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 PDF

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

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.

References

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, DOI 10.1007/978-1-4757-2467-7
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785
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
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
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
Walter G. Kelley and Allan C. Peterson, Difference equations, 2nd ed., Harcourt/Academic Press, San Diego, CA, 2001. An introduction with applications. MR 1765695
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, DOI 10.1017/S0308210500030468
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 402184, DOI 10.1016/0022-247X(76)90120-7
Pavel Řehák, Oscillatory properties of second order half-linear difference equations, Czechoslovak Math. J. 51(126) (2001), no. 2, 303–321. MR 1844312, DOI 10.1023/A:1013790713905