| Title:
|
Asymptotic properties for half-linear difference equations (English) |
| Author:
|
Cecchi, Mariella |
| Author:
|
Došlá, Zuzana |
| Author:
|
Marini, Mauro |
| Author:
|
Vrkoč, Ivo |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
131 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
347-363 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Asymptotic properties of the half-linear difference equation \[ \Delta (a_{n}|\Delta x_{n}|^{\alpha }\mathop {\mathrm sgn}\Delta x_{n} )=b_{n}|x_{n+1}|^{\alpha }\mathop {\mathrm sgn}x_{n+1} \qquad \mathrm{(*)}\] are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to $(*)$ are considered too. Our approach is based on a classification of solutions of $(*)$ and on some summation inequalities for double series, which can be used also in other different contexts. (English) |
| Keyword:
|
half-linear second order difference equation |
| Keyword:
|
nonoscillatory solutions |
| Keyword:
|
Riccati difference equation |
| Keyword:
|
summation inequalities |
| MSC:
|
39A10 |
| MSC:
|
39A11 |
| idZBL:
|
Zbl 1110.39004 |
| idMR:
|
MR2273927 |
| DOI:
|
10.21136/MB.2006.133970 |
| . |
| Date available:
|
2009-09-24T22:27:17Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133970 |
| . |
| Reference:
|
[1] R. P. Agarwal: Difference Equations and Inequalities.2nd Edition, Pure Appl. Math. 228, Marcel Dekker, New York, 2000. Zbl 0952.39001, MR 1740241 |
| Reference:
|
[2] M. Cecchi, Z. Došlá, M. Marini: Positive decreasing solutions of quasi-linear difference equations.Comput. Math. Appl. 42 (2001), 1401–1410. MR 1861536 |
| Reference:
|
[3] M. Cecchi, Z. Došlá, M. Marini: On recessive and dominant solutions for half-linear difference equations.J. Differ. Equ. Appl. 10 (2004), 797–808. MR 2074433 |
| Reference:
|
[4] M. Cecchi, Z. Došlá, M. Marini, I. Vrkoč: Summation inequalities and half-linear difference equations.J. Math. Anal. Appl. 302 (2005), 1–13. MR 2106543 |
| Reference:
|
[5] M. Cecchi, Z. Došlá, M. Marini, I. Vrkoč: Integral conditions for nonoscillation of second order nonlinear differential equations.Nonlinear Anal., T.M.A. 64 (2006), 1278–1289. MR 2200492, 10.1016/j.na.2005.06.035 |
| Reference:
|
[6] M. Cecchi, Z. Došlá, M. Marini, I. Vrkoč: Nonoscillatory solutions for Emden-Fowler type difference equations.Proc. Inf. Conf. Difference Equations. Special Functions and Applications. Munich, 2005. |
| Reference:
|
[7] Z. Došlá, I. Vrkoč: On extension of the Fubini theorem and its application to the second order differential equations.Nonlinear Anal., T.M.A. 57 (2004), 531–548. MR 2062993 |
| Reference:
|
[8] O. Došlý: Half-Linear Differential Equations.Handbook of Differential Equations, Ordinary Differential Equations I., A. Cañada, P. Drábek, A. Fonda (eds.), Elsevier, Amsterdam, 2004. Zbl 1090.34027, MR 2166491 |
| Reference:
|
[9] O. Došlý, P. Řehák: Nonoscillation criteria for half-linear second order difference equations.Comput. Math. Appl. 42 (2001), 453–464. MR 1838006 |
| Reference:
|
[10] O. Došlý, P. Řehák: Recessive solution of half-linear second order difference equations.J. Differ. Equ. Appl. 9 (2003), 49–61. MR 1958302, 10.1080/10236100309487534 |
| Reference:
|
[11] J. W. Hooker, W. T. Patula: A second-order nonlinear difference equation: oscillation and asymptotic behavior.J. Math. Anal. Appl. 91 (1983), 9–29. MR 0688528, 10.1016/0022-247X(83)90088-4 |
| Reference:
|
[12] P. Řehák: Oscillatory properties of second order half-linear difference equations.Czechoslovak Math. J. 51 (2001), 303–321. Zbl 0982.39004, MR 1844312, 10.1023/A:1013790713905 |
| . |
