Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 1, Article 19, 2004
ON A CERTAIN RETARDED INTEGRAL INEQUALITY AND ITS APPLICATIONS
B.G. PACHPATTE
57 S HRI N IKETAN C OLONY
N EAR A BHINAY TALKIES
AURANGABAD 431 001
(M AHARASHTRA ) I NDIA .
bgpachpatte@hotmail.com
Received 16 September, 2003; accepted 15 December, 2003
Communicated by D. Bainov
A BSTRACT. In the present paper explicit bound on a new retarded integral inequality in two
independent variables is established. Applications are given to illustrate the usefulness of the
inequality.
Key words and phrases: Retarded integral inequality, explicit bound, two independent variables, Volterra-Fredholm integral
equation, uniqueness of solutions, continuous dependence of solution.
2000 Mathematics Subject Classification. 26D10, 26D15.
1. I NTRODUCTION
In the study of differential, integral and finite difference equations, one has often to deal with
certain integral and finite difference inequalities, which provide explicit bounds on the unknown
functions. A detailed account on such inequalities and some of their applications can be found
in [2, 3, 6, 7, 9]. In [8] the present author has established the following useful integral inequality.
Lemma 1.1. Let u (t) ∈ C (I,R+ ), a (t, s) , b (t, s) ∈ C (D,R+ ) and a (t, s) , b (t, s) are nondecreasing in t for each s ∈ I, where I = [α, β], R+ = [0, ∞), D = {(t, s) ∈ I 2 : α ≤ s ≤ t ≤ β}
and suppose that
Z t
Z β
u (t) ≤ k +
a (t, s) u (s) ds +
b (t, s) u (s) ds,
α
α
for t ∈ I, where k ≥ 0 is a constant. If
Z s
Z β
p (t) =
b (t, s) exp
a (s, σ) dσ ds < 1,
α
α
for t ∈ I, then
k
u (t) ≤
exp
1 − p (t)
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
120-03
Z
t
a (t, s) ds ,
α
2
B.G. PACHPATTE
for t ∈ I.
A version of the above inequality when a (t, s) = a (s) , b (t, s) = b (s) is first given in [2, p.
11]. In a recent paper [10] a useful general retarded version of the above inequality is given. The
aim of the present paper is to establish a general two independent variable retarded version of
the above inequality which can be used as a tool to study the behavior of solutions of a general
retarded Volterra-Fredholm integral equation in two independent variables. Applications are
given to convey the importance of our result to the literature.
2. M AIN R ESULT
In what follows, R denotes the set of real numbers, R+ = [0, ∞), I1 = [x0 , M ] and I2 =
[y0 , N ] are the given subsets of R. Let ∆ = I1 × I2 and
E = (x, y, s, t) ∈ ∆2 : x0 ≤ s ≤ x ≤ M, y0 ≤ t ≤ y ≤ N .
Our main result is established in the following theorem.
Theorem 2.1. Let u (x, y) ∈ C (∆,R+ ) , a (x, y, s, t) , b (x, y, s, t) ∈ C (E,R+ ) and a (x, y, s, t) ,
b (x, y, s, t) be nondecreasing in x and y for each s ∈ I1 , t ∈ I2 , α ∈ C 1 (I1 , I1 ) , β ∈
C 1 (I2 , I2 ) be nondecreasing with α (x) ≤ x on I1 , β (y) ≤ y on I2 and suppose that
Z α(x) Z β(y)
a (x, y, s, t) u (s, t) dtds
(2.1) u (x, y) ≤ c +
α(x0 )
β(y0 )
α(M )
Z
Z
β(N )
b (x, y, s, t) u (s, t) dtds,
+
α(x0 )
for x ∈ I1 , y ∈ I2 , where c ≥ 0 is a constant. If
Z
Z α(M ) Z β(N )
b (x, y, s, t) exp
(2.2) p (x, y) =
α(x0 )
α(s)
!
β(t)
a (s, t, σ, τ ) dτ dσ dtds < 1,
α(x0 )
β(y0 )
Z
β(y0 )
β(y0 )
for x ∈ I1 , y ∈ I2 , then
(2.3)
Z
c
exp
u (x, y) ≤
1 − p (x, y)
α(x)
!
