Oscillation Properties for Second-Order Half-Linear Dynamic Equations on Time Scales

Abstract

Since Hilger (1990) introduced the theory of time scales, many authors have expounded on various aspects of this new theory; see the books (Bohner & Peterson, 2001, 2003) and the papers (Agarwal et al., 2007; Bohner & Saker, 2004; Chen, 2010; Chen & Liu, 2008; Doslý & Hilger 2002; Erbe et al., 2008; Hassan, 2008; Hassan, 2009; Karpuz, 2009; Medico & Kong, 2004; Saker, 2005; Zhang, 2011). A time scale T is an arbitrary nonempty closed subset of the reals R (see Hilger, 1990; Bohner & Peterson, 2001, 2003), and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential equations and of difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice–once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a time scale. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but it is also able to extend these classical cases to cases “in between,” e.g., to the so-called q-difference equations. Dynamic equations on time scales have a lot of applications in population dynamics, quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. Bohner and Peterson (2001) summarizes and organizes much of time scale calculus. For advances of dynamic equations on time scales, we refer the reader to (Bohner & Peterson, 2003).