Journal of Inequalities in Pure and
Applied Mathematics
ON AN INEQUALITY OF GRONWALL
J.A. OGUNTUASE
Department of Mathematical Sciences,
University of Agriculture,
Abeokuta, NIGERIA.
EMail: adedayo@unaab.edu.ng
volume 2, issue 1, article 9,
2001.
Received 23 May, 2000;
accepted 09 November 2000.
Communicated by: P. Cerone
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
013-00
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Abstract
In this paper, we obtain some new Gronwall-Bellman type integral inequalities,
and we give an application of our results in the study of boundedness of the
solutions of nonlinear integrodifferential equations.
2000 Mathematics Subject Classification: 26D10, 26D20, 34A40
Key words: Gronwall inequality, nonlinear integrodifferential equation, nondecreasing function, nonnegative continuous functions, partial derivatives, variational equation.
I wish to express my appreciation to the referee whose remarks and observations
have led to an improvement of this paper.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
On an Inequality of Gronwall
James Adedayo Oguntuase
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1.
Introduction
Integral inequalities play a significant role in the study of differential and integral equations. In particular, there has been a continuous interest in the following inequality.
Lemma 1.1. Let u(t) and g(t) be nonnegative continuous functions on I =
[0, ∞) for which the inequality
Z t
u(t) ≤ c +
g(s)u(s)ds, t ∈ I
a
On an Inequality of Gronwall
James Adedayo Oguntuase
holds, where c is a nonnegative constant. Then
Z t
u(t) ≤ c exp
g(s)ds ,
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t ∈ I.
a
Due to various motivations, several generalizations and applications of this
lemma have been obtained and used extensively, see the references under [1, 3].
Pachpatte [5] obtained a useful general version of this lemma. The aim of
this work is to establish some useful generalizations of the inequalities obtained
in [5]. Some consequences of our results are also given.
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2.
Statement of Results
Our main results are given in the following theorems:
Theorem 2.1. Let u(t), f (t) be nonnegative continuous functions in a real interval I = [a, b]. Suppose that k(t, s) and its partial derivatives kt (t, s) exist
and are nonnegative continuous functions for almost every t, s ∈ I. If the inequality
Z t
(2.1) u(t) ≤ c +
f (s)u(s)ds
a
Z s
Z t
+
f (s)
k(s, τ )u(τ )dτ ds, a ≤ τ ≤ s ≤ t ≤ b,
a
a
a
Proof. Define a function v(t) by the right hand side of (2.1). Then it follows
that
u(t) ≤ v(t).
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Therefore
Z
(2.4)
James Adedayo Oguntuase
a
holds, where c is a nonnegative constant, then
Z s
Z t
(2.2)
u(t) ≤ c 1 +
f (s) exp
(f (τ ) + k(τ, τ ))dτ ds .
(2.3)
On an Inequality of Gronwall
t
k(t, τ )u(τ )dτ,
v 0 (t) = f (t)u(t) + f (t)
a
Z t
≤ f (t) v(t) +
k(t, τ )v(τ )dτ .
a
v(a) = c
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(by (2.3))
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If we put
t
Z
m(t) = v(t) +
(2.5)
k(t, τ )v(τ )dτ,
a
then it is clear that
v(t) ≤ m(t).
(2.6)
Therefore
(2.7)
0
t
Z
0
m (t) = v (t) + k(t, t)v(t) +
kt (t, τ )v(τ )dτ,
m(a) = v(a) = c
a
James Adedayo Oguntuase
≤ v 0 (t) + k(t, t)v(t),
≤ f (t)m(t) + k(t, t)v(t),
≤ (f (t) + k(t, t)) m(t).
(by (2.4))
(by (2.6))
Integrate (2.7) from a to t, we obtain
Z t
(2.8)
m(t) ≤ c exp
f (s) + k(s, s) ds .
a
Substitute (2.8) into (2.4), we have
(2.9)
v 0 (t) ≤ cf (t) exp
On an Inequality of Gronwall
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Z
t
f (s) + k(s, s) ds .
a
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Integrating both sides of (2.9) from a to t, we obtain
Z s
Z t
v(t) ≤ c 1 +
f (s) exp
f (τ ) + k(τ, τ ) dτ ds .
a
By (2.3) we have the desired result.