β(y)
Z
a (x, y, s, t) dtds ,
α(x0 )
β(y0 )
for x ∈ I1 , y ∈ I2 .
Proof. Fix any arbitrary (X, Y ) ∈ ∆. Then for x0 ≤ x ≤ X, y0 ≤ y ≤ Y we have
Z α(x) Z β(y)
(2.4) u (x, y) ≤ c +
a (X, Y, s, t) u (s, t) dtds
α(x0 )
β(y0 )
Z
α(M )
Z
β(N )
+
b (X, Y, s, t) u (s, t) dtds.
α(x0 )
β(y0 )
Let
Z
(2.5)
α(M )
Z
β(N )
k =c+
b (X, Y, s, t) u (s, t) dtds,
α(x0 )
β(y0 )
then (2.4) can be restated as
Z
(2.6)
α(x)
Z
β(y)
u (x, y) ≤ k +
a (X, Y, s, t) u (s, t) dtds,
α(x0 )
J. Inequal. Pure and Appl. Math., 5(1) Art. 19, 2004
β(y0 )
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O N A C ERTAIN R ETARDED I NTEGRAL I NEQUALITY A ND I TS A PPLICATIONS
3
for x0 ≤ x ≤ X, y0 ≤ y ≤ Y . Now a suitable application of the inequality (c1 ) given in
Theorem 3 in [9, p. 51] to (2.6) yields
!
Z α(x) Z β(y)
(2.7)
u (x, y) ≤ k exp
a (X, Y, s, t) dtds ,
α(x0 )
β(y0 )
for x0 ≤ x ≤ X, y0 ≤ y ≤ Y . Since (X, Y ) ∈ ∆ is arbitrary, from (2.7) and (2.5) with X and
Y replaced by x and y we have
!
Z α(x) Z β(y)
(2.8)
u (x, y) ≤ k exp
a (x, y, s, t) dtds ,
α(x0 )
β(y0 )
where
Z
α(M )
β(N )
Z
k =c+
(2.9)
b (x, y, s, t) u (s, t) dtds,
α(x0 )
β(y0 )
for all x ∈ I1 , y ∈ I2 . Using (2.8) on the right side of (2.9) and in view of (2.2) we have
c
.
(2.10)
k≤
1 − p (x, y)
Using (2.10) in (2.8) we get the desired inequality in (2.3). The proof is complete.
By taking b (x, y, s, t) = 0 in Theorem 2.1, we get the following useful inequality.
Corollary 2.2. Let u (x, y) , a (x, y, s, t) , α (x) , β (y) and c be as in Theorem 2.1. If
Z α(x) Z β(y)
(2.11)
u (x, y) ≤ c +
a (x, y, s, t) u (s, t) dtds,
α(x0 )
β(y0 )
for x ∈ I1 , y ∈ I2 , then
Z
α(x)
Z
u (x, y) ≤ c exp
(2.12)
!
β(y)
a (x, y, s, t) dtds ,
α(x0 )
β(y0 )
for x ∈ I1 , y ∈ I2 .
The following corollaries of Theorem 2.1 and Corollary 2.2 are also useful in certain applications.
Corollary 2.3. Let u (x, y) , a (x, y, s, t) , b (x, y, s, t) and c be as in Theorem 2.1 and suppose
that
Z xZ y
Z MZ N
(2.13) u (x, y) ≤ c +
a (x, y, s, t) u (s, t) dtds +
b (x, y, s, t) u (s, t) dtds,
x0
y0
x0
y0
for x ∈ I1 , y ∈ I2 . If
Z
(2.14)
M
Z
N
q (x, y) =
Z s Z
t
b (x, y, s, t) exp
x0
a (s, t, σ, τ ) dτ dσ dtds < 1,
y0
x0
y0
for x ∈ I1 , y ∈ I2 , then
(2.15)
c
exp
u (x, y) ≤
1 − q (x, y)
Z
x
Z
y
a (x, y, s, t) dtds ,
x0
y0
for x ∈ I1 , y ∈ I2 .
