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Nonlinear Retarded Integral Inequalities with Two Variables and Applications
Journal of Inequalities and Applications volume 2010, Article number: 240790 (2010)
Abstract
We consider some new nonlinear retarded integral inequalities with two variables, which extend the results in the work of W.-S. Wang (2007), and the one in the work of Y.-H. kim (2009). These inequalities include not only a nonconstant term outside the integrals but also more than one distinct nonlinear integrals without assumption of monotonicity. Finally, we give some applications to the boundary value problem of a partial differential equation for boundedness and uniqueness.
1. Introduction
Integral inequalities that give explicit bounds on unknown functions provide a very useful and important device in the study of many qualitative as well as quantitative properties of solutions of partial differential equations, integral equations, and integrodifferential equation. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall inequality [1], which states that if and
are nonnegative continuous functions on the interval
satisfying

where is a nonnegative constant, then we have

Since the inequality (1.2) provides an explicit bound of the unknown function it furnishes a handy tool in the study of various properties of solutions of differential equations. Because of its fundamental importance, several generalizations and analogous results of Gronwall inequality [1, 2] and its applications have attracted great interests of many mathematicians (e.g., [3–5]). Some recent works can be found, for example, in [6–17] and some references therein. In 2005, Agarwal et al. [6] investigated the inequality

In 2006, Cheung [9] studied the inequality

for all , where
is a constant.
In 2007, Wang [16] discussed the retarded integral inequality

for all .
In 2008, Agarwal et al. [7] discussed the retarded integral inequality

for all , where
is a constant.
In 2009, Kim [12] obtained the explicit bound of the unknown function of the following inequality:

for all .
The purpose of the present paper is to establish some new nonlinear retarded integral inequalities of Gronwall-Bellman type with two variables. We can demonstrate that inequalities (1.4), (1.5), and (1.7), considered in [9, 12, 16], respectively, can also be solved with our results. We also apply our results to study the boundedness and uniqueness of the solutions of the boundary value problem of a partial differential equation.
2. Main Result
Throughout this paper, denotes the set of real numbers, and
are given numbers.
,
,
are the subsets of
and
. For any
, let
denote the subset
of
.
denotes the set of continuous differentiable functions of
into
.
Consider the following inequality:

Our inequality (2.1) not only includes a nonconstant term outside the integrals but also more than one distinct nonlinear integral without assumption of monotonicity. When , and
, our inequality (2.1) reduces to (1.7) studied in [12]. When
, and
our inequality (2.1) reduces to (1.5) studied in [16].
Suppose that
is a strictly increasing continuous function on
,
;
all are continuous functions on
and positive on
;
on
, and
is nondecreasing in each variable;
and
are nondecreasing such that
and
on
,
and
on
;
is a constant;
all are nonnegative functions on
.
Firstly, we technically consider a sequence of functions , which can be calculated recursively by

Moreover, we define the following functions:


Obviously, both and
are strictly increasing and continuous functions. Letting
denote
inverse function, respectively, then both
and
are also continuous and increasing functions.
Let


Then and
are nonnegative and nondecreasing in
for each fixed
and satisfy
,
,
.
Theorem 2.1.
Suppose that hold and
is a nonnegative function on
satisfying (2.1). Then

for , where


is arbitrarily given on the boundary of the planar region

Corollary 2.2.
Let and
be as defined in Theorem 2.1. Suppose that
are constants. If

for all , then

for all , where
and
are defined by (2.5) and (2.6), and


denotes the inverse function of , and
lies on the boundary of the planar region

Corollary 2.3.
Let and
be as defined in Theorem 2.1. Supposing that

for all , then

for all , where


is defined by (2.5), ,
are as defined in (2.3) and (2.4), respectively,
denote the inverse functions of
and
.
lies on the boundary of the planar region

Theorem 2.4.
Suppose that hold and
is a nonnegative function on
satisfying

Then

for , where

for , and
is arbitrarily given on the boundary of the planar region

Corollary 2.5.
Suppose that hold and
is a nonnegative function on
satisfying

where are nonnegative functions on
. Then

for , where

for , and
is arbitrarily given on the boundary of the planar region

3. Proofs and Remarks
Proof of Theorem 2.1.
Obviously, the sequence defined by
in (2.2) is nondecreasing nonnegative functions and satisfies
,
. Moreover, the ratios
,
are all nondecreasing. From (2.1), (2.2) and (2.5), (2.6), we have

