Existence, localization and stability of limit-periodic solutions to differential equations involving cubic nonlinearities - Andres, Jan; Pennequin, Denis (Topological Methods in Nonlinear Analysis, 2019)
We will prove, besides other things like localization and (in)stability, that the differential equations $x'+x^3-λ x=ε r(t)$, $λ> 0$, and $x''+x^3-x=ε r(t)$, where $r\colon\mathbb{R}→\mathbb{R}$ are uniformly limit-periodic functions, possess for sufficiently small values of $ε > 0$ uniformly limit-periodic solutions, provided $r$ in the first-order equation is strictly positive. As far as we know, these are the first nontrivial effective criteria, obtained for limit-periodic solutions of nonlinear differential equations, in the lack of global lipschitzianity restrictions. A simple illustrative example is also indicated for difference equations.