J. Inequal. Pure and Appl. Math., 5(1) Art. 19, 2004
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4
B.G. PACHPATTE
Corollary 2.4. Let u (x, y) , a (x, y, s, t) and c be as in Corollary 2.2. If
Z xZ y
(2.16)
u (x, y) ≤ c +
a (x, y, s, t) u (s, t) dtds,
x0
y0
for x ∈ I1 , y ∈ I2 , then
Z
x
Z
y
u (x, y) ≤ c exp
(2.17)
a (x, y, s, t) dtds ,
x0
y0
for x ∈ I1 , y ∈ I2 .
The proofs of Corollaries 2.3 and 2.4 follow by taking α (x) = x, β (y) = y in Theorem 2.1
and Corollary 2.2.
3. A PPLICATIONS
In this section, we present applications of Theorem 2.1 to study certain properties of solutions
of the retarded Volterra-Fredholm integral equation in two independent variables of the form
Z xZ y
(3.1) z (x, y) = f (x, y) +
A (x, y, s, t, z (s − h1 (s) , t − h2 (t))) dtds
x0
y0
Z
M
Z
N
B (x, y, s, t, z (s − h1 (s) , t − h2 (t))) dtds,
+
x0
y0
where z, f ∈ C (∆,R) , A, B ∈ C (E×R,R) and h1 ∈ C (I1 ,R+ ) , h2 ∈ C (I2 ,R+ ) , are nonincreasing, x − h1 (x) ≥ 0, y − h2 (y) ≥ 0, x − h1 (x) ∈ C 1 (I1 , I1 ) , y − h2 (y) ∈ C 1 (I2 , I2 ) ,
h01 (x) < 1, h02 (x) < 1, h1 (x0 ) = h2 (y0 ) = 0.
The following theorem gives the bound on the solution of equation (3.1).
Theorem 3.1. Suppose that the functions f, A, B in equation (3.1) satisfy the conditions
(3.2)
|f (x, y)| ≤ c,
(3.3)
|A (x, y, s, t, z)| ≤ a (x, y, s, t) |z| ,
(3.4)
|B (x, y, s, t, z)| ≤ b (x, y, s, t) |z| ,
where c, a (x, y, s, t) , b (x, y, s, t) are as in Theorem 2.1. Let
1
1
(3.5)
M1 = max
, M2 = max
,
x∈I1 1 − h01 (x)
y∈I2 1 − h02 (y)
and
!
Z φ(M ) Z ψ(N )
Z φ(s) Z ψ(t)
(3.6) p̄ (x, y) =
b̄ (x, y, s, t) exp
ā (s, t, σ, τ ) dτ dσ dtds < 1,
φ(x0 )
ψ(y0 )
φ(x0 )
ψ(y0 )
where φ (x) = x − h1 (x) , x ∈ I1 , ψ (y) = y − h2 (y) , y ∈ I2 and
ā (x, y, σ, τ ) = M1 M2 a (x, y, σ + h1 (s) , τ + h2 (t)) ,
b̄ (x, y, σ, τ ) = M1 M2 b (x, y, σ + h1 (s) , τ + h2 (t)) .
If z (x, y) is a solution of equation (3.1) on ∆, then
!
Z φ(x) Z ψ(y)
c
(3.7)
|z (x, y)| ≤
exp
ā (x, y, σ, τ ) dτ dσ ,
1 − p̄ (x, y)
φ(x0 ) ψ(y0 )
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O N A C ERTAIN R ETARDED I NTEGRAL I NEQUALITY A ND I TS A PPLICATIONS
5
for x ∈ I1 , y ∈ I2 .
Proof. Since z (x, y) is a solution of equation (3.1), from (3.1) – (3.4) we have
Z xZ y
(3.8) |z (x, y)| ≤ c +
a (x, y, s, t) |z (s − h1 (s) , t − h2 (t))| dtds
x0
y0
M
Z
Z
N
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t))| dtds.
+
x0
y0
Now by making the change of variables on the right side of (3.8) and using (3.5) we have
Z
(3.9)
φ(x)
Z
ψ(y)
|z (x, y)| ≤ c +
ā (x, y, σ, τ ) |z (σ, τ )| dτ dσ
φ(x0 )
ψ(y0 )
Z
φ(M )
Z
ψ(N )
b̄ (x, y, σ, τ ) |z (σ, τ )| dτ dσ.
