In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form $$\Delta \left[\begin{array}{c}x\left(n\right)+p\left(n\right)x(n-m)\\ y\left(n\right)+p\left(n\right)y(n-m)\end{array}\right]=\left[\begin{array}{cc}a\left(n\right)& b\left(n\right)\\ c\left(n\right)& d\left(n\right)\end{array}\right]\left[\begin{array}{c}x(n-\alpha )\\ y(n-\beta )\end{array}\right]$$ are established, where $m>0$, $\alpha \ge 0$, $\beta \ge 0$ are integers and $a\left(n\right)$, $b\left(n\right)$, $c\left(n\right)$, $d\left(n\right)$, $p\left(n\right)$ are sequences of real numbers.

We state and prove two oscillation results which deal with bounded solutions of a forced higher order differential equation. One proof involves the use of a nonlinear functional.

We obtain sufficient conditions for every solution of the differential equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ to oscillate or to tend to zero as $t$ approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when $G$ has sub-linear growth at infinity. Our results also apply to the neutral equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)G\left(y\left(g\left(t\right)\right)\right)=f\left(t\right)$$ when $q\left(t\right)$ has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.

In this paper, we study the oscillatory behavior of the solutions of the delay differential equation of the form $${\left(\frac{1}{r\left(t\right)}{y}^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)y\left(\tau \left(t\right)\right)=0.$$ The obtained results are applied to n-th order delay differential equation with quasi-derivatives of the form $$L_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0.$$

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.

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.

In the paper ordinary neutral differential equations with ?maxima? are considered. Sufficient conditions for oscillation of all solutions are obtained.