Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 1, Issue 2, Article 20, 2000
ON INTEGRAL INEQUALITIES OF GRONWALL-BELLMAN-BIHARI TYPE IN
SEVERAL VARIABLES
J. A. OGUNTUASE
D EPARTMENT OF M ATHEMATICAL S CIENCES , U NIVERSITY OF AGRICULTURE , A BEOKUTA , N IGERIA .
adedayo@unaab.edu.ng
Received 23 February, 2000; accepted 17 April 2000
Communicated by G. Anastassiou
A BSTRACT. We present some new results on the linear and non-linear integral inequalities of
Gronwall-Bellman-Bihari type to n-dimensional integrals with a kernel of the form k(x, t) where
x and t are in S ⊂ Rn .
These inequalities extend and compliment some existing results in the literature on GronwallBellman-Bihari type inequalities.
Key words and phrases: Gronwall-Bellman-Bihari inequality, good kernel, nonincreasing function, nondecreasing function,
nonnegative function.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
The results obtained in this paper originated from the celebrated Gronwall-Bellman-Bihari
inequality which has been of vital importance in the study of existence, uniqueness, continuous
dependence, comparison, perturbation, boundedness and stability of solutions of differential
and integral equations (see for example [1, 2, 3, 4, 5, 6] and the references cited therein).
In the last three decades, more than one variable generalizations of these inequalities have
been obtained and these results have generated a lot of research interests due to its usefulness
in the theory of differential and integral equations (see for example [1, 3, 6, 7, 8, 9, 10] and the
references cited therein).
The purpose of this paper is to establish some new integral inequalities in n independent
variables which will compliment the existing results in the literature on Gronwall- BellmanBihari type inequalities in several variables.
Throughout this paper, we shall assume that S is any bounded open set in the n dimensional
Euclidean space Rn and that our integrals are on Rn (n ≥ 1), unless otherwise specified.
For x = (x1 , x2 , . . . , xn ), x0 = (x01 , x02 , . . . , x0n ) ∈ S, we shall denote the integral
Z x1 Z x2
Z xn
Z x
. . . dt
. . . dtn . . . dt1
by
...
x01
x01
ISSN (electronic): 1443-5756
c 2000 Victoria University. All rights reserved.
004-00
x01
x0
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J. A. O GUNTUASE
and Di = ∂x∂ i for i = 1, 2, . . . , n.
Furthermore, for x, t ∈ Rn , we shall write t ≤ x whenever ti ≤ xi , i = 1, 2, . . . , n. Unless
otherwise specified, all functions considered are functions of n-variables which are nonnegative
and continuous on [x0 , x], x ≥ x0 ≥ 0 and x ∈ S.
2. L INEAR I NEQUALITIES
In this section, we shall obtain bounds to the linear Gronwall-Bellman-Bihari type integral
inequalities for a more general kernel k(x, t) and a product kernel k(x, t) = h(x)f (t).
Definition 2.1. A function k(x, t) of the 2n variables x1 , . . . , tn is called a good kernel if
(1) k (·, ·) ≥ 0.
(2) k (·, ·) is a continuous function of its 2n variables.
(3) k (·, ·) is monotone non-decreasing in its first n variables, i.e. k(x, t) ≥ k(y, t) whenever
x ≥ y.
Theorem 2.1. Let k(x, t) be a good kernel, u(x) is a real valued nonnegative continuous function on S and g(x) be a positive, nondecreasing continuous function on S. Suppose that the
following inequality
Z x
(2.1)
u(x) ≤ g(x) +
k(x, t)u(t)dt
x0
0
holds for all x ∈ S with x ≥ x , then
Z
(2.2)
u(x) ≤ g(x) 1 +
x
Z
s
k(s, s) exp
k(t, t)dt ds .
x0
x0
Proof. Since g(x) is positive and nondecreasing, we can write (2.1) as
Z x
u(x)
u(t)
≤1+
k(x, t)
dt.
g(x)
g(t)
x0
Setting
u(x)
g(x)
= r(x), then we have
x
Z
r(x) ≤ 1 +
k(x, t)r(t)dt.
x0
Let
Z
x
v(x) = 1 +
k(x, t)r(t)dt.
x0
Then
0
and v(x ) = 1 or xi =
x0i ,
r(x) ≤ v(x)
i = 1, 2, . . . , n. Hence
D1 . . . Dn v(x) = k(x, x)r(x) ≤ k(x, x)v(x).
(2.3)
From (2.3) we obtain
v(x)D1 . . . Dn v(x)
≤ k(x, x).
v 2 (x)
That is
Hence
v(x)D1 . . . Dn v(x)
(Dn v(x)) (D1 . . . Dn−1 v(x))
≤ k(x, x) +
.
