Journal of Inequalities in Pure and
Applied Mathematics
ON INTEGRAL INEQUALITIES OF GRONWALL-BELLMAN-BIHARI
TYPE IN SEVERAL VARIABLES
volume 1, issue 2, article 20,
2000.
JAMES ADEDAYO OGUNTUASE
Department of Mathematical Sciences
University of Agriculture, Abeokuta
NIGERIA
EMail: adedayo@unaab.edu.ng
Received 23 February, 2000;
accepted 17 April 2000.
Communicated by: G. Anastassiou
Abstract
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Victoria University
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Abstract
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 ⊂ <n .
These inequalities extend and compliment some existing results in the literature on Gronwall-Bellman-Bihari type inequalities.
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
2000 Mathematics Subject Classification: 26D15
Key words: Gronwall-Bellman-Bihari inequality, good kernel, nonincreasing function,
nondecreasing function, nonnegative function.
J.A. Oguntuase
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Non-linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
9
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1.
Introduction
The results obtained in this paper originated from the celebrated GronwallBellman-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- Bellman-Bihari type inequalities in several variables.
Throughout this paper, we shall assume that S is any bounded open set in
the n dimensional Euclidean space <n and that our integrals are on <n (n ≥ 1),
unless otherwise specified.
For x = (x1 , x2 , · · · , xn ), x0 = (x01 , x02 , · · · , x0n ) ∈ S, we shall denote the
integral
Z Z
Z
Z
x1
x2
xn
···
x01
x01
J.A. Oguntuase
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x
· · · dtn · · · dt1
x01
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
· · · dt
by
x0
and Di = ∂x∂ i for i = 1, 2, · · · , n.
Furthermore, for x, t ∈ <n , 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.
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2.
Linear Inequalities
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
holds for all x ∈ S with x ≥ x0 , then
Z s
Z x
(2.2)
u(x) ≤ g(x) 1 +
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(t)
u(x)
≤1+
k(x, t)
dt.
g(x)
g(t)
x0
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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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
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
r(x) ≤ v(x)
and v(x0 ) = 1 or xi = x0i , i = 1, 2, · · · , n.
Hence
(2.3)
J.A. Oguntuase
D1 · · · Dn v(x) = k(x, x)r(x) ≤ k(x, x)v(x).
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From (2.3) we obtain
v(x)D1 · · · Dn v(x)
≤ k(x, x).
v 2 (x)
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That is
v(x)D1 · · · Dn v(x)
(Dn v(x)) (D1 · · · Dn−1 v(x))
≤ k(x, x) +
.
2
v (x)
v 2 (x)
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Hence
Dn
D1 · · · Dn−1 v(x)
v(x)
≤ k(x, x).
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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
Z xn
D1 · · · Dn−2 v(x)
Dn−1
≤
k(x1 , x2 , . . . , xn−1 , tn , x1 , x2 , . . . , xn−1 , tn )dtn .
v(x)
x0n
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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Integrating with respect to xn−1 from x0n−1 to xn−1 , we have
D1 · · · Dn−2 v(x)
v(x)
Z xn−1 Z xn
≤
k(x1 , x2 , . . . , xn−2 , tn−1 , tn , x1 , x2 , . . . , xn−2 , tn−1 , tn )dtn dtn−1 .
x0n−1
x0n
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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
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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
(2.4)
x
v(x) ≤ exp
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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k(t, t)dt .
Contents
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
x0
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Since
u(x)
g(x)
= r(x), then we obtain
Z
x
u(x) ≤ g(x) 1 +
Z
s
k(s, s) exp
x0
k(t, t)dt ds .
x0
This completes the proof of our result.
Next, we shall consider the case in which k(x, y) = 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
n
h0 (x) = 0, where the prime denote ∂x ∂...∂x and the following inequality
1
x
u(x) ≤ g(x) + h(x)
f (t)u(t)dt
x0
holds for all x ∈ S with x ≥ x , then
Z s
Z x
(2.6)
u(x) ≤ g(x) 1 +
h(s)f (s) exp
h(t)f (t)dt ds .
x0
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
f (s) exp
x0
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0
u(x) ≤ g(x) 1 +
J.A. Oguntuase
n
Z
(2.5)
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
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f (t)dt ds .
x0
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3.
