Journal of Inequalities in Pure and
Applied Mathematics
ON SOME NEW NONLINEAR RETARDED INTEGRAL
INEQUALITIES
volume 5, issue 3, article 80,
2004.
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India
Received 28 April, 2004;
accepted 27 July, 2004.
Communicated by: J. Sándor
EMail: bgpachpatte@hotmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
164-04
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Abstract
In the present paper we establish new nonlinear retarded integral inequalities
which can be used as tools in certain applications. Some applications are also
given to illustrate the usefulness of our results.
2000 Mathematics Subject Classification: 26D15, 26D20
Key words: Retarded integral inequalities, Explicit bounds, Two independent variable
generalizations, Partial derivatives, Estimate on the solution.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 8
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
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1.
Introduction
In [3] Lipovan obtained a useful upper bound on the following inequality:
(1.1)
2
2
Z
u (t) ≤ c +
α(t)
f (s) u2 (s) + g (s) u (s) ds,
0
and its variants, under some suitable conditions on the functions involved in
(1.1). In fact, the results given in [3] are the retarded versions of the inequalities
established by Pachpatte in [4] (see also [5]). However, the bounds provided
on such inequalities in [3] (see also [1, p. 142]) are not directly applicable in
the study of certain retarded differential and integral equations. It is desirable
to find new inequalities of the above type, which will prove their importance in
achieving a diversity of desired goals. The main purpose of this paper is to establish explicit bounds on the general versions of (1.1) which can be used more
effectively in the study of certain classes of retarded differential and integral
equations. The two independent variable generalizations of the main results
and some applications are also given.
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
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2.
Statement of Results
In what follows, R denotes the set of real numbers; R+ = [0, ∞) , I = [t0 , T ) ,
I1 = [x0 , X) , I2 = [y0 , Y ) are the given subsets of R; ∆ = I1 × I2 and 0
denotes the derivative. The first order partial derivatives of a function z(x, y)
for x, y ∈ R with respect to x and y are denoted by D1 z (x, y) and D2 z (x, y)
respectively. Let C(M, N ) denote the class of continuous functions from the
set M to the set N .
Our main results are given in the following theorem.
1
Theorem 2.1. Let u, ai , bi ∈ C (I, R+ ) and αi ∈ C (I, I) be nondecreasing
with αi (t) ≤ t on I for i = 1, . . . , n. Let p > 1 and c ≥ 0 be constants.
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
(c1 ) If
(2.1)
up (t) ≤ c + p
n Z
X
i=1
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αi (t)
[ai (s) up (s) + bi (s) u (s)] ds,
Contents
αi (t0 )
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J
for t ∈ I, then
(
(2.2)
u (t) ≤
A (t) exp (p − 1)
n Z
X
i=1
1
!) p−1
αi (t)
ai (σ) dσ
αi (t0 )
A (t) = {c}
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for t ∈ I, where
(2.3)
,
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p−1
p
+ (p − 1)
n Z αi (t)
X
i=1
bi (σ) dσ,
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αi (t)
J. Ineq. Pure and Appl. Math. 5(3) Art. 80, 2004
for t ∈ I.
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(c2 ) Let w ∈ C (R+ , R+ ) be nondecreasing with w(u) > 0 on (0, ∞) . If for
t ∈ I,
n Z αi (t)
X
p
(2.4) u (t) ≤ c + p
[ai (s) u (s) w (u (s)) + bi (s) u (s)] ds,
i=1
αi (t0 )
then for t0 ≤ t ≤ t1 ,
(
(2.5)
u (t) ≤
"
G−1 G (A (t)) + (p − 1)
n Z
X
i=1
1
#) p−1
αi (t)
ai (σ) dσ
,
αi (t0 )
where A(t) is defined by (2.3), G−1 is the inverse function of
Z r
ds
1 , r > 0,
(2.6)
G (r) =
r0 w s p−1
r0 > 0 is arbitrary and t1 ∈ I is chosen so that
G (A (t)) + (p − 1)
n Z
X
i=1
αi (t)
On Some New Nonlinear
Retarded Integral Inequalities
ai (σ) dσ ∈ Dom G−1 ,
αi (t0 )
for all t lying in the interval t0 ≤ t ≤ t1 .
