Uniform stability of local extrema of an integral curve of an ODE of second order (Q1889615)

Consider the system \[ \begin{gathered} \alpha(x, y)\dot y= P(x,y),\;\dot x= y,\text{ where}\\ \alpha(x,y)\in C^{1,1},\;P(x,y)\in C^{1,1},\\ (x,y)\in M\subseteq \mathbb{R}^2,\;t\in T\subseteq \mathbb{R}.\end{gathered} \] The set \(S_1= \{(x, y)\in \mathbb{R}^2: \alpha(x, y)= 0,\,P(x,y)\neq 0\}\) is called the singular set of the first type, and the set of the form \(S_2= \{(x, y)\in\mathbb{R}^2: P(x,y)= 0,\, y\alpha(x, y)\neq 0\}\) is called the singular set of the second type. The points \((x,y)\in S_1\cup S_2\) are said to be singular. It is assumed that \(\mu(S_1\cup S_2)= 0\), where \(\mu\) is the Lebesgue measure in \(\mathbb{R}^2\). Different from previous papers of the authors, where the stability and instability of extrema on \(S_1\cup S_2\) under small perturbations of the equation were considered, in the present paper, sufficient conditions are derived for the preservation of the type of a local extremum on the finite part of \(S_1\) or \(S_2\) for the case in which the perturbation on this part does not exceed some explicitly indicated quantity.