The oscillation of an m th order perturbed nonlinear difference equation

Patricia J. Y. Wong; Ravi P. Agarwal

Archivum Mathematicum (1996)

  • Volume: 032, Issue: 1, page 13-27
  • ISSN: 0044-8753

Abstract

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We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation | Δ m y ( k ) | α - 1 Δ m y ( k ) + Q ( k , y ( k - σ k ) , Δ y ( k - σ k ) , ⋯ , Δ m - 2 y ( k - σ k ) ) = P ( k , y ( k - σ k ) , Δ y ( k - σ k ) , ⋯ , Δ m - 1 y ( k - σ k ) ) , k ≥ k 0 where α > 0 . Examples which dwell upon the importance of our results are also included.

How to cite

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Wong, Patricia J. Y., and Agarwal, Ravi P.. "The oscillation of an $m$th order perturbed nonlinear difference equation." Archivum Mathematicum 032.1 (1996): 13-27. <http://eudml.org/doc/247854>.

@article{Wong1996,
abstract = {We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation \[|\Delta ^\{m\} y(k)|^\{\alpha -1\}\Delta ^\{m\} y(k)+Q(k,y(k-\sigma \_\{k\}), \Delta y(k-\sigma \_\{k\}),\cdots , \Delta ^\{m-2\}y(k-\sigma \_\{k\}))\]$=P(k,y(k-\sigma _\{k\}),\Delta y(k-\sigma _\{k\}),\cdots , \Delta ^\{m-1\}y(k-\sigma _\{k\})),~k\ge k_\{0\}$ where $\alpha >0.$ Examples which dwell upon the importance of our results are also included.},
author = {Wong, Patricia J. Y., Agarwal, Ravi P.},
journal = {Archivum Mathematicum},
keywords = {oscillatory solutions; difference equations; oscillatory solutions; difference equations},
language = {eng},
number = {1},
pages = {13-27},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The oscillation of an $m$th order perturbed nonlinear difference equation},
url = {http://eudml.org/doc/247854},
volume = {032},
year = {1996},
}

TY - JOUR
AU - Wong, Patricia J. Y.
AU - Agarwal, Ravi P.
TI - The oscillation of an $m$th order perturbed nonlinear difference equation
JO - Archivum Mathematicum
PY - 1996
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 032
IS - 1
SP - 13
EP - 27
AB - We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation \[|\Delta ^{m} y(k)|^{\alpha -1}\Delta ^{m} y(k)+Q(k,y(k-\sigma _{k}), \Delta y(k-\sigma _{k}),\cdots , \Delta ^{m-2}y(k-\sigma _{k}))\]$=P(k,y(k-\sigma _{k}),\Delta y(k-\sigma _{k}),\cdots , \Delta ^{m-1}y(k-\sigma _{k})),~k\ge k_{0}$ where $\alpha >0.$ Examples which dwell upon the importance of our results are also included.
LA - eng
KW - oscillatory solutions; difference equations; oscillatory solutions; difference equations
UR - http://eudml.org/doc/247854
ER -

References

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  1. Agarwal R. P., Difference Equations and Inequalities, Marcel Dekker, New York, 1992. (1992) Zbl0925.39001MR1155840
  2. Agarwal R. P., Properties of solutions of higher order nonlinear difference equations I, An. St. Univ. Iasi 29(1983), 85-96. (1983) MR0739573
  3. Agarwal R. P., Difference calculus with applications to difference equations, in General Inequalities, ed. W. Walter, ISNM 71, Birkhaver Verlag, Basel (1984), 95-160. (1984) Zbl0592.39001MR0821789
  4. Agarwal R. P., Properties of solutions of higher order nonlinear difference equations II, An. St. Univ. Iasi 29(1985), 165-172. (1985) MR0858057
  5. Grace S. R., Lalli B. S., Oscillation theorems for n th order delay equations, J. Math. Anal. Appl. 91(1983), 342-366. (1983) MR0690876
  6. Grace S. R., Lalli B. S., Oscillation theorems for damped differential equations of even order with deviating arguments, SIAM J. Math. Anal. 15(1984), 308-316. (1984) Zbl0542.34057MR0731869
  7. Hooker. J. W., Patula W. T., A second order nonlinear difference equation: oscillation and asymptotic behavior, J. Math. Anal. Appl. 91(1983), 9-29. (1983) Zbl0508.39005MR0688528
  8. Lakshmikantham V., Trigiante D., Difference Equations with Applications to Numerical Analysis, Academic Press, New York, 1988. (1988) MR0939611
  9. Popenda J., Oscillation and nonoscillation theorems for second order difference equations, J. Math. Anal. Appl. 123(1987), 34-38. (1987) Zbl0612.39002MR0881528
  10. Staikos V. A., Basic results on oscillation for differential equations, Hiroshima Math. J. 10(1980), 495-516. (1980) Zbl0453.34055MR0594131
  11. Thandapani E., Oscillatory behavior of solutions of second order nonlinear difference equations, J. Math. Phys. Sci. 25(1991), 451-464. (1991) 
  12. Thandapani E., Sundaram P., Oscillation theorems for some even order nonlinear difference equations, preprint. 
  13. Wong P. J. Y., Agarwal R. P., Oscillation theorems and existence of positive monotone solutions for second order nonlinear difference equations, Math. Comp. Modelling 21(1995), 63-84. (1995) Zbl0820.39002MR1316120
  14. Wong P. J. Y., Agarwal R. P., Oscillation theorems and existence criteria of asymptotically monotone solutions for second order differential equations, Dynamic Systems and Applications, to appear. Zbl0840.34021MR1365834
  15. Wong P. J. Y., Agarwal R. P., Oscillatory behavior of solutions of certain second order nonlinear differential equations, J. Math. Anal. Applic., 198 (1996), 337-354. (198) MR1376268
  16. Wong P. J. Y., Agarwal R. P., Oscillation theorems for certain second order nonlinear difference equations, to appear. Zbl0874.39012MR1422774
  17. Wong P. J. Y., Agarwal R. P., Oscillation and monotone solutions of a second order quasilinear difference equation, to appear. 
  18. Wong P. J. Y., Agarwal R. P., On the oscillation and asymptotically monotone solutions of second order quasilinear differential equations, Appl. Math. Comp., to appear. MR1407599
  19. Wong P. J. Y., Agarwal R. P., Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments, Advances in Difference Equations II, Computers Math. Applic., to appear. MR1666122
  20. Wong Z., Yu J., Oscillation criteria for second order nonlinear difference equations, Funk. Ekv. 34(1991), 313-319. (1991) 

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