International Journal of Differential Equations

Volume 2014 (2014), Article ID 949860, 4 pages

http://dx.doi.org/10.1155/2014/949860

Research Article

On Inequality Applicable to Partial Dynamic Equations

Deepak B. Pachpatte

Department of Mathematics, Dr. B.A.M. University, Aurangabad, Maharashtra 431004, India

Received 12 February 2014; Accepted 26 March 2014; Published 15 April 2014

Academic Editor: Peiguang Wang

Copyright © 2014 Deepak B. Pachpatte. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main objective of the paper is to study new integral inequality on time scales which is used for the study of some partial dynamic equations. Some applications of our results are also given.

1. Introduction

During past few decades many authors have established various dynamic inequalities useful in the development of differential and integral equations. Mathematical inequalities on time scales play an important role in the theory of dynamic equations. The study of time scale was initiated by Hilger [1] in 1990 in his Ph.D. thesis which unifies continuous and discrete calculus. Since then, many authors have studied various properties of dynamic equations on time scales [2–9].

In what follows, let denotes the set of real numbers and let denote the arbitrary time scales. Let , , and be subsets of and . Let denote the set of rd-continuous function. The partial delta derivative of for with respect to , , and is denoted by , , and . We assume here understanding of time scales calculus and notations. Further information about time scales calculus can be found in [1, 5, 10].

We require the following lemmas given in [5, 6].

Lemma 1 (see [5], Theorem 2.6). Let , , and

for all ; then

for all .

Lemma 2 (see [6], Lemma 2.1). Let and is nondecreasing in and

for ; then

where

for .

2. Main Results

Now in this section we give our main results.

Theorem 3. Let , , , , and suppose that

for , where is a constant. If

where

for , then

for .

Proof. Define a function by

Then (6) is

It is easy to see that is nonnegative, rd-continuous, and nondecreasing function for . Treating fixed and using Lemma 1 we get

for , where is defined by (9). From (11), (12), and the fact that , we have

Define a function by right hand side of (14). Then , . One has

Define a function by

then , ,

By keeping fixed in (18), taking and delta integrating with second variable from to . Using the fact that and is nondecreasing in , we have

Let

then (20) gives

Now treating fixed in (21) and applying Lemma 1, we have

From (18), (22), and (7), it is easy to see that

Using (23) in (22) and the fact that and we get the inequality in (10).

This completes the proof.

3. Applications

Now we give some application of theorem to study properties of solutions of initial value problem:

where , for , , , , is delta differentiable with respect to .

We observe that (24) is equivalent to

where

The following theorem deals with estimate on solution (24).

Theorem 4. Suppose

where , , which are as in Theorem 3 and is rd-continuous function defined on such that . Let

where

for . If is any solution of (24), then

where .

Proof. The solution of (24) satisfies (25). Using (27) in (25) we have

Now an application of Theorem 3 (with ) to (32) yields (30).

This completes the proof.

Now we establish the uniqueness of solutions of (24).

Theorem 5. Suppose that

where , , and are as in Theorem 4. Let and be as in (28) and (30). Then (24) has at most one solution on .

Proof. Let and be two solutions of (24) on ; then we have

From (34) and (33) we obtain

Applying Theorem 3 (with , ) yields

Therefore ; there is at most one solution of (24) in .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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