2
v (x)
v 2 (x)
D1 . . . Dn−1 v(x)
Dn
≤ k(x, x).
v(x)
J. Inequal. Pure and Appl. Math., 1(2) Art. 20, 2000
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O N I NTEGRAL INEQUALITIES OF G RONWALL -B ELLMAN -B IHARI
TYPE IN SEVERAL VARIABLES
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Integrating with respect to xn from x0n to xn , we have
Z xn
D1 . . . Dn−1 v(x)
≤
k(x1 , x2 , . . . , xn−1 , tn , x1 , x2 , . . . , xn−1 , tn )dtn .
v(x)
x0n
Thus
v(x)D1 . . . Dn−1 v(x)
≤
v 2 (x)
Z
xn
k(x1 , x2 , . . . , xn−1 , tn , x1 , x2 , . . . , xn−1 , tn )dtn
x0n
+
(Dn−1 v(x)) (D1 . . . Dn−2 v(x))
.
v 2 (x)
That is
Dn−1
D1 . . . Dn−2 v(x)
v(x)
Z
xn
≤
k(x1 , x2 , . . . , xn−1 , tn , x1 , x2 , . . . , xn−1 , tn )dtn .
x0n
Integrating with respect to xn−1 from x0n−1 to xn−1 , we have
Z xn−1 Z xn
D1 . . . Dn−2 v(x)
≤
k(x1 , x2 , . . . , xn−2 , tn−1 , tn , x1 , x2 , . . . , xn−2 , tn−1 , tn )dtn dtn−1 .
v(x)
x0n−1
x0n
Continuing this process, we obtain
Z x3
Z xn
D1 D2 v(x)
≤
...
k(x1 , x2 , t3 , . . . , tn , x1 , x2 , t3 , . . . , tn )dtn . . . dt3 .
v(x)
x03
x0n
From this we obtain
Z x3
Z xn
D1 v(x)
D2
≤
...
k(x1 , x2 , t3 , . . . , tn , x1 , x2 , t3 , . . . , tn )dtn . . . dt3 .
v(x)
x03
x0n
Integrating with respect to the x2 component from x02 to x2 , we have
Z x2
Z xn
D1 v(x)
≤
...
k(x1 , t2 , t3 , . . . , tn , x1 , t2 , t3 , . . . , tn )dtn . . . dt2 .
v(x)
x02
x0n
Integrating with respect to the x1 component from x01 to x1 , we obtain
Z x
v(x)
≤
k(t, t)dt.
log
v(x01 , x2 , . . . , xn )
x0
That is
Z x
(2.4)
v(x) ≤ exp
k(t, t)dt .
x0
Substituting (2.4) into (2.3) we have
Z
x
D1 . . . Dn r(x) ≤ k(x, x)v(x) ≤ k(x, x) exp
k(t, t)dt .
x0
Integrating this inequality with respect to the xn component from x0n to xn , then with respect to
the x0n−1 to xn−1 , and continuing until finally x01 to x1 , and noting that r(x) = 1 at xi = x0i , we
have
Z s
Z x
r(x) ≤ 1 +
k(s, s) exp
k(t, t)dt ds.
x0
Since
u(x)
g(x)
x0
= r(x), then we obtain
Z
u(x) ≤ g(x) 1 +
x
x0
J. Inequal. Pure and Appl. Math., 1(2) Art. 20, 2000
Z
s
k(s, s) exp
k(t, t)dt ds .
x0
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J. A. O GUNTUASE
This completes the proof of our result.
Next, we shall consider the case in which k(x, t) = h(x)f (t). Then we have the following
result.
Theorem 2.2. Let h(x), f (t), u(x) be real valued nonnegative continuous functions on S and
g(x) be a positive, nondecreasing continuous function on S. If h0 (x) = 0, where the prime
n
denote ∂x ∂...∂x and the following inequality
n
1
Z x
(2.5)
u(x) ≤ g(x) + h(x)
f (t)u(t)dt
x0
holds for all x ∈ S with x ≥ x0 , then
Z
(2.6)
u(x) ≤ g(x) 1 +
x
Z
s
h(s)f (s) exp
x0
h(t)f (t)dt ds .
x0
Proof. Similar to the proof of Theorem 2.1 and so the details are omitted.
Remark 2.3. If we set k(x, t) = f (t) in Theorem 2.2, then our estimate reduces to
Z s
Z x
u(x) ≤ g(x) 1 +
f (s) exp
f (t)dt ds .
x0
x0
3. N ON - LINEAR I NEQUALITIES
Definition 3.1. A function φ : R+ → R+ is said to belong to the class F if it satisfies the
following conditions:
(1) φ is nondecreasing
and continuous in R+ and φ(u) > 0 for u > 0;
1
u
(2) α φ(u) ≤ φ α , u ≥ 0, α ≥ 1.
We observe from the above definition that F has the following properties:
(1) φ ∈ F if and only if φ(u)
is nonincreasing for u > 0;
u
(2) φ ∈ F implies that φ is subadditive;
(3) If φ satisfies (1) of Definition 3.1 and is concave in R+ , then φ ∈ F.
