Electron. J. Differential Equations, Vol. 2017 (2017), No. 247, pp. 1-17.

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

Irena Rachunkova, Jan Tomecek

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.

Submitted June 2, 2017. Published October 6, 2017.
Math Subject Classifications: 34A37, 34B37.
Key Words: van der Pol equation; state-dependent impulses; existence; distributional equation; periodic distributions; antiperiodic solution.

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Irena Rachunkova
Department of Mathematical Analysis
and Applications of Mathematics
Faculty of Science, Palacky University
17. Listopadu 12, 771 46 Olomouc, Czechia
email: irena.rachunkova@upol.cz
Jan Tomecek
Department of Mathematical Analysis
and Applications of Mathematics
Faculty of Science, Palacky University
17. Listopadu 12, 771 46 Olomouc, Czechia
email: jan.tomecek@upol.cz

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