On oscillation and nonoscillation properties of Emden-Fowler difference equations

Mariella Cecchi; Zuzana Došlá; Mauro Marini

Open Mathematics (2009)

  • Volume: 7, Issue: 2, page 322-334
  • ISSN: 2391-5455

Abstract

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A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation Δ ( a n Δ x n α s g n Δ x n ) + b n x n + 1 β s g n x n + 1 = 0 is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

How to cite

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Mariella Cecchi, Zuzana Došlá, and Mauro Marini. "On oscillation and nonoscillation properties of Emden-Fowler difference equations." Open Mathematics 7.2 (2009): 322-334. <http://eudml.org/doc/269183>.

@article{MariellaCecchi2009,
abstract = {A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation \[ \Delta (a\_n \left| \{\Delta x\_n \} \right|^\alpha sgn\Delta x\_n ) + b\_n \left| \{x\_\{n + 1\} \} \right|^\beta sgnx\_\{n + 1\} = 0 \] is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.},
author = {Mariella Cecchi, Zuzana Došlá, Mauro Marini},
journal = {Open Mathematics},
keywords = {Emden-Fowler type difference equation; Oscillation; Nonoscillation; Reciprocal principle; Emden-Fowler difference equation; oscillation; nonoscillation; reciprocal principle},
language = {eng},
number = {2},
pages = {322-334},
title = {On oscillation and nonoscillation properties of Emden-Fowler difference equations},
url = {http://eudml.org/doc/269183},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Mariella Cecchi
AU - Zuzana Došlá
AU - Mauro Marini
TI - On oscillation and nonoscillation properties of Emden-Fowler difference equations
JO - Open Mathematics
PY - 2009
VL - 7
IS - 2
SP - 322
EP - 334
AB - A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation \[ \Delta (a_n \left| {\Delta x_n } \right|^\alpha sgn\Delta x_n ) + b_n \left| {x_{n + 1} } \right|^\beta sgnx_{n + 1} = 0 \] is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.
LA - eng
KW - Emden-Fowler type difference equation; Oscillation; Nonoscillation; Reciprocal principle; Emden-Fowler difference equation; oscillation; nonoscillation; reciprocal principle
UR - http://eudml.org/doc/269183
ER -

References

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  1. [1] Agarwal R.P., Bohner M., Grace S.R., O’Regan D., Discrete oscillation theory, Hindawi Publishing Corporation, New York, 2005 http://dx.doi.org/10.1155/9789775945198[Crossref] Zbl1084.39001
  2. [2] Cecchi M., Došlá Z., Marini M., Nonoscillatory half-linear difference equations and recessive solutions, Adv. Difference Equ., 2005, 2, 193–204 http://dx.doi.org/10.1155/ADE.2005.193[Crossref] Zbl1111.39005
  3. [3] Cecchi M., Došlá Z., Marini M., Vrkoč I., Summation inequalities and half-linear difference equations, J. Math. Anal. Appl., 2005, 302, 1–13 http://dx.doi.org/10.1016/j.jmaa.2004.08.005[Crossref] Zbl1069.39001
  4. [4] Cecchi M., Došlá Z., Marini M., Vrkoč I., Asymptotic properties for half-linear difference equations, Math. Bohem., 2006, 131, 347–363 Zbl1110.39004
  5. [5] Cecchi M., Došlá Z., Marini M., On the growth of nonoscillatory solutions for difference equations with deviating argument, Adv. Difference Equ., 2008, Article ID 505324, 15 pp. Zbl1146.39007
  6. [6] Cecchi M., Došlá Z., Marini M., Intermediate solutions for nonlinear difference equations with p-Laplacian, Advanced Studies in Pure Mathematics, 2009, 53, 45–52 Zbl1179.39004
  7. [7] Huo H.F., Li W.T., Oscillation of certain two-dimensional nonlinear difference systems, Comput. Math. Appl., 2003, 45, 1221–1226 http://dx.doi.org/10.1016/S0898-1221(03)00089-0[Crossref] Zbl1056.39009
  8. [8] Jiang J., Li X., Oscillation and nonoscillation of two-dimensional difference systems, J. Comput. Appl. Math., 2006, 188, 77–88 http://dx.doi.org/10.1016/j.cam.2005.03.054[Crossref] 
  9. [9] Li W.T., Oscillation theorems for second-order nonlinear difference equations, Math. Comput. Modelling, 2000, 31, 71–79 http://dx.doi.org/10.1016/S0895-7177(00)00047-9[Crossref] Zbl1042.39516
  10. [10] Li W.T., Classification schemes for nonoscillatory solutions of two-dimensional nonlinear difference systems, Comput. Math. Appl., 2001, 42, 341–355 http://dx.doi.org/10.1016/S0898-1221(01)00159-6[Crossref] Zbl1006.39013
  11. [11] Wong P.J.Y., Agarwal R.P, Oscillation and monotone solutions of second order quasilinear difference equations, Funkcial. Ekvac., 1996, 39, 491–517 Zbl0871.39005
  12. [12] Zhang G., Cheng S.S., Gao Y., Classification schemes for positive solutions of a second-order nonlinear difference equation, J. Comput. Appl. Math., 1999, 101, 39–51 http://dx.doi.org/10.1016/S0377-0427(98)00189-7[Crossref] Zbl0953.39004

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