# A Note on the Integral Inequalities with Two Dependent Limits

- Allaberen Ashyralyev
^{1, 2}, - Emine Misirli
^{3}Email author and - Ozlem Mogol
^{3}

**2010**:430512

https://doi.org/10.1155/2010/430512

© Allaberen Ashyralyev et al. 2010

**Received: **4 October 2009

**Accepted: **5 July 2010

**Published: **4 August 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 integro-differential equations (see, e.g., [1–4] 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.

Note that the generalization of this integral inequality and its discrete analogies are given in papers [5–8]. In paper [9] the following useful inequality with two dependent limits was established.

Lemma 1.2.

The theory of integral inequalities with several dependent limits and its applications to differential equations has been investigated in [10–14].

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.

It is easy to see that is an even function.

- (iii)

- (iv)

The inequality (2.8) follows from (2.29), (2.55), and (2.64). Theorem 2.1 is proved.

## 3. An Application

for all Here and are real-valued nonnegative continuous functions defined on .

in a Hilbert space with a self-adjoint 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.

Proof.

Therefore, the inequality (3.4) follows from the last inequality. Theorem 3.1 is proved.

## Declarations

### Acknowledgments

The authors thank professor O. Celebi (Turkey), professor R. P. Agarwal (USA), and anonymous reviewers for their valuable comments.

## Authors’ Affiliations

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## Copyright

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