In this work, we investigate the oscillation criteria for second order neutral delay differential equations of the form (r(t)[y(t)+ p(t)y(δ (t))]′)′ +q(t)G(y(τ(t))) = 0 and (r(t)[[y(t)+ p(t)y(δ (t))]′]α )′ +q(t)(yβ (τ(t))) = 0, where α and β are the ratio of odd positive integers. Mathematics subject classification (2010): 34C10, 34C15.

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equationΔ akn · · ·Δ a2nΔ a1nΔ xn bnxn−d f n, xn−r1n , xn−r2n , . . . , xn−rsn 0, n ≥ n0, where n0 ≥ 0, d > 0, k > 0, and s > 0 are integers, {ain}n≥n0 i 1, 2, . . . , k and {bn}n≥n0 are real sequences, ⋃s j 1{rjn}n≥n0 ⊆ Z, and f : {n : n ≥ n0} × R → R is a mapping, is studied. Some sufficient...

In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation ( r(t) (( y(t)+ p(t)y(t − τ))∆)γ )∆ + f (t, y(t − δ))= 0 on a time scale T; here γ > 0 is a quotient of odd positive integers with r(t) and p(t) real-valued positive functions defined on T. To the best of our knowledge nothing is known regarding the qualitative behavior of the...

In this paper sufficient conditions are obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form ∆(y(n) + p(n)y(n−m)) + q(n)G(y(n − k)) = 0 under various ranges of p(n). The nonlinear function G,G ∈ C(R,R) is either sublinear or superlinear. Mathematics Subject classification (2000): 39 A 10, 39 A 12

Some sufficient conditions are established for the oscillation of second-order neutral differential equation x t p t x τ t ′′ q t f x σ t 0, t ≥ t0, where 0 ≤ p t ≤ p0 < ∞. The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 1985 , Grace and Lalli 1987 , Ruan 1993 , H. J. Li 1996 ...

In this paper, an oscillation theorem is established for the oscillation of second-order quasilinear neutral differential equation 0 ( )[( ( ) ( ) ( )) ] ( ) ( ( )) 0 r t x t p t x t q t f x t t t                  