Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T06:33:07.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillation results for n-th order linear differential equations with meromorphic periodic coefficients

Published online by Cambridge University Press:  22 January 2016

Shun Shimomura
Shun Shimomura*
Affiliation:
Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider n-th order linear differential equations with meromorphic periodic coefficients of the form w(n) + Rn-1(ez)w(n-1) + … + R1(ez)w′ + R0(ez)w = 0, n ≥ 2, where Rv(t) (0 ≤ ν ≤ n – 1) are rational functions of t. Under certain assumptions, we prove oscillation theorems concerning meromorphic solutions, which contain necessary conditions for the existence of a meromorphic solution with finite exponent of convergence of the zero-sequence. We also discuss meromorphic or entire solutions whose zero-sequences have an infinite exponent of convergence, and give a new zero-density estimate for such solutions.

Keywords

34M1034M0534M3030D35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 166 , June 2002 , pp. 55 - 82
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[1] Baesch, A., On the explicit determination of certain solutions of periodic differential equations of higher order, Results Math., 29 (1996), 42–55.CrossRefGoogle Scholar
[2] Baesch, A. and Steinmetz, N., Exceptional solutions of n-th order periodic linear differential equations, Complex Variables Theory Appl., 34 (1997), 7–17.Google Scholar
[3] Balser, W., Jurkat, W. B. and Lutz, D. A., Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., 71 (1979), 48–94.Google Scholar
[4] Bank, S. B., Three results in the value-distribution theory of solutions of linear differential equations, Kodai Math. J., 9 (1986), 225–240.Google Scholar
[5] Bank, S. B., On determining the location of complex zeros of solutions of certain linear differential equations, Ann. Mat. Pura Appl. (4), 151 (1988), 67–96.CrossRefGoogle Scholar
[6] Bank, S. B., On the oscillation theory of periodic linear differential equations, Appl. Anal., 39 (1990), 95–111.Google Scholar
[7] Bank, S. B., On the explicit determination of certain solutions of periodic differential equations, Complex Variables Theory Appl., 23 (1993), 101–121.Google Scholar
[8] Bank, S. B. and Laine, I., Representations of solutions of periodic second order linear differential equations, J. Reine Angew. Math., 344 (1983), 1–21.Google Scholar
[9] Bank, S. B. and Laine, I., On the zeros of meromorphic solutions of second order linear differential equations, Comment. Math. Helv., 58 (1983), 656–677.CrossRefGoogle Scholar
[10] Bank, S. B., Laine, I. and Langley, J. K., On the frequency of zeros of solutions of second order linear differential equations, Results Math., 10 (1986), 8–24.Google Scholar
[11] Bank, S. B. and Langley, J. K., Oscillation theory for higher order linear differential equations with entire coefficients, Complex Variables Theory Appl., 16 (1991), 163–175.Google Scholar
[12] Bank, S. B. and Langley, J. K., Oscillation theorems for higher order linear differential equations with entire periodic coefficients, Comment. Math. Univ. St. Paul., 41 (1992), 65–85.Google Scholar
[13] Chiang, Y. M. and Laine, I., Some oscillation results for linear differential equations in the complex plane, Japan. J. Math., 24 (1998), 367–402.CrossRefGoogle Scholar
[14] Chiang, Y. M., Laine, I. and Wang, S., An oscillation result of a third order linear differential equation with entire periodic coefficients, Complex Variables Theory Appl., 34 (1997), 25–34.Google Scholar
[15] Chiang, Y. M. and Wang, S., Oscillation results on certain higher order linear differential equations with periodic coefficients in the complex plane, J. Math. Anal. Appl., 215 (1997), 560–576.Google Scholar
[16] Gao, S., A further result on the complex oscillation theory of periodic second order linear differential equations, Proc. Edinburgh Math. Soc. (2), 33 (1990), 143–158.Google Scholar
[17] Gundersen, G. G., On the real zeros of solutions of f” + A(z)f = 0 where A(z) is entire, Ann. Acad. Sci. Fenn. Math., 11 (1986), 275–294.CrossRefGoogle Scholar
[18] Hayman, W. K., Meromorphic Functions, Clarendon, Oxford, 1964.Google Scholar
[19] Hellerstein, S. and Rossi, J., Zeros of meromorphic solutions of second order linear differential equations, Math. Z., 192 (1986), 603–612.Google Scholar
[20] Jank, G. and Volkmann, L., Einführung in die Theorie der Ganzen und Meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Boston, Stuttgart, 1985.Google Scholar
[21] Laine, I., Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin, 1993.CrossRefGoogle Scholar
[22] Laine, I. and Sorvali, T., Local solutions ofw” + A(z)w = 0 and branched polymorphic functions, Results Math., 10 (1986), 107–129.CrossRefGoogle Scholar
[23] Langley, J. K., Some oscillation theorems for higher order linear differential equations with entire coefficients of small growth, Results Math., 20 (1991), 517–529.CrossRefGoogle Scholar
[24] Sibuya, Y., Simplification of a system of linear ordinary differential equations about a singular point, Funkcial. Ekvac., 4 (1962), 29–56.Google Scholar
[25] Sibuya, Y., Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland, Amsterdam, 1975.Google Scholar
[26] Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York, London, Sydney, 1965.Google Scholar