Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order

A second-order equation can have singular sets of first and second type, $S_1$ and $S_2$ (see the introduction), where the integral curve $x(y)$ does not exist in the ordinary sense but where it can be extended by using the first integral [1–5]. Denote by $Y$ the Cartesian axis $y=0$. If the function $x(y)$ has a derivative at a point of local extremum of this function, then this point belongs to $S_1\cup Y$. The extrema at which $y'(x)$ does not exist can be placed on $S_2$. In [5–8], the stability and instability of extrema on $S_1\cup S_2$ under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of x(y) on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of $S_1$ or $S_2$ are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set.