A classical, second-order differential equation is considered with state-dependent impulses at both the position and its derivative. This means that the instants of impulsive effects depend on the solutions and they are not fixed beforehand, making the study of this problem more difficult and interesting from the real applications point of view. The existence of periodic solutions follows from a transformation of the problem into a planar system followed by a study of the Poincaré map and the use of some fixed point theorems in the plane. Some examples are presented to illustrate the main results.
"Periodic solutions of superlinear and sublinear state-dependent discontinuous differential equations." Topol. Methods Nonlinear Anal. 58 (1) 79 - 96, 2021. https://doi.org/10.12775/TMNA.2021.016