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

Došlý, O.; Graef, J.; Jaroš, J.

doi:10.1090/S0002-9939-02-06811-9

Forced oscillation of second order linear and half-linear difference equations
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by O. Došlý, J. R. Graef and J. Jaroš

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

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

Published electronically: December 30, 2002

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Abstract:

Oscillation properties of solutions of the forced second order linear difference equation \begin{equation*}\Delta (r_{k}\Delta x_{k})+c_{k}x_{k+1}=h_{k} \end{equation*} 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{equation*}\Delta \left (r_{k}\Phi (\Delta x_{k})\right )+c_{k}\Phi (x_{k+1})=h_{k}, \quad \Phi (s)=|s|^{\alpha -2}s, \quad \alpha >1, \end{equation*} are also given.

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