Abstract
Dry friction problems lead to discontinuous differential equations, e.g. to
where sgn γ=γγ for γ ≠ 0 and sgn (0)=[ − 1. 1]. We study existence of ω-periodic solutions of (1) in case ϕ is ω-periodic. Results forx> 0 are given in the book “Multivalued Differential Equations” (K. Deimling: De Gruyter 1992). and preliminary ones forx=0 are contained in K. Deimling “Multivalued differential equations and dry friction problems” (Proc. Conf. Differential & Delay Equations. World Sci. Publ. 1992). Based on the latter and considerable additional analysis, we give a complete description of the resonant casex=0.β=1. ϕ(t)=sint. In particular, it turned out that for μ ε (π 4. 1) there is a unique globally asymptotically stable 2π-periodic solutionx gm, which necessarily has deadzones (i.e.xμ (t) ≡c in certain intervals). In addition, the nonresonant case is solved by means of degree theory for multivalued maps, since in this situation a priori bounds can be found easily.
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On leave from Babeş-Bolyai University Cluj-Napoca (Roumania), supported by Alexander von Humboldt-Stiftung, which is gratefully acknowledged.
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Deimling, K., Szilágyi, P. Periodic solutions of dry friction problems. Z. angew. Math. Phys. 45, 53–60 (1994). https://doi.org/10.1007/BF00942846
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DOI: https://doi.org/10.1007/BF00942846