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

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Sufficient conditions are obtained for the oscillation of solutions of half-linear and super-half-linear elliptic inequalities with p(x)-Laplacian. The results obtained are new even for the one-dimensional case.

Oscillation criteria are established for p(x)-Laplacian elliptic inequalities with mixed variable nonlinearities of the form u ∇ • (A(x)|∇u| p(x)−2 ∇u) + b(x),|∇u| p(x)−2 ∇u − h(x,u) + g(x,u) 0, x ∈ Ω, where β (x) > p(x) > γ(x) > 1 , Ω is an exterior domain in R N , and h(x,u) =ln |u||∇u| p(x)−2 (A(x)∇u) • ∇p(x), g(x,u) =c(x)|u| p(x)−2 u + c 1 (x)|u| β (x)−2 u + c 2 (x)|u| γ(x)−2 u + f (x). The function h(x,u) recently introduced in [N. Yoshida, Nonlinear Anal. 74 (2011) 2563-2575] allows employing the Riccati transformation technique commonly used in the oscillation theory of ordinary differential equations. It should be noted that the results obtained are new for one dimensional case as well. Examples are given to illustrate the results.