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On Some Integral Inequalities on Time Scales and Their Applications
Journal of Inequalities and Applications volume 2010, Article number: 464976 (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.
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.
2. Some Preliminaries on Time Scales
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 .
For and , we define as follows (provided it exists):
we call the of at .
The following lemmas are very useful in our main results.
Lemma 2.1 (see ).
If and fix , then the exponential function is for the unique solution of the initial value problem
Lemma 2.2 (see ).
Let and be continuous at , where . 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
The following theorem is a foundational result in dynamic inequalities.
Lemma 2.3 . (Comparison Theorem ).
Suppose ; then
The following lemma is useful in our main results.
Lemma 2.4 . (see ).
Let , then
3. Main Results
In this section, we study some integral inequalities on time scales. We always assume that are constants, and .
Assume that , and are nonnegative; then
then , and (3.1) can be restated as
Using Lemma 2.1, for any , we obtain
It follows from (3.4) and (3.6) that
where are defined as in (3.3) and is regressive obviously.
From Lemma 2.3 and (3.7), noting , we obtain
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.
Let and assume that are nonnegative functions defined for . Then the inequality
Let where . We assume that and are nonnegative functions defined for . Then the inequality
Assume that are defined as in Theorem 3.1, are continuous functions, and is nondecreasing about the second variable and satisfies
for and then
then , and (3.16) can be written as (3.5).
Therefore, from (3.6) and (3.19), we have
where are defined as in (3.18) and is regressive obviously.
From Lemma 2.3 and (3.20), noting , we obtain
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.
Let and assume that are nonnegative functions defined for . satisfy
for and is nondecreasing about the second variable. Then the inequality
Assume that , and are defined as in Theorem 3.1, is defined as in Lemma 2.2 such that for with ; then
then , and (3.26) can be written as (3.5).
Therefore, from (3.6) and (3.29) we have
where are defined by (3.28) and is regressive obviously.
From Lemma 2.3 and (3.30), noting , we obtain
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.
4. Some Applications
In this section, we present some applications of Theorem 3.9 to investigate certain properties of solution of the following dynamic equation:
where is a constant, is a continuous function, and are also continuous functions.
where are constants, and are nonnegative. Then every solution of (4.1) satisfies
where are defined as in (3.3) with .
Indeed, the solution of (4.1) satisfies the following equivalent equation
It follows from (4.2) and (4.4) that
Using Theorem 3.1, the inequality (4.3) is obtained from (4.5).
, , and are defined as in Example 4.1. If and is odd, then (4.1) has at most one solution; otherwise, the two solutions of (4.1) have the relation .
Let be two solutions of (4.1). Then we have
It follows from (4.6) and (4.7) that
By Theorem 3.1, we have The results are obtained.
Consider the equation
where are constants, and are nonnegative, is defined as in Lemma 2.2 such that for with .
Then we have the estimate of the solution of (4.9) that
where are defined as in (3.28).
From (4.10) and (4.9), we have
By Theorem 3.9 and (4.12), we have that (4.11) holds.
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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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Xu, R., Meng, F. & Song, C. On Some Integral Inequalities on Time Scales and Their Applications. J Inequal Appl 2010, 464976 (2010). https://doi.org/10.1155/2010/464976
- Continuous Function
- Integral Equation
- Unique Solution
- Exponential Function
- Dynamic Equation