Antiperiodic solutions to van der Pol equations with state-dependent impulses
Date
2017-10-06
Authors
Rachunkova, Irena
Tomecek, Jan
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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), y′(t) = μ(x(t) - x3(t)/3)′ - x(t) + ƒ(t) for a. e. t ∈ ℝ,
subject to a finite number of state-dependent impulses
∆y(τi(x)) = Ji(x), i = 1, …, 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 τi and Ji need not be Lipschitz continuous nor bounded. As a direct consequence, we obtain an existence result for problem with fixed-time impulses.
Description
Keywords
van der Pol equation, State-dependent impulses, Existence, Distributional equation, Periodic distributions, Antiperiodic solution
Citation
Rachůnková, I., & Tomeček, J. (2017). Antiperiodic solutions to van der Pol equations with state-dependent impulses. <i>Electronic Journal of Differential Equations, 2017</i>(247), pp. 1-17.
Rights
Attribution 4.0 International