Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 2, Issue 1, Article 9, 2001
ON AN INEQUALITY OF GRONWALL
JAMES ADEDAYO OGUNTUASE
D EPARTMENT OF M ATHEMATICAL S CIENCES , U NIVERSITY OF AGRICULTURE , A BEOKUTA , N IGERIA .
adedayo@unaab.edu.ng
Received 23 May, 2000; accepted 09 November, 2000
Communicated by P. Cerone
A BSTRACT. 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.
Key words and phrases: Gronwall inequality, nonlinear integrodifferential equation, nondecreasing function, nonnegative
continuous functions, partial derivatives, variational equation.
1991 Mathematics Subject Classification. Primary 26D10, Secondary 26D20, 34A40.
1. I NTRODUCTION
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
holds, where c is a nonnegative constant. Then
Z t
u(t) ≤ c exp
g(s)ds , 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.
ISSN (electronic): 1443-5756
c 2001 Victoria University. All rights reserved.
I wish to express my appreciation to the referee whose remarks and observations have led to an improvement of this paper.
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JAMES A DEDAYO O GUNTUASE
2. S TATEMENT OF R ESULTS
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 s
Z t
Z t
(2.1) u(t) ≤ c +
f (s)u(s)ds +
f (s)
k(s, τ )u(τ )dτ ds, a ≤ τ ≤ s ≤ t ≤ b,
a
a
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 .
a
a
Proof. Define a function v(t) by the right hand side of (2.1). Then it follows that
u(t) ≤ v(t).
(2.3)
Therefore
t
Z
0
v (t) = f (t)u(t) + f (t)
k(t, τ )u(τ )dτ, v(a) = c
a
Z t
≤ f (t) v(t) +
k(t, τ )v(τ )dτ . (by (2.3))
(2.4)
a
If we put
t
Z
k(t, τ )v(τ )dτ,
m(t) = v(t) +
(2.5)
a
then it is clear that
v(t) ≤ m(t).
(2.6)
Therefore
(2.7)
0
Z
0
t
m (t) = v (t) + k(t, t)v(t) +
kt (t, τ )v(τ )dτ,
m(a) = v(a) = c
a
≤ v 0 (t) + k(t, t)v(t),
≤ f (t)m(t) + k(t, t)v(t), (by (2.4))
≤ (f (t) + k(t, t)) m(t). (by (2.6))
Integrate (2.7) from a to t, we obtain
Z
t
f (s) + k(s, s) ds .
m(t) ≤ c exp
(2.8)
a
Substitute (2.8) into (2.4), we have
(2.9)
Z
0
v (t) ≤ cf (t) exp
t
f (s) + k(s, s) ds .
a
Integrating both sides of (2.9) from a to t, we obtain
Z s
Z t
f (s) exp
f (τ ) + k(τ, τ ) dτ ds .
v(t) ≤ c 1 +
a
By (2.3) we have the desired result.
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Remark 2.2. If in Theorem 2.1 we set k(t, s) = g(s), our estimate reduces to Theorem 1
obtained in [5].
Theorem 2.3. 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 s
Z t
Z t
u(t) ≤ c +
f (s)u(s)ds +
f (s)h(s)
g(τ )u(τ )dτ ds a ≤ τ ≤ s ≤ t ≤ b,
a
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 .
a
Proof. This follows by similar argument as in the proof of Theorem 2.1. We omit the details.
Remark 2.4. If in Theorem 2.3, we set h(t) = 1, then our result reduces to Theorem 1 obtained
in [5].
Remark 2.5. If in Theorem 2.3, h0 (t) = 0 then our estimate is more general than Theorem 1
obtained by Pachpatte in [5].
Lemma 2.6. Let v(t) be a positive differentiable function satisfying the inequality
v 0 (t) ≤ f (t)v(t) + g(t)v p (t), t ∈ I = [a, b],
(2.10)
where the functions f (t) and g(t) are continuous in I, and p ≥ 0, p 6= 1, is a constant. Then
1q
Z s
Z t
Z t
q
f (τ )dτ ds ,
g(s) exp −q
f (s)ds v (a) + q
(2.11)
v(t) ≤ exp
a
a
a
for t, s ∈ [a, β), where q = 1 − p and β is chosen so that the expression
Z s
1q
Z t
q
v (a) + q
g(s) exp −q
f (τ )dτ ds
a
a
is positive in the subinterval [a, β).
Proof. We reduce (2.10) to a simpler differential inequality by the following substitution. Let
z(t) =
v q (t)
.
q
Then
z 0 (t) = v q−1 (t) × v 0 (t)
(2.12)
≤ 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
That is
q
Z
v (t) ≤ exp
t
Z s
Z t
q
qf (s)ds v (a) +
g(s) exp −
qf (τ )dτ ds .
