## Nagoya Mathematical Journal

### Oscillation results for {$n$}-th order linear differential equations with meromorphic periodic coefficients

Shun Shimomura

#### Abstract

Consider $n$-th order linear differential equations with meromorphic periodic coefficients of the form $w^{(n)}+R_{n-1}(e^{z})w^{(n-1)}+\cdots+ R_{1}(e^{z})w'+R_{0}(e^{z})w = 0$, $n \ge 2$, where $R_{\nu}(t)$ $(0 \le \nu \le n-1)$ 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.

#### Article information

Source
Nagoya Math. J., Volume 166 (2002), 55-82.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631733

Mathematical Reviews number (MathSciNet)
MR1908573

Zentralblatt MATH identifier
1048.34143

Subjects
Primary: 34M10: Oscillation, growth of solutions
Secondary: 30D35: Distribution of values, Nevanlinna theory

#### Citation

Shimomura, Shun. Oscillation results for {$n$}-th order linear differential equations with meromorphic periodic coefficients. Nagoya Math. J. 166 (2002), 55--82. https://projecteuclid.org/euclid.nmj/1114631733

#### References

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