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Remark 2.1. If in Theorem 2.1 we set k(t, s) = g(s), our estimate reduces to
Theorem 1 obtained in [5].
Theorem 2.2. Let u(t), f (t), h(t) and g(t) be nonnegative continuous functions
in a real interval I = [a, b]. Suppose that h0 (t) exists and is a nonnegative
continuous function. If the following inequality
Z t
u(t) ≤ c +
f (s)u(s)ds
a
Z s
Z t
+
f (s)h(s)
g(τ )u(τ )dτ ds a ≤ τ ≤ s ≤ t ≤ b,
a
a
holds, where c is a nonnegative constant, then
Z s
Z t
Z
0
u(t) ≤ c 1 +
f (s) exp
(f (τ ) + g(τ )h(τ ) + h (τ )
a
a
τ
g(σ)dσ)dτ
ds .
On an Inequality of Gronwall
James Adedayo Oguntuase
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a
Proof. This follows by similar argument as in the proof of Theorem 2.1. We
omit the details.
Remark 2.2. If in Theorem 2.2, we set h(t) = 1, then our result reduces to
Theorem 1 obtained in [5].
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Remark 2.3. If in Theorem 2.2, h (t) = 0 then our estimate is more general
than Theorem 1 obtained by Pachpatte in [5].
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Lemma 2.3. Let v(t) be a positive differentiable function satisfying the inequality
(2.10)
v 0 (t) ≤ f (t)v(t) + g(t)v p (t),
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t ∈ I = [a, b],
where the functions f (t) and g(t) are continuous in I, and p ≥ 0, p 6= 1, is a
constant. Then
Z t
(2.11) v(t) ≤ exp
f (s)ds
a
Z s
1q
Z t
q
g(s) exp −q
f (τ )dτ ds ,
× v (a) + q
a
a
for t, s ∈ [a, β), where q = 1 − p and β is chosen so that the expression
Z s
1q
Z t
v q (a) + q
g(s) exp −q
f (τ )dτ ds
a
On an Inequality of Gronwall
James Adedayo Oguntuase
a
is positive in the subinterval [a, β).
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Proof. We reduce (2.10) to a simpler differential inequality by the following
substitution. Let
v q (t)
z(t) =
.
q
Then
Contents
(2.12)
0
q−1
0
z (t) = v (t) × v (t)
≤ v q−1 (t) (f (t)v(t) + g(t)v p (t)) ,
(by (2.10))
= qf (t)z(t) + g(t)
(since q = 1 − p).
By Lemma 1.1 [1], (2.12) gives
Z t
Z t
Z t
v q (a)
z(t) ≤
exp
qf (s)ds +
g(s) exp
qf (τ )dτ ds.
q
a
a
s
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That is
q
t
Z
v (t) ≤ exp
Z s
Z t
q
qf (s)ds v (a) +
g(s) exp −
qf (τ )dτ ds .
a
a
a
From this, it follows that
Z t
Z s
1q
Z t
q
v(t) ≤ exp
f (s)ds c + q
g(s) exp −q
f (τ )dτ ds .
a
a
a
On an Inequality of Gronwall
Theorem 2.4. Let u(t), f (t) be nonnegative continuous functions in a real interval I = [a, b]. Suppose that the partial derivatives kt (t, s) exist and are
nonnegative continuous functions for almost every t, s ∈ I. If the the inequality
Z t
(2.13) u(t) ≤ c +
f (s)u(s)ds
a
Z s
Z t
p
+
f (s)
k(s, τ )u (τ )dτ ds, a ≤ τ ≤ s ≤ t ≤ b
a
a
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holds, where 0 ≤ p < 1, q = 1 − p and c > 0 are constants.
Then
Z s
Z t
(2.14) u(t) ≤ c +
f (s) exp
f (τ )dτ
a
James Adedayo Oguntuase
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a
Z s
Z
1−p
× c
+ (1 − p)
k(τ, τ ) exp −(1 − p)
a
a
τ
f (σ)dσ dτ
1
1−p
ds.
