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.


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


[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
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