+
φ(x0 )
ψ(y0 )
A suitable application of Theorem 2.1 to (3.9) yields (3.7).
The next result deals with the uniqueness of solutions of (3.1).
Theorem 3.2. Suppose that the functions A, B in equation (3.1) satisfy the conditions
(3.10)
|A (x, y, s, t, z) − A (x, y, s, t, z̄)| ≤ a (x, y, s, t) |z − z̄| ,
(3.11)
|B (x, y, s, t, z) − B (x, y, s, t, z̄)| ≤ b (x, y, s, t) |z − z̄| ,
where a (x, y, s, t) , b (x, y, s, t) are as in Theorem 2.1. Let M1 , M2 , φ, ψ, ā, b̄, p̄ be as in Theorem 3.1. Then the equation (3.1) has at most one solution on ∆.
Proof. Let z (x, y) and z̄ (x, y) be two solutions of equation (3.1) on ∆. From (3.1), (3.10),
(3.11) we have
(3.12)
|z (x, y) − z̄ (x, y)|
Z xZ y
a (x, y, s, t) |z (s − h1 (s) , t − h2 (t)) − z̄ (s − h1 (s) , t − h2 (t))| dtds
≤
x0 y 0
Z xZ y
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t)) − z̄ (s − h1 (s) , t − h2 (t))| dtds.
+
x0
y0
By making the change of variables on the right side of (3.12) and using (3.5) we have
Z
(3.13)
φ(x)
Z
ψ(y)
ā (x, y, σ, τ ) |z (σ, τ ) − z̄ (σ, τ )| dτ dσ
|z (x, y) − z̄ (x, y)| ≤
φ(x0 )
ψ(y0 )
Z
φ(M )
Z
ψ(N )
b̄ (x, y, σ, τ ) |z (σ, τ ) − z̄ (σ, τ )| dτ dσ.
+
φ(x0 )
ψ(y0 )
Now a suitable application of Theorem 2.1 to (3.13) yields
|z (x, y) − z̄ (x, y)| ≤ 0.
Therefore z (x, y) = z̄ (x, y), i.e. there is at most one solution to the equation (3.1).
J. Inequal. Pure and Appl. Math., 5(1) Art. 19, 2004
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6
B.G. PACHPATTE
The following theorem deals with the continuous dependence of solution of equation (3.1)
on the right side.
Consider the equation (3.1) and the following equation
Z xZ y
(3.14) w (x, y) = g (x, y) +
F (x, y, s, t, w (s − h1 (s) , t − h2 (t))) dtds
x0
y0
Z
M
Z
N
G (x, y, s, t, w (s − h1 (s) , t − h2 (t))) dtds,
+
x0
y0
where w, g ∈ C (∆,R) , F, G ∈ C (E×R,R) and h1 , h2 are as in equation (3.1).
Theorem 3.3. Suppose that the functions A, B in equation (3.1) satisfy the conditions (3.10),
(3.11) in Theorem 3.2 and further assume that
|f (x, y) − g (x, y)| ≤ ε,
(3.15)
Z
x
y
Z
|A (x, y, s, t, w (s − h1 (s) , t − h2 (t)))
(3.16)
x0
y0
−F (x, y, s, t, w (s − h1 (s) , t − h2 (t)))| dtds ≤ ε,
Z
M
Z
N
|B (x, y, s, t, w (s − h1 (s) , t − h2 (t)))
(3.17)
x0
y0
−G (x, y, s, t, w (s − h1 (s) , t − h2 (t)))| dtds ≤ ε,
where ε > 0 is an arbitrary small constant, and let M1 , M2 , φ, ψ, ā, b̄, p̄ be as in Theorem 3.1.
Then the solution of equation (3.1) depends continuously on the functions involved on the right
side of equation (3.1).
