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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: 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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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: https://doi.org/10.1090/S0002-9947-02-03079-9
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