Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES APPLICABLE TO CERTAIN PARTIAL
DIFFERENTIAL EQUATIONS
volume 5, issue 2, article 27,
2004.
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India.
EMail: bgpachpatte@hotmail.com
Received 24 August, 2003;
accepted 06 November, 2003.
Communicated by: J. Sándor
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
157-03
Quit
Abstract
In this paper explicit bounds on certain retarded integral inequalities involving
functions of two independent variables are established. Some applications are
also given to illustrate the usefulness of one of our results.
2000 Mathematics Subject Classification: 26D15, 26D20
Key words: Explicit bounds, retarded integral inequalities, two independent variables, non-self-adjoint, Hyperbolic partial differential equations, partial
derivatives.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Proofs of Theorems 2.1 – 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4
Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
1.
Introduction
The integral inequalities which furnish explicit bounds on unknown functions
has become a rich source of inspiration in the development of the theory of differential and integral equations. Over the years a great deal of attention has been
given to such inequalities and their applications. A detailed account related to
such inequalities can be found in [1] – [6] and the references given therein.
However, in certain situations the bounds provided by such inequalities available in the literature are inadequate and we need bounds on some new integral
inequalities in order to achieve a diversity of desired goals. In this paper, we
offer some basic integral inequalities in two independent variables which can
be used more conveniently in specific applications. Some applications are also
given to study the behavior of solutions of non-self-adjoint hyperbolic partial
differential equations with several retarded arguments.
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
2.
Statement of Results
In what follows R denotes the set of real numbers, R+ = [0, ∞), I1 = [x0 , X),
I2 = [y0 , Y ) are the subsets of R and ∆ = I1 × I2 . The partial derivatives
of a function z(x, y), x, y ∈ R with respect to x, y and xy are denoted by
D1 z (x, y) , D2 z (x, y) and D1 D2 z (x, y) (or zxy ) respectively.
Our main results are established in the following theorems.
Theorem 2.1. Let u, a, 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 and
k ≥ 0 be a constant.
(A1 ) If
B.G. Pachpatte
Z
x
(2.1) u (x, y) ≤ k +
a (s, y) u (s, y) ds
Title Page
x0
+
n Z
X
i=1
αi (x)
Z
βi (y)
bi (s, t) u (s, t) dtds,
αi (x0 )
βi (y0 )
(2.2) u (x, y) ≤ kq (x, y) exp
n Z
X
αi (x)
βi (y)
i=1
Z
II
I
!
bi (s, t) q (s, t) dtds ,
αi (x0 )
Contents
JJ
J
for x ∈ I1 , y ∈ I2 , then
βi (y0 )
Go Back
Close
for x ∈ I1 , y ∈ I2 , where
Quit
Z
(2.3)
Inequalities Applicable To
Certain Partial Differential
Equations
x
q (x, y) = exp
a (ξ, y) dξ ,
Page 4 of 27
x0
for x ∈ I1 , y ∈ I2 .
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
(A2 ) If
Z
y
(2.4) u (x, y) ≤ k +
a (x, t) u (x, t) dt
y0
+
n Z
X
i=1
αi (x)
Z
βi (y)
bi (s, t) u (s, t) dtds,
αi (x0 )
βi (y0 )
αi (x)
βi (y)
for x ∈ I1 , y ∈ I2 , then
(2.5) u (x, y) ≤ kr (x, y) exp
n Z
X
i=1
Z
!
bi (s, t) r (s, t) dtds ,
αi (x0 )
βi (y0 )
B.G. Pachpatte
for x ∈ I1 , y ∈ I2 , where
Z
(2.6)
Inequalities Applicable To
Certain Partial Differential
Equations
y
r (x, y) = exp
a (x, η) dη ,
y0
Title Page
Contents
for x ∈ I1 , y ∈ I2 .
Theorem 2.2. Let u, a, bi , αi , βi , k be as in Theorem 2.1. Let g ∈ C (R+ , R+ )
be nondecreasing and submultiplicative function with g (u) > 0 for u > 0.
