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Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part

Abstract

We study nonlinear ODE problems in the complex Euclidean space, with the right-hand side being polynomial with nonconstant periodic coefficients. As the coefficients functions, we admit only functions with vanishing Fourier coefficients for negative indices. This leads to an existence theorem which relates the number of solutions with the number of zeros of the averaged right-hand side polynomial. A priori estimates of the norms of solutions are based on the Wirtinger-Poincaré-type inequality. The proof of existence theorem is based on the continuation method of Krasnosielski et al., Mawhin et al., and the Leray-Schauder degree. We give a few applications on the complex Riccati equation and some others.

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References

  1. Andres J: A nontrivial example of application of the Nielsen fixed-point theory to differential systems: problem of Jean Leray. Proceedings of the American Mathematical Society 2000,128(10):2921–2931. 10.1090/S0002-9939-00-05324-7

    Article  MATH  MathSciNet  Google Scholar 

  2. Andres J, Górniewicz L, Jezierski J: A generalized Nielsen number and multiplicity results for differential inclusions. Topology and its Applications 2000,100(2–3):193–209. 10.1016/S0166-8641(98)00092-3

    Article  MATH  MathSciNet  Google Scholar 

  3. Borisovich A, Kucharski Z, Marzantowicz W: Nielsen numbers and lower estimates for the number of solutions to a certain system of nonlinear integral equations. In Applied Aspects of Global Analysis, Novoe Global. Anal.. Volume 14. Voronezh University Press, Voronezh; 1994:3–10, 99.

    Google Scholar 

  4. Borisovich A, Kucharski Z, Marzantowicz W: Some applications of the Nielsen number to algebraic sets. In Proceedings of the Conference "Topological Methods in Nonlinear Analysis", December 1995, 1997, Gdańsk. Gdańsk Scientific Society Press; 78–90.

  5. Borisovich A, Kucharski Z, Marzantowicz W: A multiplicity result for a system of real integral equations by use of the Nielsen number. In Nielsen Theory and Reidemeister Torsion (Warsaw, 1996), Banach Center Publ.. Volume 49. Polish Academy of Sciences, Warsaw; 1999:9–18.

    Google Scholar 

  6. Borisovich A, Marzantowicz W: Multiplicity of periodic solutions for the planar polynomial equation. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2001,43(2):217–231.

    Article  MATH  MathSciNet  Google Scholar 

  7. Brown RF: Retraction methods in Nielsen fixed point theory. Pacific Journal of Mathematics 1984,115(2):277–297.

    Article  MATH  MathSciNet  Google Scholar 

  8. Brown RF: Topological identification of multiple solutions to parametrized nonlinear equations. Pacific Journal of Mathematics 1988,131(1):51–69.

    Article  MATH  MathSciNet  Google Scholar 

  9. Campos J: Möbius transformations and periodic solutions of complex Riccati equations. The Bulletin of the London Mathematical Society 1997,29(2):205–215. 10.1112/S0024609396002160

    Article  MATH  MathSciNet  Google Scholar 

  10. Campos J, Ortega R: Nonexistence of periodic solutions of a complex Riccati equation. Differential and Integral Equations. An International Journal for Theory & Applications 1996,9(2):247–249.

    MATH  MathSciNet  Google Scholar 

  11. Fečkan M: Nielsen fixed point theory and nonlinear equations. Journal of Differential Equations 1993,106(2):312–331. 10.1006/jdeq.1993.1110

    Article  MATH  MathSciNet  Google Scholar 

  12. Hassan HS: On the set of periodic solutions of differential equations of Riccati type. Proceedings of the Edinburgh Mathematical Society. Series II 1984,27(2):195–208.

    MATH  MathSciNet  Google Scholar 

  13. Lloyd NG: The number of periodic solutions of the equation. Proceedings of the London Mathematical Society. Third Series 1973, 27: 667–700. 10.1112/plms/s3-27.4.667

    Article  MATH  MathSciNet  Google Scholar 

  14. Lloyd NG: On a class of differential equations of Riccati type. Journal of the London Mathematical Society. Second Series 1975, 10: 1–10. 10.1112/jlms/s2-10.1.1

    Article  MATH  MathSciNet  Google Scholar 

  15. Manásevich R, Mawhin J, Zanolin F: Hölder inequality and periodic solutions of some planar polynomial differential equations with periodic coefficients. In Inequalities and Applications, World Sci. Ser. Appl. Anal.. Volume 3. World Scientific, New Jersey; 1994:459–466.

    Chapter  Google Scholar 

  16. Marzantowicz W: Periodic solutions of nonlinear problems with positive oriented periodic coefficients. In Variational and Topological Methods in the Study of Nonlinear Phenomena (Pisa, 2000), Progr. Nonlinear Differential Equations Appl.. Volume 49. Birkhäuser Boston, Massachusetts; 2002:43–63.

    Chapter  Google Scholar 

  17. Mawhin J: Periodic solutions of some planar nonautonomous polynomial differential equations. Differential and Integral Equations. An International Journal for Theory and Applications 1994,7(3–4):1055–1061.

    MATH  MathSciNet  Google Scholar 

  18. Mawhin J: Continuation theorems and periodic solutions of ordinary differential equations. In Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.. Volume 472. Kluwer Academic, Dordrecht; 1995:291–375.

    Chapter  Google Scholar 

  19. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York; 1989:xiv+277.

    Book  Google Scholar 

  20. Miklaszewski D: An equationwith no-periodic solutions. Bulletin of the Belgian Mathematical Society. Simon Stevin 1996,3(2):239–242.

    MATH  MathSciNet  Google Scholar 

  21. Srzednicki R: On periodic solutions of planar polynomial differential equations with periodic coefficients. Journal of Differential Equations 1994,114(1):77–100. 10.1006/jdeq.1994.1141

    Article  MATH  MathSciNet  Google Scholar 

  22. Żołądek H: The method of holomorphic foliations in planar periodic systems: the case of Riccati equations. Journal of Differential Equations 2000,165(1):143–173. 10.1006/jdeq.1999.3721

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Andrei Borisovich.

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Borisovich, A., Marzantowicz, W. Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part. J Inequal Appl 2006, 42908 (2006). https://doi.org/10.1155/JIA/2006/42908

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