Remark 2.1. If we take p = 2, n = 1, α1 = α, a1 = f, b1 = g in Theorem 2.1,
then we recapture the inequalities given in [3] (see Corollary 2 and Theorem
1).
B.G. Pachpatte
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The following theorem deals with the two independent variable versions of
the inequalities established in Theorem 2.1 which can be used in certain applications.
Theorem 2.2. Let u, ai , bi ∈ C (∆, R+ ) and αi ∈ C 1 (I1 , I1 ) , βi ∈ C 1 (I2 , I2 )
be nondecreasing with αi (x) ≤ x on I1 , βi (y) ≤ y on I2 for i = 1, . . . , n. Let
p > 1 and c ≥ 0 be constants.
(d1 ) If
On Some New Nonlinear
Retarded Integral Inequalities
p
(2.7) u (x, y)
n Z
X
≤c+p
i=1
αi (x)
Z
B.G. Pachpatte
βi (y)
p
[ai (s, t) u (s, t) + bi (s, t) u (s, t)] dtds,
αi (x0 )
βi (y0 )
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for (x, y) ∈ ∆, then
Contents
(2.8) u (x, y)
(
≤
B (x, y) exp (p − 1)
n Z
X
i=1
αi (x)
Z
βi (y)
ai (σ, τ ) dτ dσ
αi (x0 )
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1
!) p−1
βi (y0 )
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for (x, y) ∈ ∆, where
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(2.9)
B (x, y) = {c}
p−1
p
+ (p − 1)
n Z
X
i=1
αi (x)
αi (x0 )
Z
βi (y)
bi (σ, τ ) dτ dσ,
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βi (y0 )
J. Ineq. Pure and Appl. Math. 5(3) Art. 80, 2004
for (x, y) ∈ ∆.
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(d2 ) Let w be as in Theorem 2.1, part (c2 ). If for (x, y) ∈ ∆,
n Z αi (x) Z βi (y)
X
p
(2.10) u (x, y) ≤ c + p
[ai (s, t) u (s, t) w (u (s, t))
i=1
αi (x0 )
βi (y0 )
+ bi (s, t) u (s, t)]dtds,
then, for x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 ,
(2.11) u (x, y)
(
"
≤
G−1 G (B (x, y)) + (p − 1)
n Z αi (x)
X
i=1
αi (x0 )
Z
ai (σ, τ ) dτ dσ
αi (x0 )
,
B.G. Pachpatte
βi (y0 )
where B(x, y) is defined by (2.9), G, G−1 are as in Theorem 2.1, part (c2 )
and x1 ∈ I1 , y1 ∈ I2 are chosen so that
n Z αi (x) Z βi (y)
X
G (B (x, y)) + (p − 1)
ai (σ, τ ) dτ dσ ∈ Dom G−1 ,
i=1
On Some New Nonlinear
Retarded Integral Inequalities
1
#) p−1
βi (y)
βi (y0 )
for all x, y lying in the interval x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 .
Remark 2.2. We note that the inequalities established in Theorem 2.2 can be
extended very easily for functions involving more than two independent variables (see [5]). If we take p = 2, n = 1, α1 = α, β1 = β, a1 = f, b1 = g in
Theorem 2.2, then we get the two independent variable generalizations of the
inequalities given in [3] (see Corollary 2 and Theorem 1). For a slight variant
of the inequality in Theorem 2.2 given in [3] and its two independent variable
version, see [6].
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3.
Proofs of Theorems 2.1 and 2.2
We give the details of the proofs for (c1 ) and (d2 ) only; the proofs of (c2 ) and
(d1 ) can be completed by following the proofs of the above mentioned inequalities.
