×

Oscillation properties of an Emden-Fowler type equation on discrete time scales. (English) Zbl 1038.39009

The authors consider the second order dynamic equation \(u^{\Delta ^{2}}+p(t)u^{\gamma }(\sigma (t))=0\), where \( \gamma\) is a quotient of odd positive integers. They establish some necessary and sufficient conditions for oscillations. When \(\gamma >1\) they prove that every solution oscillates if and only if \(\int_{a}^{\infty }\sigma (l)p(l)\Delta l=\infty \) and when \(0<\gamma <1\) they prove that every solution oscillates if and only if \(\int_{a}^{\infty }\left[ \sigma (l)\right] ^{\gamma }p(l)\Delta l=\infty \). The results are applied only to discrete time scales. Some examples illustrating the main results are given.

MSC:

39A12 Discrete version of topics in analysis
93C70 Time-scale analysis and singular perturbations in control/observation systems
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Agarwal R. P. Difference Equations and Inequalities Marcel Dekker Inc. New York 1992
[2] Akin-Bohner E. Hoffacker J. Solution properties on discrete time scales J. Difference Equ. Appl.9 · Zbl 1032.39006
[3] Atkinson F. V. On second order nonlinear oscillations Pacific J. Math. 5 1995 643 647
[4] Belohorec S. Oscillatory solutions of certain nonlinear differential equations of the second order Mat.Fyz. Casopsis Sloven. Akad. Vied. 11 1961 250 255
[5] Belohorec S. On some properties of the equationyxfxy^{ \mgreek{a} }x, 0 < \mgreek{a} Mat.Fyz. Casopsis Sloven. Akad. Vied. 17 1967 10 19 · Zbl 0166.07702
[6] Bohner M. Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications Birkhäuser Boston 2001 · Zbl 0978.39001
[7] Erbe L. Muldowney J. Nonoscillation results for second order nonlinear differential equations Rocky Mt. J. Math. 12 4 1982 635 642 · Zbl 0516.34030
[8] Hilger S. Ein Ma{\(\beta\)}kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten Universität Würzburg 1988 · Zbl 0695.34001
[9] Hilger S. Analysis on measure chains-a unified approach to continuous and discrete calculus Result. Math. 18 1990 19 56 · Zbl 0722.39001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.