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Note on Some Nonlinear Integral Inequalities and Applications to Differential Equations
Using ideas from Boukerrioua and Guezane-Lakoud (2008), some nonlinear integral inequalities are established.
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 ). 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 ). 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
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
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
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
(2) For (3) For and , where
(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
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.
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).
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