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

Address. Dept. of Math., Faculty of Mechanical Engineering, Slovak Technical University, Namestie Slobody 17,
                812 31 Bratislava, Slovak Republic


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