Lipschitz perturbations of differentiable implicit functions

TL;DR

This paper demonstrates that small Lipschitz perturbations of differentiable implicit functions result in Lipschitz solutions that closely approximate the original functions, with applications to analyzing nonautonomous discontinuous ODE systems.

Contribution

It establishes the stability of implicit functions under Lipschitz perturbations and applies this to study differentiability intervals in discontinuous ODEs, advancing the theory for such systems.

Findings

Lipschitz solutions approximate original implicit functions as perturbation vanishes

Application to estimating differentiability intervals in nonautonomous discontinuous ODEs

Supports generalization of Bogolyubov's theorem for discontinuous systems

Abstract

Let $y=f(x)$ be a continuously differentiable implicit function solving the equation $F(x,y)=0$ with continuously differentiable $F.$ In this paper we show that if $F_\eps$ is a Lipschitz function such that the Lipschitz constant of $F_\eps-F$ goes to 0 as $\eps\to 0$ then the equation $F_\eps(x,y)=0$ has a Lipschitz solution $y=f_\eps(x)$ such that the Lipschitz constant of $f_\eps-f$ goes to 0 as $\eps\to 0$ either. As an application we evaluate the length of time intervals where the right hand parts of some nonautonomous discontinuous systems of ODEs are continuously differentiable with respect to state variables. The latter is done as a preparatory step toward generalizing the second Bogolyubov's theorem for discontinuous systems.