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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter


Authors: Kusano Takasi and Manabu Naito
Journal: Trans. Amer. Math. Soc. 354 (2002), 4751-4767
MSC (2000): Primary 34C10; Secondary 34B16
Published electronically: July 8, 2002
MathSciNet review: 1926835
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the following half-linear ordinary differential equation is considered:

\begin{displaymath}({H}_{\lambda})\quad\quad\quad\quad\quad\quad (\vert x'\vert^... ...atorname{sgn}\:x =0,\qquad t \geq a,\quad\quad\quad\quad\quad \end{displaymath}

where $\alpha > 0$ is a constant, $\lambda > 0$ is a parameter, and $p(t)$ is a continuous function on $[a, \infty)$, $a > 0$, and $p(t) > 0$ for $t \in [a, \infty)$. The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions $x(t; \lambda)$ of (H $_{\lambda})$ such that

\begin{displaymath}\lim_{t\to\infty}\frac{x(t; \lambda)}{\sqrt{t}} = 0. \end{displaymath}

It is shown that, if $\alpha \geq 1$ and if (H $_{\lambda})$ is strongly nonoscillatory, then there exists a sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ such that $0=\lambda_{0}<\lambda_{1}<\cdots< \lambda_{n}<\cdots$,   $\lambda_{n} \to +\infty$ as $n \to \infty$; and $x(t; \lambda)$ with $\lambda = \lambda_n$ has exactly $n-1$ zeros in the interval $(a,\infty)$ and $x(a; \lambda_n) = 0$; and $x(t; \lambda)$ with $\lambda \in (\lambda_{n-1}, \lambda_n)$ has exactly $n-1$ zeros in $(a,\infty)$ and $x(a; \lambda_n) \neq 0$. For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.


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  • 1. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. MR 16:1022b
  • 2. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965. MR 32:7875
  • 3. M. Del Pino, M. Elgueta and R. Manasevich, Generalizing Hartman's oscillation result for $(\vert x'\vert^{p-2}x')'+c(t)\vert x\vert^{p-2}x=0, p>1$, Houston J. Math. 17 (1991), 63-70. MR 92e:34040
  • 4. Á. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. J. Bolyai 30: Qualitative Theory of Differential Equations (Szeged) (1979), 153-180. MR 84g:34008
  • 5. Á. Elbert and T. Kusano, Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations, Acta Math. Hungar. 56 (1990), 325-336. MR 93b:34039
  • 6. Á. Elbert, T. Kusano and M. Naito, On the number of zeros of nonoscillatory solutions to second-order half-linear differential equations, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 42 (1999), 101-131. MR 2001f:34056
  • 7. P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964. MR 30:1270
  • 8. H. Hoshino, R. Imabayashi, T. Kusano and T. Tanigawa , On second-order half-linear oscillations, Adv. Math. Sci. Appl. 8 (1998), 199-216. MR 99c:34059
  • 9. T. Kusano and M. Naito, A singular eigenvalue problem for the Sturm-Liouville equation, Differentsial'nye Uravneniya 34 (1998), 303-312; English transl., Differential Equations 34 (1998), 302-311. MR 99i:34039
  • 10. T. Kusano and M. Naito, Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations, Rocky Mountain J. Math. 31 (2001), 1039-1054.
  • 11. T. Kusano, M. Naito and T. Tanigawa, Second-order half-linear eigenvalue problems, Fukuoka University Science Reports 27 (1997), 1-7. MR 98f:34025
  • 12. T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 81-99. MR 98f:34071
  • 13. T. Kusano, Y. Naito and A. Ogata, Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations and Dynamical Systems 2 (1994), 1-10. MR 97d:34030
  • 14. T. Kusano and N. Yoshida, Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. Appl. 189 (1995), 115-127. MR 97f:34019
  • 15. H. J. Li and C. C. Yeh, Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh 125A (1995), 1193-1204. MR 96i:34067
  • 16. J. D. Mirzov, On some analogs of Sturm's and Kneser's theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418-425. MR 53:6005
  • 17. Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428-445. MR 19:415a

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Additional Information

Kusano Takasi
Affiliation: Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan
Email: tkusano@cis.fukuoka-u.ac.jp

Manabu Naito
Affiliation: Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan
Email: mnaito@math.sci.ehime-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03079-9
PII: S 0002-9947(02)03079-9
Keywords: Half-linear equations, zeros of nonoscillatory solutions
Received by editor(s): January 5, 2001
Published electronically: July 8, 2002
Article copyright: © Copyright 2002 American Mathematical Society