Nonlinear Integral Inequalities in Two Independent Variables on Time Scales

Abstract

We investigate some nonlinear integral inequalities in two independent variables on time scales. Our results unify and extend some integral inequalities and their corresponding discrete analogues which established by Pachpatte. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.

1. Introduction

The theory of dynamic equations on time scales unifies existing results in differential and finite difference equations and provides powerful new tools for exploring connections between the traditionally separated fields. During the last few years, more and more scholars have studied this theory. For example, we refer the reader to [1, 2] and the references cited therein. At the same time, some integral inequalities used in dynamic equations on time scales have been extended by many authors [3–11].

On the other hand, a few authors have focused on the theory of partial dynamic equations on time scales [12–17]. However, only [10, 11] have studied integral inequalities useful in the theory of partial dynamic equations on time scales, as far as we know. In this paper, we investigate some nonlinear integral inequalities in two independent variables on time scales, which can be used as handy tools to study the properties of certain partial dynamic equations on time scales.

Throughout this paper, a knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to [1, 2].

2. Main Results

In what follows, is an arbitrary time scale, denotes the set of rd-continuous functions, denotes the set of all regressive and rd-continuous functions, for all, denotes the set of real numbers, , and denotes the set of nonnegative integers. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume that and are time scales, , , , , , and we write for the partial delta derivatives of with respect to , and for the partial delta derivatives of with respect to .

The following two lemmas are useful in our main results.

Lemma 2.1 (see [18]).

If , and with , then

(2.1)

with equality holding if and only if .

Lemma 2.2 (Comparison Theorem [1]).

Suppose , . Then,

(2.2)

implies

(2.3)

Next, we establish our main results.

Theorem 2.3.

Assume that , , , , and are nonnegative functions defined for that are right-dense continuous for , and is a real constant. Then,

(2.4)

implies

(2.5)

where

(2.6)

(2.7)

Proof.

Define a function by

(2.8)

Then, (2.4) can be written as

(2.9)

From (2.9), by Lemma 2.1, we have

(2.10)

It follows from (2.8)–(2.10) that

(2.11)

where is defined by (2.6). It is easy to see that is nonnegative, right-dense continuous, and nondecreasing for . Let be given, and from (2.11), we obtain

(2.12)

Define a function by

(2.13)

It follows from (2.12) and (2.13) that

(2.14)

From (2.13), a delta derivative with respect to yields

(2.15)

where is defined by (2.7). Noting that , , and using Lemma 2.2, from (2.15), we obtain

(2.16)

It follows from (2.9), (2.14), and (2.16) that

(2.17)

Letting in (2.17), we immediately obtain the required (2.5). The proof of Theorem 2.3 is complete.

Remark 2.4.

Letting and , respectively, we easily see that Theorem 2.3 reduces to Theorem 2.3.3 and Theorem 5.2.4 in [19].

Theorem 2.5.

Assume that all assumptions of Theorem 2.3 hold. If and is nondecreasing for , then

(2.18)

implies

(2.19)

where

(2.20)

Proof.

Noting that and is nondecreasing for , from (2.18), we have

(2.21)

By Theorem 2.3, from (2.21), we easily obtain the desired (2.19). This completes the proof of Theorem 2.5.

Remark 2.6.

If in Theorem 2.5, then we easily obtain Theorem 2.3.3 in [19].

Theorem 2.7.

Assume that , , and are nonnegative functions defined for that are right-dense continuous for , and is a real constant. If is right-dense continuous on and continuous on such that

(2.22)

for , , where is right-dense continuous on and continuous on , then

(2.23)

implies

(2.24)

where

(2.25)

(2.26)

Proof.

Define a function by

(2.27)

As in the proof of Theorem 2.3, from (2.23), we easily see that (2.9) and (2.10) hold. Combining (2.10), (2.27) and noting the assumptions on , we have

(2.28)

where is defined by (2.25). It is easy to see that is nonnegative, right-dense continuous, and nondecreasing for . The remainder of the proof is similar to that of Theorem 2.3 and we omit it.

Remark 2.8.

Letting and in Theorem 2.7, respectively, we can obtain Theorem 2.3.4 and Theorem 5.2.4 in [19].

Theorem 2.9.

Assume that , , and are nonnegative functions defined for that are right-dense continuous for , and is a real constant. If is right-dense continuous on and continuous on , and such that

(2.29)

for , , where is right-dense continuous on and continuous on , is the inverse function of , and

(2.30)

then

(2.31)

implies

(2.32)

where is defined by (2.25), and

(2.33)

Proof.

Define a function by (2.27). Similar to the proof of Theorem 2.3, we have

(2.34)

(2.35)

From (2.27), (2.35) and the assumptions on and , we obtain

(2.36)

where is defined by (2.25). Obviously, is nonnegative, right-dense continuous, and nondecreasing for . The remainder of the proof is similar to that of Theorem 2.3, and we omit it here. This completes the proof of Theorem 2.9.

Remark 2.10.

We note that when , Theorem 2.9 reduces to Theorem 2.3.4 in [19].

Remark 2.11.

Using our main results, we can obtain many integral inequalities for some peculiar time scales. For example, letting , , from Theorem 2.3, we easily obtain the following result.

Corollary 2.12.

Assume that , , , and are nonnegative functions defined for , that are continuous for , and is a real constant. Then,

(2.37)

implies

(2.38)

where

(2.39)

3. Some Applications

In this section, we present two applications of our main results.

Example 3.1.

Consider the following partial dynamic equation on time scales

(3.1)

with the initial boundary conditions

(3.2)

where is a constant, is right-dense continuous on and continuous on , is right-dense continuous on , and are right-dense continuous, and is a constant.

Assume that

(3.3)

where and are nonnegative right-dense continuous functions for . If is a solution of (3.1), (3.2), then satisfies

(3.4)

where

(3.5)

In fact, the solution of (3.1), (3.2) satisfies

(3.6)

Therefore,

(3.7)

It follows from (3.3) and (3.7) that

(3.8)

Using Theorem 2.3, from (3.8), we easily obtain (3.4).

Example 3.2.

Consider the following dynamic equation on time scales:

(3.9)

where , are constants, is right-dense continuous on and continuous on .

Assume that

(3.10)

where is a nonnegative right-dense continuous function for . If is a solution of (3.9), then

(3.11)

where

(3.12)

In fact, if is a solution of (3.9), then

(3.13)

It follows from (3.10) and (3.13) that

(3.14)

Therefore, by Theorem 2.5, from (3.14), we immediately obtain (3.11).