Properties of Some Partial Dynamic Equations on Time Scales

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 [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