Antiperiodic solutions to van der Pol equations with state-dependent impulses

Abstract

In this article we give sufficient conditions for the existence of an antiperiodic solution to the van der Pol equation $$ x'(t) = y(t), \quad y'(t) = \mu \Big(x(t) - \frac{x^3(t)}{3}\Big)' - x(t) + f(t) \text{for a. e. }t \in \mathbb{R}, $$ subject to a finite number of state-dependent impulses $$ \Delta y(\tau_i(x)) = \mathcal{J}_i(x), \quad i = 1,\ldots,m\,. $$ Our approach is based on the reformulation of the problem as a distributional differential equation and on the Schauder fixed point theorem. The functionals $\tau_i$ and $\mathcal{J}_i$ need not be Lipschitz continuous nor bounded. As a direct consequence, we obtain an existence result for problem with fixed-time impulses.

Keywords

• van der Pol equation
• state-dependent impulses
• existence
• distributional equation
• periodic distributions
• antiperiodic solution