Forced oscillation of second order linear and halflinear difference equations
Authors:
O. Dosly, J. R. Graef and J. Jaros
Journal:
Proc. Amer. Math. Soc. 131 (2003), 28592867
MSC (2000):
Primary 39A11
Published electronically:
December 30, 2002
MathSciNet review:
1974343
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Abstract 
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Abstract: Oscillation properties of solutions of the forced second order linear difference equation
are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general halflinear equation are also given.
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W. A. Coppel, Disconjugacy, Lecture Notes in Math. No. 220, SpringerVerlag, New York, 1971. MR 57:778
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Á. Elbert, A halflinear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), 153180. MR 84g:34008
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J. R. Graef, S. M. Rankin, P. W. Spikes, Oscillation results for nonlinear functional differential equations, Funkcial. Ekvac. 27 (1984), 255260. MR 86c:34133
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J. Jaros, T. Kusano, A Picone type identity for halflinear differential equations, Acta Math. Univ. Comenianae 68 (1999), 127151. MR 2000h:34060
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W. G. Kelley, A. Peterson, Difference Equations: An Introduction with Applications, Second Edition, Academic Press, New York, 2001. MR 2001i:39001
 7.
H. J. Li, C. C. Yeh, Sturmian comparison theorem for halflinear second order differential equations, Proc. Roy. Soc. Edinburgh 125A (1996), 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
 9.
P. Rehák, Oscillatory properties of second order halflinear difference equations, Czech. Math. J. 51 (2001), 303321. MR 2002d:39008
 1.
 C. D. Ahlbrandt, A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic Publishers, DordrechtBostonLondon, 1996. MR 98m:39043
 2.
 W. A. Coppel, Disconjugacy, Lecture Notes in Math. No. 220, SpringerVerlag, New York, 1971. MR 57:778
 3.
 Á. Elbert, A halflinear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), 153180. MR 84g:34008
 4.
 J. R. Graef, S. M. Rankin, P. W. Spikes, Oscillation results for nonlinear functional differential equations, Funkcial. Ekvac. 27 (1984), 255260. MR 86c:34133
 5.
 J. Jaros, T. Kusano, A Picone type identity for halflinear differential equations, Acta Math. Univ. Comenianae 68 (1999), 127151. MR 2000h:34060
 6.
 W. G. Kelley, A. Peterson, Difference Equations: An Introduction with Applications, Second Edition, Academic Press, New York, 2001. MR 2001i:39001
 7.
 H. J. Li, C. C. Yeh, Sturmian comparison theorem for halflinear second order differential equations, Proc. Roy. Soc. Edinburgh 125A (1996), 11931204. MR 96i:34067
 8.
 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
 9.
 P. Rehák, Oscillatory properties of second order halflinear difference equations, Czech. Math. J. 51 (2001), 303321. MR 2002d:39008
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Additional Information
O. Dosly
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Žižkova 22, CZ–61662 Brno, Czech Republic
Email:
dosly@math.muni.cz
J. R. Graef
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403
Email:
johngraef@utc.edu
J. Jaros
Affiliation:
Department of Mathematical Analysis, Comenius University, 842 15 Bratislava, Slovakia
Email:
jaros@alpha.dcs.fmph.uniba.sk
DOI:
http://dx.doi.org/10.1090/S0002993902068119
PII:
S 00029939(02)068119
Keywords:
Linear difference equation,
halflinear difference equation,
variational principle,
forced oscillation
Received by editor(s):
December 30, 1999
Received by editor(s) in revised form:
January 18, 2002, and April 10, 2002
Published electronically:
December 30, 2002
Additional Notes:
The first author was supported by Grant No. 201/98/0677 of the Czech Grant Agency (Prague).
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2002
American Mathematical Society