We first discuss the case that for all
. Consider the auxiliary inequality

for all , where
and
are chosen arbitrarily. Let
denote the function on the right-hand side of (3.2), which is a nonnegative and nondecreasing function on
and
. Then, we get the equivalent form of (3.2)

Since is nondecreasing and satisfies
for
. By the definition of
, hypothesis
, the monotonicity of
and
, and (3.3), we have

From (3.4), we have

Keeping fixed in (3.5), setting
, integrating both sides of (3.5) with respect to
from
to
, and using the definition of
in (2.3), we have

for all , where

Let

From (3.6), we have

for all . We claim that the unknown function
in (3.9) satisfies

for all , where


Now, we prove (3.10) by induction. For , let
denote the function on the right-hand side of (3.9), which is a nonnegative and nondecreasing function on
,
and
. Then we have

for all . From (3.13), we have

Keeping fixed in (3.14), setting
, integrating both sides of (3.14) with respect to
from
to
, and using the definition of
in (2.4), we have

for all . Using
, from (3.15), we obtain

for all . This proves that (3.10) is true for
.
Next, we make the inductive assumption that (3.10) is true for . Now, we consider

for all . Let
denote the nonnegative and nondecreasing function on the right-hand side of (3.17). Then
and

Let

By (2.2), we see that each ,
, is a nondecreasing function. Then, we have

for all . Keeping
fixed in (3.20), setting
, integrating both sides of (3.20) with respect to
from
to
, and using the definition of
in (2.4), we have

for all . Let


Using (3.22) and (3.23), from (3.21), we have

It has the same form as (3.9). Let . Since
, and
are continuous, nondecreasing, and positive on
, each
is continuous, nondecreasing, and positive on
. Moreover,

which are also continuous, nondecreasing, and positive on . Therefore, the inductive assumption for (3.9) can be used to (3.24), and then we have

for all , where



We note that

Thus, from (3.18), (3.22), (3.26), and (3.29), we have

for all . We can prove that the term of
in (3.30) is just the same as
defined in (3.12). Let
. By (3.23), we have

Then using (3.28) and (3.29), we get

This proves that in (3.30) is just the same as
defined in (3.12). Therefore, from (3.29), (3.30), and (3.32), we obtain


. The relations of (3.33) imply that in (3.26) and (3.28)


. Hence, (3.30) can be equivalently written as

for all The claim in (3.10) is proved by induction.
Therefore, by (3.3), (3.8), and (3.10), we have

for all . Hence, we obtain the estimation of the unknown function
in the auxiliary inequality (3.2).
Letting ,
, from (3.36), we have

for all ,
. Since
and
and
are arbitrarily chosen, this proves (2.7).
The remainder case is that for some
. Let

where is an arbitrary small number. Obviously,
, for all
. Using the same arguments as above, where
is replaced with
, we get

for all . Letting
, we obtain (2.7) because of continuity of
in
and continuity of
, and
for
. This completes the proof.
The proofs of Corollary 2.2, 2.3, 2.5 and Theorem 2.4 are similar to the argument in the proofs of Theorem 2.1 with appropriate modification. We omit the details here.
Remark 3.1.
When and
, Corollary 2.2 reduces to Theorem
in [9].
Remark 3.2.
When and
, Corollary 2.3 reduces to Theorem
of Wang [16].
Remark 3.3.
When , Theorem 2.4 reduces to Theorem
of Kim [12].
4. Applications
In this section, we apply our results to study the boundedness and uniqueness of the solutions of boundary value problem to a partial differential equation. We consider the partial differential equation with the initial boundary conditions:


for all , where
is defined as in Section 2,
,
are defined in
,
is a continuous and strictly increasing odd function on
, satisfying
,
for
,
,
,
, and
are nondecreasing continuous functions, and the ratio
is also nondecreasing, and
for
.
In the following corollary, we firstly apply our result to discuss boundedness on the solution of problem (4.1).
Corollary 4.1.
Assume that is a continuous function for which there exist a constant
, nonnegative functions
,
, such that


for all , and
is nondecreasing in each variable. If
is any solution of problem (4.1) with condition (4.2), then

for all , where

and are as defined in Theorem 2.1.
lies on the boundary of the planar region

Proof.
It is easy to see that the solution of (4.1) satisfies the following equivalent integral equation:

