Our aim in this paper is to investigate some new integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as handy tools to study the properties of certain dynamic equations on time scales.
1. Introduction
The study of dynamic equations on time scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. For example, we refer the reader to literatures [2–8] and the references cited therein. At the same time, some fundamental integral inequalities used in analysis on time scales have been extended by many authors [9–14]. In this paper, we investigate some new nonlinear integral inequalities on time scales, which unify and extend some continuous inequalities and their corresponding discrete analogues. Our results can be used as handy tools to study the properties of certain dynamic equations 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 [2, 3].
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 rdcontinuous functions, denotes the set of all regressive and rdcontinuous 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 .
We firstly introduce the following lemmas, which are useful in our main results.
Lemma 2.1 ([15] (Bernoulli's inequality)).
Let and . Then
Lemma 2.2 ([2]).
Let and be continuous at , where with . Assume that is rdcontinuous 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 ([2] (Comparison Theorem)).
Suppose , . Then
implies
Lemma 2.4 (see [13]).
Let , , , and be nonnegative. If is nondecreasing, then
implies
Next, we establish our main results.
Theorem 2.5.
Assume that , , , , and are nonnegative. Then
implies
where
Proof.
Define a function by
Then (2.8) can be restated as
Using Lemma 2.1, from the above inequality, we easily obtain
It follows from (2.12) and (2.15) that
where and are defined as in (2.10) and (2.11), respectively. Using Lemma 2.3 and noting , from (2.16) we have
Therefore, the desired inequality (2.9) follows from (2.14) and (2.17). This completes the proof of Theorem 2.5.
Corollary 2.6.
Assume that , , and are nonnegative. If is a constant, then
implies
where
Proof.
Letting , , and in Theorem 2.5, we obtain
Therefore,
The proof of Corollary 2.6 is complete.
Remark 2.7.
The result of Theorem 2.5 holds for an arbitrary time scale. Therefore, using Theorem 2.5, we can 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 , and . Then the inequality
implies
where and are defined as in Theorem 2.5.
Corollary 2.9.
Let and assume that , , , , and are nonnegative functions defined for . Then the inequality
implies
where and are defined as in Theorem 2.5.
Investigating the proof procedure of Theorem 2.5 carefully, we easily obtain the following more general result.
Theorem 2.10.
Assume that , , , , , and are nonnegative, If there exists a series of positive real numbers such that , then
implies
where
Theorem 2.11.
Assume that , , , , , and are nonnegative. If is defined as in Lemma 2.2 such that and for with , then
implies
where
Proof.
Define a function by
Then . As in the proof of Theorem 2.5, we easily obtain (2.14) and (2.15). Using Lemma 2.2 and combining (2.34) and (2.15), we have
where and are defined as in (2.32) and (2.33), respectively. Therefore, in the above inequality, using Lemma 2.3 and noting , we get
It is easy to see that the desired inequality (2.31) follows from (2.14) and (2.36). This completes the proof of Theorem 2.11.
Corollary 2.12.
Let and assume that , . If and its partial derivative are real–valued nonnegative continuous functions for with , then the inequality
implies
where
Corollary 2.13.
Let and assume that , , , , and are nonnegative functions defined for . If and are realvalued nonnegative functions for with , then the inequality
implies
where for with ,
Corollary 2.14.
Suppose that , and are defined as in Theorem 2.11. If is nondecreasing for , then
implies
where
Proof.
Letting , , and in Theorem 2.11, we obtain
where the inequality holds because is nondecreasing for . Therefore, using Theorem 2.11 and noting (2.46), we easily have
The proof of Corollary 2.14 is complete.
By Theorem 2.11, we can establish the following more general result.
Theorem 2.15.
Assume that , , , , , , and are nonnegative, and there exists a series of positive real numbers such that , . If is defined as in Lemma 2.2 such that and for with , then
implies
where
Theorem 2.16.
Let , and be nonnegative, , and be nondecreasing. Assume that there exists a series of positive real numbers such that , . If is a continuous function such that
for and , where is a nonnegative continuous function, , then
implies
where
Proof.
Let
Then (2.52) can be restated as
It is easy to see that , , and is nondecreasing. Using Lemma 2.4, from (2.58), we have
where is defined as in (2.54). It follows from (2.57) and (2.59) that
Using Lemma 2.1 to the above inequality, we obtain
Noting the hypotheses on , from (2.62), we get
where is defined by (2.55). Clearly, and are nondecreasing. Therefore, for any , from (2.63), we obtain
Let
and define by the right hand of (2.64). Then , , , and
where is defined by (2.56). Using Lemma 2.3 and noting , from (2.66), we have
Therefore,
It follows from (2.61) and (2.68) that
Letting in (2.69), we immediately obtain the desired inequality (2.53). This completes the proof of Theorem 2.16.
Corollary 2.17.
Let , , , and be nondecreasing. Assume that there exists a series of positive real numbers such that , . If is a continuous function such that
for and , where is a continuous function, , then
implies
where is defined as in (2.56),
Corollary 2.18.
Let , , be nondecreasing, and be nonnegative functions defined for . Assume that there exists a series of positive real numbers such that , . If such that
for and , where , , then
implies
where is defined as in (2.56),
Remark 2.19.
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. Some Applications
In this section, we present two applications of our main results.
Example 3.1.
Consider the inequality as in (2.25) with , , , , , , and we compute the values of from (2.25) and also we find the values of by using the result (2.26). In our computations we use (2.25) and (2.26) as equations and as we see in Table 1 the computation values as in (2.25) are less than the values of the result (2.26).
From Table 1, we easily find that the numerical solution agrees with the analytical solution for some discrete inequalities. The program is written in the programming Matlab 7.0.
Example 3.2.
Consider the following initial value problem on time scales:
where and are constants, and is a continuous function.
Assume that
where is defined as in Corollary 2.6, is a constant. If is a solution of IVP (3.1), then
where
In fact, the solution of IVP (3.1) satisfies the following equation:
Using the assumption (3.2), from (3.5), we have
Now a suitable application of Corollary 2.6 to (3.6) yields (3.2).
Acknowledgments
This work is supported by the Natural Science Foundation of Shandong Province (Y2007A08), the National Natural Science Foundation of China (60674026, 10671127), 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).
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