Abstract
The object of this paper is to establish some nonlinear integrodifferential integral inequalities in independent variables. These new inequalities represent a generalization of the results obtained by Pachpatte in the case of a function with one and two variables. Our results can be used as tools in the qualitative theory of a certain class of partial integrodifferential equation.
1. Introduction
It is well known that the integral inequalities involving functions of one and more than one independent variables, which provide explicit bounds on unknown functions, play a fundamental role in the development of the theory of differential equations. In the past few years, a number of integral inequalities had been established by many scholars, which are motivated by certain applications. For example, we refer the reader to (see [1–5]) and the references therein.
The study of integrodifferential inequalities for functions of one or independent variables is also a very important tool in the study of stability, existence, bounds, and other qualitative properties of differential equation solutions, integrodifferential equations, and in the theory of hyperbolic partial differential equations (see [6–9]).
One of the most useful inequalities is given in the following lemma (see [1, 10]).
Lemma 1.1 (see [1]). Let and be nonnegative continuous functions defined for , for which the inequality holds for , , where ; are continuous functions defined for , . Then for , .
Wendroff's inequality has recently evoked a lively interest, as may be seen from the papers of Pachpatte [10]. In [10], Pachpatte considered some new integrodifferential inequalities of the Wendroff type for functions of two independent variables. Our aim in this paper is to establish some integrodifferential inequalities in independent variables, an application of our results is also given.
2. Results
Throughout this paper, we will assume that in any bounded open set in the dimensional Euclidean space and that our integrals are on ().
For , , , we will denote
Furthermore, for , we will write whenever , and, for .
We note , wherefor.
We use the usual convention of writing if is the empty set.
Our main results are given in the following theorems.
Theorem 2.1. Let and be nonnegative continuous functions defined on , for which the inequality
holds for all with , where are continuous functions defined for for all . Then
for with , where
Proof. We define the function by the right member of (2.2), Then
Using in (2.5), we have
From (2.8), we observe that
that is
hence
Integrating (2.11) with respect to from to , we have
thus
that is
Integrating (2.14) with respect to from to , we have
Continuing this process, we obtain
from this we obtain
Integrating (2.17) with respect to from to and by (2.7), we have
Integrating (2.18) with respect to from to and by (2.6), we have
that is
By (2.20) and , we obtain the desired bound in (2.3).
Remark 2.2. We note that in the special case , and in Theorem 2.1. our estimate reduces to Lemma 1.1 (see [10]).
Theorem 2.3. Let , , , and be nonnegative continuous functions for all defined for , and for any . If
holds for , where ; are continuous functions defined for for all . Then
For with , where is defined in (2.4).
Proof. We define the function
Then, (2.21) can be restated as
Differentiating (2.23),
Integrating both sides of (2.26) to from to , we have
Now, using (2.27) and (2.25) in (2.26) we obtain
If we put
then by (2.29), we have
Using the facts that and , we have
Which, by following an argument similar to that in the proof of Theorem 2.1, yields the estimate for such that
By (2.33) and (2.28), we have
Integrating both sides of (2.34) to from to and by (2.35), we have
By (2.23), we have
Continuing this process, and by (2.37), we obtain
By (2.23), we have
Integrating both sides of (2.38) to from to and by (2.39), we have
Integrating (2.40) with respect to from to , and by (2.24), we have
By (2.41) and (2.25), we obtain the desired bound in (2.22).
Remark 2.4. We note that in the special case , and in Theorem 2.3, then our result reduces to Theorem 1 obtained in [10].
Theorem 2.5. Let , , , and be nonnegative continuous functions for all defined for , and for any . If
holds for , where ; are continuous functions defined for for all . and is constant. Then
for , with , where is defined in (2.4).
Proof. We define the function
with
Differentiating (2.44), we have
Using the fact that and , we have
by (2.47), we have
where is defined in (2.4).
By (2.48) and using the fact that from (2.42), we obtain the desired bound in (2.43).
Remark 2.6. We note that in the special case , and in Theorem 2.5, then our result reduces to Theorem 2 obtained in [10].
Theorem 2.7. Let , , and be nonnegative continuous functions defined for . If
holds for , where ; are continuous functions defined for for all . Then
for all , where
with defined in (2.4).
Proof. The proof of this Theorem follows by an argument similar to that in Theorem 2.1, We omit the details.
Remark 2.8. We note that in the special case , and in Theorem 2.7, our result reduces to Theorem 2 obtained in [10].
3. Nonlinear Integrodifferential in Independents Variables
In this section, we will give some new nonlinear integrodifferential inequalities for the functions of n-independent variables.
We can also give the following lemma.
Lemma 3.1 (see [2, 11]). Let , , and be nonnegative continuous functions, defined for .
Assume that is a positive, continuous function and nondecreasing in each of the variables . If
holds for all , with . Then
Theorem 3.2. Let , , , , , , and be nonnegative continuous functions for all defined for , and for any . Let be a real-valued, positive, continuous, strictly nondecreasing, subadditive, and submultiplicative function for , and let be a real-valued, continuous positive, and nondecreasing function defined for . Assume that and are positive and nondecreasing in each of the variables . If
holds, for with . Then
for , where
where is the inverse function of , and
is in the domain of for .
Proof. We define the function
then (2.4) can be restated as
Clearly, is a positive, continuous function and nondecreasing in each of the variables , using (3.1) of Lemma 3.1 to (3.8), we have
Integrating to from to , we have
where
By (3.7), we have
where
By (3.10) and (3.13), we have
From (3.14) and (3.13) and since is a subadditive and submultiplicative function, we notice that
We define as the right side of (3.14), then
is positive and nondecreasing in each of the variables then
Dividing both sides of (3.18) by , we get
We note that
Thus, it follows that
From (3.19), (3.20), and (3.21), we have
Now, setting in (3.22) and then integrating with respect from to , we obtain
by (3.23), we have
The required inequality in (3.4) follows from the fact (3.9), (3.12), (3.17), and (3.24).
Many interesting corollaries can be obtained from Theorem 3.2.
Corollary 3.3. Let , , , , , , and be as defined in Theorem 3.2. If
holds, for with . Then
for with , where
where is the inverse function of and
is in the domain of for .
Proof. The proof of this Corollary follows by an argument similar to that in Theorem 3.2. We omit the details.
Corollary 3.4. Let , , , , and be as defined in Theorem 3.2. If
holds, for with , where is a constant, then
for with , where
Proof. Setting and in Corollary 3.3, we obtain our result in this Corollary. We omit the details.
Similarly, we can obtain many other kinds of estimates.
4. An Application
In this section, we present an immediate simple example of application of Theorem 3.2 to the study of boundedness of the solution of a partial integrodifferential equation.
Consider the nonlinear partial integrodifferential equation for , where , are continuous functions.
Assume that functions are defined and continuous on their respective domains of definition, such that for , where is a constant and and are nonnegative, continuous functions defined for . If is any solution of boundary value problem (4.1), then for , by (4.2), we have Now, by a suitable application of Corollary 3.4 of Theorem 3.2, we obtain the bound on the solution of (4.1). or , where
Remark 4.1. Using a similar method of those in the proof of the theorems above, we can also obtain new reversed inequalities of our results. Our results also can be generalized to integrodifferential inequalities with a time delay for functions of one or independent variables, this is under study and will be addressed in a forthcoming work. Among these integrodifferential inequalities with a delay, we can quote: