Some Sublinear Dynamic Integral Inequalities on Time Scales

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.

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.

Define the forward (backward) jump operator at for (resp. at for ) by

(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.

Fix and let . Define to be the number (if it exists) with the property that given there is a neighborhood of such that, for all ,

(2.2)

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.

If then define the (Cauchy) delta integral by

(2.3)

Definition 2.5.

Say is regressive provided that for all . Denote by the set of all regressive and rd-continuous functions satisfying on . For , define the cylinder transformation by , where Log is the principal logarithm function, , and . For , define . Define the exponential function by

(2.4)

3. Main Results

In the sequel, we always assume that is a constant, is a time scale with . The following sublinear integral inequalities on time scales will be considered:

(I)

(II)

(III)

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]).

Let , C and . Then

(3.1)

Implies that

(3.2)

Lemma 3.2.

Let and are nonnegative functions, is a constant. Then, for any positive function ,

(3.3)

holds, where and are nonnegative constants satisfying .

Proof.

For nonnegative constants and , positive constants and with , the basic inequality in [17]

(3.4)

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]).

Suppose that for each there exists a neighborhood of , independent of , such that

(3.5)

where is continuous at , with and (the derivative of with respect to the first variable) is rd-continuous on . Then

(3.6)

implies that

(3.7)

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

Theorem 3.5.

Assume that are rd-continuous functions. Then, for any rd-continuous function on , any nonnegative constants and satisfying , inequality (I) implies that

(3.8)

where

(3.9)

Proof.

Set

(3.10)

Then, and (I) can be restated as

(3.11)

Based on a straightforward computation and Lemma 3.2, we have

(3.12)

Combining (3.11) and (3.12) yields

(3.13)

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.

Assume that are rd-continuous functions. Let be defined as in Lemma 3.4 such that for and (3.5) holds. Then, for any rd-continuous function , any nonnegative constants and satisfying , inequality (II) implies that

(3.14)

where

(3.15)

and are the same as in Theorem 3.5.

Proof.

Define a function

(3.16)

where

(3.17)

Then, , is nondecreasing, and

(3.18)

Similar to the arguments in Theorem 3.5, by Lemmas 3.2 and 3.4 we have

(3.19)

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

Theorem 3.9.

Assume that are nonnegative rd-continuous functions defined on . Let be a continuous function satisfying

(3.20)

for and , where is a continuous function. Then, for any rd-continuous function , any nonnegative constants and satisfying , inequality (III) implies that

(3.21)

where

(3.22)

Proof.

Define a function by

(3.23)

Then, and . According to the straightforward computation, from (3.20) we get

(3.24)

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.

Let and let be continuous. Then, for any continuous function on , any nonnegative constants and satisfying , inequality (I) implies that

(4.1)

where and are defined as in Theorem 3.5.

Corollary 4.2.

Let and . Then, for any function on , any nonnegative constants and satisfying , inequality (I) implies that

(4.2)

where and are defined as in Theorem 3.5.

Corollary 4.3.

Assume that and are continuous. Let be defined as in Lemma 3.4 such that for and (3.5) holds. Then, for any continuous function on , any nonnegative constants and satisfying , inequality (II) implies that

(4.3)

where and are the same as in Theorem 3.8.

Corollary 4.4.

Assume that and . Let be defined as in Lemma 3.4 such that for and (3.5) holds. Then, for any function on , any nonnegative constants and satisfying , inequality (II) implies that

(4.4)

where and are the same as in Theorem 3.8.

Corollary 4.5.

Assume that and are nonnegative continuous functions. Let be a continuous function satisfying (3.20). Then, for any continuous function on , any nonnegative constants and satisfying , inequality (III) implies that

(4.5)

where and are defined as in Theorem 3.9.

Corollary 4.6.

Assume that and are nonnegative functions on . Let be a function satisfying (3.20). Then, for any function on , any nonnegative constants and satisfying , inequality (III) implies that

(4.6)

where and are defined as in Theorem 3.9.

Remark 4.7.

It is not difficult to provide similar results for other specific time scales of interest. For example, consider the time scale with . Note that and for any ; we have

(4.7)

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

Finally, we apply Theorem 3.5 to a numerical example. Consider the following initial value problem on time scales:

(4.8)

where is a continuous function satisfying

(4.9)

where and are nonnegative rd-continuous functions on . Then, by Theorem 3.5, we see that the solution of (4.8) satisfies

(4.10)

where

(4.11)

are nonnegative constants, and .

In fact, the solution of (4.8) satisfies the following integral inequality:

(4.12)

It yields

(4.13)

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