JJ
J
II
I
Go Back
(B1 ) If
Z
Close
x
(2.7) u (x, y) ≤ k +
a (s, y) u (s, y) ds
Quit
x0
+
n Z
X
i=1
αi (x)
αi (x0 )
Z
Page 5 of 27
βi (y)
bi (s, t) g (u (s, t)) dtds,
βi (y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 ; then for x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 ; x, x1 ∈ I1 , y, y1 ∈
I2 ,
(2.8) u (x, y) ≤ q (x, y)
"
× G−1 G (k) +
n Z
X
i=1
αi (x)
Z
#
βi (y)
bi (s, t) g (q (s, t)) dtds ,
αi (x0 )
βi (y0 )
where q (x, y) is given by (2.3) and G−1 is the inverse function of
Z r
ds
(2.9)
G (r) =
, r > 0,
r0 g (s)
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
r0 > 0 is arbitrary and x1 ∈ I1 , y1 ∈ I2 are chosen so that
G (k) +
n Z
X
i=1
αi (x)
βi (y)
Z
αi (x0 )
Title Page
bi (s, t) g (q (s, t)) dtds ∈ Dom(G−1 ) ,
βi (y0 )
for all x and y lying in [x0 , x1 ] and [y0 , y1 ] respectively.
(B2 ) If
Contents
JJ
J
II
I
Go Back
Z
Close
y
(2.10) u (x, y) ≤ k +
a (x, t) u (x, t) dt
Quit
y0
+
n Z αi (x)
X
i=1
αi (x0 )
Z
βi (y)
bi (s, t) g (u (s, t)) dtds,
Page 6 of 27
βi (y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 ; then for x0 ≤ x ≤ x2 , y0 ≤ y ≤ y2 ; x, x2 ∈ I1 , y, y2 ∈
I2 ,
(2.11) u (x, y) ≤ r (x, y)
"
× G−1 G (k) +
n Z
X
i=1
αi (x)
Z
#
βi (y)
bi (s, t) g (r (s, t)) dtds ,
αi (x0 )
βi (y0 )
where G, G−1 are as in part (B1 ), r (x, y) is given by (2.6) and x2 ∈
I1 , y2 ∈ I2 are chosen so that
n Z αi (x) Z βi (y)
X
G (k) +
bi (s, t) g (r (s, t)) dtds ∈ Dom(G−1 ),
i=1
αi (x0 )
βi (y0 )
for all x and y lying in [x0 , x2 ] and [y0 , y2 ] respectively.
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
The inequalities in the following theorems can be used in the qualitative analysis of certain partial integrodifferential equations involving several retarded
arguments.
Theorem 2.3. Let u, a, bi , αi , βi , k be as in Theorem 2.1.
Contents
JJ
J
II
I
Go Back
(C1 ) If c ∈ C (∆, R+ ) and
Z
x
(2.12) u (x, y) ≤ k +
Z
a (s, y) u (s, y) +
x0
+
n Z
X
i=1
αi (x)
αi (x0 )
Z
Close
s
c (σ, y) u (σ, y) dσ ds
x0
βi (y)
Quit
Page 7 of 27
bi (s, t) u (s, t) dtds,
βi (y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 , then
(2.13) u (x, y) ≤ kp (x, y)
× exp
n Z
X
αi (x)
Z
bi (s, t) p (s, t) dtds ,
αi (x0 )
i=1
!
βi (y)
βi (y0 )
for x ∈ I1 , y ∈ I2 , where
Z ξ
Z x
(2.14) p (x, y) = 1 +
a (ξ, y) exp
[a (σ, y) + c (σ, y)] dσ dξ,
x0
x0
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
for x ∈ I1 , y ∈ I2 .
(C2 ) If c ∈ C (∆, R+ ) and
Title Page
Z
y
(2.15) u (x, y) ≤ k +
Z
t
a (x, t) u (x, t) +
c (x, τ ) u (x, τ ) dτ
y0
dt
y0
+
n Z
X
i=1
αi (x)
Z
βi (y)
bi (s, t) u (s, t) dtds,
αi (x0 )
βi (y0 )
JJ
J
II
I
Go Back
for x ∈ I1 , y ∈ I2 , then
Close
Quit
(2.16) u (x, y) ≤ kw (x, y)
× exp
Contents
n Z
X
i=1
αi (x)
αi (x0 )
Z
βi (y)
!
Page 8 of 27
bi (s, t) w (s, t) dtds ,
βi (y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 , where
Z
Z y
(2.17) w (x, y) = 1 +
a (x, η) exp
y0
η
[a (x, τ ) + c (x, τ )] dτ
dη,
y0
for x ∈ I1 , y ∈ I2 .