Proof. Let z (x, y) and w (x, y) be the solutions of (3.1) and (3.14) respectively. Then we have
(3.18)
z (x, y) − w (x, y)
= f (x, y) − g (x, y)
Z xZ y
{A (x, y, s, t, z (s − h1 (s) , t − h2 (t)))
+
y0
x0
− A (x, y, s, t, w (s − h1 (s) , t − h2 (t)))}dtds
Z
x
y
Z
{A (x, y, s, t, w (s − h1 (s) , t − h2 (t)))
+
x0
y0
− F (x, y, s, t, w (s − h1 (s) , t − h2 (t)))}dtds
Z
M
Z
N
{B (x, y, s, t, z (s − h1 (s) , t − h2 (t)))
+
x0
y0
−B (x, y, s, t, w (s − h1 (s) , t − h2 (t)))}dtds
Z
M Z
N
{B (x, y, s, t, w (s − h1 (s) , t − h2 (t)))
+
x0
y0
− G (x, y, s, t, w (s − h1 (s) , t − h2 (t)))}dtds.
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O N A C ERTAIN R ETARDED I NTEGRAL I NEQUALITY A ND I TS A PPLICATIONS
7
Using (3.10), (3.11), (3.15) – (3.17) in (3.18) we get
|z (x, y) − w (x, y)|
Z xZ y
≤ 3ε +
a (x, y, s, t) |z (s − h1 (s) , t − h2 (t)) − w (s − h1 (s) , t − h2 (t))| dtds
(3.19)
Z
x0
M
y0
N
Z
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t)) − w (s − h1 (s) , t − h2 (t))| dtds.
+
x0
y0
By making the change of variables on the right side of (3.19) and using (3.5) we get
Z φ(x) Z ψ(y)
(3.20) |z (x, y) − w (x, y)| ≤ 3ε +
ā (x, y, s, t) |z (σ, τ ) − w (σ, τ )| dτ dσ
φ(x0 )
ψ(y0 )
φ(M )
Z
Z
ψ(N )
b̄ (x, y, s, t) |z (σ, τ ) − w (σ, τ )| dτ dσ.
+
φ(x0 )
ψ(y0 )
Now a suitable application of Theorem 2.1 to (3.20) yields
"
!#
Z φ(x) Z ψ(y)
1
(3.21) |z (x, y) − w (x, y)| ≤ 3ε
exp
ā (x, y, σ, τ ) dτ dσ
,
1 − p̄ (x, y)
φ(x0 ) ψ(y0 )
for x ∈ I1 , y ∈ I2 . On the compact set, the quantity in square brackets in (3.21) is bounded by
some positive constant M . Therefore |z (x, y) − w (x, y)| ≤ 3M ε on the set, so the solution
to equation (3.1) depends continuously on the functions involved on the right side of equation
(3.1). If ε → 0, then |z (x, y) − w (x, y)| → 0 on the set.
We next consider the following retarded Volterra-Fredholm integral equations
Z xZ y
(3.22) z (x, y) = f (x, y) +
A (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ) dtds
x0
y0
Z
M
Z
N
B (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ) dtds,
+
x0
Z
x
y0
y
Z
A (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ0 ) dtds
(3.23) z (x, y) = f (x, y) +
x0
y0
Z
M
Z
N
B (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ0 ) dtds,
+
x0
y0
where z, f ∈ C (∆,R) , A, B ∈ C (E×R×R,R) and µ, µ0 are real parameters.
The following theorem shows the dependency of solutions of equations (3.22) and (3.23) on
parameters.
Theorem 3.4. Suppose that
(3.24)
|A (x, y, s, t, z, µ) − A (x, y, s, t, z̄, µ)| ≤ a (x, y, s, t) |z − z̄| ,
(3.25)
|A (x, y, s, t, z̄, µ) − A (x, y, s, t, z̄, µ0 )| ≤ r (x, y, s, t) |µ − µ0 | ,
(3.26)
|B (x, y, s, t, z, µ) − B (x, y, s, t, z̄, µ)| ≤ b (x, y, s, t) |z − z̄| ,
J. Inequal. Pure and Appl. Math., 5(1) Art. 19, 2004
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8
(3.27)
B.G. PACHPATTE
|B (x, y, s, t, z̄, µ) − B (x, y, s, t, z̄, µ0 )| ≤ e (x, y, s, t) |µ − µ0 | ,
where a (x, y, s, t) , b (x, y, s, t) are as in Theorem 2.1 and r, e ∈ C (E,R+ ) are such that
Z xZ y
(3.28)
r (x, y, s, t) dtds ≤ k1 ,
x0
Z
M
y0
Z
N
e (x, y, s, t) dtds ≤ k2 ,
(3.29)
x0
y0
where k1 , k2 are positive constants. Let M1 , M2 , φ, ψ, ā, b̄, p̄ be as in Theorem 3.1. Let z1 (x, y)
and z2 (x, y) be the solutions of (3.22) and (3.23) respectively. Then
!
