Oscillation of even order nonlinear delay dynamic equations on time scales

Lynn H. Erbe; Raziye Mert; Allan Peterson; Ağacık Zafer

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 265-279
  • ISSN: 0011-4642

Abstract

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One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.

How to cite

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Erbe, Lynn H., et al. "Oscillation of even order nonlinear delay dynamic equations on time scales." Czechoslovak Mathematical Journal 63.1 (2013): 265-279. <http://eudml.org/doc/252521>.

@article{Erbe2013,
abstract = {One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.},
author = {Erbe, Lynn H., Mert, Raziye, Peterson, Allan, Zafer, Ağacık},
journal = {Czechoslovak Mathematical Journal},
keywords = {time scale; even order; delay; oscillation; Taylor monomial; time scale; even order; delay; oscillation; Taylor monomial},
language = {eng},
number = {1},
pages = {265-279},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Oscillation of even order nonlinear delay dynamic equations on time scales},
url = {http://eudml.org/doc/252521},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Erbe, Lynn H.
AU - Mert, Raziye
AU - Peterson, Allan
AU - Zafer, Ağacık
TI - Oscillation of even order nonlinear delay dynamic equations on time scales
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 265
EP - 279
AB - One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.
LA - eng
KW - time scale; even order; delay; oscillation; Taylor monomial; time scale; even order; delay; oscillation; Taylor monomial
UR - http://eudml.org/doc/252521
ER -

References

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  1. Agarwal, R. P., Bohner, M., 10.1007/BF03322019, Result. Math. 35 (1999), 3-22. (1999) Zbl0927.39003MR1678096DOI10.1007/BF03322019
  2. Bohner, M., Peterson, A., Dynamic Equations on Time Scales, Birkhäuser, Boston (2001). (2001) Zbl0993.39010MR1843232
  3. Çakmak, D., Tiryaki, A., 10.1016/j.camwa.2005.02.005, Comput. Math. Appl. 49 (2005), 1647-1653. (2005) Zbl1093.34552MR2154674DOI10.1016/j.camwa.2005.02.005
  4. El-Sayed, M. A., An oscillation criterion for a forced second-order linear differential equation, Proc. Am. Math. Soc. 118 (1993), 813-817. (1993) Zbl0777.34023MR1154243
  5. Erbe, L., Hassan, T. S., Peterson, A., Oscillation of second order neutral delay differential equations, Adv. Dyn. Syst. Appl. 3 (2008), 53-71. (2008) MR2547661
  6. Han, Z., Sun, S., Shi, B., 10.1016/j.jmaa.2007.01.004, J. Math. Anal. Appl. 334 (2007), 847-858. (2007) Zbl1125.34047MR2338632DOI10.1016/j.jmaa.2007.01.004
  7. Kartsatos, A. G., 10.1016/0022-0396(71)90058-1, J. Differ. Equations 10 (1971), 355-363. (1971) MR0288358DOI10.1016/0022-0396(71)90058-1
  8. Lakshmikantham, V., Sivasundaram, S., Kaymakçalan, B., Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht (1996). (1996) Zbl0869.34039MR1419803
  9. Mert, R., Zafer, A., Eventually positive solutions of second-order superlinear dynamic equations, Further progress in analysis. Proceedings of the 6th International ISAAC Congress, 13-18 August, 2007, Ankara, Turkey, World Scientific (2009), 535-544 H. G. W. Begehr et. al. (2009) Zbl1185.34146MR2581655
  10. Mert, R., Zafer, A., A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations, Discrete Contin. Dyn. Syst. Supplement Volume (2011), 1061-1067. (2011) MR3012907
  11. Naito, M., 10.32917/hmj/1206133990, Hiroshima Math. J. 11 (1981), 553-560. (1981) Zbl0512.34056MR0635038DOI10.32917/hmj/1206133990
  12. Onose, H., 10.1017/S0004972700024217, Bull. Aust. Math. Soc. 13 (1975), 13-19. (1975) Zbl0307.34034MR0393732DOI10.1017/S0004972700024217
  13. Ou, C. H., Wong, J. S. W., 10.1006/jmaa.2001.7614, J. Math. Anal. Appl. 262 (2001), 722-732. (2001) Zbl0997.34059MR1859335DOI10.1006/jmaa.2001.7614
  14. Sun, Y. G., Wong, J. S. W., 10.1016/j.jmaa.2004.03.076, J. Math. Anal. Appl. 298 (2004), 114-119. (2004) Zbl1064.34020MR2086536DOI10.1016/j.jmaa.2004.03.076
  15. Yílmaz, Y. Şahiner, Zafer, A., Oscillation of even order nonlinear neutral differential equations with damping, Math. Inequal. Appl. 1 (1998), 445-451. (1998) MR1629420
  16. Yang, Q., 10.1016/S0096-3003(01)00307-1, Appl. Math. Comput. 135 (2003), 49-64. (2003) Zbl1030.34034MR1934314DOI10.1016/S0096-3003(01)00307-1

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