Journal of Inequalities in Pure and
Applied Mathematics
ON A CERTAIN RETARDED INTEGRAL INEQUALITY
AND ITS APPLICATIONS
volume 5, issue 1, article 19,
2004.
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India.
EMail: bgpachpatte@hotmail.com
Received 16 September, 2003;
accepted 15 December, 2003.
Communicated by: D. Bainov
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
120-03
Quit
Abstract
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.
2000 Mathematics Subject Classification: 26D10, 26D15.
Key words: Retarded integral inequality, explicit bound, two independent variables,
Volterra-Fredholm integral equation, uniqueness of solutions, continuous
dependence of solution.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
1.
Introduction
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
B.G. Pachpatte
Title Page
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
On A Certain Retarded Integral
Inequality And Its Applications
α
for t ∈ I, then
Contents
JJ
J
II
I
Go Back
k
exp
u (t) ≤
1 − p (t)
Z
t
a (t, s) ds ,
α
for t ∈ I.
Close
Quit
Page 3 of 22
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
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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.
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
2.
Main Result
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
a (x, y, s, t) u (s, t) dtds
α(x0 )
Contents
β(y0 )
Z
α(M )
Z
β(N )
+
b (x, y, s, t) u (s, t) dtds,
α(x0 )
β(y0 )
for x ∈ I1 , y ∈ I2 , where c ≥ 0 is a constant. If
Z
α(M )
Z
II
I
Close
b (x, y, s, t)
α(x0 )
JJ
J
Go Back
β(N )
(2.2) p (x, y) =
B.G. Pachpatte
Title Page
β(y)
(2.1) u (x, y) ≤ c +
On A Certain Retarded Integral
Inequality And Its Applications
Quit
β(y0 )
Z
α(s)
Z
β(t)
× exp
!
Page 5 of 22
a (s, t, σ, τ ) dτ dσ dtds < 1,
α(x0 )
β(y0 )
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 , then
(2.3)
Z
c
u (x, y) ≤
exp
1 − p (x, y)
α(x)
Z
!
β(y)
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
B.G. Pachpatte
β(N )
+
b (X, Y, s, t) u (s, t) dtds.
α(x0 )
β(y0 )
Let
Z
(2.5)
α(M )
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
a (X, Y, s, t) u (s, t) dtds,
β(y0 )
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 )
JJ
J
II
I
Go Back
β(y)
u (x, y) ≤ k +
α(x0 )
Title Page
Contents
β(N )
Z
On A Certain Retarded Integral
Inequality And Its Applications
β(y0 )
Close
Quit
Page 6 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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
Z
(2.8)
α(x)
β(y)
α(x0 )
β(y0 )
u (x, y) ≤ k exp
a (x, y, s, t) dtds ,
where
Z
(2.9)
α(M )
Z
β(N )
k =c+
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
k≤
.
1 − p (x, y)
(2.10)
Using (2.10) in (2.8) we get the desired inequality in (2.3). The proof is complete.
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
By taking b (x, y, s, t) = 0 in Theorem 2.1, we get the following useful
inequality.
Go Back
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
Quit
Z
(2.11)
α(x)
Z
Page 7 of 22
β(y)
u (x, y) ≤ c +
a (x, y, s, t) u (s, t) dtds,
α(x0 )
Close
β(y0 )
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 , then
Z
(2.12)
α(x)
!
β(y)
Z
u (x, y) ≤ c exp
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
(2.13) u (x, y) ≤ c +
a (x, y, s, t) u (s, t) dtds
x0
y0
M
Z
+
b (x, y, s, t) u (s, t) dtds,
x0
x0
JJ
J
N
Z s Z
t
b (x, y, s, t) exp
y0
a (s, t, σ, τ ) dτ dσ dtds < 1,
x0
c
u (x, y) ≤
exp
1 − q (x, y)
for x ∈ I1 , y ∈ I2 .
II
I
Go Back
y0
Close
Quit
for x ∈ I1 , y ∈ I2 , then
(2.15)
Contents
y0
for x ∈ I1 , y ∈ I2 . If
(2.14) q (x, y)
Z MZ
=
B.G. Pachpatte
Title Page
N
Z
On A Certain Retarded Integral
Inequality And Its Applications
Z
x
Z
y
Page 8 of 22
a (x, y, s, t) dtds ,
x0
y0
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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
(2.17)
x
Z
y
u (x, y) ≤ c exp
a (x, y, s, t) dtds ,
x0
y0
for x ∈ I1 , y ∈ I2 .
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
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.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
3.
Applications
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
(3.1) z (x, y)
Z
x
Z
y
A (x, y, s, t, z (s − h1 (s) , t − h2 (t))) dtds
= f (x, y) +
x0
Z
y0
M
Z
On A Certain Retarded Integral
Inequality And Its Applications
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,
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
(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| ,
Page 10 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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
,
0
x∈I1 1 − h1 (x)
y∈I2 1 − h02 (y)
and
Z φ(M ) Z ψ(N )
(3.6) p̄ (x, y) =
b̄ (x, y, s, t)
φ(x0 )
ψ(y0 )
Z
φ(s)
Z
ψ(t)
× exp
!
ā (s, t, σ, τ ) dτ dσ dtds < 1,
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 )
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
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 y 0
M Z N
Close
Quit
Page 11 of 22
Z
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t))| dtds.
+
x0
y0
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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| ,
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
(3.11)
|B (x, y, s, t, z) − B (x, y, s, t, z̄)| ≤ b (x, y, s, t) |z − z̄| ,
II
I
Go Back
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 ∆.