From the hypotheses we observe that αi0 (t) ≥ 0 for t ∈ I, αi0 (x) ≥ 0 for
x ∈ I1 , βi (y) ≥ 0 for y ∈ I2 .
(c1 ) Let c > 0 and define a function z(t) by the right hand side of (2.1). Then
1
z(t) > 0, z (t0 ) = c, z(t) is nondecreasing for t ∈ I, u (t) ≤ {z (t)} p and
0
z (t) = p
n
X
B.G. Pachpatte
p
[ai (αi (t)) u (αi (t)) +
bi (αi (t)) u (αi (t))]αi0
(t)
i=1
≤p
=p
≤p
n h
X
i=1
n h
X
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ai (αi (t)) z (αi (t)) + bi (αi (t)) {z (αi (t))}
i
αi0 (t)
1− p1
i
1
+ bi (αi (t)) {z (αi (t))} p αi0 (t)
p−1
p
i
1
+ bi (αi (t)) {z (t)} p αi0 (t)
ai (αi (t)) {z (αi (t))}
i=1
n h
X
1
p
ai (αi (t)) {z (αi (t))}
i=1
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i.e.
(3.1)
On Some New Nonlinear
Retarded Integral Inequalities
z 0 (t)
1
{z (t)} p
≤p
n h
X
i=1
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ai (αi (t)) {z (αi (t))}
p−1
p
i
+ bi (αi (t)) αi0 (t) .
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By taking t = s in (3.1) and integrating it with respect to s from t0 to t we
get
p−1
p−1
{z (t)} p ≤ {c} p
Z tX
n h
i
p−1
p
+ (p − 1)
ai (αi (s)) {z (αi (s))}
+ bi (αi (s)) αi0 (s) ds.
(3.2)
t0 i=1
Making the change of variables on the right hand side in (3.2)and rewriting
we get
{z (t)}
p−1
p
≤ A (t) + (p − 1)
n Z
X
i=1
αi (t)
ai (σ) {z (σ)}
p−1
p
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
dσ.
αi (t0 )
Title Page
Clearly A(t) is a continuous, positive and nondecreasing function for t ∈
I. Now by following the idea used in the proof of Theorem 1 in [3] (see
also [6]) we get
!
n Z αi (t)
X
p−1
(3.3)
{z (t)} p ≤ A (t) exp (p − 1)
ai (σ) dσ .
i=1
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αi (t0 )
1
p
Using (3.3) in u (t) ≤ {z (t)} we get the desired inequality in (2.2).
If c ≥ 0 we carry out the above procedure with c + ε instead of c, where
ε > 0 is an arbitrary small constant, and subsequently pass the limit ε → 0
to obtain (2.2).
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(d2 ) Let c > 0 and define a function z(x, y) by the right hand side of (2.10).
Then z(x, y) > 0, z (x0 , y) = z (x, y0 ) = c, z(x, y) is nondecreasing in
1
(x, y) ∈ ∆, u (x, y) ≤ {z (x, y)} p and
(3.4) D2 D1 z (x, y)
n
X
[ai (αi (x) , βi (y)) u (αi (x) , βi (y)) w (u (αi (x) , βi (y)))
=p
i=1
+bi (αi (x) , βi (y)) u (αi (x) , βi (y))] βi0 (y) αi0 (x)
≤p
n h
X
On Some New Nonlinear
Retarded Integral Inequalities
1
ai (αi (x) , βi (y)) {z (αi (x) , βi (y))} p
B.G. Pachpatte
i=1
1
× w {z (αi (x) , βi (y))} p
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+bi (αi (x) , βi (y)) {z (αi (x) , βi (y))}
≤p
n
X
1
p
i
βi0 (y) αi0 (x)
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1
[ai (αi (x) , βi (y))w {z (αi (x) , βi (y))} p
i=1
II
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1
+bi (αi (x) , βi (y))] {z (x, y)} p βi0 (y) αi0 (x) .
