In this paper we compare the asymptotic behaviour of the advanced functional equation $$L_{n}u\left(t\right)-F(t,u\left[g\left(t\right)\right])=0$$ with the asymptotic behaviour of the set of ordinary functional equations $${\alpha }_{i}u\left(t\right)-F(t,u\left(t\right))=0.$$ On the basis of this comparison principle the sufficient conditions for property (B) of equation (*) are deduced.

The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of systems of two first order linear two by two dimensional matrix differential equations. An integral and an interval oscillatory criteria are obtained. Two non oscillatory criteria are obtained as well. On an example, one of the obtained oscillatory criteria is compared with some well known results.

The Riccati equation method is used to study the oscillatory and non-oscillatory behavior of solutions of linear four-dimensional Hamiltonian systems. One oscillatory and three non-oscillatory criteria are proved. Examples of the obtained results are compared with some well known ones.

In this paper we discuss the existence of oscillatory and nonoscillatory solutions of first order impulsive differential inclusions. We shall rely on a fixed point theorem of Bohnenblust-Karlin combined with lower and upper solutions method.

In this paper we discuss the existence of oscillatory and nonoscillatory solutions for first order impulsive dynamic inclusions on time scales. We shall rely of the nonlinear alternative of Leray-Schauder type combined with lower and upper solutions method.

In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation $${(y\left(t\right)-\sum _{i=1}^k{p}_i\left(t\right)y\left(r_i\left(t\right)\right))}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ oscillates or tends to zero as $t\to \infty$, where, $n\ge 1$ is any positive integer, $p_i$, $r_i\in C^{\left(n\right)}([0,\infty ),ℝ)$  and $p_i$ are bounded for each $i=1,2,\cdots ,k$. Further, $f\in C\left(\right[0,\infty ),ℝ)$, $g$, $h$, $v$, $u\in C\left(\right[0,\infty ),[0,\infty \left)\right)$, $G$ and $H\in C(ℝ,ℝ)$. The functional delays $r_i\left(t\right)\le t$, $g\left(t\right)\le t$ and $h\left(t\right)\le t$ and all of them approach $\infty$ as $t\to \infty$. The results hold when $u\equiv 0$ and $f\left(t\right)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.

Sufficient conditions are obtained for oscillation of all solutions of a class of forced nth order linear and nonlinear neutral delay differential equations. Also, asymptotic behaviour of nonoscillatory solutions of a class of forced first order neutral equations is studied.

Oscillation and nonoscillation criteria for the self-adjoint linear differential equation $${\left(t^{\alpha }{y}^{\text{'}\text{'}}\right)}^{\text{'}\text{'}}-\frac{{\gamma }_{2,\alpha }}{t^{4-\alpha }}y=q\left(t\right)y,\alpha \notin \{1,3\},$$ where $${\gamma }_{2,\alpha }=\frac{{(\alpha -1)}^2{(\alpha -3)}^2}{16}$$ and $q$ is a real and continuous function, are established. It is proved, using these criteria, that the equation $${\left(t^{\alpha }{y}^{\text{'}\text{'}}\right)}^{\text{'}\text{'}}-\left(\frac{{\gamma }_{2,\alpha }}{t^{4-\alpha }}+\frac{\gamma }{t^{4-\alpha }{ln}^{2}t}\right)y=0$$ is nonoscillatory if and only if $\gamma \le \frac{{\alpha }^2-4\alpha +5}{8}$.