Journal of Inequalities in Pure and
Applied Mathematics
ON SOME RETARDED INTEGRAL INEQUALITIES
AND APPLICATIONS
volume 3, issue 2, article 18,
2002.
B.G. PACHPATTE
57, Shri Niketen Colony
Aurangabad - 431 001,
(Maharashtra) India.
EMail: bgpachpatte@hotmail.com
Received 11 July, 2001;
accepted 13 November, 2001.
Communicated by: S.S. Dragomir
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
056-01
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Abstract
The aim of this paper is to establish explicit bounds on certain retarded integral
inequalities which can be used as convenient tools in some applications. The
two independent variable generalizations of the main results and some applications are also given.
2000 Mathematics Subject Classification: 26D15, 26D20.
Key words: Retarded Integral Inequalities, Explicit Bounds, Two Independent Variable Generalizations, Qualitative Study, Nondecreasing, Change of Variable, Integrodifferential Equation, Uniqueness of Solutions.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
On Some Retarded Integral
Inequalities and Applications
B.G. Pachpatte
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1.
Introduction
Integral inequalities which provide explicit bounds on unknown functions have
played a fundamental role in the development of the theory of differential and
integral equations. Over the years, various investigators have discovered many
useful integral inequalities in order to achieve a diversity of desired goals, see
[1] – [6] and the references given therein. In a recent paper [5] Lipovan has
given a useful nonlinear generalisation of the celebrated Gronwall inequality
and presented some of its applications. However, the integral inequalities available in the literature do not apply directly in certain general situations and it is
desirable to find integral inequalities useful in some new applications. The main
purpose of the present paper is to establish explicit bounds on more general retarded integral inequalities which can be used as tools in the qualitative study
of certain retarded integrodifferential equations. Some immediate applications
of one of the result to convey the importance of our results to the literature are
also given.
On Some Retarded Integral
Inequalities and Applications
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 ),
J1 = [x0 , X), J2 = [y0 , Y ) are the given subsets of R, ∆ = J1 × J2 and 0
denotes the derivative. The partial derivatives of a function z (x, y), x, y ∈ R
with respect to x and y are denoted by D1 z (x, y) and D2 z (x, y) respectively.
Our main results are given in the following theorems.
Theorem 2.1. Let u (t) , a (t) ∈ C (I, R+ ) , b (t, s) ∈ C (I 2 , R+ ) for t0 ≤ s ≤
t ≤ T and α (t) ∈ C 1 (I, I) be nondecreasing with α (t) ≤ t on I and k ≥ 0 be
a constant.
On Some Retarded Integral
Inequalities and Applications
B.G. Pachpatte
(a1 ) If
α(t)
Z
(2.1)
u (t) ≤ k +
Z
a (s) u (s) +
α(t0 )
s
b (s, σ) u (σ) dσ ds,
α(t0 )
u (t) ≤ k exp (A (t)) ,
Close
Z
α(t)
A (t) =
α(t0 )
for t ∈ I.
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for t ∈ I, where
(2.3)
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for t ∈ I, then
(2.2)
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Z
a (s) +
s
α(t0 )
b (s, σ) dσ ds,
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(a2 ) Let g ∈ C (R+ , R+ ) be a nondecreasing function with g (u) > 0 for u > 0.
If
Z α(t) Z s
(2.4) u (t) ≤ k +
a (s) g (u (s)) +
b (s, σ) g (u (σ)) dσ ds,
α(t0 )
α(t0 )
for t ∈ I, then for t0 ≤ t ≤ t1 ,
(2.5)
u (t) ≤ G−1 [G (k) + A (t)] ,
where A (t) is defined by (2.3), G−1 is the inverse function of
Z r
ds
(2.6)
G (r) =
, r > 0, r0 > 0,
r0 g (s)
On Some Retarded Integral
Inequalities and Applications
B.G. Pachpatte
and t1 ∈ I is chosen so that
G (k) + A (t) ∈ Dom G−1 ,
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for all t lying in the interval [t0 , t1 ] .
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Theorem 2.2. Let u (x, y) , a (x, y) ∈ C (∆, R+ ) , b (x, y, s, t) ∈ C (∆2 , R+ ) ,
for x0 ≤ s ≤ x ≤ X, y0 ≤ t ≤ y ≤ Y, α (x) ∈ C 1 (J1 , J1 ) , β (y) ∈ C 1 (J2 , J2 )
be nondecreasing with α (x) ≤ x on J1 , β (y) ≤ y on J2 and k ≥ 0 be a
constant.
