- Research Article
- Open access
- Published:
Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs
Journal of Inequalities and Applications volume 2009, Article number: 258569 (2009)
Abstract
We establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. These inequalities generalize former results and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of differential equations. Example of applying these inequalities to derive the properties of BVPs is also given.
1. Introduction
The Gronwall-Bellman inequality states that if and
are nonnegative continuous functions on an interval
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ1_HTML.gif)
for some constant , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ2_HTML.gif)
Inequality (1.2) provides an explicit bound to the unknown function and hence furnishes a handy tool in the study of quantitative and qualitative properties of solutions of differential and integral equations. Because of its fundamental importance over the years many generalizations and analogous results of (1.2) have been established (see, e.g., [1–20]). Among various of Gronwall-Bellman-type inequalities, a very useful one is the following.
Theorem 1.1 (see [21]).
If and
are nonnegative continuous functions defined on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ3_HTML.gif)
for all , where
is a constant, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ4_HTML.gif)
for all .
Inequality (1.4) is called Ou-Iang's inequality, which was established by Ou-Iang during his study of the boundedness of certain kinds of second-order differential equations.
Recently, Pachpatte established the following generalization of Ou-Iang-type inequality.
Theorem 1.2 (see [20]).
Let ,
,
be nonnegative continuous functions defined on
, and let
be a continuous nondecreasing function on
with
for
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ5_HTML.gif)
for all , where
is a constant, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ6_HTML.gif)
for all , where
is the inverse function of
and
is chosen such that
for all
.
Bainov-Simeonov and Lipovan gave the following Gronwall-Bellman-type inequalities, which are useful in the study of global existence of solutions of certain integral equations and functional differential equations.
Theorem 1.3 (see [1]).
Let ,
, where, for the sake of covenience, we allow
to be
(in this case we mean interval
). Let
be a constant, let
be nondecreasing with
for
, and
. If
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ7_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ8_HTML.gif)
for all , where
is defined as in Theorem 1.2 and
is chosen such that
for all
.
Theorem 1.4 (see [13]).
Suppose ,
are nonnegative continuous functions defined on
,
with
for
, and
with
on
are nondecreasing functions. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ9_HTML.gif)
for all , where
is a constant, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ10_HTML.gif)
for all , where
is defined as in Theorem 1.2, and
is chosen such that
for all
.
Very recently, the above results have been further generalized by Cheung to the following.
Theorem 1.5 (see [7]).
Let and
be constants. Let
,
,
, and
be functions satisfying
(i) are nondecreasing and
;
(ii) is nondecreasing with
for
.
If satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ11_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ12_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_IEq74_HTML.gif)
is the inverse function of , and
is chosen such that
for all
.
In this paper, we intend to establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. The setup is basically along the line of [7] but this is by all means a nontrivial improvement of the results there. Examples are also given to illustrate the usefulness of these inequalities in the study of qualitative as well as the quantitative properties of solutions of BVPs.
2. Main Results
Throughout this paper, are two fixed numbers. Let
,
, and
, here we allow
or
to be
. We denote by
the set of all
-times continuously differentiable functions of
into
, and
. Partial derivatives of a function
are denoted by
, and so forth. The identity function will be denoted as
and so, in particular,
is the identity function of
onto itself.
Let and for any
, define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ14_HTML.gif)
Note that we allow and
to be
here.
Theorem 2.1.
Suppose . Let
be a constant. If
,
,
, and
are functions satisfying
(i) are nondecreasing and
;
(ii) is nondecreasing with
for
;
(iii) is strictly increasing with
and
as
;
(iv)for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ15_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ16_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_IEq118_HTML.gif)
is the inverse function of , and
is chosen such that
for all
.
Proof.
It suffices to only consider the case , since the case
can then be arrived at by continuity argument. If we let
denote the right-hand side of (2.2), then we have
on
, and
is nondecreasing in each variable. Hence for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ18_HTML.gif)
By the definition of ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ19_HTML.gif)
Integrating with respect to over
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ20_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ21_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ22_HTML.gif)
for all
Corollary 2.2.
Suppose . Let
be a constant. If
,
,
, and
are functions satisfying
(i) are nondecreasing and
;
(ii)h is strictly increasing with and
as
;
(iii)for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ23_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ24_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ25_HTML.gif)
Proof.