Z φ(x) Z ψ(y)
(k1 + k2 ) |µ − µ0 |
exp
(3.30) |z1 (x, y) − z2 (x, y)| ≤
ā (x, y, s, t) dtds ,
1 − p̄ (x, y)
φ(x0 ) ψ(y0 )
for x ∈ I1 , y ∈ I2 .
Proof. Let z (x, y) = z1 (x, y) − z2 (x, y) , (x, y) ∈ ∆. Then
Z xZ y
(3.31) z (x, y) =
{A (x, y, s, t, z1 (s − h1 (s) , t − h2 (t)) , µ)
x0
y0
−A (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ)} dtds
Z xZ y
+
{A (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ)
x0
y0
−A (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ0 )} dtds
Z MZ N
+
{B (x, y, s, t, z1 (s − h1 (s) , t − h2 (t)) , µ)
x0
y0
−B (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ)} dtds
Z MZ N
+
{B (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ)
x0
y0
−B (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ0 )} dtds.
Using (3.24) – (3.29) in (3.31) we get
(3.32)
|z (x, y)| ≤ |µ − µ0 | k1 + |µ − µ0 | k2
Z xZ y
+
a (x, y, s, t) |z (s − h1 (s) , t − h2 (t))| dtds
x0
y0
Z
M
Z
N
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t))| dtds.
+
x0
y0
By using the change of variables on the right side of (3.32) and (3.5) we get
Z φ(x) Z ψ(y)
(3.33) |z (x, y)| ≤ (k1 + k2 ) |µ − µ0 | +
ā (x, y, σ, τ ) |z (σ, τ )| dτ dσ
φ(x0 )
ψ(y0 )
Z
φ(M )
Z
ψ(N )
b̄ (x, y, σ, τ ) |z (σ, τ )| dτ dσ.
+
φ(x0 )
J. Inequal. Pure and Appl. Math., 5(1) Art. 19, 2004
ψ(y0 )
http://jipam.vu.edu.au/
O N A C ERTAIN R ETARDED I NTEGRAL I NEQUALITY A ND I TS A PPLICATIONS
9
Now a suitable application of Theorem 2.1 to (3.33) yields (3.30), which shows the dependency
of solutions of (3.22) and (3.23) on parameters.
In conclusion, we note that the results given in this paper can be extended very easily to
functions involving many independent variables. Since the formulations of such results are
quite straightforward in view of the results given above (see also [6]) and hence we omit the
details. For the study of behavior of solutions of Volterra-Fredholm integral equations involving
functions of one independent variable, see [1, 4, 5].
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[3] O. LIPOVAN, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl., 252
(2000), 389–401.
[4] R.K. MILLER, J.A. NOHEL AND J.S.W. WONG, A stability theorem for nonlinear mixed integral
equations, J. Math. Anal. Appl., 25 (1969), 446–449.
[5] B.G. PACHPATTE, On the existence and uniqueness of solutions of Volterra-Fredholm integral
equations, Math. Seminar Notes, 10 (1982), 733–742.
[6] B.G. PACHPATTE, Inequalities for Differential and Integral Equations, Academic Press, New
York, 1998.
[7] B.G. PACHPATTE, Inequalities for Finite Difference Equations, Marcel Dekker, Inc. New York,
2002.
[8] B.G. PACHPATTE, A note on certain integral inequality, Tamkang J. Math., 33 (2002), 353–358.
[9] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl., 267
(2002), 48–61.
[10] B.G. PACHPATTE, Explicit bound on a retarded integral inequality, Math. Inequal. Appl., 7 (2004),
7–11.
J. Inequal. Pure and Appl. Math., 5(1) Art. 19, 2004
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