| Title:
|
Phases of linear difference equations and symplectic systems (English) |
| Author:
|
Došlá, Zuzana |
| Author:
|
Škrabáková, Denisa |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
128 |
| Issue:
|
3 |
| Year:
|
2003 |
| Pages:
|
293-308 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The second order linear difference equation \[ \Delta (r_k\triangle x_k)+c_kx_{k+1}=0, \qquad \mathrm{(1)}\] where $r_k\ne 0$ and $k\in \mathbb{Z}$, is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Borůvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investigated in connection with phases and trigonometric systems. Some applications to summation of number series are given, too. (English) |
| Keyword:
|
second order linear difference equation |
| Keyword:
|
symplectic system |
| Keyword:
|
phase |
| Keyword:
|
oscillation |
| Keyword:
|
nonoscillation |
| Keyword:
|
trigonometric transformation |
| MSC:
|
39A05 |
| MSC:
|
39A10 |
| MSC:
|
39A11 |
| MSC:
|
39A12 |
| idZBL:
|
Zbl 1055.39026 |
| idMR:
|
MR2012606 |
| DOI:
|
10.21136/MB.2003.134182 |
| . |
| Date available:
|
2009-09-24T22:09:56Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134182 |
| . |
| Reference:
|
[1] C. D. Ahlbrandt, A. C. Peterson: Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations.Kluwer Academic Publ., Boston, 1996. MR 1423802 |
| Reference:
|
[2] M. Bohner, O. Došlý: Disconjugacy and transformations for symplectic systems.Rocky Mountain J. Math. 27 (1997), 707–743. MR 1490271, 10.1216/rmjm/1181071889 |
| Reference:
|
[3] M. Bohner, O. Došlý: Trigonometric transformations of symplectic difference systems.J. Differential Equations 163 (2000), 113–129. MR 1755071, 10.1006/jdeq.1999.3728 |
| Reference:
|
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| Reference:
|
[5] O. Borůvka: Lineare Differentialtransformationen 2. Ordnung.Hochschulbücher für Mathematik. Band 67. VEB, Berlin, 1967; Linear Differential Transformations of the Second Order, The English Univ. Press, London, 1971. MR 0236448 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[8] F. Neuman: Global Properties of Linear Ordinary Differential Equations.Mathematics and Its Applications (East European Series), Kluwer Acad. Publ., Dordrecht, 1991. Zbl 0784.34009, MR 1192133 |
| Reference:
|
[9] S. Staněk: On transformation of solutions of the differential equation $y^{\prime \prime }=Q(t)y$ with a complex coefficient of a real variable.Acta Univ. Palack. Olomucensis, F.R.N. 88 Math. 26 (1987), 57–83. MR 1033331 |
| Reference:
|
[10] P. Šarmanová: Otakar Borůvka and Differential Equations.PhD. thesis, MU, Brno, 1998. |
| . |