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From (3.4) we observe that
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D2 D1 z (x, y)
{z (x, y)}
1
p
≤p
n
X
1
[ai (αi (x) , βi (y))w {z (αi (x) , βi (y))} p
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h
i
1
D1 z (x, y) D2 {z (x, y)} p
+bi (αi (x) , βi (y))] βi0 (y) αi0 (x) +
,
h
i
1 2
p
{z (x, y)}
i.e.
(3.5) D2
D1 z (x, y)
!
1
{z (x, y)} p
n
X
1
p
≤p
[ai (αi (x) , βi (y))w {z (αi (x) , βi (y))}
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
i=1
+bi (αi (x) , βi (y))] βi0 (y) αi0 (x) ,
for (x, y) ∈ ∆. By keeping x fixed in (3.5), we set y = t and then, by integrating with respect to t from y0 to y and using the fact that D1 z (x, y0 ) =
0, we have
(3.6)
D1 z (x, y)
1
{z (x, y)} p
Z yX
n
1
[ai (αi (x) , βi (t))w {z (αi (x) , βi (t))} p
≤p
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y0 i=1
+bi (αi (x) , βi (t))] βi0
(t) αi0
(x) dt.
Now by keeping y fixed in (3.6) and setting x = s and integrating with
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respect to s from x0 to x we have
p−1
(3.7)
p−1
{z (x, y)} p ≤ {c} p + (p − 1)
Z xZ yX
n
1
×
[ai (αi (s) , βi (t))w {z (αi (s) , βi (t))} p
x0
y0 i=1
+bi (αi (s) , βi (t))] βi0 (t) αi0 (s) dtds.
By making the change of variables on the right hand side of (3.7) and
rewriting we have
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
(3.8)
{z (x, y)}
p−1
p
≤ B (x, y) + (p − 1)
n Z
X
αi (x)
Z
αi (x0 )
i=1
βi (y)
1
ai (σ, τ ) w {z (σ, τ )} p dτ dσ.
Contents
βi (y0 )
Now fix λ ∈ I1 , µ ∈ I2 such that x0 ≤ x ≤ λ ≤ x1 , y0 ≤ y ≤ µ ≤ y1 .
Then from (3.8) we observe that
(3.9)
{z (x, y)}
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p−1
p
≤ B (λ, µ) + (p − 1)
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n Z
X
i=1
αi (x)
αi (x0 )
Z
βi (y)
ai (σ, τ ) w {z (σ, τ )}
1
p
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dτ dσ,
βi (y0 )
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for x0 ≤ x ≤ λ, y0 ≤ y ≤ µ. Define a function v(x, y) by the right hand
side of (3.9). Then v(x, y) > 0, v (x0 , y) = v (x, y0 ) = B (λ, µ) , v(x, y)
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is nondecreasing for x0 ≤ x ≤ λ, y0 ≤ y ≤ µ, {z (x, y)}
and
p−1
p
≤ v (x, y)
v (x, y)
≤ B (λ, µ)+(p − 1)
n Z
X
i=1
αi (x)
αi (x0 )
Z
β(y)
1
ai (σ, τ ) w {v (σ, τ )} p−1 dτ dσ,
βi (y0 )
for x0 ≤ x ≤ λ, y0 ≤ y ≤ µ. Now by following the proof of Theorem 2.2,
part (B1 ) given in [7] (see also [6]) we get
On Some New Nonlinear
Retarded Integral Inequalities
B.G. Pachpatte
(3.10) v (x, y)
"
≤ G−1 G (B (λ, µ)) + (p − 1)
n Z
X
i=1
αi (x)
αi (x0 )
Z
#
β(y)
ai (σ, τ ) dτ dσ ,
βi (y0 )
for x0 ≤ x ≤ λ ≤ x1 , y0 ≤ y ≤ µ ≤ y1 . Since (λ, µ) is arbitrary, we get
the desired inequality in (2.11) from (3.10) and the fact that
n
o p1
p
1
1
u (x, y) ≤ {z (x, y)} p ≤ [v (x, y)] p−1
= {v (x, y)} p−1 .
