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Periodic solutions of dissipative systems revisited

Fixed Point Theory and Applications volume 2006, Article number: 65195 (2006) Cite this article

Abstract

We reprove in an extremely simple way the classical theorem that time periodic dissipative systems imply the existence of harmonic periodic solutions, in the case of uniqueness. We will also show that, in the lack of uniqueness, the existence of harmonics is implied by uniform dissipativity. The localization of starting points and multiplicity of periodic solutions will be established, under suitable additional assumptions, as well. The arguments are based on the application of various asymptotic fixed point theorems of the Lefschetz and Nielsen type.

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References

  1. Andres J, Gaudenzi M, Zanolin F: A transformation theorem for periodic solutions of nondissipative systems. Università e Politecnico di Torino. Seminario Matematico. Rendiconti 1990,48(2):171–186 (1992).

    MathSciNet  MATH  Google Scholar 

  2. Andres J, Górniewicz L: Topological Fixed Point Principles for Boundary Value Problems, Topological Fixed Point Theory and Its Applications. Volume 1. Kluwer Academic, Dordrecht; 2003:xvi+761.

    Book  MATH  Google Scholar 

  3. Andres J, Górniewicz L, Jezierski J: Relative versions of the multivalued Lefschetz and Nielsen theorems and their application to admissible semi-flows. Topological Methods in Nonlinear Analysis 2000,16(1):73–92.

    MathSciNet  MATH  Google Scholar 

  4. Andres J, Górniewicz L, Lewicka M: Partially dissipative periodic processes. In Topology in Nonlinear Analysis (Warsaw, 1994), Banach Center Publ.. Volume 35. Edited by: Gęba K, Górniewicz L. Polish Acad. Sci., Warsaw; 1996:109–118.

    Google Scholar 

  5. Andres J, Wong P: Relative Nielsen theory for noncompact spaces and maps. to appear in Topology and Its Applications

  6. Billotti JE, LaSalle JP: Dissipative periodic processes. Bulletin of the American Mathematical Society (New Series) 1971, 77: 1082–1088. 10.1090/S0002-9904-1971-12879-3

    Article  MathSciNet  MATH  Google Scholar 

  7. Browder FE: On a generalization of the Schauder fixed point theorem. Duke Mathematical Journal 1959,26(2):291–303. 10.1215/S0012-7094-59-02629-8

    Article  MathSciNet  MATH  Google Scholar 

  8. Burton TA, Zhang SN: Unified boundedness, periodicity, and stability in ordinary and functional-differential equations. Annali di Matematica Pura ed Applicata. Serie Quarta 1986, 145: 129–158. 10.1007/BF01790540

    Article  MathSciNet  MATH  Google Scholar 

  9. Cartwright ML: Forced oscillations in nonlinear systems. In Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, no. 20. Princeton University Press, New Jersey; 1950:149–241.

    Google Scholar 

  10. Chueshov ID: Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics. AKTA, Kharkiv; 1999:436.

    MATH  Google Scholar 

  11. Fournier G: Généralisations du théorème de Lefschetz pour des espaces non-compacts. I. Applications éventuellement compactes. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1975,23(6):693–699.

    MathSciNet  MATH  Google Scholar 

  12. Fournier G: Généralisations du théorème de Lefschetz pour des espaces non-compacts. II. Applications d'attraction compacte. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1975,23(6):701–706.

    MathSciNet  MATH  Google Scholar 

  13. Fournier G: Généralisations du théorème de Lefschetz pour des espaces non-compacts. III. Applications asymptotiquement compactes. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1975,23(6):707–711.

    MathSciNet  MATH  Google Scholar 

  14. Geršteĭn VM: On the theory of dissipative differential equations in a Banach space. Funkcional'nyi Analiz i ego Priloženija 1970,4(3):99–100.

    MathSciNet  Google Scholar 

  15. Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications. Volume 495. Kluwer Academic, Dordrecht; 1999:x+399.

    Book  MATH  Google Scholar 

  16. Górniewicz L: On the Lefschetz fixed point theorem. Mathematica Slovaca 2002,52(2):221–233.

    MathSciNet  MATH  Google Scholar 

  17. Górniewicz L, Granas A: On a theorem of C. Bowszyc concerning the relative version of the Lefschetz fixed point theorem. Bulletin of the Institute of Mathematics. Academia Sinica 1985,13(2):137–142.

    MathSciNet  MATH  Google Scholar 

  18. Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York; 2003:xvi+690.

    Book  MATH  Google Scholar 

  19. Hale JK: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs. Volume 25. American Mathematical Society, Rhode Island; 1988:x+198.

    Google Scholar 

  20. Hale JK, LaSalle JP, Slemrod M: Theory of a general class of dissipative processes. Journal of Mathematical Analysis and Applications 1972,39(1):177–191. 10.1016/0022-247X(72)90233-8

    Article  MathSciNet  MATH  Google Scholar 

  21. Hale JK, Lopes O: Fixed point theorems and dissipative processes. Journal of Differential Equations 1973,13(2):391–402. 10.1016/0022-0396(73)90025-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Jones GS: Stability and asymptotic fixed-point theory. Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1262–1264. 10.1073/pnas.53.6.1262

    Article  MathSciNet  MATH  Google Scholar 

  23. Levinson N: Transformation theory of non-linear differential equations of the second order. Annals of Mathematics. Second Series 1944,45(4):723–737. 10.2307/1969299

    Article  MathSciNet  MATH  Google Scholar 

  24. Nussbaum RD: Some asymptotic fixed point theorems. Transactions of the American Mathematical Society 1972, 171: 349–375.

    Article  MathSciNet  MATH  Google Scholar 

  25. Nussbaum RD: The Fixed Point Index and Some Applications, Séminaire de Mathématiques Supérieures. Volume 94. Presses de l'Université de Montréal, Quebec; 1985:145.

    Google Scholar 

  26. Pavel N: On dissipative systems. Bollettino della Unione Matematica Italiana. Serie IV 1971, 4: 701–707.

    MathSciNet  MATH  Google Scholar 

  27. Pliss VA: Nonlocal Problems of the Theory of Oscillations. Academic Press, New York; 1966:xii+306.

    MATH  Google Scholar 

  28. Richeson D, Wiseman J: A fixed point theorem for bounded dynamical systems. Illinois Journal of Mathematics 2002,46(2):491–495.

    MathSciNet  MATH  Google Scholar 

  29. Scholz UK: The Nielsen fixed point theory for noncompact spaces. Rocky Mountain Journal of Mathematics 1974, 4: 81–87. 10.1216/RMJ-1974-4-1-81

    Article  MathSciNet  MATH  Google Scholar 

  30. Yoshizawa T: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences. Volume 14. Springer, New York; 1975:vii+233.

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics Analysis, Faculty of Science, Palacký University, Tomkova 40, Olomouc-Hejčín, 779 00, Czech Republic

    Jan Andres

  2. Faculty of Mathematics and Informatics, Nicolaus Copernicus University, Chopina 12/18, Torun, 87-100, Poland

    Lech Górniewicz

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  1. Jan Andres
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  2. Lech Górniewicz
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Correspondence to Jan Andres.

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Andres, J., Górniewicz, L. Periodic solutions of dissipative systems revisited. Fixed Point Theory Appl 2006, 65195 (2006). https://doi.org/10.1155/FPTA/2006/65195

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  • DOI: https://doi.org/10.1155/FPTA/2006/65195

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