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JAMES A DEDAYO O GUNTUASE
From this, it follows that
Z t
Z s
1q
Z t
v(t) ≤ exp
f (s)ds cq + q
g(s) exp −q
f (τ )dτ ds .
a
a
a
Theorem 2.7. 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
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
a
Z s
Z
1−p
× c
+ (1 − p)
k(τ, τ ) exp −(1 − p)
a
τ
1
1−p
f (σ)dσ dτ
ds.
a
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
0
Z
t
Z
t
v (t) = f (t)u(t) + f (t)
k(t, τ )up (τ )dτ, v(a) = c
a
Z t
p
≤ f (t) v(t) +
k(t, τ )v (τ )dτ . (by (2.15))
(2.16)
a
If we put
m(t) = v(t) +
(2.17)
k(t, τ )v p (τ )dτ,
a
then it is clear that
v(t) ≤ m(t).
(2.18)
Hence
(2.19)
0
0
p
Z
m (t) = v (t) + k(t, t)v (t) +
t
kt (t, τ )v p (τ )dτ,
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), (by (2.16))
≤ f (t)m(t) + k(t, t)mp (t). (by (2.18))
By Lemma 2.6 we have
Z
(2.20)
m(t) ≤ exp
t
Z s
1q
Z s
q
(f (s)ds m + q
k(s, s) exp −q
f (τ )dτ ds .
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Substituting (2.20) into (2.16), we have
Z s
1q
Z t
Z s
(f (s)ds mq + q
k(s, s) exp −q
f (τ )dτ ds .
(2.21)
v 0 (t) ≤ f (t) exp
a
a
a
Integrate both sides of (2.21) from a to t and using (2.15), we obtain
Z s
Z t
u(t) ≤ c +
f (s) exp
f (τ )dτ c1−p
a
a
Z
+ (1 − p)
s
Z
τ
k(τ, τ ) exp −(1 − p)
a
1
1−p
ds.
f (σ)dσ dτ
a
This completes the proof of the theorem
Remark 2.8. If in Theorem 2.7, we put k(t, s) = g(s), then our result reduces to Theorem 2
obtained in [5].
Theorem 2.9. 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 s
Z t
Z t
p
(2.22) u(t) ≤ c +
f (s)u(s)ds +
f (s)h(s)
g(τ )u (τ )dτ ds a ≤ τ ≤ s ≤ t ≤ b,
a
a
a
holds, where 0 ≤ p < 1, q = 1 − p and c > 0 are nonnegative constant. Then
Z s
Z t
Z s
1−p
(2.23) u(t) ≤ c +
f (s) exp
f (τ )dτ
c
+ (1 − p)
(h(τ )f (τ )
a
a
0
a
Z
τ
Z
f (σ)dσ exp −(1 − p)
+ h (τ )
a
τ
f (σ)dσ dτ
1
1−p
ds.
a
Proof. This follows by similar argument as in the proof of Theorem 2.7. We also omit the
details.
Remark 2.10. If in Theorem 2.9, we set h(t) = 1 then our result reduces to the estimate in
Theorem 2 obtained by Pachpatte in [5].
Remark 2.11. If in Theorem 2.9, h0 (t) = 0 then our result is more general than Theorem 2
obtained in [5].
3. A PPLICATIONS
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
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 ,
J. Inequal. Pure and Appl. Math., 2(1) Art. 9, 2001
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JAMES A DEDAYO O GUNTUASE
g, k : R+ ×R+ ×Rn → Rn , gx : R+ ×R+ ×Rn → Rn×n and h : R+ ×R+ ×Rn → Rn are contin∂x
uous functions in their respective domains. Then we have by [2] that ∂x
(t, t0 , x0 ) = Φ(t, t0 , x0 )
0
exists and satisfies the variational equation
Z t
0
(3.3)
x (t) = fx (t, x(t; t0 , x0 ))z(t) +
gx (t, s, x(s; t0 , x0 )) z(s)ds, z(t0 ) = I
t0
and
(3.4)
∂x
(t; t0 , x0 ) + Φ(t, t0 , x0 )f (t0 , x0 )
∂t0
Z
t
Φ(t, s, x0 )g(s, t0 , x0 )ds = 0.
t0
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+ .
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
Hence by (3.9), we easily see that |u(t)| is bounded and the proof is complete.
R EFERENCES
[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 Gronwall-Bellman lemma in partially ordered Banach spaces, J. Math. Anal. Appl., 31 (1970), 668 – 681.
[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.
J. Inequal. Pure and Appl. Math., 2(1) Art. 9, 2001
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