%0 Journal Article
%T Interval Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales
%A Yang-Cong Qiu
%A Qi-Ru Wang
%J Discrete Dynamics in Nature and Society
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/952932
%X Using functions in some function classes and a generalized Riccati technique, we establish interval oscillation criteria for second-order nonlinear dynamic equations on time scales of the form . The obtained interval oscillation criteria can be applied to equations with a forcing term. An example is included to show the significance of the results. 1. Introduction In this paper, we study the second-order nonlinear dynamic equation on a time scale . Throughout this paper we will assume that(C1) ; (C2) , where is an arbitrary positive constant;(C3) . Preliminaries about time scale calculus can be found in [1¨C3] and hence we omit them here. Without loss of generality, we assume throughout that . Definition 1.1. A solution of (1.1) is said to have a generalized zero at if , and it is said to be nonoscillatory on if there exists such that for all . Otherwise, it is oscillatory. Equation (1.1) is said to be oscillatory if all solutions of (1.1) are oscillatory. It is well-known that either all solutions of (1.1) are oscillatory or none are, so (1.1) may be classified as oscillatory or nonoscillatory. The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [4] in 1988 in order to unify continuous and discrete analysis, see also [5]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales, for example, see [1¨C27] and the references therein. In Do£¿ly and Hilger¡¯s study [10], the authors considered the second-order dynamic equation and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded time scales. In Del Medico and Kong¡¯s study [8, 9], the authors employed the following Riccati transformation: and gave sufficient conditions for Kamenev-type oscillation criteria of (1.2) on a measure chain. And in Yang¡¯s study [27], the author considered the interval oscillation criteria of solutions of the differential equation In Wang¡¯s study [24], the author considered second-order nonlinear differential equation used the following generalized Riccati transformations: where , and gave new oscillation criteria of (1.5). In Huang and Wang¡¯s study [16], the authors considered second-order nonlinear dynamic equation on time scales By using a similar generalized Riccati transformation which is more general than (1.3) where , , the authors extended the results in Del Medico and Kong [8, 9] and Yang [27], and established some new Kamenev-type oscillation criteria and interval oscillation
%U http://www.hindawi.com/journals/ddns/2012/952932/