(b1 ) If
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Z
α(x)
Z
β(y)
"
(2.7) u (x, y) ≤ k +
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a (s, t) u (s, t)
α(x0 )
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β(y0 )
Z
s
Z
t
+
b (s, t, σ, η) u (σ, η) dηdσ dtds,
α(x0 )
β(y0 )
J. Ineq. Pure and Appl. Math. 3(2) Art. 18, 2002
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for (x, y) ∈ ∆, then
u (x, y) ≤ k exp (A (x, y)) ,
(2.8)
for (x, y) ∈ ∆, where
(2.9) A (x, y)
Z α(x) Z
=
α(x0 )
β(y)
Z
a (s, t) +
β(y0 )
s
Z
t
b (s, t, σ, η) dηdσ dtds,
α(x0 )
β(y0 )
On Some Retarded Integral
Inequalities and Applications
for (x, y) ∈ ∆.
B.G. Pachpatte
(b2 ) Let g be as in Theorem 2.1, part (a2 ) . If
"
Z
Z
α(x)
β(y)
(2.10) u (x, y) ≤ k +
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a (s, t) g (u (s, t))
α(x0 )
Z
s
β(y0 )
Z
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t
+
b (s, t, σ, η) g (u (σ, η)) dηdσ dtds,
α(x0 )
β(y0 )
for (x, y) ∈ ∆, then for x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 ,
(2.11)
u (x, y) ≤ G−1 [G (k) + A (x, y)] ,
where A (x, y) is defined by (2.9), G, G−1 are as defined in Theorem 2.1,
part (a2 ) and x1 ∈ J1 , y1 ∈ J2 are chosen so that
G (k) + A (x, y) ∈ Dom G−1 ,
for all x and y lying in [x0 , x1 ] and [y0 , y1 ] respectively.
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3.
Proofs of Theorems 2.1 and 2.2
From the hypotheses, we observe that α0 (t) ≥ 0 for t ∈ I, α0 (x) ≥ 0 for
x ∈ J1 , β 0 (y) ≥ 0 for y ∈ J2 .
(a1 ) Let k > 0 and define a function z (t) by the right hand side of (2.1). Then
z (t) > 0, z (t0 ) = k, u (t) ≤ z (t) and
"
#
Z
α(t)
(3.1) z 0 (t) = a (α (t)) u (α (t)) +
[b (α (t) , σ) u (σ) dσ] α0 (t)
α(t0 )
"
Z
#
α(t)
≤ a (α (t)) z (α (t)) +
On Some Retarded Integral
Inequalities and Applications
[b (α (t) , σ) z (σ) dσ] α0 (t) .
B.G. Pachpatte
α(t0 )
From (3.1) it is easy to observe that
"
#
Z α(t)
0
z (t)
≤ a (α (t)) +
b (α (t) , σ) dσ α0 (t) .
(3.2)
z (t)
α(t0 )
Integrating (3.2) from t0 to t, t ∈ I and by making the change of variables
yields
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(3.3)
z (t) ≤ k exp (A (t)) ,
for t ∈ I. Using (3.3) in u (t) ≤ z (t) we get the inequality in (2.2). 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 to the limit as
ε → 0 to obtain (2.2).
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(a2 ) Let k > 0 and define a function z (t) by the right hand side of (2.4). Then
z (t) > 0, z (t0 ) = k, u (t) ≤ z (t) and as in the proof of (a1 ) we get
"
#
Z α(t)
z 0 (t)
(3.4)
≤ a (α (t)) +
b (α (t) , σ) dσ α0 (t) .
g (z (t))
α(t0 )
From (2.6) and (3.4) we have
(3.5)
d
G (z (t))
dt
On Some Retarded Integral
Inequalities and Applications
"
#
Z α(t)
z 0 (t)
=
≤ a (α (t)) +
b (α (t) , σ) dσ α0 (t) .
g (z (t))
α(t0 )
Integrating (3.5) from t0 to t, t ∈ I and by making the change of variables
we have
(3.6)
G (z (t)) ≤ G (k) + A (t) .
Since G−1 (z) is increasing, from (3.6) we have
B.G. Pachpatte
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z (t) ≤ G−1 [G (k) + A (t)] .
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Using (3.7) in u (t) ≤ z (t) we get (2.5). The case k ≥ 0 can be completed
as mentioned in the proof of (a1 ) . The subinterval t0 ≤ t ≤ t1 for t is
obvious.
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(3.7)
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(b1 ) Let k > 0 and define a function z (x, y) by the right hand side of (2.7).
