×

Oscillation for a nonlinear dynamic system on time scales. (English) Zbl 1232.34124

The authors study oscillation properties for a system of two first-order nonlinear dynamic equations \[ x^\Delta=a_1(t)f_1(y),\quad y^\Delta=-a_2(t)f_2(x^\sigma)\tag{\(\ast\)} \] on time scales being unbounded above. The coefficients \(a_1,a_2\) are supposed to be rd-continuous, \(a_1\) is nonnegative and \(f_1,f_2\) are continuous and satisfy \(uf_i(u)>0\) for all \(u\neq 0\), beyond further assumptions. This form includes the classical Emden-Fowler equations and various extensions.
In the superlinear, sublinear, delay-dynamic and Waltman’s case referring to the assumptions on \(f_i\), sufficient conditions are given for the solutions to \((\ast)\) to be oscillatory. They explicitly require certain integrals over the coefficients \(a_i\) to be unbounded.
Concrete examples for differential and difference equations illustrate the results.

MSC:

34N05 Dynamic equations on time scales or measure chains
34K11 Oscillation theory of functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Akın-Bohner E., Electron. Trans. Numer. Anal. 27 pp 1– (2007)
[2] DOI: 10.1080/1023619021000053575 · Zbl 1038.39009
[3] Atkinson F.V., Pacific J. Math. 5 pp 643– (1955)
[4] Belohorec S., Mat. Fyz. Časopis Sloven Akad. Vied. 11 pp 250– (1961)
[5] DOI: 10.1016/j.jmaa.2004.07.038 · Zbl 1061.34018
[6] Bohner M., Dynamic Equations on Time Scales: An Introduction with Applications (2001) · Zbl 0978.39001
[7] DOI: 10.1007/978-0-8176-8230-9
[8] DOI: 10.4153/CMB-1973-011-1 · Zbl 0272.34095
[9] Erbe L., Adv. Dynam. Syst. Appl. 3 pp 73– (2008)
[10] L. Erbe and R. Mert, Some new oscillation results for a nonlinear dynamic system on time scales, Appl. Math. Comput. (to appear) · Zbl 1185.34141
[11] DOI: 10.1090/S0002-9939-03-07061-8 · Zbl 1055.39007
[12] DOI: 10.1112/S0024610703004228 · Zbl 1050.34042
[13] Erbe L., Dynam. Systems Appl. 15 pp 65– (2006)
[14] DOI: 10.1016/S0898-1221(99)00246-1 · Zbl 0964.39012
[15] DOI: 10.1016/0022-247X(83)90088-4 · Zbl 0508.39005
[16] DOI: 10.1016/S0898-1221(03)00089-0 · Zbl 1056.39009
[17] DOI: 10.1016/j.camwa.2005.10.020 · Zbl 1148.39005
[18] DOI: 10.1016/j.cam.2005.03.054 · Zbl 1117.39004
[19] S. Keller, Asymptotisches Verhalten Invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Ph.D. thesis, Universität Augsburg, Augsburg, Germany, 1999
[20] DOI: 10.1090/S0002-9939-98-04503-1 · Zbl 0897.34033
[21] Kwong M.K., Differ. Integral Equ. 1 pp 133– (1988)
[22] Mert R., Proceedings of the sixth international ISAAC Congress pp 535– (2007)
[23] R. Mert and A. Zafer, A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations (submitted) · Zbl 1306.34138
[24] DOI: 10.1016/S0377-0427(01)00450-2 · Zbl 1011.34045
[25] DOI: 10.1016/0022-247X(65)90138-1 · Zbl 0131.08902
[26] DOI: 10.1016/j.cam.2008.06.010 · Zbl 1161.39012
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.