Abstract and Applied Analysis
Volume 2016 (2016), Article ID 3520236, 6 pages
http://dx.doi.org/10.1155/2016/3520236

Research Article

Integrodifferential Inequalities Arising in the Theory of Differential Equations

Zareen A. Khan

Department of Mathematics, Princess Nora Bint Abdul Rahman University, Riyadh, Saudi Arabia

Received 16 June 2016; Accepted 18 October 2016

Academic Editor: Hatem Mejjaoli

Copyright © 2016 Zareen A. Khan. 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.

The goal of this paper is to achieve some new results related to integrodifferential inequalities of one independent variable which can be applied as a study of qualitative and quantitative properties of solutions of some nonlinear integral equations.

1. Introduction

Integral and integrodifferential inequalities play a significant role in recent years by many authors [1–11], which provide an explicit bounds on the solutions of a class of differential and integral equations.

Lemma 1. Pachpatte (1995) studied the following useful integral inequality: Let , , and be nonnegative continuous functions defined on and and be positive constants. If

for , where

for , then

for all .

2. Main Results

Here by using Lemma 1, we establish some new results in the form of integrodifferential inequalities instead of integral inequality.

Theorem 2. Let , , , and be nonnegative real valued continuous functions defined for . Let and be positive constants. If

for all , then

where .
, , and are defined as in (2), (3), and (4), respectively, for all .

Proof. Define a function by the right-hand side of (6), such that

where

From (6) and (8), we get

By differentiating (8) and using the fact that

we observe

or

Let

Differentiating (14) with respect to , we get

where

By substituting (14) and (15) in (13), we have

Inequality (17) implies the estimation for and by using (16), we observe that

where and are defined as in (3) and (4) and by applying (11) and (14) it is noticed that

where is defined as in (2). This completes the proof.

Theorem 3. Let , , , , , and be defined as in Theorem 2 for . If

for all , then

where and

for all .

Proof. Define a function by the right-hand side of (20), such that

where

From (20) and (24), we get

By differentiating (24) and since is monotone nondecreasing function for and using the fact that

we observe that

or

Let

By repeating the same steps from (14)–(18) in (29) with suitable modifications, the estimation for implies

From (27) and (30) in (31), we get

for all , where , , and are defined as in (3), (22), and (23), respectively. This completes the proof.

Theorem 4. Let , , , , , and be defined as in Theorem 2 for . If

for all , then

where and .

for all .

Proof. Define a function by the right-hand side of (33), such that

where

From (33) and (37), we get

or

By differentiating (37) and since is monotone nondecreasing function for , we observe that

or

Let

By repeating the same steps from (14)–(18) in (42) with suitable modifications, the estimation for implies

From (40) and (43) in (44), we get

for all , where , , and are defined as in (3), (35), and (36), respectively. This completes the proof.

Theorem 5. Let , , , , , and be defined as in Theorem 2 for . If

for all , then

where and

for all .

Proof. Define a function by the right-hand side of (46), such that

where

From (46) and (50), we get

or

By differentiating (50) and since is monotone nondecreasing function for , we observe that

or

Let

By repeating the same steps from (14)–(18) in (55) with suitable modifications, the estimation for implies

From (53) and (56) in (57), we get

for all , where , , and are defined as in (3), (48), and (49), respectively. This completes the proof.

Theorem 6. Let , , , , , and be defined as in Theorem 2 for . If

for all , then

where , , and .

for all .

Proof. Define a function by the right-hand side of (59), such that

where

From (59) and (62), we get

or

By differentiating (62) and since is monotone nondecreasing function for , we observe that

or

Let

By repeating the same steps from (14)–(18) in (67) with suitable modifications, the estimation for implies

From (65) and (68) in (69), we get

for all , where and are defined as in (3) and (61), respectively. This completes the proof.

3. Application

As an application, the explicit bounds of some of the integral inequalities can be found by some examples.

Example 1. Let us consider the explicit bound on the solution of the nonlinear integrodifferential equation

where is a nonnegative real valued continuous function and every solution of of (71) exists for .
By using the application of Theorem 4 to (71), we observe that

where

Therefore the right-hand side of (74) provides the bound of the solution of (75) of known quantities

for .

Example 2. Let us consider the nonlinear integrodifferential equation of the form

where is a nonnegative real valued continuous function and every solution of of (77) exists for .
By using the application of Theorem 5 to (77), we observe that

where

Therefore the right-hand side of (80) provides the bound of the solution of (77) of known quantities

for .

Example 3. Now let us consider the boundedness and asymptotic behaviour of the solutions of nonlinear Volterra integrodifferential inequality of the form

is nonnegative real valued continuous function defined on and , , , are real valued continuous function defined on .
We assume that every solution of in (81) exists on , and and are defined as in Theorem 6. Define the following hypotheses on the function of (81) as

Also

for all , . , , , are nonnegative real constants and , are nonnegative real valued continuous function defined on .

Proof. For the boundedness of the solution of nonlinear integrodifferential equation (83), let us suppose that the hypotheses (84), (85), and (86) are satisfied and let be a solution of (83); then we observe that

Replacing by and by and applying the same proof with some modifications of Theorem 6 in (88) and with and being the same as defined in Theorem 6, we noticed that every solution of of (88) that exists on is bounded; that is,

For the asymptotic behaviour of the solution of nonlinear integrodifferential equation (83), assume the following hypotheses

Also

are satisfied. Let be a solution of (83); then

Let be for in (92) and by applying the same proof with some changes of Theorem 6 in (92), we get

Therefore the solution of (83) is asymptotically stable.

Competing Interests

There is no conflict of interests regarding the publication of this paper.

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