Close
Proof. Let z (x, y) and z̄ (x, y) be two solutions of equation (3.1) on ∆. From
Page 12 of 22
Quit
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
(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))
x0
Z
x
+
x0
y0
−z̄ (s − h1 (s) , t − h2 (t))| dtds
Z y
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t))
y0
−z̄ (s − h1 (s) , t − h2 (t))| dtds.
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
By making the change of variables on the right side of (3.12) and using (3.5)
we have
(3.13)
|z (x, y) − z̄ (x, y)|
Z φ(x) Z ψ(y)
≤
ā (x, y, σ, τ ) |z (σ, τ ) − z̄ (σ, τ )| dτ dσ
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).
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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
(3.14) w (x, y)
Z
x
Z
y
F (x, y, s, t, w (s − h1 (s) , t − h2 (t))) dtds
= g (x, y) +
x0
Z
y0
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
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
|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,
JJ
J
II
I
Go Back
Close
Quit
Z
M Z
N
|B (x, y, s, t, w (s − h1 (s) , t − h2 (t)))
(3.17)
x0
Page 14 of 22
y0
−G (x, y, s, t, w (s − h1 (s) , t − h2 (t)))| dtds ≤ ε,
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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)))
x0
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
y0
− 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
Contents
− 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
Title Page
y0
− G (x, y, s, t, w (s − h1 (s) , t − h2 (t)))}dtds.
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
Using (3.10), (3.11), (3.15) – (3.17) in (3.18) we get
(3.19)
|z (x, y) − w (x, y)|
Z xZ y
≤ 3ε +
a (x, y, s, t) |z (s − h1 (s) , t − h2 (t))
x0
Z
M
+
y0
−w (s − h1 (s) , t − h2 (t))| dtds
Z N
b (x, y, s, t) |z (s − h1 (s) , t − h2 (t))
x0
y0
−w (s − h1 (s) , t − h2 (t))| dtds.
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
By making the change of variables on the right side of (3.19) and using (3.5)
we get
Title Page
(3.20)
|z (x, y) − w (x, y)|
Z φ(x) Z ψ(y)
≤ 3ε +
ā (x, y, s, t) |z (σ, τ ) − w (σ, τ )| dτ dσ
φ(x0 )
Z
ψ(y0 )
φ(M ) 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
Contents
JJ
J
II
I
Go Back
Close
Quit
(3.21)
|z (x, y) − w (x, y)|
"
!#
Z φ(x) Z ψ(y)
1
≤ 3ε
exp
ā (x, y, σ, τ ) dτ dσ
,
1 − p̄ (x, y)
φ(x0 ) ψ(y0 )
Page 16 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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
On A Certain Retarded Integral
Inequality And Its Applications
(3.22) z (x, y)
Z
x
Z
y
A (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ) dtds
= f (x, y) +
x0
Z
y0
M Z N
B (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ) dtds,
+
x0
y0
x
Z
y
A (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ0 ) dtds
= f (x, y) +
x0 y0
Z MZ N
B (x, y, s, t, z (s − h1 (s) , t − h2 (t)) , µ0 ) dtds,
+
x0
Title Page
Contents
JJ
J
(3.23) z (x, y)
Z
B.G. Pachpatte
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.
II
I
Go Back
Close
Quit
Page 17 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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| ,
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
(3.27)
|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
y0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Z
M 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.
Quit
Page 18 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
Then
(3.30)
|z1 (x, y) − z2 (x, y)|
(k1 + k2 ) |µ − µ0 |
≤
exp
1 − p̄ (x, y)
Z
φ(x)
Z
!
ψ(y)
ā (x, y, s, t) dtds ,
φ(x0 )
ψ(y0 )
for x ∈ I1 , y ∈ I2 .
Proof. Let z (x, y) = z1 (x, y) − z2 (x, y) , (x, y) ∈ ∆. Then
Z
x
Z
y
{A (x, y, s, t, z1 (s − h1 (s) , t − h2 (t)) , µ)
(3.31) z (x, y) =
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
B.G. Pachpatte
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
On A Certain Retarded Integral
Inequality And Its Applications
y0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 22
−B (x, y, s, t, z2 (s − h1 (s) , t − h2 (t)) , µ0 )} dtds.
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
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
(3.33)
|z (x, y)| ≤ (k1 + k2 ) |µ − µ0 |
Z φ(x) Z ψ(y)
+
ā (x, y, σ, τ ) |z (σ, τ )| dτ dσ
φ(x0 )
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
ψ(y0 )
Z
φ(M )
Z
Contents
ψ(N )
b̄ (x, y, σ, τ ) |z (σ, τ )| dτ dσ.
+
φ(x0 )
ψ(y0 )
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].
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
References
[1] S. ASIROV AND Ja.D. MAMEDOV, Investigation of solutions of nonlinear Volterra-Fredholm operator equations, Dokl. Akad. Nauk SSSR, 229
(1976), 982–986.
[2] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications,
Kluwer Academic Publishers, Dordrecht, 1992.
[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.
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au
[10] B.G. PACHPATTE, Explicit bound on a retarded integral inequality, Math.
Inequal. Appl., 7 (2004), 7–11.
On A Certain Retarded Integral
Inequality And Its Applications
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 22
J. Ineq. Pure and Appl. Math. 5(1) Art. 19, 2004
http://jipam.vu.edu.au