- Research Article
- Open Access
On Some Integral Inequalities on Time Scales and Their Applications
© Run Xu et al 2010
- Received: 15 January 2010
- Accepted: 18 March 2010
- Published: 6 April 2010
The purpose of this paper is to investigate some new dynamic inequalities on time scales. We establish some new dynamic inequalities; the results unify and extend some continuous inequalities and their corresponding discrete analogues. The inequalities given here can be used as tools in the qualitative theory of certain dynamic equations. Some examples are given in the end of this paper.
- Continuous Function
- Integral Equation
- Unique Solution
- Exponential Function
- Dynamic Equation
The theory of time scales was introduced by Hilger  in 1988 in order to contain both difference and differential calculus in a consistent way. Recently, many authors have extended some fundamental integral inequalities used in the theory of differential and integral equations on time scales. For example, we refer the reader to the papers [2–9, 11–13] and the references cited there in.
In this paper, we investigate some nonlinear integral inequalities on time scales, which extend some inequalities established by Li and Sheng  and Li . The obtained inequalities can be used as important tools in the study of dynamic equations on time scales.
Throughout this paper, let us assume that we have already acquired the knowledge of time scales and time scales notation; for an excellent introduction to the calculus on time scales, we refer the reader to Bohner and Peterson  for general overview.
In what follows, denotes the set of real numbers, denotes the set of integers, denotes the set of nonnegative integers, denotes the set of complex numbers, and denotes the class of all continuous functions defined on set with range in the set . is an arbitrary time scale. If has a right-scattered maximum , then the set ; otherwise, . denotes the set of rd-continuous functions; denotes the set of all regressive and rd-continuous functions. We define the set of all positively regressive functions by . Obviously, if and for , then .
we call the of at .
The following lemmas are very useful in our main results.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
The following theorem is a foundational result in dynamic inequalities.
Lemma 2.3 . (Comparison Theorem ).
The following lemma is useful in our main results.
Lemma 2.4 . (see ).
In this section, we study some integral inequalities on time scales. We always assume that are constants, and .
where are defined as in (3.3) and is regressive obviously.
Therefore, the desired inequality (3.2) follows from (3.5) and (3.8).
The result of Theorem 3.1 holds for an arbitrary time scale. If , then Theorem 3.1 becomes the Theorem established by Yuan et al. . If , we can have the following Corollary.
then , and (3.16) can be written as (3.5).
where are defined as in (3.18) and is regressive obviously.
Therefore, the desired inequality (3.17) follows from (3.5) and (3.21).
If , then Theorem 3.6 becomes [10, Theorem ]. If , we can have the following Corollary.
then , and (3.26) can be written as (3.5).
where are defined by (3.28) and is regressive obviously.
Therefore, the desired inequality (3.27) follows from (3.5) and (3.31).
If then Theorem 3.9 reduces to [8, Theorem ].
Using our results, we can also obtain many dynamic inequalities for some peculiar time scales; here, we omit them.
where is a constant, is a continuous function, and are also continuous functions.
where are defined as in (3.3) with .
Using Theorem 3.1, the inequality (4.3) is obtained from (4.5).
By Theorem 3.1, we have The results are obtained.
where are constants, and are nonnegative, is defined as in Lemma 2.2 such that for with .
where are defined as in (3.28).
By Theorem 3.9 and (4.12), we have that (4.11) holds.
The authors thank the referees very much for their careful comments and valuable suggestions on this paper. This research is supported by the National Natural Science Foundation of China (10771118), the Natural Science Foundation of Shandong Province (ZR2009AM011), and the Science Foundation of the Education Department of Shandong Province, China (J07yh05).
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