By(4.3),(4.4), and (4.8), we have

Since , (4.9) is the form of (2.14). Applying Corollary 2.3 to inequality (4.9), using the relation

we obtain the estimation of as given in (4.5).
Corollary 4.1 gives a condition of boundedness for solutions, concretely. If there is a ,

for all . Then every solution
of (4.1) is bounded on
.
Next, we discuss the uniqueness of the solutions of (4.1).
Corollary 4.2.
Additionally, assume that

for and
, where
is defined as in Section 2,
is a constant,
,
,
are continuous nondecreasing with the nondecreasing ratio
such that
for all
, and
,
, and
is a strictly increasing odd function satisfying
for all
. Then, (4.1) has at most one solution on
Proof.
Let and
be two solutions of (4.1). By (4.8) and (4.12), we have

for all , which is an inequality of the form (2.14), where
is an arbitrary small number. Applying Corollary 2.2, we obtain an estimation of the difference
in the form (4.5). Namely,

for all , where

and denotes the inverse function of
.
Furthermore, by the definition of , we conclude that

Letting , it follows that

Thus, from (4.5), we deduce that , implying that
, for all
, since
is strictly increasing. The uniqueness is proved.
References
Gronwall TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Annals of Mathematics 1919, 20(4):292–296. 10.2307/1967124
Bellman R: The stability of solutions of linear differential equations. Duke Mathematical Journal 1943, 10: 643–647. 10.1215/S0012-7094-43-01059-2
Baĭnov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.
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: 81–94. 10.1007/BF02022967
Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611.
Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications. Applied Mathematics and Computation 2005, 165(3):599–612. 10.1016/j.amc.2004.04.067
Agarwal RP, Kim Y-H, Sen SK: New retarded integral inequalities with applications. Journal of Inequalities and Applications 2008, 2008:-15.
Chen C-J, Cheung W-S, Zhao D: Gronwall-Bellman-type integral inequalities and applications to BVPs. Journal of Inequalities and Applications 2009, 2009:-15.
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
Choi SK, Deng S, Koo NJ, Zhang W: Nonlinear integral inequalities of Bihari-type without class H . Mathematical Inequalities & Applications 2005, 8(4):643–654.
Dragomir SS, Kim Y-H: Some integral inequalities for functions of two variables. Electronic Journal of Differential Equations 2003, 2003(10):1–13.
Kim Y-H: Gronwall, Bellman and Pachpatte type integral inequalities with applications. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(12):e2641-e2656. 10.1016/j.na.2009.06.009
Lipovan O: Integral inequalities for retarded Volterra equations. Journal of Mathematical Analysis and Applications 2006, 322(1):349–358. 10.1016/j.jmaa.2005.08.097
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
Wang W-S: A generalized retarded Gronwall-like inequality in two variables and applications to BVP. Applied Mathematics and Computation 2007, 191(1):144–154. 10.1016/j.amc.2007.02.099
Zhang W, Deng S: Projected Gronwall-Bellman's inequality for integrable functions. Mathematical and Computer Modelling 2001, 34(3–4):393–402. 10.1016/S0895-7177(01)00070-X
Acknowledgments
The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This work is supported by the Natural Science Foundation of Guangxi Autonomous Region (0991265), the Scientfic Research Foundation of the Education Department of Guangxi Autonomous Region (200707MS112), the Key Discipline of Applied Mathematics (200725) and the Key Project (2009YAZ-N001) of Hechi University of China.
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Wang, WS., Li, Z., Li, Y. et al. Nonlinear Retarded Integral Inequalities with Two Variables and Applications. J Inequal Appl 2010, 240790 (2010). https://doi.org/10.1155/2010/240790
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DOI: https://doi.org/10.1155/2010/240790
Keywords
- Partial Differential Equation
- Unknown Function
- Planar Region
- Nonnegative Function
- Nondecreasing Function