Theorem 2.4. Let u, a, bi , αi , βi , k be as in Theorem 2.1 and g be as in Theorem
2.2.
(D1 ) If c ∈ C (∆, R+ ) and
Z
x
(2.18) u (x, y) ≤ k +
Z
s
a (s, y) u (s, y) +
x0
c (σ, y) u (σ, y) dσ ds
x0
+
n Z
X
αi (x)
Z
B.G. Pachpatte
βi (y)
bi (s, t) g (u (s, t)) dtds,
αi (x0 )
i=1
Title Page
βi (y0 )
for x ∈ I1 , y ∈ I2 ; then for x0 ≤ x ≤ x3 , y0 ≤ y ≤ y3 ; x, x3 ∈ I1 , y, y3 ∈
I2 ,
(2.19) u (x, y) ≤ p (x, y)
"
× G−1 G (k) +
n Z
X
i=1
αi (x)
αi (x0 )
Z
βi (y)
#
bi (s, t) g (p (s, t)) dtds ,
βi (y0 )
−1
where p (x, y) is given by (2.14), G, G are as in part (B1 ) in Theorem
2.2 and x3 ∈ I1 , y3 ∈ I2 are chosen so that
n Z αi (x) Z βi (y)
X
G (k) +
bi (s, t) g (p (s, t)) dtds ∈ Dom(G−1 ) ,
i=1
αi (x0 )
βi (y0 )
Inequalities Applicable To
Certain Partial Differential
Equations
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for all x and y lying in [x0 , x3 ] and [y0 , y3 ] respectively.
(D2 ) If c ∈ C (∆, R+ ) and
y
Z
(2.20) u (x, y) ≤ k +
Z
t
a (x, t) u (x, t) +
y0
c (x, τ ) u (x, τ ) dτ
dt
y0
+
n Z
X
i=1
αi (x)
Z
βi (y)
bi (s, t) g (u (s, t)) dtds,
αi (x0 )
βi (y0 )
for x ∈ I1 , y ∈ I2 ; then for x0 ≤ x ≤ x4 , y0 ≤ y ≤ y4 ; x, x4 ∈ I1 , y, y4 ∈
I2 ,
(2.21) u (x, y) ≤ w (x, y)
"
× G−1 G (k) +
n Z
X
i=1
αi (x)
αi (x0 )
Z
#
βi (y)
bi (s, t) g (w (s, t)) dtds ,
βi (y0 )
where w (x, y) is given by (2.17), G, G−1 are as in part (B1 ) in Theorem
2.2 and x4 ∈ I1 , y4 ∈ I2 are chosen so that
G (k) +
n Z
X
i=1
αi (x)
αi (x0 )
Z
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
βi (y)
−1
bi (s, t) g (w (s, t)) dtds ∈ Dom(G ),
Close
βi (y0 )
Quit
for all x and y lying in [x0 , x4 ] and [y0 , y4 ] respectively.
Page 10 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
3.
Proofs of Theorems 2.1 – 2.4
We give the details of the proofs of (A1 ) , (B1 ) and (C1 ) only. The proofs of the
remaining inequalities can be completed by closely looking at the proofs of the
above mentioned inequalities with suitable modifications.
(A1 ) Define a function z (x, y) by
(3.1)
z (x, y) = k +
n Z
X
i=1
αi (x)
Z
βi (y)
bi (s, t) u (s, t) dtds.
αi (x0 )
βi (y0 )
Then (2.1) can be restated as
B.G. Pachpatte
Z
(3.2)
x
u (x, y) ≤ z (x, y) +
a (s, y) u (s, y) ds.
x0
It is easy to observe that z (x, y) is a nonnegative, continuous and nondecreasing function for x ∈ I1 , y ∈ I2 . Treating y, y ∈ I2 fixed in (3.2) and
using Lemma 2.1 in [4] (see also [3, Theorem 1.3.1]) to (3.2), we get
(3.3)
u (x, y) ≤ q (x, y) z (x, y) ,
for x ∈ I1 , y ∈ I2 , where q (x, y) is defined by (2.3). From (3.1) and (3.3)
we have
n Z αi (x) Z βi (y)
X
(3.4)
z (x, y) ≤ k +
bi (s, t) q (s, t) z (s, t) dtds.
i=1
Inequalities Applicable To
Certain Partial Differential
Equations
αi (x0 )
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 27
βi (y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
Let k > 0 and define a function v (x, y) by the right hand side of (3.4).
