One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.
Necessary and sufficient conditions are obtained for every solution of
\[ \Delta (y_{n}+p_{n}y_{n-m})\pm q_{n}G(y_{n-k})=f_{n} \] to oscillate or tend to zero as $n\rightarrow \infty $, where $p_{n}$, $q_{n}$ and $f_{n}$ are sequences of real numbers such that $q_{n}\ge 0$. Different ranges for $p_{n}$ are considered.
We study oscillatory behavior of a class of fourth-order quasilinear differential equations without imposing restrictive conditions on the deviated argument. This allows applications to functional differential equations with delayed and advanced arguments, and not only these. New theorems are based on a thorough analysis of possible behavior of nonoscillatory solutions; they complement and improve a number of results reported in the literature. Three illustrative examples are presented.
In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form \[ y^{\prime }_i(t)-p_i(t)y_{i+1}(t)=0, \quad i=1,2,\dots , n-2, y^{\prime }_{n-1}(t)-p_{n-1}(t)|y_n(h_n(t))|^\alpha \mathop {\mathrm sgn}[y_n(h_n(t))]=0, y^{\prime }_n(t) \mathop {\mathrm sgn}[y_1(h_1(t))]+p_n(t)|y_1(h_1(t))|^\beta \, \le 0, \] where $ n\ge 3 $ is odd, $ \alpha >0$, $ \beta >0$.
In this paper the three-dimensional nonlinear difference system ∆xn = anf(yn−l ), ∆yn = bng(zn−m), ∆zn = δcnh(xn−k), is investigated. Sufficient conditions under which the system is oscillatory or almost oscillatory are presented.
We establish some new oscillation criteria for the second order neutral delay differential equation [r(t)|[x(t) + p(t)x[τ (t)]]′ | α−1 [x(t) + p(t)x[τ (t)]]′ ] ′ + q(t)f(x[σ(t)]) = 0. The obtained results supplement those of Dzurina and Stavroulakis, Sun and Meng, Xu and Meng, Baculíková and Lacková. We also make a slight improvement of one assumption in the paper of Xu and Meng.
In the paper we offer criteria for oscillation of the third order Euler differential equation with delay y ′′′(t) + k 2 ⁄ t 3 y(ct) = 0. We provide detail analysis of the properties of this equation, we fill the gap in the oscillation theory and provide necessary and sufficient conditions for oscillation of equation considered.
We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$ x'''(t)+q(t)x'(t)+r(t)|x|^{\lambda }(t)\mathop {\rm sgn} x(t)=0 ,\quad t\geq 0. $$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \leq 1$ and if the corresponding second order differential equation $h''+q(t)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.
Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form ∆(an(∆2 (xn + bnxn−δ))α ) + qnx α n+1−τ = 0 and ∆(an(∆2 (xn − bnxn−δ))α ) + qnx α n+1−τ = 0 are established. Some examples are presented to illustrate the main results.
The aim of this paper is to present new oscillatory criteria for the second order neutral differential equation with mixed argument (x(t) − px(t − τ ))'' − q(t)x(σ(t)) = 0. The results include also sufficient conditions for bounded and unbounded oscillation of the equations considered.