On Some Integral Inequalities on Time Scales and Their Applications

Abstract

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.

1. Introduction

The theory of time scales was introduced by Hilger [1] 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 [8] and Li [9]. 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 [4] 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):

(2.1)

we call the of at .

The following lemmas are very useful in our main results.

Lemma 2.1 (see [4]).

If and fix , then the exponential function is for the unique solution of the initial value problem

(2.2)

Lemma 2.2 (see [4]).

Let and be continuous at , where . Assume that is rd-continuous on . If for any , there exists a neighborhood of , independent of , such that

(2.3)

where denotes the derivative of with respect to the first variable, then

(2.4)

implies

(2.5)

The following theorem is a foundational result in dynamic inequalities.

Lemma 2.3 . (Comparison Theorem [4]).

Suppose ; then

(2.6)

implies

(2.7)

The following lemma is useful in our main results.

Lemma 2.4 . (see [7]).

Let , then

(2.8)

3. Main Results

In this section, we study some integral inequalities on time scales. We always assume that are constants, and .

Theorem 3.1.

Assume that , and are nonnegative; then

(3.1)

implies

(3.2)

where

(3.3)

Proof.

Define by

(3.4)

then , and (3.1) can be restated as

(3.5)

Using Lemma 2.1, for any , we obtain

(3.6)

It follows from (3.4) and (3.6) that

(3.7)

where are defined as in (3.3) and is regressive obviously.

From Lemma 2.3 and (3.7), noting , we obtain

(3.8)

Therefore, the desired inequality (3.2) follows from (3.5) and (3.8).

Remark 3.2.

Theorem 3.1 extends some known inequalities on time scales. If then Theorem 3.1 reduces to [7, Theorem ]. If then Theorem 3.1 reduces to [8, Theorem ].

Remark 3.3.

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. [10]. If , we can have the following Corollary.

Corollary 3.4.

Let and assume that are nonnegative functions defined for . Then the inequality

(3.9)

implies

(3.10)

where

(3.11)

Corollary 3.5.

Let where . We assume that and are nonnegative functions defined for . Then the inequality

(3.12)

implies

(3.13)

where

(3.14)

Theorem 3.6.

Assume that are defined as in Theorem 3.1, are continuous functions, and is nondecreasing about the second variable and satisfies

(3.15)

for and then

(3.16)

implies

(3.17)

where

(3.18)

Proof.

Define by

(3.19)

then , and (3.16) can be written as (3.5).

Therefore, from (3.6) and (3.19), we have

(3.20)

where are defined as in (3.18) and is regressive obviously.

From Lemma 2.3 and (3.20), noting , we obtain

(3.21)

Therefore, the desired inequality (3.17) follows from (3.5) and (3.21).

Remark 3.7.

If , then Theorem 3.6 becomes [10, Theorem ]. If , we can have the following Corollary.

Corollary 3.8.

Let and assume that are nonnegative functions defined for . satisfy

(3.22)

for and is nondecreasing about the second variable. Then the inequality

(3.23)

implies

(3.24)

where

(3.25)

Theorem 3.9.

Assume that , and are defined as in Theorem 3.1, is defined as in Lemma 2.2 such that for with ; then

(3.26)

implies

(3.27)

where

(3.28)

Proof.

Define by

(3.29)

then , and (3.26) can be written as (3.5).

Therefore, from (3.6) and (3.29) we have

(3.30)

where are defined by (3.28) and is regressive obviously.

From Lemma 2.3 and (3.30), noting , we obtain

(3.31)

Therefore, the desired inequality (3.27) follows from (3.5) and (3.31).

Remark 3.10.

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:

(4.1)

where is a constant, is a continuous function, and are also continuous functions.

Example 4.1.

Assume that

(4.2)

where are constants, and are nonnegative. Then every solution of (4.1) satisfies

(4.3)

where are defined as in (3.3) with .

Indeed, the solution of (4.1) satisfies the following equivalent equation

(4.4)

It follows from (4.2) and (4.4) that

(4.5)

Using Theorem 3.1, the inequality (4.3) is obtained from (4.5).

Example 4.2.

Assume that

(4.6)

, , 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 .

Proof.

Let be two solutions of (4.1). Then we have

(4.7)

It follows from (4.6) and (4.7) that

(4.8)

By Theorem 3.1, we have The results are obtained.

Example 4.3.

Consider the equation

(4.9)

If

(4.10)

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

(4.11)

where are defined as in (3.28).

Proof.

From (4.10) and (4.9), we have

(4.12)

By Theorem 3.9 and (4.12), we have that (4.11) holds.