Then it is easy to observe that
v (x, y) > 0,
v (x0 , y) = v (x, y0 ) = k,
z (x, y) ≤ v (x, y)
and
D1 v (x, y) =
≤
Z
n
X
i=1
n
X
βi (y)
!
bi (αi (x) , t) q (αi (x) , t) z (αi (x) , t) dt αi0 (x)
βi (y0 )
Z
i=1
≤ v (x, y)
βi (y)
!
bi (αi (x) , t) q (αi (x) , t) v (αi (x) , t) dt αi0 (x)
βi (y0 )
B.G. Pachpatte
Z
n
X
i=1
βi (y)
!
bi (αi (x) , t) q (αi (x) , t) dt αi0 (x)
βi (y0 )
Title Page
Contents
i.e.
n
(3.5)
Inequalities Applicable To
Certain Partial Differential
Equations
D1 v (x, y) X
≤
v (x, y)
i=1
Z
βi (y)
!
bi (αi (x) , t) q (αi (x) , t) dt αi0 (x) .
βi (y0 )
JJ
J
II
I
Go Back
Keeping y fixed in (3.5) , setting x = σ and integrating it with respect to σ
from x0 to x, x ∈ I1 , and making the change of variables we get
!
n Z αi (x) Z βi (y)
X
(3.6)
v (x, y) ≤ k exp
bi (s, t) q (s, t) dtds ,
i=1
αi (x0 )
Close
Quit
Page 12 of 27
βi (y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 . Using (3.6) in z (x, y) ≤ v (x, y) we get
(3.7)
n Z
X
z (x, y) ≤ k exp
i=1
αi (x)
Z
!
βi (y)
bi (s, t) q (s, t) dtds .
αi (x0 )
βi (y0 )
Using (3.7) in (3.3) we get the required inequality in (2.5).
If k ≥ 0 we carry out the above procedure with k + ε instead of k, where
ε > 0 is an arbitrary small constant, and subsequently pass the limit ε → 0
to obtain (2.5).
(B1 ) Define a function z (x, y) by
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
(3.8)
z (x, y) = k +
n Z
X
i=1
αi (x)
βi (y)
Z
bi (s, t) g (u (s, t)) dtds.
αi (x0 )
βi (y0 )
Contents
Then (2.7) can be stated as
Z
(3.9)
Title Page
x
u (x, y) ≤ z (x, y) +
a (s, y) u (s, y) ds.
x0
As in the proof of part (A1 ), using Lemma 2.1 in [4] to (3.9) we have
JJ
J
II
I
Go Back
Close
(3.10)
u (x, y) ≤ q (x, y) z (x, y) ,
for x ∈ I1 , y ∈ I2 , where q (x, y) and z (x, y) are defined by (2.3) and
Quit
Page 13 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
(3.8). From (3.8) and (3.10) and the hypotheses on g we have
(3.11) z (x, y)
≤k+
n Z
X
i=1
≤k+
βi (y)
Z
bi (s, t) g (q (s, t) z (s, t)) dtds
αi (x0 )
n Z
X
i=1
αi (x)
αi (x)
βi (y0 )
βi (y)
Z
bi (s, t) g (q (s, t)) g (z (s, t)) dtds.
αi (x0 )
βi (y0 )
Let k > 0 and define a function v (x, y) by the right hand side of (3.11).
Then, it is easy to observe that v (x, y) > 0, v (x0 , y) = v (x, y0 ) = k,
z (x, y) ≤ v (x, y) and
(3.12) D1 v (x, y)
!
n Z βi (y)
X
=
bi (αi (x), t) g (q (αi (x), t)) g (z (αi (x), t)) dt αi0 (x)
≤
i=1
n
X
i=1
βi (y0 )
Z
!
βi (y)
bi (αi (x), t) g (q (αi (x), t)) g (v (αi (x), t)) dt
αi0 (x)
βi (y0 )
≤g (v (x, y))
n Z
X
i=1
βi (y)
!
bi (αi (x), t) g (q (αi (x), t)) dt
βi (y0 )
αi0 (x) .
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
From (2.9) and (3.12) we have
(3.13) D1 G (v (x, y)) =
D1 v (x, y)
g (v (x, y))
Page 14 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
≤
Z
n
X
!
βi (y)
bi (αi (x) , t) g (q (αi (x) , t)) dt αi0 (x) .