Then z (x, y) > 0, z (x0 , y) = z (x, y0 ) = k, u (x, y) ≤ z (x, y) and
(3.8)D1 z (x, y)
"Z
"
β(y)
=
a (α (x) , t) u (α (x) , t)
β(y0 )
Z
α(x)
Z
#
t
b (α (x) , t, σ, η) u (σ, η) dηdσ dt α0 (x)
+
α(x0 )
"Z
β(y)
≤
#
On Some Retarded Integral
Inequalities and Applications
β(y0 )
"
a (α (x) , t) z (α (x) , t)
B.G. Pachpatte
β(y0 )
Z
α(x)
Z
#
t
b (α (x) , t, σ, η) z (σ, η) dηdσ dt α0 (x) .
+
α(x0 )
#
β(y0 )
Contents
From (3.8) it is easy to observe that
"Z
"
β(y)
D1 z (x, y)
(3.9)
≤
a (α (x) , t)
z (x, y)
β(y0 )
# #
Z α(x) Z t
+
b (α (x) , t, σ, η) dηdσ dt α0 (x) .
α(x0 )
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β(y0 )
Keeping y fixed in (3.9), setting x = ξ and integrating it with respect to ξ
from x0 to x and making the change of variables we get
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(3.10)
z (x, y) ≤ k exp (A (x, y)) .
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Using (3.10) in u (x, y) ≤ z (x, y), we get the required inequality in (2.8).
The case k ≥ 0 follows as mentioned in the proof of (a1 ) .
(b2 ) The proof can be completed by following the proof of (a2 ) and closely
looking at the proof of (b1 ) . Here we omit the details.
On Some Retarded Integral
Inequalities and Applications
B.G. Pachpatte
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4.
Some Applications
In this section, we present some immediate applications of the inequality (a1 )
in Theorem 2.1 to study certain properties of solutions of the integrodifferential
equation
Z t
0
(P )
x (t) = F t, x (t − h (t)) ,
f (t, σ, x (σ − h (σ))) dσ ,
t0
On Some Retarded Integral
Inequalities and Applications
with the given initial condition
x (t0 ) = x0 ,
(P0 )
where f ∈ C (I 2 × R, R), F ∈ C (I × R2 , R), x0 is a real constant and h ∈
C 1 (I, I) be nondecreasing with t − h (t) ≥ 0, h0 (t) < 1, h (t0 ) = 0.
The following theorem deals with the estimate on the solution of (P ) – (P0 ).
Theorem 4.1. Suppose that
(4.1)
(4.2)
|f (t, s, x)| ≤ b (t, s) |x| ,
|F (t, x, w)| ≤ a (t) |x| + |w| ,
B.G. Pachpatte
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where a (t) , b (t, s) are as defined in Theorem 2.1 and let
(4.3)
M = max
t∈I
1
.
1 − h0 (t)
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If x (t) is any solution of (P ) – (P0 ), then
Z
(4.4)
t−h(t)
|x (t)| ≤ |x0 | exp
"
M a (s + h (η))
t0
Z
s
M b (s + h (η) , σ + h (τ )) dσ ds ,
2
+
t0
for t, η, τ in I.
Proof. The solution x (t) of (P ) – (P0 ) can be written as
Z t Z s
(4.5) x (t) = x0 +
F s, x (s − h (s)) ,
f (s, σ, x (σ − h (σ))) dσ ds.
t0
t0
t−h(t)
|x (t)| ≤ |x0 | +
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"
M a (s + h (η)) |x (s)|
t0
Z
s
2
M b (s + h (η) , σ + h (τ )) |x (σ)| dσ ds,
+
t0
for t, η, τ in I. Now a suitable application of the inequality in (a1 ) given in
Theorem 2.1 yields the required estimate in (4.4).
Next, we shall prove the uniqueness of the solutions of (P ) – (P0 ).
B.G. Pachpatte
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Using (4.1) – (4.3) in (4.5) and making the change of variables we have
Z
On Some Retarded Integral
Inequalities and Applications
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Theorem 4.2. Suppose that the functions f, F in (P ) satisfy the conditions
(4.6)
(4.7)
|f (t, s, x) − f (t, s, y)| ≤ b (t, s) |x − y| ,
|F (t, x, x̄) − F (t, y, ȳ)| ≤ a (t) |x − y| + |x̄ − ȳ| ,
where a (t) , b (t, s) are as defined in Theorem 2.1 and let M be as in (4.3). Then
the problem (P ) – (P0 ) has at most one solution on I.
