Some Generalized Gronwall-Bellman Type Impulsive Integral Inequalities and Their Applications

Abstract

This paper investigates some generalized Gronwall-Bellman type impulsive integral inequalities containing integration on infinite intervals. Some new results are obtained, which generalize some existing conclusions. Our result is also applied to study a boundary value problem of differential equations with impulsive terms.

1. Introduction

It is well known that Gronwall-Bellman type integral inequalities involving functions of one and more than one independent variables play important roles in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of the theory of differential and integral equations. A lot of contributions to its generalization have been archived by many researchers (see [1–14]). Pachpatte [15] especially studied the following inequality: containing integration on infinite integral and used it in the study of terminal value problems for Gronwall-Bellman type differential equations. Then, Cheung and Ma [16] generalized it into two independent variables with a nonlinear term:

Along the development of the theory of impulsive differential systems, more and more attention is paid to generalizations of Gronwall-Bellman’s results for discontinuous functions (that is, impulsive integral inequalities) and their applications (see [17–25]). Among them, one of the important things is that Samoilenko and Perestyuk [17] considered about the nonnegative piecewise continuous function where , are nonnegative constants, is a positive function, and are the first kind discontinuity points of the function . Then Borysenko [18] investigated integral inequalities with two independent variables: Here is an unknown nonnegative continuous function with the exception of the points where there is a finite jump for . In 2013, Zheng [25] considered the following delay integral inequalities containing integration on infinite intervals: with one general nonlinear term . They assumed that where the class consists of all nonnegative, nondecreasing, and continuous functions on such that and for all and . Actually, when we study behaviors of solutions of differential equations with impulsive terms, may not satisfy the following condition: . For example, does not belong to the class for any and large . Thus, it is very interesting to avoid such conditions. Our main aim here, motivated by the work above, is to discuss the following much more general integral inequality: with two nonlinear terms and where we do not restrict and to the class . Moreover, our main results are applied to estimate the bounds of solutions of differential equations with impulsive terms.

2. Main Results

In what follows, denotes the set of real numbers, , and denotes the first-order partial derivative of with respect to .

Consider (7) and assume that() is a continuous and nonnegative function for and is bounded in for each fixed ;() and are continuous and nonnegative functions on and positive on such that is nondecreasing;() is a nonnegative and continuous function defined on with the first kind of discontinuities at the points where and ;() is a continuous and bounded function for and ; is a nonnegative constant for any positive integer ;() and are continuous and nonnegative functions on such that and for , , and .

Let for and where is a given positive constant. Clearly, is strictly increasing so its inverse is well defined, continuous, and increasing in its corresponding domain.

Theorem 1. Suppose that hold and satisfies (7) for a positive constant . If one lets for ,  , then the estimate of is recursively given by for and , where provided that

Proof. From the assumptions, we know that and are well defined. Moreover, is nonnegative and nonincreasing in and is nonnegative and nonincreasing in and satisfies , .
Case  1. If (in fact, ), from the definition of , we have . According to (7) and (10) we get Take any fixed , and we investigate the following inequality: for . Let and let . Hence, . Clearly, is a nonnegative, nonincreasing, and differentiable function for . The assumption yields that . Thus Integrating both sides of the above inequality from to , we obtain for , where , so or, equivalently, where It is easy to check that , and is differentiable, positive, and nonincreasing on . Since is nondecreasing, from the assumption , we have Note that Integrating both sides of (20) from to , we obtain Thus, We have by (11) Since the inequality above is true for any , we obtain Replacing by and by yields This means that (9) is true for if we replace with .
Case  2. If , (7) becomes where the definition of is given in (10), which is similar to (12). Then, we obtain This implies that (9) is true for if we replace by .
Case  3. If (7) is true for , that is, then, for , (7) becomes where we use the fact that the estimate of is already known for . By assumption (29), again (30) is the same as (27) if we replace by and by . Thus, by (28), we have This completes the proof of Theorem 1 by mathematical induction.

Remark 2. Zheng [25] investigated (5) which is the special case of (7). His results are under the assumptions that , , are decreasing in for every fixed and . In our result, these assumptions are avoided.

Consider the inequality which looks much more complicated than (7).

Corollary 3. In addition to the assumptions , suppose that is positive on , is positive and strictly increasing on , and satisfies (32). If one lets for , , then the estimate of is recursively given by where , , and are given in Theorem 1 and is defined as follows: provided that

Proof. Let . Since the function is strictly increasing on , its inverse is well defined. And (32) becomes Let and ; (36) becomes It is easy to see that , and are continuous and nonnegative functions on , and is nondecreasing on . Even though is much more general, using the same way in Theorem 1, for , we can obtain the estimate of : This completes the proof of Corollary 3.

