#### Abstract

Using ideas from Boukerrioua and Guezane-Lakoud (2008), some nonlinear integral inequalities are established.

#### 1. Introduction

Integral inequalities provide a very useful and handy device for the study of qualitative as well as quantitative properties of solutions of differential equations. The Gronwall-Bellman type (see, e.g., [1–4]) is particularly useful in that they provide explicit bounds for the unknown functions. One of the most useful inequalities in the development of the theory of diferential equations is given in the following theorem.

Theorem 1.1 (see [3]). *If and are non-nonnegative continuous functions on satisfying
**
for some constant , then
*

The importance of this inequality lies in its successful utilization of the situation for which the other available inequalities do not apply directly. It has been frequently used to obtain global existence, uniqueness, stability, boundedness, and other properties of the solution for wide classes of nonlinear differential equations. The aim of this paper is to give other results on nonlinear integral inequalities and their applications.

#### 2. Main Results

In this section, we begin by giving some material necessary for our study. We denote by , the set of real numbers and the nonnegative real numbers.

Lemma 2.1. *For , one has
*

Lemma 2.2 (see [1]). *Let and be continuous functions for , let be a differentiable function for and suppose
**
Then for ,
*

Now we state the main results of this work

Theorem 2.3. *Let be real-valued nonnegative continuous functions and there exists a series of positive real numbers and satisfy the following integral inequality,
**
for then
**
for .*

*Proof. *Define a function by
then and (2.4) can be written as
By (2.7) and Lemma 2.1, we get
Differentiating (2.6), we get
Using (2.8) and (2.9), it yields
where
By Lemma 2.2, we have
Using (2.7) and (2.12), we get

This achieves the proof of the theorem.

*Remark 2.4. *if we take , then the inequality established in Theorem 2.3 become the inequality given in [4,Theorem ].

Theorem 2.5. *Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function is nondecreasing and
**
for then
**
for and where
*

*Proof. *For

By (2.7) and the fact that , one gets:
Differentiating (2.6) and using (2.17), we obtain
then
Since the function is nondecreasing, for then,
where
Consequently
For , we can see that
then the function can be estimated as
Let
Now we estimate the expression by using (2.24) to get
Remarking that
we integrate (2.27) from 0 to to get
replacing by its value in (2.28), we obtain
then
Using (2.7), (2.23), and (2.30) we have,
This achieves the proof of the theorem.

Theorem 2.6. *Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function , is nondecreasing and
**
for then
**
for and ,
*

*Proof. *for .

Using (2.7), the fact that the function is nondecreasing, and , we have
Differentiating (2.6) and using (2.35), we obtain
then
from the proof of Theorem 2.5, we get the required inequality in (2.33).

*Remark 2.7. *if we take , then the inequalities established in Theorems 2.5 and 2.6 become the inequalities given in [5, Theorem 1.2].

Theorem 2.8. *Suppose that the hypothesis of Theorem 2.3 holds and moreover the function is decreasing. Let be a real valued nonnegative continuous and nondecreasing function for . If
**
then*(1)* **
for , where
*(2)* **
for andwhere
*(3)* **
for and where
*

*Proof. *Since is a nonnegative, continuous, and nondecreasing function, for , from (2.38) we observe that
we put
then, we have

Then a direct application of the inequalities established in Theorems 2.3, 2.5, and 2.6 gives the required results.

Theorem 2.9. *Suppose that the hypothesis of Theorem 2.3 holds. Assume that the functions are nondecreasing and let and its derivative partial be real-valued nonnegative continuous functions, for . If
**
then*(1)* **
where
**
for .*

(2)* **
For *(3)* **
For and , where
*

*Proof. *Let
(1) for .

Differentiating(2.57) we get
Using (2.8) and (2.58) and the fact that is nondecreasing, we obtain, for ,
then

By Lemma 2.2, we have
where
Finally using (2.61) in (2.7), we get the required inequality.(2)For . Using (2.17) and (2.58), we get
For , and the fact that is nondecreasing, we have
then,
where
From the proof of Theorem 2.5, we get the required inequality.(3)Using (2.35) and (2.58), we get
taking account the fact that is nondecreasing and from the proof of Theorem 2.6, we get the required inequality.

*Remark 2.10. *if we take , then the inequality established in Theorem 2.9 (part 1) becomes the inequality given in [4, Theorem ].

#### 3. Further Results

In this section, we investigate some Gronwall-type inequalities.

Theorem 3.1. *Assume that and are non-nonnegative continuous functions on and is a constant. If is defined as in Theorem 2.9, then
**
implies
**
where
**
where , and .*

*Proof. *Define a function by the right side of (3.1) then
Define a function by
Then
and is nondecreasing for .

Then,
its follow from (3.4), (3.6), and (3.7) that
then (3.8) can be written as
where
by Lemma 2.2 we obtain
from (3.6) and (3.11), it follows that
integrating (3.12), we obtain
but
then the result required is found.

#### 4. Application

In this section we present some applications of Theorems 2.3, 2.5, 2.6 and 3.1 to investigate certain properties of solutions of differential equation.

*Example 4.1. *We consider a nonlinear differential equation
Assume that , are fixed real numbers, is a real constant, and ,
are continuous functions.

Integrating (4.1), from 0 to and using (4.2) we obtain where . By applying Theorems 2.3, 2.5, and 2.6, we estimate the solution of the equation, that gives us a bound of the solution.

*Example 4.2. *Consider the following initial value problem:
where and are as defined in Theorem 3.1, and and is a constant.

Theorem 4.3. *Assume is a solution of (4.4), then
**
where and are defined in (3.10).*

*Proof. *The solution of (4.4) satisfies the following equivalent equation:
It follows from (4.6) that

Using Theorem 3.1, we obtain (4.5).