Forced oscillation of second order linear and half-linear difference equations

Authors:
O. Dosly, J. R. Graef and J. Jaros

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2859-2867

MSC (2000):
Primary 39A11

Published electronically:
December 30, 2002

MathSciNet review:
1974343

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 half-linear equation

are also given.

**1.**Calvin D. Ahlbrandt and Allan C. Peterson,*Discrete Hamiltonian systems*, Kluwer Texts in the Mathematical Sciences, vol. 16, Kluwer Academic Publishers Group, Dordrecht, 1996. Difference equations, continued fractions, and Riccati equations. MR**1423802****2.**W. A. Coppel,*Disconjugacy*, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR**0460785****3.**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****4.**John R. Graef, Samuel M. Rankin III, and Paul W. Spikes,*Oscillation results for nonlinear functional-differential equations*, Funkcial. Ekvac.**27**(1984), no. 2, 255–260. MR**775209****5.**J. Jaroš and T. Kusano,*A Picone type identity for second order half-linear differential equations*, Acta Math. Univ. Comenian. (N.S.)**68**(1999), no. 1, 137–151. MR**1711081****6.**Walter G. Kelley and Allan C. Peterson,*Difference equations*, 2nd ed., Harcourt/Academic Press, San Diego, CA, 2001. An introduction with applications. MR**1765695****7.**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**, 10.1017/S0308210500030468**8.**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****9.**Pavel Řehák,*Oscillatory properties of second order half-linear difference equations*, Czechoslovak Math. J.**51(126)**(2001), no. 2, 303–321. MR**1844312**, 10.1023/A:1013790713905

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
39A11

Retrieve articles in all journals with MSC (2000): 39A11

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:
john-graef@utc.edu

**J. Jaros**

Affiliation:
Department of Mathematical Analysis, Comenius University, 842 15 Bratislava, Slovakia

Email:
jaros@alpha.dcs.fmph.uniba.sk

DOI:
https://doi.org/10.1090/S0002-9939-02-06811-9

Keywords:
Linear difference equation,
half-linear 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