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International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 345697, 9 pages
http://dx.doi.org/10.1155/2013/345697
Research Article

Properties of Some Partial Dynamic Equations on Time Scales

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

Received 23 February 2013; Revised 6 June 2013; Accepted 16 June 2013

Academic Editor: Anjan Biswas

Copyright © 2013 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 the properties of the solution of a certain partial dynamic equation on time scales. The tools employed are based on the application of the Banach fixed-point theorem and a certain integral inequality with explicit estimates on time scales.

1. Introduction

Recently, there has been a lot of interest in shown studying various properties of dynamic equations on time scales by various authors [111]. In this paper, we study some partial dynamic equations on time scales. Let denote the dimensional Euclidean space with appropriate norm . In this, let denote the set of real numbers, the set of integers, and the arbitrary time scales. Let and be two time scales, and let . Let denote the set of rd-continuous function. We assume basic understanding of time scales and notations. More information about time scales calculus can be found in [1214]. The partial delta derivative of for with respect to , , and is denoted by , , and .

Many physical systems can be modeled using dynamical systems on time scales. As response to the needs of diverse applications, many authors have studied qualitative properties of various equations on time scales [49, 11]. Motivated by the results in this paper, I consider the partial dynamic equation of the form with the initial boundary conditions for , where , , , and .

2. Preliminaries and Basic Inequality

We now give some basic definitions and notations about time scales. Define the jump operators by If and , then the point is left dense and left scattered. If and , then the point is right-dense and right-scattered. If has a right scattered minimum , define ; otherwise, . If has a left-scattered maximum , define ; otherwise, . The graininess function is defined by . We say that is regressive provided for all . For and , the delta derivative of at denoted by is the number (provided it exists) with the property that given any , there is a neighborhood of such that for all . For , the usual derivative; for , the delta derivative is the forward difference operator, . A function is right-dense continuous or rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) and at left-dense points in and its left-sided limit exists (finite) at left-dense points in . If , then is rd-continuous if and only if is continuous. It is known [12, Theorem 1.74] that is right-dense continuous, there is a function such that and where . Note that when , , , , and , while then , , , , and . We denote by the set of all regressive and rd-continuous functions and . For , we define (see [12, Theorem 2.35]) the exponential function on time scale as the unique solution to the scalar initial value problem If , then for all (see [12, Theorem 2.41]). As usual, the set of rd-continuous functions is denoted by .

Denote by . For ; the notation ; then there exists a constant such that right hand neighbourhood. Let be the space with function which are rd-continuous for and satisfy the condition for , where is a constant. In space , define the norm The norm defined in (9) is clearly a Banach space. Then, (8) implies that there is a constant such that Using (10), we observe that The solution of (1) and (2) means a function satisfying (1) and (2). It is easy to observe that solution of initial boundary value problem (1) and (2) satisfies following dynamic integral equations on time scales: for .

I need the following lemma proved in [15].

Lemma 1. Let with and nondecreasing in each of the variables and , , . If for , then for .

Now I prove the following integral inequality which is used in our results.

Lemma 2. Let , , , , , , and let be a constant. If for , then for , where

Proof. Define a function by the right hand side of (15), then , , and is nondecreasing in and : where is given by (18). Now, by application of Lemma 1, we get Using (20) in , we get the required inequality (16).

3. Existence and Uniqueness

Now our main results are as follows.

Theorem 3. Suppose that(i) the function in (1) satisfies the conditions
where , .(ii) for as in (10),(a) there exists a nonnegative constant such that , where (b) there exists a nonnegative constant such that
then, the initial boundary value problem (1) and (2) has a unique solution on .

Proof. Let , and define the operator by By delta differentiating (25) with respect to and , we get Now, we show that maps into itself and is rd-continuous on .
From (25) and (26) and hypothesis, we have From (27), it follows that , thus proving that maps into itself.
Now, we verify that is a contraction map. Let . From (25) and (26) and using the hypotheses, we have From (28), we have
Since , it follows from Banach fixed-point theorem that has unique fixed point in . The fixed point of is the solution of (1) and (2). This completes the proof.

Theorem 4. Suppose that function in (1) satisfies the condition where is a nonnegative constant such that and , , , , . Then, initial boundary value problem (1) and (2) has at most one solution on .

