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Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 571795, 7 pages

http://dx.doi.org/10.1155/2011/571795

## Bounds of Solutions of Integrodifferential Equations

Department of Mathematics, Faculty of Electrical Engineering and Communication, TechnickΓ‘ 8, Brno University of Technology, 61600 Brno, Czech Republic

Received 20 January 2011; Accepted 24 February 2011

Academic Editor: MiroslavaΒ RΕ―ΕΎiΔkovΓ‘

Copyright Β© 2011 ZdenΔk Ε marda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: , , where , , are continuous functions, .

#### 1. Introduction

Ou Yang [1] established and applied the following useful nonlinear integral inequality.

Theorem 1.1. *Let and be nonnegative and continuous functions defined on and let be a constant. Then, the nonlinear integral inequality **
implies
*

This result has been frequently used by authors to obtain global existence, uniqueness, boundedness, and stability of solutions of various nonlinear integral, differential, and integrodifferential equations. On the other hand, Theorem 1.1 has also been extended and generalized by many authors; see, for example, [2β19]. Like Gronwall-type inequalities, Theorem 1.1 is also used to obtain *a priori* bounds to unknown functions. Therefore, integral inequalities of this type are usually known as Gronwall-Ou Yang type inequalities.

In the last few years there have been a number of papers written on the discrete inequalities of Gronwall inequality and its nonlinear version to the Bihari type, see [13, 16, 20]. Some applications discrete versions of integral inequalities are given in papers [21β23].

Pachpatte [11, 12, 14β16] and Salem [24] have given some new integral inequalities of the Gronwall-Ou Yang type involving functions and their derivatives. Lipovan [7] used the modified Gronwall-Ou Yang inequality with logarithmic factor in the integrand to the study of wave equation with logarithmic nonlinearity. Engler [5] used a slight variant of the Haraux's inequality for determination of global regular solutions of the dynamic antiplane shear problem in nonlinear viscoelasticity. Dragomir [3] applied his inequality to the stability, boundedness, and asymptotic behaviour of solutions of nonlinear Volterra integral equations.

In this paper, we present new integral inequalities which come out from above-mentioned inequalities and extend Pachpatte's results (see [11, 16]) especially. Obtained results are applied to certain classes of integrodifferential equations.

#### 2. Integral Inequalities

Lemma 2.1. *Let , , and be nonnegative continuous functions defined on . If the inequality
**
holds where is a nonnegative constant, , then
**
for .*

* Proof. * Define a function by the right-hand side of (2.1)
Then, , and
Define a function by
then , ,
Integrating (2.7) from 0 to , we have
Using (2.8) in (2.6), we obtain
Integrating from 0 to and using , we get inequality (2.2). The proof is complete.

Lemma 2.2. *Let , , and be nonnegative continuous functions defined on , be a positive nondecreasing continuous function defined on . If the inequality
**
holds, where is a nonnegative constant, , then
**
where .*

* Proof. *Since the function is positive and nondecreasing, we obtain from (2.10)
Applying Lemma 2.1 to inequality (2.12), we obtain desired inequality (2.11).

Lemma 2.3. *Let , , , and be nonnegative continuous functions defined on , and let be a nonnegative constant. **If the inequality **
holds for , then
**
where
*

* Proof. * Define a function by the right-hand side of (2.13)
Then , and
Differentiating and using (2.17), we get
Integrating inequality (2.18) from 0 to , we have
where is defined by (2.15), is positive and nondecreasing for . Now, applying Lemma 2.2 to inequality (2.19), we get
Using (2.20) and the fact that , we obtain desired inequality (2.14).

#### 3. Application of Integral Inequalities

Consider the following initial value problem where , , are continuous functions. We assume that a solution of (3.1) exists on .

Theorem 3.1. *Suppose that
**
where , are nonnegative continuous functions defined on . Then, for the solution of (3.1) the inequality
**
holds on .*

* Proof. * Multiplying both sides of (3.1) by and integrating from 0 to we obtain
From (3.2) and (3.4), we get
Using inequality (2.14) in Lemma 2.3, we have
where
which is the desired inequality (3.3).

*Remark 3.2. *It is obvious that inequality (3.3) gives the bound of the solution of (3.1) in terms of the known functions.

#### Acknowledgment

This author was supported by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30529 and by the Grant FEKTS-11-2-921 of Faculty of Electrical Engineering and Communication.

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