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International Journal of Differential Equations
Volume 2014 (2014), Article ID 949860, 4 pages
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.
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.
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  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].
Lemma 1 (see , Theorem 2.6). Let , , and for all ; then for all .
Lemma 2 (see , 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
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.
Now we give some application of theorem to study properties of solutions of initial value problem: where , for , , , , is delta differentiable with respect to .
Now we establish the uniqueness of solutions of (24).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001.