On the number of zeros of nonoscillatory solutions to halflinear ordinary differential equations involving a parameter
Authors:
Kusano Takasi and Manabu Naito
Journal:
Trans. Amer. Math. Soc. 354 (2002), 47514767
MSC (2000):
Primary 34C10; Secondary 34B16
Published electronically:
July 8, 2002
MathSciNet review:
1926835
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Abstract: In this paper the following halflinear 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.
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P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964. MR 30:1270
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H. Hoshino, R. Imabayashi, T. Kusano and T. Tanigawa , On secondorder halflinear oscillations, Adv. Math. Sci. Appl. 8 (1998), 199216. MR 99c:34059
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T. Kusano and M. Naito, A singular eigenvalue problem for the SturmLiouville equation, Differentsial'nye Uravneniya 34 (1998), 303312; English transl., Differential Equations 34 (1998), 302311. MR 99i:34039
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T. Kusano and M. Naito, SturmLiouville eigenvalue problems for halflinear ordinary differential equations, Rocky Mountain J. Math. 31 (2001), 10391054.
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T. Kusano, M. Naito and T. Tanigawa, Secondorder halflinear eigenvalue problems, Fukuoka University Science Reports 27 (1997), 17. MR 98f:34025
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T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 8199. MR 98f:34071
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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), 110. MR 97d:34030
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T. Kusano and N. Yoshida, Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. Appl. 189 (1995), 115127. MR 97f:34019
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H. J. Li and C. C. Yeh, Sturmian comparison theorem for halflinear secondorder differential equations, Proc. Roy. Soc. Edinburgh 125A (1995), 11931204. MR 96i:34067
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J. D. Mirzov, On some analogs of Sturm's and Kneser's theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418425. MR 53:6005
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Z. Nehari, Oscillation criteria for secondorder linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428445. MR 19:415a
 1.
 E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrawHill, 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 , Houston J. Math. 17 (1991), 6370. MR 92e:34040
 4.
 Á. Elbert, A halflinear second order differential equation, Colloq. Math. Soc. J. Bolyai 30: Qualitative Theory of Differential Equations (Szeged) (1979), 153180. 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), 325336. MR 93b:34039
 6.
 Á. Elbert, T. Kusano and M. Naito, On the number of zeros of nonoscillatory solutions to secondorder halflinear differential equations, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 42 (1999), 101131. 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 secondorder halflinear oscillations, Adv. Math. Sci. Appl. 8 (1998), 199216. MR 99c:34059
 9.
 T. Kusano and M. Naito, A singular eigenvalue problem for the SturmLiouville equation, Differentsial'nye Uravneniya 34 (1998), 303312; English transl., Differential Equations 34 (1998), 302311. MR 99i:34039
 10.
 T. Kusano and M. Naito, SturmLiouville eigenvalue problems for halflinear ordinary differential equations, Rocky Mountain J. Math. 31 (2001), 10391054.
 11.
 T. Kusano, M. Naito and T. Tanigawa, Secondorder halflinear eigenvalue problems, Fukuoka University Science Reports 27 (1997), 17. MR 98f:34025
 12.
 T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 8199. 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), 110. 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), 115127. MR 97f:34019
 15.
 H. J. Li and C. C. Yeh, Sturmian comparison theorem for halflinear secondorder differential equations, Proc. Roy. Soc. Edinburgh 125A (1995), 11931204. 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), 418425. MR 53:6005
 17.
 Z. Nehari, Oscillation criteria for secondorder linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428445. MR 19:415a
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Additional Information
Kusano Takasi
Affiliation:
Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 8140180, Japan
Email:
tkusano@cis.fukuokau.ac.jp
Manabu Naito
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 7908577, Japan
Email:
mnaito@math.sci.ehimeu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994702030799
PII:
S 00029947(02)030799
Keywords:
Halflinear equations,
zeros of nonoscillatory solutions
Received by editor(s):
January 5, 2001
Published electronically:
July 8, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
