Some sufficient conditions for oscillation of a first order nonautonomuous delay differential equation with several positive and negative coefficients are obtained.

Our aim in this paper is to present the relationship between property (B) of the third order equation with delay argument y'''(t) - q(t)y(τ(t)) = 0 and the oscillation of the second order delay equation of the form y''(t) + p(t)y(τ(t)) = 0.

Consider the first-order linear delay (advanced) differential equation$$x^{\text{'}}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0(x^{\text{'}}\left(t\right)-q\left(t\right)x\left(\sigma \left(t\right)\right)=0),t\ge t_0,$$ where $p$$\left(q\right)$ is a continuous function of nonnegative real numbers and the argument $\tau \left(t\right)$$...$

We study oscillatory properties of solutions of systems $$\begin{array}{ccc}\hfill {[y_1\left(t\right)-a\left(t\right)y_1\left(g\left(t\right)\right)]}^{\text{'}}=& p_1\left(t\right)y_2\left(t\right),y_2^{\text{'}}\left(t\right)=\hfill & \hfill -p_2\left(t\right)f\left(y_1\left(h\left(t\right)\right)\right),t\ge t_0.\end{array}$$

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...

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 paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…. $$\left\{\begin{array}{c}{t}_k{D}^{\alpha }\left(p\left(t\right)\left[t_k{D}^{\alpha }x\left(t\right)+r\left(t\right)x\left(t\right)\right]\right)+q\left(t\right)x\left(t\right)=0,t\ge t_0,t\ne t_k,\hfill \\ x\left(t_k^{+}\right)=a_{k}x\left(t_k^{-}\right),t_k{D}^{\alpha }x\left(t_k^{+}\right)=b_{k{t}_{k-1}}{D}^{\alpha }x\left(t_k^{-}\right),k=1,2,....\hfill \end{array}\right.$$ Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.