βi (y0 )
i=1
Keeping y fixed in (3.13), setting x = σ and integrating it with respect to
σ from x0 to x, x ∈ I1 and making the change of variables we get
(3.14) G (v (x, y)) ≤ G (k) +
n Z
X
i=1
Since G
−1
αi (x)
Z
βi (y)
bi (s, t) g (q (s, t)) dtds.
αi (x0 )
βi (y0 )
Inequalities Applicable To
Certain Partial Differential
Equations
(v) is increasing, from (3.14) we have
B.G. Pachpatte
(3.15) v (x, y)
"
≤ G−1 G (k) +
n Z
X
i=1
αi (x)
αi (x0 )
Z
#
βi (y)
bi (s, t) g (q (s, t)) dtds .
βi (y0 )
Title Page
Contents
Using (3.15) in z (x, y) ≤ v (x, y) and then the bound on z (x, y) in (3.10)
we get the required inequality in (2.8). The case k ≥ 0 can be completed
as mentioned in the proof of (A1 ).
JJ
J
II
I
Go Back
(C1 ) Define a function z (x, y) by (3.1) . Then (2.12) can be stated as
Close
(3.16) u (x, y)
Quit
Z
x
≤ z (x, y) +
Z
s
a (s, y) u (s, y) +
x0
c (σ, y) u (σ, y) dσ ds.
Page 15 of 27
x0
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
Clearly, z (x, y) is nonnegative, continuous and nondecreasing function for
x ∈ I1 , y ∈ I2 . Treating y, y ∈ I2 fixed in (3.16) and applying Theorem
1.7.4 given in [3, p. 39] to (3.16) yields
u (x, y) ≤ p (x, y) z (x, y) ,
where p (x, y) and z (x, y) are defined by (2.14) and (3.1). Now by following the proof of (A1 ) with suitable changes we get the desired inequality
in (2.13).
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
4.
Some Applications
In this section, we present applications of the inequality (A1 ) given in Theorem
2.1 which display the importance of our results to the literature. Consider the
following retarded non-self-adjoint hyperbolic partial differential equation
(4.1) zxy (x, y) = D2 (a (x, y) z (x, y))
+ f (x, y, z (x − h1 (x) , y − g1 (y)) ,
. . . , z (x − hn (x) , y − gn (y))) ,
with the given initial boundary conditions
(4.2)
z (x, y0 ) = a1 (x) , z (x0 , y) = a2 (y) , a1 (x0 ) = a2 (y0 ) = 0,
where f ∈ C (∆ × Rn , R) , a1 ∈ C 1 (I1 , R) , a2 ∈ C 1 (I2 , R) , and a ∈
C (∆, R) is differentiable with respect to y; 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 (x) < 1, gi0 (y) < 1 , hi (x0 ) =
gi (y0 ) = 0 for i = 1, . . . , n; x ∈ I1 , y ∈ I2 and
(4.3)
1
1
, Ni = max
.
0
y∈I2 1 − gi0 (y)
x∈I1 1 − hi (x)
Mi = max
Our first result gives the bound on the solution of the problem (4.1) – (4.2).
|f (x, y, u1 , . . . , un )| ≤
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Theorem 4.1. Suppose that
(4.4)
Inequalities Applicable To
Certain Partial Differential
Equations
n
X
i=1
Page 17 of 27
bi (x, y) |ui | ,
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
|e (x, y)| ≤ k,
(4.5)
where bi (x, y) , k are as in Theorem 2.1 and
Z
(4.6)
e (x, y) = a1 (x) + a2 (y) −
x
a (s, y0 ) a1 (s) ds.
x0
If z (x, y) is any solution of (4.1) – (4.2), then
n Z φi (x) Z
X
(4.7) |z (x, y)| ≤ k q̄ (x, y) exp
i=1
φi (x0 )
ψ(y)
!
b̄i (σ, τ ) q̄ (σ, τ ) dτ dσ ,
ψi (y0 )
for x ∈ I1 , y ∈ I2 , where φi (x) = x − hi (x) , x ∈ I1 , ψi (y) = y − gi (y) ,
y ∈ I2 , b̄i (σ, τ ) = Mi Ni bi (σ + hi (s) , τ + gi (t)) for σ, s ∈ I1 , τ, t ∈ I2 and
Z x
|a (ξ, y)| dξ ,
(4.8)
q̄ (x, y) = exp
x0
for x ∈ I1 , y ∈ I2 .