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Proof. Define a function v(t) by the right hand side of (2.13) from which it
follows that
u(t) ≤ v(t).
(2.15)
Then
Z
0
t
(2.16) v (t) = f (t)u(t) + f (t)
k(t, τ )up (τ )dτ,
a
Z t
p
≤ f (t) v(t) +
k(t, τ )v (τ )dτ .
v(a) = c
(by (2.15))
a
On an Inequality of Gronwall
James Adedayo Oguntuase
If we put
Z
m(t) = v(t) +
(2.17)
t
k(t, τ )v p (τ )dτ,
a
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then it is clear that
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v(t) ≤ m(t).
(2.18)
Hence
(2.19)
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0
0
p
m (t) = v (t) + k(t, t)v (t)
Z t
+
kt (t, τ )v p (τ )dτ,
Close
m(a) = v(a) = c
a
≤ v 0 (t) + k(t, t)v p (t),
≤ f (t)m(t) + k(t, t)v p (t),
≤ f (t)m(t) + k(t, t)mp (t).
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(by (2.16))
(by (2.18))
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By Lemma 2.3 we have
Z
t
(2.20) m(t) ≤ exp
(f (s)ds
a
Z s
1q
Z s
q
× m +q
k(s, s) exp −q
f (τ )dτ ds .
a
a
Substituting (2.21) into (2.16), we have
On an Inequality of Gronwall
(2.21) v 0 (t) ≤ f (t) exp
Z
t
James Adedayo Oguntuase
(f (s)ds
a
Z s
1q
Z s
× mq + q
k(s, s) exp −q
f (τ )dτ ds .
a
a
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Integrate both sides of (2.22) from a to t and using (2.15), we obtain
Z
u(t) ≤ c +
t
Z
f (s) exp
s
f (τ )dτ
a
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1−p
c
+ (1 − p)
s
Z
k(τ, τ ) exp −(1 − p)
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a
Z
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τ
f (σ)dσ dτ
1
1−p
ds.
a
This completes the proof of the theorem
Remark 2.4. If in Theorem 2.4, we put k(t, s) = g(s), then our result reduces
to Theorem 2 obtained in [5].
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Theorem 2.5. Let u(t), f (t), h(t) and g(t) be nonnegative continuous functions
in a real interval I = [a, b]. Suppose that h0 (t) exists and is a nonnegative
continuous function. If the following inequality
t
Z
(2.22) u(t) ≤ c +
f (s)u(s)ds
Z s
Z t
p
+
f (s)h(s)
g(τ )u (τ )dτ ds a ≤ τ ≤ s ≤ t ≤ b,
a
a
a
On an Inequality of Gronwall
holds, where 0 ≤ p < 1, q = 1 − p and c > 0 are nonnegative constant. Then
Z
t
(2.23) u(t) ≤ c +
Z
f (s) exp
a
0
s
Z s
1−p
f (τ )dτ
c
+ (1 − p)
(h(τ )f (τ )
a
Z
τ
a
Z
f (σ)dσ exp −(1 − p)
+ h (τ )
a
τ
f (σ)dσ dτ
1
1−p
James Adedayo Oguntuase
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ds.
a
Proof. This follows by similar argument as in the proof of Theorem 2.4. We
also omit the details.
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Remark 2.5. If in Theorem 2.5, we set h(t) = 1 then our result reduces to the
estimate in Theorem 2 obtained by Pachpatte in [5].
Close
Remark 2.6. If in Theorem 2.5, h0 (t) = 0 then our result is more general than
Theorem 2 obtained in [5].
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3.
Applications
There are many applications of the inequalities obtained in Section 2. Here we
shall give an application which is just sufficient to convey the importance of our
results. We shall consider the nonlinear integrodifferential equation
Z t
0
(3.1)
x (t) = f (t, u(t)) +
g (t, s, x(s)) ds,
t0
and the corresponding perturbed equation
Z t
Z t
0
(3.2) u (t) = f (t, u(t))+
g (t, s, u(s)) ds+h t, u(t),
k(t, s, u(s))ds
t0
On an Inequality of Gronwall
James Adedayo Oguntuase
t0
for all t0 , t ∈ R+ and x, u, f, g, h ∈ Rn .