Let , then
. The corollary now follows immediately from Theorem 2.1.
Remark 2.3.
It is easily seen that Theorem 2.1 generalizes Theorems 1.3 and 1.4.
Theorem 2.4.
Suppose . Let
be a constant. If
,
,
, and
are functions satisfying
(i) are nondecreasing and
;
(ii) is nondecreasing with
for
;
(iii) and
are strictly increasing with
as
;
(iv)for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ26_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ27_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_IEq166_HTML.gif)
is the inverse function of , and
is chosen such that
for all
.
Proof.
It suffices to consider the case . If we let
denote the right-hand side of (2.13), then we have
on
, and
is nondecreasing in each variable. Hence for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ29_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ30_HTML.gif)
Integrating with respect to over
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ31_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_IEq179_HTML.gif)
is strictly increasing and is nondecreasing with respect to each variable. Together with (2.18), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ33_HTML.gif)
for all Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ34_HTML.gif)
for all
For any fixed , by the fact that
is nondecreasing in each variable, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ35_HTML.gif)
for all . Now by applying Theorem 2.1 to the strictly increasing function
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ36_HTML.gif)
for all . In particular, this leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ37_HTML.gif)
Since is arbitrary, this concludes the proof of the theorem.
Corollary 2.5.
Suppose . Let
be a constant. If
,
, and
are functions satisfying
(i) are nondecreasing and
;
(ii) is nondecreasing with
for
;
(iii) and
are strictly increasing with
as
;
(iv)for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ38_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ39_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_IEq205_HTML.gif)
is the inverse function of , and
is chosen such that
for all
Remark 2.6.
If we choose then Corollary 2.5 reduces to Theorem 1.2.
Theorem 2.7.
Suppose . Let
be a constant. If
,
,
, and
are functions satisfying
(i) are nondecreasing and
;
(ii) is nondecreasing with
for
;
(iii),
,
and
, are strictly increasing with
when
and
as
;
(iv)for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ41_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ42_HTML.gif)
for all , where
are the same as in Theorem 2.4,
, and
is chosen such that
for all
Proof.
It suffices to consider only the case . For any
, define
. Then clearly
satisfies condition (ii) of Theorem 2.4. Let
, from (2.18), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ43_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ44_HTML.gif)
From Theorem 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ45_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ46_HTML.gif)
for all .
Remark 2.8.
Theorem 2.7 is a generalization of Theorem 1.5.
3. Application to Boundary Value Problems
In this section, we use the results obtained in Section 2 to study certain properties of positive solutions of the following boundary value problem (BVP):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ47_HTML.gif)
where is defined as in Theorem 2.1,
,
, and
are given.
Our first result deals with the boundedness of solutions.
Theorem 3.1.
Consider BVP (3.1). If
(i) for some
;
(ii) for some
,
then all positive solutions to BVP (3.1) satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ48_HTML.gif)
where , and
are defined as in Theorem 2.1, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ49_HTML.gif)
In particular, if is bounded on
, then every solution to BVP (3.1) is bounded on
.
Proof.
It is easily seen that solves BVP (3.1) if and only if it satisfies the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ50_HTML.gif)
Hence by (i) and (ii),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ51_HTML.gif)
Changing variables by letting , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ52_HTML.gif)
From Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ53_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ54_HTML.gif)
The next result is about the quantitative property of solutions.
Theorem 3.2.
Consider BVP (3.1). If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ55_HTML.gif)
for some , then (BVP) (3.1) has at most one solution on
.
Proof.
Assume that and
are two solutions to BVP (3.1). By (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equa_HTML.gif)
Changing variables by letting , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ56_HTML.gif)
From Corollary 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ57_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ58_HTML.gif)
Finally, we investigate the continuous dependence of the solutions of BVP (3.1) on the functional and the boundary data
and
. For this, we consider the following variation of BVP (3.1):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ59_HTML.gif)
where is defined as in Theorem 2.1,
,
, and
are given.
Theorem 3.3.
Consider BVP (3.1) and BVP (3.13). If
(i) for some
;
(ii);
(iii)for all solutions of BVP (3.13),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ60_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ61_HTML.gif)
where is as defined in Theorem 3.1. Hence
depends continuously on
,
, and
.