Proof. It is easy to see that, the solution z(x, y) of the problem (4.1) – (4.2)
satisfies the equivalent integral equation
Z x
(4.9) z (x, y) = e (x, y) +
a (s, y) z (s, y) ds
x0
Z xZ y
+
f (s, t, z (s − h1 (s) , t − g1 (t)) ,
x0
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 27
y0
. . . , z (s − hn (s) , t − gn (t))) dtds,
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
where e(x, y) is given by (4.6). From (4.9), (4.4), (4.5), (4.3) and making the
change of variables we have
Z x
(4.10) |z (x, y)| ≤ k +
|a (s, y)| |z (s, y)| ds
x0
x
Z
+
x0
Z x
≤k+
+
Z
y
n
X
bi (s, t) |z (s − hi (s) , t − gi (t))| dtds
y0 i=1
|a (s, y)| |z (s, y)| ds
x0
n Z φi (x)
X
i=1
φi (x0 )
Z
Inequalities Applicable To
Certain Partial Differential
Equations
ψi (y)
b̄i (σ, τ ) |z (σ, τ )| dτ dσ.
Now a suitable application of the inequality (A1 ) given in Theorem 2.1 to (4.10)
yields (4.7).
The next theorem deals with the uniqueness of solutions of (4.1) – (4.2).
Theorem 4.2. . Suppose that the function f in (4.1) satisfies the condition
(4.11)
B.G. Pachpatte
ψi (y0 )
|f (x, y, u1 , . . . , un ) − f (x, y, v1 , . . . , vn )| ≤
n
X
Title Page
Contents
JJ
J
II
I
Go Back
bi (x, y) |ui − vi | ,
i=1
where bi (x, y) are as in Theorem 2.1. Let Mi , Ni , φi , ψi , b̄i be as in Theorem
4.1. Then the problem (4.1) – (4.2) has at most one solution on ∆ .
Close
Quit
Page 19 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
Proof. Let u (x, y) and v (x, y) be two solutions of (4.1) – (4.2) on ∆ , then
Z
x
(4.12) u (x, y) − v (x, y) =
a (s, y) {u (s, y) − v (s, y)} ds
x0
Z xZ y
+
{f (s, t, u(s − h1 (s), t − g1 (t)),
x0
y0
. . . , u(s − hn (s), t − gn (t)))
− f (s, t, v(s − h1 (s), t − g1 (t)),
. . . , v(s − hn (s), t − gn (t)))}dtds.
From (4.12), (4.11), making the change of variables and in view of (4.3) we
have
(4.13)
|u (x, y) − v (x, y)|
Z x
≤
|a (s, y)| |u (s, y) − v (s, y)| ds
x0
Z
x
y
Z
+
x0
n
X
bi (s, t) |u (s − hi (s) , t − gi (t))
y0 i=1
x
≤
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
−v (s − hi (s) , t − gi (t))| dtds
Z
Inequalities Applicable To
Certain Partial Differential
Equations
|a (s, y)| |u (s, y) − v (s, y)| ds
Close
Quit
x0
+
n Z
X
i=1
φi (x)
φi (x0 )
Z
Page 20 of 27
ψ(y)
b̄i (σ, τ ) |u (σ, τ ) − v (σ, τ )| dτ dσ.
ψ(y0 )
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
A suitable application of the inequality (A1 ) in Theorem 2.1 to (4.13) yields
|u (x, y) − v (x, y)| ≤ 0.
Therefore u(x, y) = v(x, y) i.e. there is at most one solution of the problem
(4.1) – (4.2).
The following theorem shows the dependency of solutions of equation (4.1)
on given initial boundary data.
Theorem 4.3. Let u (x, y) and v (x, y) be the solutions of (4.1) with the given
initial boundary data
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
(4.14)
u (x, y0 ) = c1 (x) , u (x0 , y) = c2 (y) , c1 (x0 ) = c2 (y0 ) = 0,
Title Page
and
Contents
(4.15)
v (x, y0 ) = d1 (x) , v (x0 , y) = d2 (y) , d1 (x0 ) = d2 (y0 ) = 0,
1
1
respectively, where c1 , d1 ∈ C (I1 , R) , c2 , d2 ∈ C (I2 , R) . Suppose that the
function f satisfies the condition (4.11) in Theorem 4.2. Let
Z x
(4.16)
e1 (x, y) = c1 (x) + c2 (y) −
a (s, y0 ) c1 (s) ds,
x0
Z
(4.17)
II
I
Go Back
Close
Quit
Page 21 of 27
x
e2 (x, y) = d1 (x) + d2 (y) −
JJ
J
a (s, y0 ) d1 (s) ds,
x0
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
for x ∈ I1 , y ∈ I2 and
|e1 (x, y) − e2 (x, y)| ≤ k,
(4.18)
where k is as in Theorem 2.1. Let Mi , Ni , φi , ψi , b̄i , q̄ be as in Theorem 4.1.
