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

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

Published electronically:
July 8, 2002

MathSciNet review:
1926835

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

where is a constant, is a parameter, and is a continuous function on , , and for . The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions of (H such that

It is shown that, if and if (H is strongly nonoscillatory, then there exists a sequence such that , as ; and with has exactly zeros in the interval and ; and with has exactly zeros in and . For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.

**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*, D. C. Heath and Co., Boston, Mass., 1965. MR**0190463****3.**Manuel del Pino, Manuel Elgueta, and Raúl Manásevich,*Generalizing Hartman’s oscillation result for (|𝑥’|^{𝑝-2}𝑥’)’+𝑐(𝑡)|𝑥|^{𝑝-2}𝑥=0,𝑝>1*, Houston J. Math.**17**(1991), no. 1, 63–70. MR**1107187****4.**Á. Elbert,*A half-linear second order differential equation*, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), Colloq. Math. Soc. János Bolyai, vol. 30, North-Holland, Amsterdam-New York, 1981, pp. 153–180. MR**680591****5.**Á. Elbert and T. Kusano,*Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations*, Acta Math. Hungar.**56**(1990), no. 3-4, 325–336. MR**1111319**, https://doi.org/10.1007/BF01903849**6.**Árpád Elbert, Kusano Takaŝi, and Manabu 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 (2000). MR**1771815****7.**Philip Hartman,*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0171038****8.**H. Hoshino, R. Imabayashi, T. Kusano, and T. Tanigawa,*On second-order half-linear oscillations*, Adv. Math. Sci. Appl.**8**(1998), no. 1, 199–216. MR**1623342****9.**T. Kusano and M. Naito,*A singular eigenvalue problem for the Sturm-Liouville equation*, Differ. Uravn.**34**(1998), no. 3, 303–312, 428 (Russian, with Russian summary); English transl., Differential Equations**34**(1998), no. 3, 302–311. MR**1668257****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.**Takaŝi Kusano, Manabu Naito, and Tomoyuki Tanigawa,*Second-order half-linear eigenvalue problems*, Fukuoka Univ. Sci. Rep.**27**(1997), no. 1, 1–7. MR**1448784****12.**Deming Zhu,*Transversal heteroclinic orbits in general degenerate cases*, Sci. China Ser. A**39**(1996), no. 2, 113–121. MR**1397473****13.**Takaŝi Kusano, Yūki Naito, and Akio Ogata,*Strong oscillation and nonoscillation of quasilinear differential equations of second order*, Differential Equations Dynam. Systems**2**(1994), no. 1, 1–10. MR**1386034****14.**Takaŝi Kusano and Norio Yoshida,*Nonoscillation theorems for a class of quasilinear differential equations of second order*, J. Math. Anal. Appl.**189**(1995), no. 1, 115–127. MR**1312033**, https://doi.org/10.1006/jmaa.1995.1007**15.**Horng Jaan Li and Cheh Chih Yeh,*Sturmian comparison theorem for half-linear second-order differential equations*, Proc. Roy. Soc. Edinburgh Sect. A**125**(1995), no. 6, 1193–1204. MR**1362999**, https://doi.org/10.1017/S0308210500030468**16.**J. D. Mirzov,*On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems*, J. Math. Anal. Appl.**53**(1976), no. 2, 418–425. MR**0402184**, https://doi.org/10.1016/0022-247X(76)90120-7**17.**Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc.**85**(1957), 428-445. MR**19:415a**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
34C10,
34B16

Retrieve articles in all journals with MSC (2000): 34C10, 34B16

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