Proof. Let x (t) and x̄ (t) be two solutions of (P ) – (P0 ) on I, then we have
(4.8) x (t) − x̄ (t)
Z t Z s
=
F s, x (s − h (s)) ,
f (s, σ, x (σ − h (σ))) dσ
t0
t0
Z s
− F s, x̄ (s − h (s)) ,
f (s, σ, x̄ (σ − h (σ))) dσ
ds.
t0
t−h(t)
≤
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|x (t) − x̄ (t)|
Z
B.G. Pachpatte
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Using (4.6), (4.7) in (4.8) and making the change of variables we have
(4.9)
On Some Retarded Integral
Inequalities and Applications
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M a (s + h (η)) |x (s) − x̄ (s)|
t0
Close
Z
s
+
M 2 b (s + h (η) , σ + h (τ )) |x (σ) − x̄ (σ)| dσ ds
t0
for t, η, τ in I. A suitable application of the inequality in (a1 ) given in Theorem
2.1 yields |x (t) − x̄ (t)| ≤ 0. Therefore x (t) = x̄ (t) , i.e., there is at most one
solution of (P ) – (P0 ).
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Our next result shows the dependency of solutions of (P ) – (P0 ) on initial
values.
Theorem 4.3. Let x1 (t) and x2 (t) be the solutions of (P ) with the given initial
conditions
(P1 )
x1 (t0 ) = x1 ,
and
(P2 )
x2 (t0 ) = x2 ,
respectively, where x1 , x2 , are real constants. Suppose that the functions f and
F in (P ) satisfy the conditions (4.6) and (4.7) in Theorem 4.2 and let M be as
in (4.3). Then
Z t−h(t) "
(4.10) |x1 (t) − x2 (t)| ≤ |x1 − x2 | exp
M a (s + h (η))
t0
Z
s
+
M b (s + h (η) , σ + h (τ )) dσ ds ,
2
t0
for t, η, τ in I.
Proof. By using the facts that x1 (t) and x2 (t) are the solutions of (P ) – (P1 )
and (P ) – (P2 ) respectively, we have
(4.11) x1 (t) − x2 (t)
Z t Z s
= x1 − x2 +
F s, x1 (s − h (s)) ,
f (s, σ, x1 (σ − h (σ))) dσ
t0
t0
Z s
f (s, σ, x2 (σ − h (σ))) dσ
ds.
− F s, x2 (s − h (s)) ,
t0
On Some Retarded Integral
Inequalities and Applications
B.G. Pachpatte
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Using (4.6), (4.7) in (4.11) and by making the change of variables, we have
(4.12)
|x1 (t) − x2 (t)|
t−h(t)
Z
≤ |x1 − x2 | +
"
M a (s + h (η)) |x1 (s) − x2 (s)|
t0
Z
s
2
M b (s + h (η) , σ + h (τ )) |x1 (σ) − x2 (σ)| dσ ds,
+
t0
for t, η, τ in I. Now a suitable application of the inequality in (a1 ) given in
Theorem 2.1 to (4.12) yields the required estimate in (4.10).
In concluding we note that the inequality in (b1 ) given in Theorem 2.2 can be
used to study the similar properties as in Theorems 4.1 – 4.3 for the hyperbolic
partial integrodifferential equation
(4.13)
D1 D2 z (x, y) = F (x, y, z (x − h1 (x) , y − h2 (y)) , T z (x, y)) ,
with the given initial boundary conditions
(4.14)
z (x, y0 ) = a1 (x) , z (x0 , y) = a2 (y) , a1 (x0 ) = a2 (y0 ) ,
B.G. Pachpatte
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where
Z
(4.15)
On Some Retarded Integral
Inequalities and Applications
xZ
y
K (x, y, s, t, z (s − h1 (s) , t − h2 (t))) dtds,
T z (x, y) =
x0
y0
under some suitable conditions on the functions involved in (4.13) – (4.15).
Since the formulations of these results are very close to those given above, we
omit it here. 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] R. BELLMAN, The stability of solutions of linear differential equations,
Duke Math. J., 10 (1943), 643–647.
[3] I. BIHARI, A generalization of a lemma of Bellman and its application
to uniqueness problems of differential equations, Acta. Math. Acad. Sci.
Hungar., 7 (1965), 81–94.
On Some Retarded Integral
Inequalities and Applications
[4] T.H. GRONWALL, Note on the derivatives with respect to a parameter of
the solutions of a system of differential equations, Ann. Math., 20 (1919),
292–296.
B.G. Pachpatte
[5] O. LIPOVAN, A retarded Gronwall-like inequality and its applications, J.
Math. Anal. Appl., 252 (2000), 389–401.
Contents
[6] B.G. PACHPATTE, Inequalities for Differential and Integral Equations,
Academic Press, New York, 1998.
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