results for boundary value problems of second order impulsive differential equations, J. Math. Anal. Appl. 149(1990), 56-69.
[6] A. Lakmeche, A. Boucherif; Boundary value problems for impulsive second order differential
equations, Dynam. Cont. Discr. Impul. Syst. Series A: Math. Anal. 9 (2002), 313-319.
[7] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov; Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[8] Y. Lee, X. Liu; Study of singular boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl. 331 (2007), 159-176.
[9] J. J. Nieto; Periodic boundary value problems for first-order impulsive ordinary differential
equations, NonlinearAnal. 51 (2002), 1223–1232.
[10] J. J. Nieto; Basic theory for nonresonance impulsive periodic problems of first order, J. Math.
Anal. Appl. 205(1997), 423–433.
[11] S. G. Pandit, S. G. Deo; Differential Systems Involving Impulses, Lecture Notes in Math.
Vol. 954, Springer-Verlag, Berlin, 1982.
[12] I. Rachůnková,J Tomeček; Singular Dirichlet problem for ordinary differential equations with
impulses, Non. Anal. 65 (2006), 210-229.
[13] Y. V. Rogovchenko; Impulsive evolution systems:Main results and new trends, Dyn. Contin.
Discrete Impuls.Syst. 3 (1997) 57–88.
10
A. BOUCHERIF, A. S. AL-QAHTANI, B. CHANANE
EJDE-2012/24
[14] A. M. Samoilenko, N. A. Perestyuk; Impulsive Differential Equations, World Scientific, Singapore, 1995.
[15] J. Tomeček; Nonlinear boundary value problem for nonlinear second order differential equations with impulses, Electronic J. Qual. Theo. Diff. Equ. Vol 2005 (2005), No. 10, 1-22.
Abdelkader Boucherif
King Fahd University of Petroleum and Minerals, Department of Mathematics and
Statistics, P.O. Box 5046, Dhahran 31261, Saudi Arabia
E-mail address: aboucher@kfupm.edu.sa
Ali S. Al-Qahtani
King Fahd University of Petroleum and Minerals, Department of Mathematics and
Statistics, P.O. Box 5046, Dhahran 31261, Saudi Arabia
E-mail address: alitalhan@hotmail.com
Bilal Chanane
King Fahd University of Petroleum and Minerals, Department of Mathematics and
Statistics, P.O. Box 5046, Dhahran 31261, Saudi Arabia
E-mail address: chanane@kfupm.edu.sa