Non-linear Inequalities
Definition 3.1. A function φ : <+ → <+ is said to belong to the class F if it
satisfies the following conditions:
1. φ is nondecreasing and continuous in <+ and φ(u) > 0 for u > 0;
2. α1 φ(u) ≤ φ αu , 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)
u
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
is nonincreasing for u > 0;
2. φ ∈ F implies that φ is subadditive;
J.A. Oguntuase
+
3. If φ satisfies (1) of Definition 3.1 and is concave in < , 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 x
−1
(3.2)
u(x) ≤ g(x)G
G(1) +
k(t, t)dt ,
x0
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where
Z
z
G(z) =
z0
ds
, z ≥ z 0 > 0,
φ(s)
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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
Proof. Since g(x) is positive and nondecreasing, we can write (3.1) as
Z x
Z x
u(x)
φ(u(t))
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)
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
= v(x), then we have
Z
J.A. Oguntuase
x
v(x) ≤ 1 +
k(x, t)φ(v(t))dt.
x0
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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
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D1 · · · Dn r(x) = k(x, x)φ(r(x)).
That is
D1 · · · Dn r(x)
≤ k(x, x).
φ(r(x))
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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)
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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It thus follows that
D1 G(r(x)) =
D1 r(x)
,
φ(r(x))
so that
Z
x2
D1 G(r(x)) ≤
k(x1 , t2 , . . . , tn , x1 , t2 , . . . , tn )dtn . . . dt2 .
x02
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
Z
−1
r(x)) ≤ G
G(1) +
x
k(t, t)dt .
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)
u(x) ≤ g(x)G
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= v(x), we have
−1
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
Z
x
G(1) +
k(t, t)dt
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x0
which is required and the proof is complete.
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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
G(z) =
z0
G
−1
ds
, z ≥ z 0 > 0,
φ(s)
∗
is the inverse of G and x is chosen so that
Z x
G(1) + h(x)
f (t)dt ∈ Dom(G−1 ).
x0
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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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
u(x) ≤ g(x)G−1 G(1) +
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f (t)dt .
x0
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References
[1] O. AKINYELE, On Gronwall-Bellman-Bihari type integral inequalities in
several variables with retardation, J. Math. Anal. Appl., 104 (1984), 1–24.
[2] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications,
Kluwer Academic Publishers, Dordrecht, Netherlands, 1992
[3] J. CONLAN AND C-L. WANG, Higher Dimensional Gronwall- Bellman
type inequalities, Appl. Anal., 18 (1984), 1–12.
[4] F.M. DANNAN, Submultiplicative and subadditive functions and integral
inequalities of Bellman-Bihari type, J. Math. Anal. Appl., 120 (1986),
631–646.
[5] U.D. DHONGADE AND S.G. DEO, Some generalization of BellmanBihari integral inequalities, J. Math. Anal. Appl., 120 (1973), 218–226.
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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[6] J.A. OGUNTUASE, Higher dimensional integral inequality of GronwallBellman type, Ann. Stiint. Univ. ‘Al. I. Cuza’ din Iasi, Ser. T.45 (1999), in
press.
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[7] J.A. OGUNTUASE, Gronwall-Bellman type integral inequalities for
multi-distribution, Riv. Mat. Univ. Parma, 6(2) (1999), in press.
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[8] C.C. YEH, Bellman-Bihari integral inequality in n independent variables,
J. Math. Anal. Appl., 87 (1982), 311–321.
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[9] C.C. YEH, On some integral inequalities in n independent variables and
their applications, J. Math. Anal. Appl., 87 (1982), 387–410.
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[10] C.C. YEH AND M-H. SHIH, The Gronwall-Bellman inequality in several
variables, J. Math. Anal. Appl., 86 (1982), 157–167.
On Integral inequalities of
Gronwall-Bellman-Bihari type in
several variables
J.A. Oguntuase
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