### A classification scheme for positive solutions to second order nonlinear iterative differential equations.

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In this paper property (A) of the linear delay differential equation $${L}_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0,$$ is to deduce from the oscillation of a set of the first order delay differential equations.

We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the...

We give an equivalence criterion on property A and property B for delay third order linear differential equations. We also give comparison results on properties A and B between linear and nonlinear equations, whereby we only suppose that nonlinearity has superlinear growth near infinity.

This paper deals with property A and B of a class of canonical linear homogeneous delay differential equations of $n$-th order.

Sufficient conditions are established for the oscillation of proper solutions of the system $$\begin{array}{cc}\hfill {u}_{1}^{\text{'}}\left(t\right)& =p\left(t\right){u}_{2}\left(\sigma \left(t\right)\right)\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {u}_{2}^{\text{'}}\left(t\right)& =-q\left(t\right){u}_{1}\left(\tau \left(t\right)\right)\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$ where $p,\phantom{\rule{0.166667em}{0ex}}q:{R}_{+}\to {R}_{+}$ are locally summable functions, while $\tau $ and $\sigma :{R}_{+}\to {R}_{+}$ are continuous and continuously differentiable functions, respectively, and $\underset{t\to +\infty}{lim}\tau \left(t\right)=+\infty $, $\underset{t\to +\infty}{lim}\sigma \left(t\right)=+\infty $.

We study asymptotic properties of solutions of the system of differential equations of neutral type.

This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasilinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as $t\to \infty $.

The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation ${\left[p\left(t\right){\left[{\left(r\left(t\right){x}^{\Delta}\left(t\right)\right)}^{\Delta}\right]}^{\gamma}\right]}^{\Delta}+q\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right)=0$, t ≥ t₀, on a time scale , where γ > 0 is a quotient of odd positive integers, and p, q, r and τ are positive right-dense continuous functions defined on . We classify the nonoscillatory solutions into certain classes ${C}_{i}$, i = 0,1,2,3, according to the sign of the Δ-quasi-derivatives and obtain sufficient conditions in order that ${C}_{i}=\varnothing $. Also, we establish...

This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form $$\dot{x}\left(t\right)=-c\left(t\right)x(t-\tau \left(t\right))\phantom{\rule{2.0em}{0ex}}{(}^{*})$$ with positive function $c\left(t\right).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation $$\dot{y}\left(t\right)=\beta \left(t\right)[y\left(t\right)-y(t-\tau \left(t\right))]$$ where the function $\beta \left(t\right)$ is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of...