If we let x(t) = x(t; t0 , x0 ) and u(t) = u(t; t0 , x0 ) be the solutions of (3.1)
and (3.2) respectively with x(t0 ) = u(t0 ) = x0 and f : R+ × Rn → Rn , fx :
R+ ×Rn → Rn×n , g, k : R+ ×R+ ×Rn → Rn , gx : R+ ×R+ ×Rn → Rn×n and
h : R+ × R+ × Rn → Rn are continuous functions in their respective domains.
∂x
Then we have by [2] that ∂x
(t, t0 , x0 ) = Φ(t, t0 , x0 ) exists and satisfies the
0
variational equation
(3.3) x0 (t) = fx (t, x(t; t0 , x0 ))z(t)
Z t
+
gx (t, s, x(s; t0 , x0 )) z(s)ds,
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z(t0 ) = I
t0
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and
(3.4)
∂x
(t; t0 , x0 ) + Φ(t, t0 , x0 )f (t0 , x0 )
∂t0
Z
t
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Φ(t, s, x0 )g(s, t0 , x0 )ds = 0.
t0
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Thus the solutions x(t) and u(t) are related by
Z t
Z t
(3.5)
u(t) = x(t)
Φ(t, s, u(s))h s, u(s),
k(s, τ, u(τ ))dτ ds.
t0
t0
Theorem 3.1. Let f , fx , g, gx , k, h, as earlier defined, be nonnegative continuous functions. Suppose that the following inequalities hold:
(3.6)
(3.7)
(3.8)
|Φ(t, s, u)| ≤ M e−α(t−s) ,
|Φ(t, s, u)h(s, u, z)| ≤ p(s) (|u| + |z|) ,
|k(t, s, u)| ≤ q(s, s) |y|
for 0 ≤ s ≤ t, u, z ∈ Rn , M ≥ 1 and α > 0 are constants. If p(t) and q(t, t)
are continuous and nonnegative and
Z ∞
Z ∞
(3.9)
p(s)ds < ∞,
q(s, s)ds < ∞.
Then for any bounded solution x(t; t0 , x0 ) of (3.1) in R+ , then the corresponding
solutions of (3.2) is bounded in R+ .
On an Inequality of Gronwall
James Adedayo Oguntuase
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Proof. We have from (3.6)– (3.8) that equation (3.2) gives
Z t
Z t
Z t
|u(t)| ≤ M |x0 | +
p(s) |u(s)| ds +
p(s)
q(τ, τ ) |u(τ )| dτ ds.
t0
t0
t0
Hence by Theorem 2.1, we have
Z s
Z t
|u(t)| ≤ M |x0 | 1 +
p(s) exp
(p(τ ) + q(τ, τ ))dτ ds .
t0
s0
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Hence by (3.9), we easily see that |u(t)| is bounded and the proof is complete.
On an Inequality of Gronwall
James Adedayo Oguntuase
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References
[1] D. BAINOV AND P. SIMEONOV, Integral inequalities and Applications,
Academic Publishers, Dordrecht, 1992.
[2] F. BRAUER, A nonlinear variation of constants formula for Volterra equations, Mat. Systems Th., 6 (1972), 226 – 234.
[3] J. CHANDRA AND B.A. FLEISHMAN, On a generalization of GronwallBellman lemma in partially ordered Banach spaces, J. Math. Anal. Appl.,
31 (1970), 668 – 681.
On an Inequality of Gronwall
James Adedayo Oguntuase
[4] J.A. OGUNTUASE, Remarks on Gronwall type inequalities, An. Stiint.
Univ. “Al. I. Cuza”, t.45 (1999), in press
[5] B.G. PACHPATTE, A note on Gronwall-Bellman inequality, J. Math. Anal.
Appl., 44 (1973), 758– 762.
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