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The proof of the case when c ≥ 0 can be completed as mentioned in the
proof of Theorem 2.1, part (c1 ). The domain x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 is
obvious.
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4.
Applications
In this section, we present some model applications which demonstrate the importance of our results to the literature.
First consider the differential equation involving several retarded arguments
(4.1)
xp−1 (t) x0 (t) = f (t, x (t − h1 (t)) , . . . , x (t − hn (t))) ,
for t ∈ I, with the given initial condition
On Some New Nonlinear
Retarded Integral Inequalities
x (t0 ) = x0
(4.2)
B.G. Pachpatte
n
where p > 1 and x0 are constants, f ∈ C (I × R , R) and for i = 1, . . . , n,
let hi ∈ C (I, R+ ) be nonincreasing and such that t − hi (t) ≥ 0, t − hi (t) ∈
C 1 (I, I) , h0i (t) < 1, hi (t0 ) = 0. For the theory and applications of differential
equations with deviating arguments, see [2].
The following theorem deals with the estimate on the solution of the problem
(4.1) – (4.2).
Theorem 4.1. Suppose that
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(4.3)
|f (t, u1 , . . . , un )| ≤
n
X
bi (t) |ui | ,
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i=1
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where bi (t) are as in Theorem 2.1. Let
(4.4)
Qi = max
t∈I
1
,
1 − h0i (t)
Close
i = 1, . . . , n.
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If x(t) is any solution of the problem (4.1) – (4.2), then
1
) p−1
(
n Z t−hi (t)
X
(4.5)
|x (t)| ≤ |x0 |p−1 + (p − 1)
b̄i (σ) dσ
,
i=1
t0
for t ∈ I, where b̄i (σ) = Qi bi (σ + hi (s)) , σ, s ∈ I.
Proof. The solution x(t) of the problem (4.1) – (4.2) can be written as
Z t
p
p
(4.6)
x (t) = x0 + p
f (s, x (s − h1 (s)) , . . . , x (s − hn (s))) ds.
t0
B.G. Pachpatte
From (4.6), (4.3), (4.4) and making the change of variables we have
Z tX
n
p
p
(4.7)
bi (s) |x (s − hi (s))| ds
|x (t)| ≤ |x0 | + p
≤ |x0 |p + p
On Some New Nonlinear
Retarded Integral Inequalities
t0 i=1
n
X Z t−hi (t)
i=1
b̄i (σ) |x (σ)| dσ,
t0
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for t ∈ I. Now a suitable application of the inequality in Theorem 2.1, part (c1 )
(when ai = 0 ) to (4.7) yields the required estimate in (4.5).
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Next, we obtain an explicit bound on the solution of a retarded partial differential equation of the form
(4.8) D2 z p−1 (x, y) D1 z (x, y)
= F (x, y, z (x − h1 (x) , y − g1 (y)) , . . . , z (x − hn (x) , y − gn (y))) ,
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for (x, y) ∈ ∆, with the given initial boundary conditions
(4.9)
z (x, y0 ) = e1 (x) ,
z (x0 , y) = e2 (y) ,
e1 (x0 ) = e2 (y0 ) = 0,
where p > 1 is a constant, F ∈ C (∆ × Rn , R) , e1 ∈ C 1 (I1 , R), e2 ∈
C 1 (I2 , R), and hi ∈ C (I1 , R+ ) , gi ∈ C (I2 , R+ ) are nonincreasing and such
that x − hi (x) ≥ 0, x − hi (x) ∈ C 1 (I1 , I1 ) , y − gi (y) ≥ 0, y − gi (y) ∈
C 1 (I2 , I2 ) , h0i (t) < 1, gi0 (t) < 1, hi (x0 ) = gi (y0 ) = 0 for i = 1, . . . , n;
x ∈ I1 , y ∈ I2 . For the study of special versions of equation (4.8), we refer
interested readers to [8].
