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A Note on the Integral Inequalities with Two Dependent Limits
Journal of Inequalities and Applications volumeÂ 2010, ArticleÂ number:Â 430512 (2010)
Abstract
The theorem on the Gronwall's type integral inequalities with two dependent limits is established. In application, the boundedness of the solutions of nonlinear differential equations is presented.
1. Introduction
Integral inequalities play a significant role in the study of qualitative properties of solutions of integral, differential and integrodifferential equations (see, e.g., [1â€“4] and the references given therein). One of the most useful inequalities in the development of the theory of differential equations is given in the following lemma (see [5]).
Lemma 1.1.
Let and be realvalued nonnegative continuous functions for all . If
for all , where is a real constant, then
for all .
Note that the generalization of this integral inequality and its discrete analogies are given in papers [5â€“8]. In paper [9] the following useful inequality with two dependent limits was established.
Lemma 1.2.
Let be a realvalued nonnegative continuous function defined on and let and be nonnegative constants. Then the inequality
implies that
The theory of integral inequalities with several dependent limits and its applications to differential equations has been investigated in [10â€“14].
The present study involves some Gronwall's type integral inequalities with two dependent limits. Section 2 includes some new integral inequalities with two dependent limits and relevant proofs. Subsequently, Section 3 includes an application on the boundedness of the solutions of nonlinear differential equations.
2. A Main Statement
Our main statement is given by the following theorem.
Theorem 2.1.
Let , , , , , and be realvalued nonnegative continuous functions defined on .

(i)
Let be a nonnegative constant. If
(2.1)
for then
for all

(ii)
Let be a real constant. If
(2.3)
for then
for all

(iii)
Let be a realvalued positive continuous and nondecreasing function defined on and be a real constant. If
(2.5)
for then
for all

(iv)
Let and its partial derivative be realvalued nonnegative continuous functions on and let be even function in If
(2.7)
for then
for all . Here
where
Proof.

(i)
Define a function by
(2.12)
Note that is a nonnegative function and Then (2.1) can be rewritten as
It is easy to see that is an even function.
First, let ; then (2.12) can be rewritten as
Differentiating (2.14) and using (2.13), we get
Dividing both sides of (2.15) by , we get
Integrating the last inequality from to , we get
Second, let . Then, (2.12) can be written as
Differentiating (2.18) and using (2.13), we get
Dividing both sides of (2.19) by , we get
Integrating (2.20) from to 0, we get
Finally, using (2.17) and (2.21), we obtain
The inequality (2.2) follows from (2.13) and (2.22).

(ii)
Define a function by
(2.23)
It is evident that is an even and nonnegative function. We have that
Using Young's inequality (see, e.g., [2]), we obtain that
Let . Then
Differentiating (2.26), we get
Using (2.24) and (2.25), we get
Denoting
we get
From that it follows that
for any Integrating the last inequality from to and using , we get
It is easy to see that
Then
Since , we have that
Applying (2.24), we obtain
From (2.36), and (2.29) it follows (2.4) for Let ; then
Using (2.24) and (2.25), we get
From that it follows that
for any Integrating the last inequality from to and using , we get
It is easy to see that
Then
Since , we have that
Applying (2.43) and (2.24), we obtain (2.36) for Then from (2.36) and (2.29), (2.4) follows for

(iii)
Since is a positive, continuous, and nondecreasing function for , we have that
(2.44)
Now the application of the inequality proven in (ii) yields the desired result in (2.6).

(iv)
We define a function by
(2.45)
Evidently, the function is a nonnegative, monotonic, and nondecreasing in and We have that
Let . Then
Differentiating (2.47), we get
Using (2.46) and Young's inequality, we obtain that
Using (2.29), we get
Applying the differential inequality, we get
Since , we have that
Using (2.33), we get
Since , we have that
Using (2.9) and (2.11), we get
Let . Then
Differentiating (2.56), we get
Using (2.46) and Young's inequality, we obtain that
Using (2.29), we get
Applying the differential inequality, we get
Since , we have that
Using (2.41), we get
Since , we have that
Using (2.9) and (2.10), we get
The inequality (2.8) follows from (2.29), (2.55), and (2.64). Theorem 2.1 is proved.
3. An Application
In this section, we indicate an application of Theorem 2.1 (part (ii)) to obtain the explicit bound on the solution of the following boundary value problem for one dimensional partial differential equations:
where is a fixed real number and . Let , , , , , be smooth functions and problem (3.1) has a unique smooth solution Assume that
for all Here and are realvalued nonnegative continuous functions defined on .
This allows us to reduce the nonlocal boundaryvalue (3.1) to the initialvalue problem
in a Hilbert space with a selfadjoint positive definite operator defined by the formula with the domain (see, e.g., [15, 16]).
Let us give a corollary of Theorem 2.1.
Theorem 3.1.
The solution of problem (3.1) satisfies the estimates
for all Here
Proof.
It is known thatthe formula (see, e.g., [15, 16])
gives a solution of problem ( 3.3 ). Here
Applying the triangle inequality, condition (3.2), formula (3.5), and estimates (see, e.g., [17])
we get
Since
we have that
Denote that Then
for Applying the integral inequality (2.4), we get
We have that
Therefore, the inequality (3.4) follows from the last inequality. Theorem 3.1 is proved.
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Acknowledgments
The authors thank professor O. Celebi (Turkey), professor R. P. Agarwal (USA), and anonymous reviewers for their valuable comments.
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Ashyralyev, A., Misirli, E. & Mogol, O. A Note on the Integral Inequalities with Two Dependent Limits. J Inequal Appl 2010, 430512 (2010). https://doi.org/10.1155/2010/430512
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DOI: https://doi.org/10.1155/2010/430512