If where is a constant, we can study the inequality According to Corollary 3, we have the following result.

Corollary 4. In addition to the assumptions , suppose that is positive on and satisfies (39). If one lets for , then the estimate of is recursively given by where , , , and are given in Corollary 3.

Let for , ,  and  .

Consider (8) and assume that is continuous and nonnegative on and bounded in for each fixed and satisfies if , for arbitrary ; and are continuous and nonnegative functions on and are positive on such that is nondecreasing; is nonnegative and continuous on with the exception of the points where there is a finite jump: ; is continuous and bounded for and ; is a nonnegative constant for any positive integer ; and are continuous and nonnegative such that and for , and and for .

Theorem 5. Suppose that hold and satisfies (8) for a positive constant . If one lets for , then the estimate of is recursively given by for , where provided that

Proof. Obviously, for any ,   is positive and nonincreasing with respect to and ; is nonnegative and nonincreasing with respect to and for each fixed and . They satisfy and .
Case  1. If , we have from (8) Take any fixed , , and for arbitrary , , we get Let and let . Hence, . Clearly, is a nonnegative, nonincreasing, and differentiable function for and . Since and , we have Integrating both sides of the above inequality from to , we obtain Thus, for and , where , or equivalently where It is easy to check that , , and is differentiable, positive, and nonincreasing on and . Since is nondecreasing, from assumption (), we have Note that Integrating both sides of (53) from to , we obtain Thus, Hence, Since the above inequality is true for any , , we obtain Replacing , , and by , , and , respectively, yields This means that (42) is true for and if we replace with .
Case  2. If , (8) becomes where the definition of is given in (43). Note that the estimate of is known. Clearly, (60) is the same as (45) if we replace and by and . Thus, by (59), we have This implies that (42) is true for and if we replace by .
Case  3. Assume that (42) is true for . Then for , (8) becomes where we use the fact that the estimate of is already known for ,  . Again, (62) is the same as (60) if we replace and by and . Thus, by (61), we have This yields that (42) is true for if we replace by . By mathematical induction, we know that (42) holds for for any nonnegative integer . This completes the proof of Theorem 5.

Remark 6. (1) If is nonincreasing in each variable and we take , , , , and being continuous on , then (8) reduces to (2) and Theorem 1 becomes Theorem  2.2 in [16].
(2) Zheng [25] investigated (6) which is the special case of (8). His results are under the assumptions that , , and . In our results, these assumptions are avoided.
Consider the inequality which looks much more complicated than (8).

Corollary 7. In addition to the assumptions , suppose that is positive on , is positive and strictly increasing on , and satisfies (64) for a positive constant . If one lets for , then the estimate of is recursively given by where ,  , and are given in Theorem 5; is defined as follows: provided that

The proof is similar to Corollary 3.

If , where is a constant, we can study the inequality According to Corollary 7, we have the following result.

Corollary 8. In addition to the assumptions , suppose that is positive on and satisfies (68) for a positive constant . If one lets for , then the estimate of is recursively given by where ,  ,  , and are given in Corollary 7.

3. Applications

Example 9. Consider the following impulsive differential equation: where , , and , . Here, is a constant.
Assume that where and are nonnegative, bounded, and continuous on ; where and are nonnegative constants.

Theorem 10. Suppose that and hold. If one lets for , , then the solution of (70) has an estimate for : where and

Proof. Integrating (70) from to and using the initial conditions (71), we get which implies that Let Thus, (75) is the same as (7). It is easy to obtain that for any positive constants and Therefore, for any nonnegative and provided that

Example 11. Consider the following partial differential equation with an impulsive term: where , , , and .
Assume that where are nonnegative, bounded, and continuous on , , for ,  ,  ; where and are nonnegative constants.

Theorem 12. Suppose that and hold. If one lets for , then the solution of system (80) has an estimate for : where

Proof. The solution of (80) with an initial value is given by which implies that Similar to Theorem 10, we can obtain, for ,

Remark 13. From Examples 9 and 11, we know that . Clearly, does not hold for large . Thus, does not belong to class in [25]. Hence, the results in [25] can not be applied to inequality (75).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by National Natural Science Foundation of China (no. 11371314), Guangdong Natural Science Foundation (no. S2013010015957), and the Project of the Department of Education of Guangdong Province, China (no. 2012KJCX0074).