About this Journal Submit a Manuscript Table of Contents
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

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 [29].

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.

References

  1. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. András and A. Mészáros, “Wendroff type inequalities on time scales via Picard operators,” Mathematical Inequalities & Applications, vol. 16, no. 1, pp. 159–174, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. R. Anderson, “Dynamic double integral inequalities in two independent variables on time scales,” Journal of Mathematical Inequalities, vol. 2, no. 2, pp. 163–184, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. D. R. Anderson, “Nonlinear dynamic integral inequalities in two independent variables on time scale pairs,” Advances in Dynamical Systems and Applications, vol. 3, no. 1, pp. 1–13, 2008. View at MathSciNet
  5. E. Akin-Bohner, M. Bohner, and F. Akin, “Pachpatte inequalities on time scales,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 1, article 6, 2005. View at Zentralblatt MATH · View at MathSciNet
  6. R. A. C. Ferreira and D. F. M. Torres, “Some linear and nonlinear integral inequalities on time scales in two independent variables,” Nonlinear Dynamics and Systems Theory, vol. 9, no. 2, pp. 161–169, 2009. View at Zentralblatt MATH · View at MathSciNet
  7. D. B. Pachpatte, “Explicit estimates on integral inequalities with time scale,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 4, article 143, 2006. View at Zentralblatt MATH · View at MathSciNet
  8. D. B. Pachpatte, “Properties of solutions to nonlinear dynamic integral equations on time scales,” Electronic Journal of Differential Equations, vol. 2008, no. 136, pp. 1–8, 2008. View at Zentralblatt MATH · View at MathSciNet
  9. D. B. Pachpatte, “Integral inequalitys for partial dynamic equations on time scales,” Electronic Journal of Differential Equations, vol. 2012, no. 50, pp. 1–7, 2012. View at Zentralblatt MATH · View at MathSciNet
  10. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet