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

Abstract

A second-order equation can have singular sets of first and second type, S1 and S2 (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 S1∪Y. The extrema at which y'(x) does not exist can be placed on S2. In [5–-8], the stability and instability of extrema on S1∪ S2 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 S1 or S2 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.