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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:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A11Stability of difference equations (MSC2000)
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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) · 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.: Oscillation criteria for second order matrix dynamic equations on a time scale. 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) · Zbl 46.0702.01
[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.: Chain rule and invariance principle on measure chains. 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