| Title:
|
Some notes on oscillation of two-dimensional system of difference equations (English) |
| Author:
|
Opluštil, Zdeněk |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
417-428 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Oscillatory properties of solutions to the system of first-order linear difference equations $$ \begin {aligned} \Delta u_k & = q_k v_k \\ \Delta v_k & = -p_k u_{k+1}, \end {aligned} $$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. \endgraf We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011). (English) |
| Keyword:
|
two-dimensional system |
| Keyword:
|
linear difference equation |
| Keyword:
|
oscillatory solution |
| MSC:
|
39A10 |
| MSC:
|
39A21 |
| idZBL:
|
Zbl 06362270 |
| idMR:
|
MR3238851 |
| DOI:
|
10.21136/MB.2014.143866 |
| . |
| Date available:
|
2014-07-14T08:49:19Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143866 |
| . |
| Reference:
|
[1] Agarwal, R. P.: Difference Equations and Inequalities: Theory, Methods and Applications.Pure and Appl. Math. Marcel Dekker, New York (1992). Zbl 0925.39001, MR 1155840 |
| Reference:
|
[2] Chantladze, T., Kandelaki, N., Lomtatidze, A.: Oscillation and nonoscillation criteria for a second order linear equation.Georgian Math. J. 6 (1999), 401-414. Zbl 0944.34025, MR 1692963, 10.1023/A:1022911815254 |
| Reference:
|
[3] Hartman, P.: Ordinary Differential Equations.John Wiley, New York (1964). Zbl 0125.32102, MR 0171038 |
| Reference:
|
[4] Hille, E.: Non-oscillation theorems.Trans. Am. Math. Soc. 64 (1948), 234-252. Zbl 0031.35402, MR 0027925, 10.1090/S0002-9947-1948-0027925-7 |
| Reference:
|
[5] Lomtatidze, A.: Oscillation and nonoscillation criteria for second-order linear differential equations.Georgian Math. J. 4 (1997), 129-138. Zbl 0877.34029, MR 1439591, 10.1023/A:1022978000000 |
| Reference:
|
[6] Lomtatidze, A., Partsvania, N.: Oscillation and nonoscillation criteria for two-dimensional systems of first order linear ordinary differential equations.Georgian Math. J. 6 (1999), 285-298. Zbl 0930.34025, MR 1679448, 10.1023/A:1022187214750 |
| Reference:
|
[7] Nehari, Z.: Oscillation criteria for second-order linear differential equations.Trans. Am. Math. Soc. 85 (1957), 428-445. Zbl 0078.07602, MR 0087816, 10.1090/S0002-9947-1957-0087816-8 |
| Reference:
|
[8] Opluštil, Z.: Oscillatory criteria for two-dimensional system of difference equations.Tatra Mt. Math. Publ. 48 (2011), 153-163. Zbl 1265.39019, MR 2841115 |
| Reference:
|
[9] Polák, L.: Oscillation and nonoscillation criteria for two-dimensional systems of linear ordinary differential equations.Georgian Math. J. 11 (2004), 137-154. Zbl 1064.34019, MR 2065547 |
| Reference:
|
[10] Wintner, A.: A criterion of oscillatory stability.Q. Appl. Math. 7 (1949), 115-117. Zbl 0032.34801, MR 0028499, 10.1090/qam/28499 |
| Reference:
|
[11] Wintner, A.: On the non-existence of conjugate points.Am. J. Math. 73 (1951), 368-380. MR 0042005, 10.2307/2372182 |
| . |
