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

- Andrei Borisovich
^{1}Email author and - Wacław Marzantowicz
^{2}

**2006**:42908

**DOI: **10.1155/JIA/2006/42908

© Hindawi Publishing Corporation. 2006

**Received: **8 February 2004

**Accepted: **12 March 2004

**Published: **16 January 2006

## 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.

## Authors’ Affiliations

## References

- 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/S0002-9939-00-05324-7MATHMathSciNetView ArticleGoogle Scholar - 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)00092-3MATHMathSciNetView ArticleGoogle Scholar - 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 - 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.Google Scholar
- 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 - 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.MATHMathSciNetView ArticleGoogle Scholar - Brown RF:
**Retraction methods in Nielsen fixed point theory.***Pacific Journal of Mathematics*1984,**115**(2):277–297.MATHMathSciNetView ArticleGoogle Scholar - Brown RF:
**Topological identification of multiple solutions to parametrized nonlinear equations.***Pacific Journal of Mathematics*1988,**131**(1):51–69.MATHMathSciNetView ArticleGoogle Scholar - 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/S0024609396002160MATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetGoogle Scholar - Fečkan M:
**Nielsen fixed point theory and nonlinear equations.***Journal of Differential Equations*1993,**106**(2):312–331. 10.1006/jdeq.1993.1110MATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetGoogle Scholar - 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/s3-27.4.667MATHMathSciNetView ArticleGoogle Scholar - 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/s2-10.1.1MATHMathSciNetView ArticleGoogle Scholar - 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.View ArticleGoogle Scholar - 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.View ArticleGoogle Scholar - 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.MATHMathSciNetGoogle Scholar - 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.View ArticleGoogle Scholar - Mawhin J, Willem M:
*Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences*.*Volume 74*. Springer, New York; 1989:xiv+277.View ArticleGoogle Scholar - Miklaszewski D:
**An equationwith no-periodic solutions.***Bulletin of the Belgian Mathematical Society. Simon Stevin*1996,**3**(2):239–242.MATHMathSciNetGoogle Scholar - 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.1994.1141MATHMathSciNetView ArticleGoogle Scholar - 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.1999.3721MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.