Then
|u (x, y) − v (x, y)|
(4.19)
≤ k q̄ (x, y) exp
n Z
X
i=1
φi (x)
φi (x0 )
Z
!
ψ(y)
b̄i (σ, τ ) q̄ (σ, τ ) dτ dσ ,
ψ(y0 )
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
for x ∈ I1 , y ∈ I2 .
Proof. Since u(x, y) and v(x, y) are the solutions of (4.1) – (4.14) and (4.1) –
(4.15) respectively, we have
Title Page
Contents
(4.20) u (x, y) − v (x, y)
Z
x
= e1 (x, y) − e2 (x, y) +
a (s, y) {u(s, y) − v (s, y)} ds
x0
Z
x
Z
y
{f (s, t, u (s − h1 (s) , t − g1 (t)) , . . . , u (s − hn (s) , t − gn (t)))
+
x0
y0
−f (s, t, v (s − h1 (s) , t − g1 (t)) , . . . , v (s − hn (s) , t − gn (t)))} dtds ,
for x ∈ I1 , y ∈ I2 . From (4.20), (4.18), (4.11), making the change of variables
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
and in view of (4.3) we have
x
Z
(4.21)
|u (x, y) − v (x, y)| ≤ k +
|a (s, y)| |u (s, y) − v (s, y)| ds
x0
+
n Z
X
i=1
φi (x)
φi (x0 )
Z
ψ(y)
b̄i (σ, τ ) |u (σ, τ ) − v (σ, τ )| dτ dσ,
ψ(y0 )
for x ∈ I1 , y ∈ I2 . Now a suitable application of the inequality (A1 ) in Theorem 2.1 to (4.21) yields the required estimate in (4.19), which shows the dependency of solutions of (4.1) on given initial boundary data.
We next consider the following retarded non-self-adjoint hyperbolic partial
differential equations
(4.22) zxy (x, y) = D2 (a (x, y) z (x, y))
+ f (x, y, z (x − h1 (x) , y − g1 (y)) ,
. . . , z (x − hn (x) , y − gn (y)) , µ),
(4.23) zxy (x, y) = D2 (a (x, y) z (x, y))
+ f (x, y, z (x − h1 (x) , y − g1 (y)) ,
. . . , z (x − hn (x) , y − gn (y)) , µ0 ),
with the given initial boundary conditions (4.2), where f ∈ C (∆ × Rn × R, R) ,
hi , gi are as in (4.1) and µ, µ0 are real parameters.
The following theorem shows the dependency of solutions of problems (4.22)
– (4.2) and (4.23) – (4.2) on parameters.
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
Theorem 4.4. Suppose that
(4.24)
|f (x, y, u1 , . . . , un , µ) − f (x, y, v1 , . . . , vn , µ)|
n
X
bi (x, y) |ui − vi | ,
≤
i=1
(4.25) |f (x, y, u1 , . . . , un , µ) − f (x, y, u1 , . . . , un , µ)| ≤ m (x, y) |µ − µ0 | ,
where bi (x, y) are as in Theorem 2.1 and m : ∆ → R is a continuous function
such that
Z xZ y
(4.26)
m (s, t) dtds ≤ M,
x0
y0
where M ≥ 0 is a real constant . Let Mi , Ni , φi , ψi , b̄i be as in Theorem 4.1.
If z1 (x, y) and z2 (x, y) are the solutions of (4.22) – (4.2) and (4.23) =- (4.2),
then
(4.27)
|z1 (x, y) − z2 (x, y)|
≤ k̄ q̄ (x, y) exp
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
n Z
X
i=1
φi (x) Z
φi (x0 )
!