Proof. Let and be any two solutions of (1) and (2) and ; then by hypotheses we have From (31), we have
Now, a suitable application of Lemma 2 to (32) yields which implies that for . Therefore, there is at most one solution of (1) and (2) on .

4. Properties of Solutions

The following theorem contains estimates of solutions of (1) and (2).

Theorem 5. Assume that where are nonnegative constants such that and , , , , . If for is any solution of (1) and (2); then where for ,

Proof. Since is a solution of (1) and (2) and by hypothesis, we have From (40), we have Now, a suitable application of Lemma 2 to (41) yields (37).

Remark 6. If the estimate obtained in (37) is bounded, then solution of (1) and (2) and also is bounded on .

The following result deals with the continuous dependence of solution of (1) and (2).

Theorem 7. Assume that the functions in (1) and (2) satisfy the conditions (30). Let and be the solutions of (1) with the given initial boundary conditions respectively where , where is a constant. Then for , where where is defined by the right hand side of (18) replacing and by .

Proof. Let for . We have From (47), we have
Now, a suitable application of Lemma 2 to (48) yields the bound (45) which shows the dependency of solution of (1) on the initial boundary conditions.
Now, we consider initial boundary value problem (1) and (2) and the corresponding initial boundary value problem with the initial boundary conditions where , , and .
Now, we present the result which deals with continuous dependence of solution of initial boundary value problem (1) and (2) on the functions involved therein.

Theorem 8. Assume that the functions in (1) satisfy the conditions (30) and where and are as in initial boundary value problem (1) and (2) and initial boundary value problem (49), (50), is a constant, and is a solution of initial boundary value problem (49) and (50). Then, solution (1) and (2) depends continuously on the functions involved therein.

Proof. Let for . We have From (53), we have Now, a suitable application of Lemma 2 yields for , where is given by (43). From (35), it follows that solution of initial boundary value problem (1) and (2) depends continuously on the functions involved therein.

Remark 9. In this paper, we have studied the existence and uniqueness of solution of (1) using Banach fixed-point theorem. Here, we have offered simple and concise proofs of properties of solutions of (1). I believeed results given will serve as a model for further investigations.

Acknowledgment

The author is grateful to the anonymous referee whose comments and suggestions helped the author to improve the text.

References

  1. E. A. 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.
  2. M. Bohner and G. S. Guseinov, “Partial differentiation on time scales,” Dynamic Systems and Applications, vol. 13, no. 3-4, pp. 351–379, 2004. View at Scopus
  3. M. Bohner and G. S. Guseinov, “Multiple integration on time scales,” Dynamic Systems and Applications, vol. 14, no. 3-4, pp. 579–606, 2005. View at Scopus
  4. J. Hoffacker, “Basic partial dynamic equations on time scales,” Journal of Difference Equations and Applications, vol. 8, no. 4, pp. 307–319, 2002. View at Publisher · View at Google Scholar
  5. B. Jackson, “Partial dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 186, no. 2, pp. 391–415, 2006. View at Publisher · View at Google Scholar
  6. T. Kulik and C. C. Tisdell, “Volterra integral equations on time scales: basic qualitative and quantitative results with applications to initial value problems on unbounded domains,” International Journal of Difference Equations, vol. 3, no. 1, pp. 103–133, 2008.
  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, 2005. View at Scopus
  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 Scopus
  9. D. B. Pachpatte, “Integral inequalitys for partial dynamic equations on time scales,” Electronic Journal of Differential Equations, vol. 2012, no. 50, pp. 1–17, 2012. View at Scopus
  10. M. H. Sarikaya, N. A. Yildirim, and K. I. Larslan, “Partial Δ-differentiation for multivariable functions on n-dimensional time scales,” Journal of Mathematical Inequalities, vol. 3, no. 2, pp. 277–291, 2009.
  11. C. C. Tisdell and A. Zaidi, “Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling,” Nonlinear Analysis: Theory, Methods and Applications, vol. 68, no. 11, pp. 3504–3524, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäauser, Boston, Mass, USA, 2001.
  13. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäauser, Boston, Mass, USA, 2003.
  14. S. Hilger, “Analysis on measure chain—a unified approch to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Publisher · View at Google Scholar
  15. 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 Scopus