# Some Sublinear Dynamic Integral Inequalities on Time Scales

- Yuangong Sun
^{1}Email author

**2010**:983052

https://doi.org/10.1155/2010/983052

© Yuangong Sun. 2010

**Received: **7 July 2010

**Accepted: **15 October 2010

**Published: **20 October 2010

## Abstract

We study some nonlinear dynamic integral inequalities on time scales by introducing two adjusting parameters, which provide improved bounds on unknown functions. Our results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales.

## Keywords

## 1. Introduction

Following Hilger's landmark paper [1], there have been plenty of references focused on the theory of time scales in order to unify continuous and discrete analysis, where a time scale is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist; for example, for (which has important applications in quantum theory), with , and the space of the harmonic numbers.

Recently, many authors have extended some continuous and discrete integral inequalities to arbitrary time scales. For example, see [2–14] and the references cited therein. The purpose of this paper is to further investigate some sublinear integral inequalities on time scales that have been studied in a recent paper [6]. By introducing two adjusting parameters and , we first generalize a basic inequality that plays a fundamental role in the proofs of the main results in [6]. Then, we provide improved bounds on unknown functions, which include many existing results in [6, 14] as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales.

## 2. Time Scale Essentials

The definitions below merely serve as a preliminary introduction to the time scale calculus; they can be found in the context of a much more robust treatment than is allowed here in the text [15, 16] and the references therein.

Definition 2.1.

Also define , if , and , if . The graininess functions are given by and . The set is derived from as follows: if has a left-scattered maximum , then ; otherwise, .

Throughout this paper, the assumption is made that inherits from the standard topology on the real numbers . The jump operators and allow the classification of points in a time scale in the following way. If the point is right-scattered, while if then is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. If and the point is right-dense; if and then is left-dense. Points that are right-dense and left-dense at the same time are called dense. The composition is often denoted .

Definition 2.2.

A function is said to be rd-continuous (denoted C ) if it is continuous at each right-dense point and if there exists a finite left limit in all left-dense points.

Every right-dense continuous function has a delta antiderivative [15, Theorem ]. This implies that the delta definite integral of any right-dense continuous function exists. Likewise every left-dense continuous function on the time scale, denoted , has a nabla antiderivative [15, Theorem ]

Definition 2.3.

Call the (delta) derivative of at . It is easy to see that is the usual derivative for and the usual forward difference for .

Definition 2.4.

Definition 2.5.

## 3. Main Results

where are rd-continuous functions, is continuous, and is continuous.

If we let and , then inequalities (I)–(III) reduce to those inequalities studied in [6]. We say inequalities (I)–(III) are sublinear since . In the sequel, some generalized and improved bounds on unknown functions will be provided by introducing two adjusting parameters and .

Before establishing our main results, we need the following lemmas.

Lemma 3.1 ([15, Theorem , page 255]).

Lemma 3.2.

holds, where and are nonnegative constants satisfying .

Proof.

holds. Let , , and . Then, inequality (3.3) is valid.

Remark 3.3.

When , Lemma 3.2 reduces to Lemma with in [6].

Lemma 3.4 ([15, Theorem , page 46]).

Now, let us give the main results of this paper.

Theorem 3.5.

Proof.

Note that C and . By Lemma 3.1, (3.11), and (3.13), we get (3.8).

Remark 3.6.

For given , by choosing different constants and , some improved bounds on can be obtained. For example, when is sufficiently large, we may set since the value of changes drastically. Similarly, we may set for sufficiently small .

Remark 3.7.

When , , Theorem 3.5 reduces to Theorem in [6]. For some particular cases of , , , and , Theorem 3.5 reduces to Corollary , Corollary in [6], Theorem , and Theorem in [14].

Theorem 3.8.

Proof.

Note that C and . By Lemma 3.1, we get (3.14).

Theorem 3.9.

Proof.

Note that C and . By Lemma 3.1, we get (3.21).

Remark 3.10.

For some particular cases of , , and , Theorems 3.8 and 3.9 include Theorem , Theorem , Corollary , Corollary in [6], Theorem , Theorem and Theorem in [14] as special cases.

Remark 3.11.

Some other integral inequalities on time scales were studied in [8, 9] by using Lemma in [6]. Since Lemma 3.1 generalizes and improves Lemma , similar to the arguments in this paper, the results in [8, 9] can also be generalized and improved based on Lemma 3.1.

## 4. Applications

To illustrate the usefulness of the results, we state the corresponding theorems in the previous section for the special cases and .

Corollary 4.1.

where and are defined as in Theorem 3.5.

Corollary 4.2.

where and are defined as in Theorem 3.5.

Corollary 4.3.

where and are the same as in Theorem 3.8.

Corollary 4.4.

where and are the same as in Theorem 3.8.

Corollary 4.5.

where and are defined as in Theorem 3.9.

Corollary 4.6.

where and are defined as in Theorem 3.9.

Remark 4.7.

for and . Thus, Theorems 3.5–3.9 can be easily applied.

Using Theorem 3.5 with , and , we see that (4.13) implies (4.10).

## Declarations

### Acknowledgment

The author thanks the referees for their valuable suggestions and helpful comments on this paper. This work was supported by the National Natural Science Foundation of China under the grant 60704039.

## Authors’ Affiliations

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