Proof.
Let and
be solutions to BVP (3.1) and BVP (3.13), respectively. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ62_HTML.gif)
Hence from assumption (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ63_HTML.gif)
From Corollary 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ64_HTML.gif)
If we restrict to any compact subset of
, then
is bounded, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ65_HTML.gif)
for some for all
. Therefore,
depends continuously on
,
, and
.
Remark 3.4.
The uniqueness of solution is often a direct consequence of the continuous dependence on parameters. In fact, let BVP (3.13) be coincide with (3.1), then according to Theorem 3.3,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F258569/MediaObjects/13660_2008_Article_1923_Equ66_HTML.gif)
thus , and so Theorem 3.2 can be viewed as a corollary of Theorem 3.3.
References
Bainov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.
Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge: Heft 30. Springer, Berlin, Germany; 1961:xii+198.
Bellman R: The stability of solutions of linear differential equations. Duke Mathematical Journal 1943,10(4):643–647. 10.1215/S0012-7094-43-01059-2
Bihari I: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 1956,7(1):81–94. 10.1007/BF02022967
Cheung W-S: On some new integrodifferential inequalities of the Gronwall and Wendroff type. Journal of Mathematical Analysis and Applications 1993,178(2):438–449. 10.1006/jmaa.1993.1317
Cheung W-S: Some discrete nonlinear inequalities and applications to boundary value problems for difference equations. Journal of Difference Equations and Applications 2004,10(2):213–223. 10.1080/10236190310001604238
Cheung W-S: Some new nonlinear inequalities and applications to boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2006,64(9):2112–2128. 10.1016/j.na.2005.08.009
Cheung W-S, Ma Q-H: Nonlinear retarded integral inequalities for functions in two independent variables. Journal of Concrete and Applicable Mathematics 2004,2(2):119–134.
Cheung W-S, Ma Q-H, Pecaric J: Some discrete nonlinear inequalities and applications to difference equations. Acta Mathematica Scientia 2008,28(2):417–430. 10.1016/S0252-9602(08)60044-2
Cheung W-S, Ren J: Discrete non-linear inequalities and applications to boundary value problems. Journal of Mathematical Analysis and Applications 2006,319(2):708–724. 10.1016/j.jmaa.2005.06.064
Dafermos CM: The second law of thermodynamics and stability. Archive for Rational Mechanics and Analysis 1979,70(2):167–179.
Haraux A: Nonlinear Evolution Equations: Global Behavior of Solutions, Lecture Notes in Mathematics. Volume 841. Springer, Berlin, Germany; 1981:xii+313.
Lipovan O: A retarded Gronwall-like inequality and its applications. Journal of Mathematical Analysis and Applications 2000,252(1):389–401. 10.1006/jmaa.2000.7085
Ma Q-H, Yang E-H: On some new nonlinear delay integral inequalities. Journal of Mathematical Analysis and Applications 2000,252(2):864–878. 10.1006/jmaa.2000.7134
Ma Q-H, Yang E-H: Some new Gronwall-Bellman-Bihari type integral inequalities with delay. Periodica Mathematica Hungarica 2002,44(2):225–238. 10.1023/A:1019600715281
Mitrinovic DS: Analytic Inequalities, Die Grundlehren der Mathematischen Wissenschaften, Band 16. Springer, New York, NY, USA; 1970:xii+400.
Mitrinovic DS, Pecaric J, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications. Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.
Pachpatte BG: Explicit bounds on certain integral inequalities. Journal of Mathematical Analysis and Applications 2002,267(1):48–61. 10.1006/jmaa.2001.7743
Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611.
Pachpatte BG: On some new inequalities related to certain inequalities in the theory of differential equations. Journal of Mathematical Analysis and Applications 1995,189(1):128–144. 10.1006/jmaa.1995.1008
Ou-Iang L: The boundedness of solutions of linear differential equations . Shuxue Jinzhan 1957, 3: 409–415.
Acknowledgments
The authors express their gratitude to the referees for their careful reading and many useful comments and suggestions. Research is supported in part by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Chen, CJ., Cheung, WS. & Zhao, D. Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs. J Inequal Appl 2009, 258569 (2009). https://doi.org/10.1155/2009/258569
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/258569