Theorem 4.2. Suppose that
(4.10)
|F (x, y, u1 , . . . , un )| ≤
n
X
B.G. Pachpatte
bi (x, y) |ui | ,
i=1
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|ep1 (x) + ep2 (y)| ≤ c,
(4.11)
where bi (x, y) and c are as in Theorem 2.2. Let
(4.12)
Mi = max
x∈I1
1
,
1 − h0i (x)
Ni = max
y∈I2
1
,
1 − gi0 (y)
i = 1, . . . , n.
If z(x, y) is any solution of the problem (4.8) – (4.9), then
1
(
) p−1
n Z φi (x) Z ψi (y)
X
p−1
b̄i (σ, τ ) dτ dσ
(4.13) |z (x, y)| ≤ {c} p + (p − 1)
,
i=1
On Some New Nonlinear
Retarded Integral Inequalities
φi (x0 )
ψi (y0 )
for x ∈ I1 , y ∈ I2 , where φi (x) = x − hi (x) , x ∈ I1 , ψi (y) = y − ψi (y) ,
y ∈ I2 , b̄i (σ, τ ) = Mi Ni bi (σ + hi (s) , τ + gi (t)) for σ, s ∈ I1 ; τ, t ∈ I2 .
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Proof. It is easy to see that the solution z(x, y) of the problem (4.8) – (4.9)
satisfies the equivalent integral equation
(4.14) z p (x, y)
=
ep1
(x) +
ep2
Z
x
Z
y
F (s, t, z (s − h1 (s) , t − g1 (t)) ,
(y) + p
x0
y0
. . . , z (s − hn (s) , t − gn (t)))dtds.
From (4.14), (4.10)-(4.12) and making the change of variables we have
Z
p
x
(4.15) |z (x, y)| ≤ c + p
≤c+p
Z
y
n
X
B.G. Pachpatte
bi (s, t) |z (s − hi (s) , t − gi (t))|dtds
x0 y0 i=1
n
X Z φi (x) Z ψi (y)
i=1
φi (x0 )
On Some New Nonlinear
Retarded Integral Inequalities
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b̄i (σ, τ ) |z (σ, τ )|dτ dσ.
ψi (y0 )
Now a suitable application of the inequality given in Theorem 2.2, part (d1 )
(when ai = 0) to (4.15) yields (4.13).
Remark 4.1. From Theorem 4.1, it is easy to observe that the inequalities given
in [3] cannot be used to obtain an estimate on the solution of the problem (4.1) –
(4.2). Various other applications of the inequalities given here is left to another
work.
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References
[1] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications,
Kluwer Academic Publishers, Dordrecht, 1992.
[2] L.E. EL’SGOL’TS AND S.B. NORKIN, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic
Press, New York, 1973.
[3] O. LIPOVAN, A retarded integral inequality and its applications, J. Math.
Anal. Appl., 285 (2003), 436–443.
On Some New Nonlinear
Retarded Integral Inequalities
[4] B.G. PACHPATTE, On some new inequalities related to certain inequalities
in the theory of differential equations, J. Math. Anal. Appl., 189 (1995),
128–144.
B.G. Pachpatte
[5] B.G. PACHPATTE, Inequalities for Differential and Integral Equations,
Academic Press, New York, 1998.
Contents
[6] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math.
Anal. Appl., 267 (2002), 48–61.
[7] B.G. PACHPATTE, Inequalities applicable to certain partial differential
equations, J. Inequal. Pure and Appl. Math., 5(1) (2004), Art.18. [ONLINE
http://jipam.vu.edu.au/article.php?sid=371]
[8] W. WALTER, Differential and Integral Inequalities, Springer-Verlag, New
York, 1970.
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