''Homogeneous'' second order differential equation: zeros separation principles
DOI:
https://doi.org/10.24193/subbmath.2018.2.08Keywords:
Second order differential equation, first order system of differential equations, zero separation, Sturm theorem, Nicolescu theorem, Butlewski theorem, Markov theorem, zeros of special functions defined by differential equations, zero distance function,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.