ψ(y)
b̄i (σ, τ ) q̄ (σ, τ ) dτ dσ ,
ψ(y0 )
for x ∈ I1 , y ∈ I2 ,where k̄ = |µ − µ0 | M and q̄ (x, y) is defined by (4.8).
Close
Quit
Page 24 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
Proof. Let z (x, y) = z1 (x, y) − z2 (x, y) for x ∈ I1 , y ∈ I2 . As in the proof of
Theorem 4.2, from the hypotheses we have
Z
x
(4.28) z (x, y) =
a (s, y) z (s, y) ds
x0
Z xZ y
+
{f (s, t, z1 (s − h1 (s) , t − g1 (t)) , . . . , z1 (s − hn (s) , t − gn (t)) , µ)
x0
y0
− f (s, t, z2 (s − h1 (s) , t − g1 (t)) , . . . , z2 (s − hn (s) , t − gn (t)) , µ)
+ f (s, t, z2 (s − h1 (s) , t − g1 (t)) , . . . , z2 (s − hn (s) , t − gn (t)) , µ)
−f (s, t, z2 (s − h1 (s) , t − g1 (t)) , . . . , z2 (s − hn (s) , t − gn (t)) , µ0 )} dtds.
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
From (4.28), (4.24) – (4.26), making the change of variables and in view of
(4.3) we have
Z x
(4.29)
|z (x, y)| ≤
|a (s, y)| |z (s, y)| ds
x0
Z
x
+
x0
Z
y
n
X
bi (s, t) |z1 (s − hi (s) , t − gi (t))
y0 i=1
−z2 (s − hi (s) , t − gi (t))| dtds
Z xZ y
+
m (s, t) |µ − µ0 | dtds
x0 y 0
Z x
≤ k̄ +
|a (s, y)| |z (s, y)| ds
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 27
x0
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
+
n Z
X
i=1
φi (x)
Z
ψ(y)
b̄i (σ, τ ) |z (σ, τ )| dτ dσ.
φi (x0 )
ψi (y0 )
A suitable application of the inequality (A1 ) in Theorem 2.1 to (4.29) yields
(4.27), which shows the dependency of solutions of problems (4.22) – (4.2) and
(4.23) – (4.2) on parameters µ and µ0 .
We note that the inequality given in Theorem 2.1 part (A2 ) can be used to
study the similar properties as in Theorems 4.1 – 4.4 by replacing
D2 (a (x, y) z (x, y)) by D1 (a (x, y) z (x, y)) in the equations (4.1), (4.22), (4.23)
with the corresponding given initial-boundary conditions, under some suitable
conditions on the functions involved therein. We also note that the inequalities
given in Theorem 2.3 can be used to establish similar results as in Theorems 4.1
– 4.4 by replacing D2 (a (x, y) z (x, y)) by
Z x
D2 Q1 x, y, z (x, y) ,
k1 (σ, y, z (σ, y)) dσ
x0
or
Z
y
D1 Q2 x, y, z (x, y) ,
k2 (x, τ, z (x, τ )) dτ
y0
in the equations (4.1), (4.22), (4.23) with the corresponding given initial-boundary
conditions and under some suitable conditions on the functions involved therein.
Further it is to be noted that the inequalities and their applications given here
can be extended very easily to functions involving many independent variables.
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au
References
[1] D. BAINOV AND P. SIMENOV, Integral Inequalities and Applications,
Kluwer Academic Publishers, Dordrecht, 1992 .
[2] O. LIPOVAN, A retarded Gronwall-like inequality and its applications, J.
Math. Anal. Appl., 252 (2000), 389–401.
[3] B.G. PACHPATTE, Inequalities for Differential and Integral Equations,
Academic Press, New York, 1998.
[4] B.G. PACHPATTE, On some fundamental integral inequalities and their
discrete analogues, J. Inequal. Pure and Appl. Math., 2(2) (2001), Art.15.
[ONLINE http://jipam.vu.edu.au]
[5] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math.
Anal. Appl., 267 (2002), 48–61.
Inequalities Applicable To
Certain Partial Differential
Equations
B.G. Pachpatte
Title Page
Contents
[6] B.G. PACHPATTE, On some retarded integral inequalities and applications,
J. Inequal. Pure and Appl. Math., 3(2) (2002), Art. 18. [ONLINE http:
//jipam.vu.edu.au]
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 27
J. Ineq. Pure and Appl. Math. 5(2) Art. 27, 2004
http://jipam.vu.edu.au