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On meromorphic solutions of certain nonlinear differential equations

Published online by Cambridge University Press:  17 April 2009

J. Heittokangas ,
R. Korhonen  and
I. Laine
J. Heittokangas
Affiliation:
Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland, e-mail: heittoka@cc.joensuu.fi, rkorhone@cc.joensuu.fi, ilpo.laine@joensuu.fi
R. Korhonen
Affiliation:
Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland, e-mail: heittoka@cc.joensuu.fi, rkorhone@cc.joensuu.fi, ilpo.laine@joensuu.fi
I. Laine
Affiliation:
Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland, e-mail: heittoka@cc.joensuu.fi, rkorhone@cc.joensuu.fi, ilpo.laine@joensuu.fi
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In this paper, we consider the growth of meromorphic solutions of nonlinear differential equations of the form L (f) + P (z, f) = h (z), where L (f) denotes a linear differential polynomial in f, P (z, f) is a polynomial in f, both with small meromorphic coefficients, and h (z) is a meromorphic function. Specialising to L (f) − p (z) fn = h (z), where p (z) is a small meromorphic function, we consider the uniqueness of meromorphic solutions with few poles only. Our results complement earlier ones due to C.-C. Yang.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 2 , October 2002 , pp. 331 - 343
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Gundersen, G.G. and Laine, I., ‘On the meromorphic solutions of some algebraic differential equations’, J. Math. Anal. Appl. 111 (1985), 281–300.Google Scholar
[2]Hayman, W.K., Meromorphic functions (Clarendon Press, Oxford, 1964).Google Scholar
[3]Hellerstein, S. and Rubel, L.A., ‘Subfields that are algebraically closed in the field of meromorphic functions’, J. Anal. Math. 12 (1964), 105–111.Google Scholar
[4]Laine, I., Nevanlinna theory and complex differential equations (Walter de Gruyter, Berlin, 1993).Google Scholar
[5]Mohon'ko, A.Z., ‘The Nevanlinna characteristic of certain meromorphic functions’, (Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen 14 (1971), 83–87.Google Scholar
[6]Steinmetz, N., ‘Value distribution of the Painlevé transcendents’, Israel J. Math. 128 (2002), 29–52.CrossRefGoogle Scholar
[7]Yang, C.-C., ‘On entire solutions of a certain type of nonlinear differential equation’, Bull. Austral. Math. Soc. 64 (2001), 377–380.Google Scholar