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Proceedings of the American Mathematical Society

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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
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Abstract: Oscillation properties of solutions of the forced second order linear difference equation

\begin{displaymath}\Delta (r_{k}\Delta x_{k})+c_{k}x_{k+1}=h_{k} \end{displaymath}

are investigated. The authors show that if the forcing term $h$ 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

\begin{displaymath}\Delta \left (r_{k}\Phi (\Delta x_{k})\right )+c_{k}\Phi (x_... ...k}, \quad \Phi (s)=\vert s\vert^{\alpha -2}s, \quad \alpha >1, \end{displaymath}

are also given.

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Additional Information

O. Dosly
Affiliation: Mathematical Institute, Czech Academy of Sciences, Žižkova 22, CZ–61662 Brno, Czech Republic

J. R. Graef
Affiliation: Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403

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

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