Bounds for Certain New Integral Inequalities on Time Scales
© Wei Nian Li. 2009
Received: 31 March 2009
Accepted: 10 June 2009
Published: 14 July 2009
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.
The study of dynamic equations on time scales, which goes back to its founder Hilger , 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 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 .
We firstly introduce the following lemmas, which are useful in our main results.
Lemma 2.1 ( (Bernoulli's inequality)).
Let and . Then
Lemma 2.2 ().
Lemma 2.3 ( (Comparison Theorem)).
Lemma 2.4 (see ).
Next, we establish our main results.
Therefore, the desired inequality (2.9) follows from (2.14) and (2.17). This completes the proof of Theorem 2.5.
The proof of Corollary 2.6 is complete.
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.
where and are defined as in Theorem 2.5.
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.
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.
The proof of Corollary 2.14 is complete.
By Theorem 2.11, we can establish the following more general result.
Letting in (2.69), we immediately obtain the desired inequality (2.53). This completes the proof of Theorem 2.16.
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.
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).
where and are constants, and is a continuous function.
Now a suitable application of Corollary 2.6 to (3.6) yields (3.2).
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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