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International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 345697, 9 pages
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.
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.
Recently, there has been a lot of interest in shown studying various properties of dynamic equations on time scales by various authors [1–11]. 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 [12–14]. 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 [4–9, 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 .
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.
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
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.
The author is grateful to the anonymous referee whose comments and suggestions helped the author to improve the text.
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