Theorem 3.1. Let k(x, t) be a good kernel and u(x) be a real valued nonnegative continuous
function on S. If g(x) be a positive, nondecreasing continuous function on S and φ belong to
class F for which the following inequality
Z x
(3.1)
u(x) ≤ g(x) +
k(x, t)φ(u(t))dt
x0
holds for all x ∈ S with x ≥ x0 , then for x0 ≤ x ≤ x∗ ,
Z
−1
(3.2)
u(x) ≤ g(x)G
G(1) +
x
k(t, t)dt ,
x0
where
Z
z
G(z) =
z0
ds
, z ≥ z 0 > 0,
φ(s)
G−1 is the inverse of G and x∗ is chosen so that
Z x
G(1) +
k(t, t)dt ∈ Dom(G−1 ).
x0
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O N I NTEGRAL INEQUALITIES OF G RONWALL -B ELLMAN -B IHARI
TYPE IN SEVERAL VARIABLES
5
Proof. Since g(x) is positive and nondecreasing, we can write (3.1) as
Z x
Z x
u(x)
u(t)
φ(u(t))
≤1+
k(x, t)
dt ≤ 1 +
k(x, t)φ
dt.
g(x)
g(t)
g(t)
x0
x0
Setting
u(x)
g(x)
= v(x), then we have
Z
x
v(x) ≤ 1 +
k(x, t)φ(v(t))dt.
x0
Let
Z
x
r(x) = 1 +
k(x, t)φ(v(t))dt.
x0
Then
v(x) ≤ r(x)
and v(x0 ) = 1 or xi = x0i , i = 1, 2, . . . , n and
D1 . . . Dn r(x) = k(x, x)φ(r(x)).
That is
D1 . . . Dn r(x)
≤ k(x, x).
φ(r(x))
Since
Dn
D1 . . . Dn−1 r(x)
φ(v(x))
=
D1 . . . Dn r(x) Dn φ(r(x))D1 . . . Dn−1 r(x)
−
φ(r(x))
φ2 (r(x))
and
Dn φ(r(x)) = φ0 (r(x))Dn r(x) ≥ 0, D1 . . . Dn−1 r(x) ≥ 0.
The above inequality implies
D1 . . . Dn−1 r(x)
Dn
≤ k(x, x)
φ(r(x))
provided φ0 (r(x)) ≥ 0 for r(x) ≥ 0.
Integrating with respect to xn from x0n to xn and taking into account the fact that D1 . . . Dn−1 r(x) =
0 for xn = x0n , we have
Z xn
D1 . . . Dn−1 r(x)
k(x1 , x2 , . . . , xn−1 , tn , x1 , x2 , . . . , xn−1 , tn )dtn .
≤
φ(v(x))
x0n
Repeating this, we find (after n − 1 steps) that
Z xn
Z x1
D1 r(x)
≤
...
k(x1 , . . . , xn−1 , tn , x1 , . . . , xn−1 , tn )dtn . . . dt2 .
φ(r(x))
x01
x0n
We note that for
Z
s
G(s) =
s0
dz
, s ≥ s0 > 0.
φ(z)
It thus follows that
D1 G(r(x)) =
so that
Z
D1 r(x)
,
φ(r(x))
x2
D1 G(r(x)) ≤
k(x1 , t2 , . . . , tn , x1 , t2 , . . . , tn )dtn . . . dt2 .
x02
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J. A. O GUNTUASE
Integrating both sides of the above inequality with respect to the component
Z x
G(r(x1 , . . . , xn )) − G(r(t1 , x2 , . . . , xn )) ≤
k(t, t)dt.
x0
Since r(t1 , x2 , . . . , xn ) = 1 we have
r(x)) ≤ G
−1
Z
x
k(t, t)dt .
G(1) +
x0
From this we obtain
v(x) ≤ r(x)) ≤ G
−1
Z
x
G(1) +
k(t, t)dt .
x0
Using the fact that
u(x)
g(x)
= v(x), we have
u(x) ≤ g(x)G
−1
Z
x
G(1) +
k(t, t)dt
x0
which is required and the proof is complete.
If we set k(x, t) = h(x)f (t), then we shall obtain the following result
Theorem 3.2. Let h(x), f (t), u(x) be real valued nonnegative continuous functions on S and
g(x) be a positive, nondecreasing continuous function on S, and φ belong to class F .If h0 (x) =
0 and the following inequality
Z x
(3.3)
u(x) ≤ g(x) + h(x)
f (t)φ(u(t))dt
x0
holds for all x ∈ S with x ≥ x0 , then for x0 ≤ x ≤ x∗ , then
Z x
−1
(3.4)
u(x) ≤ g(x)G
G(1) + h(x)
f (t)dt ,
x0
where
Z
z
ds
, z ≥ z 0 > 0,
φ(s)
0
z
G−1 is the inverse of G and x∗ is chosen so that
Z x
G(1) + h(x)
f (t)dt ∈ Dom(G−1 ).
G(z) =
x0
Proof. Similar to the proof of Theorem 3.1 and so the details are omitted.
Remark 3.3. If we set k(x, t) = f (t) in Theorem 3.2, then our estimate reduces to
Z x
−1
u(x) ≤ g(x)G
G(1) +
f (t)dt .
x0
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TYPE IN SEVERAL VARIABLES
7
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J. Inequal. Pure and Appl. Math., 1(2) Art. 20, 2000
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