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Stability of periodic solutions for Lipschitz systems obtained via the averaging method

Authors: Adriana Buica and Aris Daniilidis
Journal: Proc. Amer. Math. Soc. 135 (2007), 3317-3327
MSC (2000): Primary 34C29, 34C25; Secondary 49J52
Published electronically: May 17, 2007
MathSciNet review: 2322764
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Abstract: Existence and asymptotic stability of the periodic solutions of the Lipschitz system $ x^{\prime}(t)=\varepsilon F(t,x,\varepsilon )$ is hereby studied via the averaging method. The traditional $ C^{1}$ dependence of $ F(s,\cdot,\varepsilon)$ on $ z$ is relaxed to the mere strict differentiability of $ F(s,\cdot,0)$ at $ z=z_{0}$ for $ \varepsilon=0$, giving room to potential applications for structured nonsmooth systems.

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Additional Information

Adriana Buica
Affiliation: Department of Applied Mathematics, Babeş-Bolyai University, Cluj-Napoca 400084, Romania

Aris Daniilidis
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain

Keywords: Periodic solution, averaging method, nonsmooth Lipschitz system, Poincar\'{e}--Andronov mapping, fixed point
Received by editor(s): August 1, 2006
Published electronically: May 17, 2007
Additional Notes: The first author was supported by the “Agence universitaire de la Francophonie” (France)
The second author was supported by the MEC Grant No. MTM2005-08572-C03-03 (Spain)
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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