Property $A$ of the $ (n+1)^{th}$ Order Differential Equation $\left[ \frac 1 {r_1(t)} \ \left( x^{(n)}(t) + p(t) x(t) \right)\ \right]' = f(t, x (t), \cdots , x^{(n)}(t))$

Monika Kovacova

E-mail: kovacova_v@sjf.stuba.sk

Abstract. The aim of this contribution is to study properties of solutions of the $n +1 ^{th}$-order differential equation of the form \begin{equation} \left[ \frac 1 {r_1(t)} \ \left( x^{(n)}(t) + p(t) x(t) \right)\ \right]' = f(t, x (t), \cdots , x^{(n)}(t))\,.\label{moni1} \end{equation} where $ n\ge 2 $ is a natural number. A new approach using ``submersivity'' of a solution of an equation is presented, by means of it a sufficient condition for the property A is proved. This approach can be also used to prove necessary condition for the property A.

AMSclassification. 34C10, 34C15

Keywords. Property $A$, oscillatory solutions