Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales

Abstract

We consider the equation (r(t)(y Δ (t)) γ ) Δ + f (t,x(δ(t))) = 0, t ∈ T, where y(t) = x(t) + p(t) x(τ(t)) and γ is a quotient of positive odd integers. We present some sufficient conditions for neutral delay and mixed-type dynamic equations to be oscillatory, depending on deviating arguments τ(t) and δ(t), t ∈ T.Copyright © 2006 Y. Ş ahiner. 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. Some preliminaries on time scalesA time scale T is an arbitrary nonempty closed subset of the real numbers. The theory of time scales was introduced by Hilger [6] in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis. Several authors have expounded on various aspects of this new theory, see [7] and the monographs by Bohner and Peterson [3,4], and the references cited therein.First, we give a short review of the time scales calculus extracted from [3]. For any t ∈ T, we define the forward and backward jump operators by σ(t) := inf{s ∈ T : s > t}, ρ(t) := sup{s ∈ T : s < t}, ( 1.1) respectively. The graininess function μ :A point t ∈ T is said to be right dense if t < supT and σ(t) = t, left dense if t > inf T and ρ(t) = t. Also, t is said to be right scattered if σ(t) > t, left scattered if t > ρ(t). A function f : T → R is called rd-continuous if it is continuous at right dense points in T and its left-sided limit exists (finite) at left dense points in T.For a function f : T → R, if there exists a number α ∈ R such that for all ε > 0 there exists a neighborhood U of t with | f (σ(t)) − f (s) − α(σ(t) − s)| ≤ ε|σ(t) − s|, for all s ∈ U, then f is Δ-differentiable at t, and we call α the derivative of f at t and denote

New Oscillation Criteria of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

“…In 2007, Erbe, Peterson and Saker [3] considered the general nonlinear delay dynamic equations (4) and obtained some new oscillation criteria, which improved the results given by Sahiner [2]. Saker [4] in 2005 and Grace, Bohner and Agarwal [5] in 2009 considered the half-linear dynamic equations (5), and established some sufficient conditions for oscillation of (5).…”

“…After the careful consideration of the linear delay dynamic equations by Agarwal, Bohner and Saker in 2005 [1] (7) and the nonlinear delay dynamic equations by Sahiner [2] (6), some sufficient conditions for oscillation of (7) and (6) have been established. In 2007, Erbe, Peterson and Saker [3] considered the general nonlinear delay dynamic equations (4) and obtained some new oscillation criteria, which improved the results given by Sahiner [2].…”

Oscillation criteria for nonlinear neutral functional dynamic equations on time scales

“…For oscillation of neutral dynamic equations we refer the reader to the papers [3,27,34,37,38,43] and for oscillation of different types of equations we refer the reader to the book by the second author [30]. We note that (1.1) in its general form involve some different types of differential and difference equations depending on the choice of the time scale T. For example, when T = R, we have σ(t) = t, µ(t) = 0, f ∆ (t) = f (t) and (1.1) becomes the second-order neutral differential equation…”

Integral Oscillation Criteria for Second-Order Linear Neutral Delay Dynamic Equations on Time Scales

“…¢ www.ccsenet.org/jmr ISSN: 1916-9795 Lemma 2 (Sahiner, 2005 Suppose that the following conditions hold:…”