''Homogeneous'' second order differential equation: zeros separation principles

Abstract

In this paper we study the following problems:
\medskip

\noindent\underline{Problem 1}. Let $I\subset \mathbb{R}$ be an open interval and $F:\mathbb{R}^3\times I\to \mathbb{R}$ be a continuous function with, $F(0,0,0,x)=0$, for all $x\in I$. We consider the following differential equations
\begin{equation}\label{equ1}
F(y^{\prime\prime},y^{\prime},y,x)=0.
\end{equation}
Let $y\in C^2(I)$ be a nontrivial solution of \eqref{equ1}. In which conditions we have that:

(1) the zeros of $y$ and $y^\prime$ separate each other ?

(2) the zeros of $y$ and $y^{\prime\prime}$ separate each other ?

(3) the zeros of $y^\prime$ and $y^{\prime\prime}$ separate each other ?
\medskip

\noindent\underline{Problem 2}. Let $y_1, y_2\in C^2(I)$ be two linearly independent solutions of \eqref{equ1}. In which conditions we have that:

(1) the zeros of $y_1$ and $y_2$ separate each other ?

(2) the zeros of $y_1^\prime$ and $y_2^{\prime}$ separate each other ?

(3) the zeros of $y_1^{\prime\prime}$ and $y_2^{\prime\prime}$ separate each other ?
\medskip

\noindent\underline{Problem 3}. Let $F,G:\mathbb{R}^3\times I\to\mathbb{R}$ be two continuous functions with, $F(0,0,0,x)=0$, $G(0,0,0,x)=0$, for all $x\in I$. We consider the following system of differential equations,
\begin{equation}\label{equ2}
\begin{split}
F(y^\prime,y,z,x)=0, \\
G(z^\prime,y,z,x)=0.
\end{split}
\end{equation}
Let $(y,z)\in C^1(I,\mathbb{R}^2)$ be a nontrivial solution of \eqref{equ2}. In which conditions we have that:

(1) the zeros of $y$ and $z$ separate each other ?

(2) the zeros of $y^\prime$ and $z^{\prime}$ separate each other ?
\medskip

\noindent\underline{Problem 4}. Let $(y_1,z_1)$ and $(y_2,z_2)$ be two linearly independent solutions of \eqref{equ2}. In which conditions we have that:

(1) the zeros of $y_1$ and $y_2$ separate each other ?

(2) the zeros of $z_1$ and $z_2$ separate each other ?

(3) the zeros of $y_1^\prime$ and $y_2^{\prime}$ separate each other ?

(4) the zeros of $z_1^\prime$ and $z_2^{\prime}$ separate each other ?
\medskip

\noindent Some other problems are formulated.