Abstract
We investigate some new nonlinear dynamic inequalities on time scales. Our results unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used to investigate the properties of certain dynamic equations on time scales.
1. Introduction
To unify the theory of continuous and discrete dynamic systems, in 1988, Hilger [1] first introduced the calculus on time scales. Motivated by the paper [1], many authors have expounded on various aspects of the theory of dynamic equations on time scales. For example, we refer the reader to the literatures [2–7] and the references cited therein. At the same time, a few papers [8–13] have studied the theory of dynamic inequalities on time scales.
The main purpose of this paper is to investigate some nonlinear dynamic inequalities on time scales, which unify and extend some integral inequalities and their corresponding discrete analogues. Our work extends some known results of dynamic inequalities on time scales.
Throughout this paper, a knowledge and understanding of time scales and time-scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to monographes [6, 7].
2. Main Results
In what follows, denotes the set of real numbers, , denotes the set of integers, denotes the set of nonnegative integers, denotes the class of all continuous functions defined on set with range in the set , is an arbitrary time scale, denotes the set of rd-continuous functions, denotes the set of all regressive and rd-continuous functions, and . We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume that , and are real constants, and .
Firstly, we introduce the following lemmas, which are useful in our main results.
Lemma 2.1. Let . Then
Proof. If , then we easily see that the inequality holds. Thus we only prove that the inequality holds in the case of .
Letting we haveIt is easy to see thatTherefore, The proof of Lemma 2.1 is complete.
Lemma 2.2 (see [6]). Let and be continuous at , with . Assume that is rd-continuous on . If, for any , there exists a neighborhood of , independent of , such that where denotes the derivative of with respect to the first variable, then implies
Lemma 2.3 (Comparison theorem [6]). Suppose , . Then implies
Next, we establish our main results.
Theorem 2.4. Assume that , and and are nonnegative. Thenimplieswhereand also
Proof. Obviously, if , then the inequality holds. Therefore, in the next proof, we always assume that .
Define a function byThen can be restated asUsing Lemma 2.1, from , for any , we easily obtainIt follows from and that where and are defined as in and , respectively. Using Lemma 2.3 and noting , from we have Therefore, the desired inequality follows from and . This completes the proof of Theorem 2.4.
Remark 2.5. By letting in Theorem 2.4, it is easy to observe that the bound obtained in reduces to the bound obtained in [9, Theorem ].
As a particular case of Theorem 2.4, we immediately obtain the following result.
Corollary 2.6. Assume that , and and are nonnegative. If is a constant, then implieswhere
Remark 2.7. The result of Theorem 2.4 holds for an arbitrary time scale. Therefore, using Theorem 2.4, we immediately obtain many results for some peculiar time scales. For example, letting and , respectively, we have the following two results.
Corollary 2.8. Let and assume that . Then the inequality implieswhere and are defined as in Theorem 2.4.
Corollary 2.9. Let and assume that , and are nonnegative functions defined for . Then the inequalityimplies where and are defined as in Theorem 2.4.
Investigating the proof procedure of Theorem 2.4 carefully, we can obtain the following result.
Theorem 2.10. Assume that , and , and are nonnegative, If there exists a series of positive real numbers such that , then implieswhere
Theorem 2.11. Assume that , , and are nonnegative, and is defined as in Lemma 2.2 such that and for with . If, for any , there exists a neighborhood of , independent of , such that for all , thenimplies whereand also
Proof. Define a function by whereThen . As in the proof of Theorem 2.4, we easily obtain and .
It follows from (2.25) that and alsoTherefore, noting the condition , using Lemma 2.2 and combining –, , and , we havewhere and are defined as in and , respectively. Therefore, using Lemma 2.3 and noting , we getIt is easy to see that the desired inequality follows from and . This completes the proof of Theorem 2.11.
Remark 2.12. Letting , in Theorem 2.11, we easily obtain [9, Theorem ].
The following two corollaries are easily established by using Theorem 2.11.
Corollary 2.13. Let and assume that . If and its partial derivative are real-valued nonnegative continuous functions for with , then the inequalityimplies where and also
Remark 2.14. Letting , in Corollary 2.13, we easily obtain [14, Theorem ].
Corollary 2.15. Let and assume that and are nonnegative functions defined for . If and are real-valued nonnegative functions for with , then the inequalityimplieswhere for with ,
Remark 2.16. By letting , in Corollary 2.15, it is very easy to obtain [15, Theorem ].
Corollary 2.17. Suppose that , and are defined as in Theorem 2.11, and let be nondecreasing for all . If, for any , there exists a neighborhood of , independent of , such that for all , thenimplies where
Proof. Letting , and in Theorem 2.11, we obtain and alsowhere the inequality holds because is nondecreasing for all . Therefore, using Theorem 2.11 and noting and , we easily have The proof of Corollary 2.17 is complete.
Remark 2.18. In Corollary 2.17, letting , we immediately obtain [12, Theorem ].
From the proof procedure of Theorem 2.11, we can obtain the following result.
Theorem 2.19. Assume that , , and are nonnegative, , and there exists a series of positive real numbers such that . Let be defined as in Lemma 2.2 such that and for with . If, for any , there exists a neighborhood of , independent of , such that for all ,thenimplieswhere
Remark 2.20. Using our main results, we can obtain many dynamic inequalities for some peculiar time scales. Due to limited space, their statements are omitted here.
3. An Application
In this section, we present an application of Corollary 2.6 to obtain the explicit estimates on the solutions of a dynamic equation on time scales.
Example 3.1. Consider the dynamic equation where and are constants, , and is a continuous function.
Assume that where , is nonnegative, and is a constant. If is a solution of (3.1), then where is defined as in (2.12), and In fact, the solution of (3.1) satisfies the following equivalent equation: Using the assumption (3.2), we have Now a suitable application of Corollary 2.6 to yields .
Acknowledgments
This work is supported by the National Natural Science Foundation of China (10971018, 10971139), the Natural Science Foundation of Shandong Province (Y2009A05), China Postdoctoral Science Foundation Funded Project (20080440633), Shanghai Postdoctoral Scientific Program (09R21415200), the Project of Science and Technology of the Education Department of Shandong Province (J08LI52), and the Doctoral Foundation of Binzhou University (2006Y01).