×

zbMATH — the first resource for mathematics

Oscillation for nonlinear second order dynamic equations on a time scale. (English) Zbl 1061.34018
Summary: We obtain some oscillation criteria for solutions to the nonlinear dynamic equation \[ x^{\Delta\Delta}+ q(t)x^{\Delta^\sigma} +p(t) (f\circ x^\sigma)=0 \] on time scales. In particular, no explicit sign assumptions are made with respect to the coefficients \(p(t)\), \(q(t)\). We illustrate the results by several examples, including a superlinear Emden-Fowler dynamic equation.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A11 Stability of difference equations (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atkinson, F.V., On second-order nonlinear oscillations, Pacific J. math., 5, 643-647, (1955) · Zbl 0065.32001
[2] E. Akın-Bohner, M. Bohner, S.H. Saker, Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations, preprint · Zbl 1177.34047
[3] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001
[4] M. Bohner, S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math., in press · Zbl 1075.34028
[5] M. Bohner, S.H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comput. Modelling, in press · Zbl 1112.34019
[6] Erbe, L., Oscillation theorems for second order nonlinear differential equations, Proc. amer. math. soc., 24, 811-814, (1970) · Zbl 0194.12102
[7] Erbe, L., Oscillation criteria for second order linear equations on a time scale, Canad. appl. math. quart., 9, 1-31, (2001) · Zbl 1050.39024
[8] Erbe, L., Oscillation criteria for second order nonlinear differential equations, Ann. mat. pura appl., 44, 257-268, (1972) · Zbl 0296.34026
[9] Erbe, L.; Peterson, A.; Agarwal, R.P.; Bohner, M.; O’Regan, D., Oscillation criteria for second order matrix dynamic equations on a time scale, Special issue on “dynamic equations on time scales”, J. comput. appl. math., 141, 169-185, (2002) · Zbl 1017.34030
[10] Erbe, L.; Peterson, A., Boundedness and oscillation for nonlinear dynamic equations on a time scale, Proc. amer. math. soc., 132, 735-744, (2003) · Zbl 1055.39007
[11] L. Erbe, A. Peterson, An oscillation result for a nonlinear dynamic equation on a time scale, Canad. Appl. Math. Quart., in press · Zbl 1086.39004
[12] Erbe, L.; Peterson, A.; Rehak, P., Comparison theorems for linear dynamic equations on time scales, J. math. anal. appl., 275, 418-438, (2002) · Zbl 1034.34042
[13] Erbe, L.; Peterson, A.; Saker, S.H., Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London math. soc., 67, 701-714, (2003) · Zbl 1050.34042
[14] Fite, W.B., Concerning the zeros of solutions of certain differential equations, Trans. amer. math. soc., 19, 341-352, (1917)
[15] S. Keller, Asymptotisches Verhalten Invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Ph.D. thesis, Universität Augsburg, 1999
[16] Leighton, W., On self-adjoint differential equations of second order, J. London math. soc., 27, 37-47, (1952) · Zbl 0048.06503
[17] Pötzsche, C.; Agarwal, R.P.; Bohner, M.; O’Regan, D., Chain rule and invariance principle on measure chains, Special issue on “dynamic equations on time scales”, J. comput. appl. math., 141, 249-254, (2002) · Zbl 1011.34045
[18] Waltman, P., An oscillation criterion for a nonlinear second order equation, J. math. anal. appl., 10, 439-441, (1965) · Zbl 0131.08902
[19] Wintner, A., On the nonexistence of conjugate points, Amer. J. math., 73, 368-380, (1951) · Zbl 0043.08703
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.