Phases of linear difference equations and symplectic systems

Zuzana Došlá; Denisa Škrabáková

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 3, page 293-308
  • ISSN: 0862-7959

Abstract

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The second order linear difference equation Δ ( r k ▵ x k ) + c k x k + 1 = 0 , ( 1 ) where r k ≠ 0 and k ∈ 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.

How to cite

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Došlá, Zuzana, and Škrabáková, Denisa. "Phases of linear difference equations and symplectic systems." Mathematica Bohemica 128.3 (2003): 293-308. <http://eudml.org/doc/249239>.

@article{Došlá2003,
abstract = {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.},
author = {Došlá, Zuzana, Škrabáková, Denisa},
journal = {Mathematica Bohemica},
keywords = {second order linear difference equation; symplectic system; phase; oscillation; nonoscillation; trigonometric transformation; second order linear difference equation; symplectic system; phase; oscillation; nonoscillation; trigonometric transformation},
language = {eng},
number = {3},
pages = {293-308},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Phases of linear difference equations and symplectic systems},
url = {http://eudml.org/doc/249239},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Došlá, Zuzana
AU - Škrabáková, Denisa
TI - Phases of linear difference equations and symplectic systems
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 3
SP - 293
EP - 308
AB - 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.
LA - eng
KW - second order linear difference equation; symplectic system; phase; oscillation; nonoscillation; trigonometric transformation; second order linear difference equation; symplectic system; phase; oscillation; nonoscillation; trigonometric transformation
UR - http://eudml.org/doc/249239
ER -

References

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  6. Phase matrix of linear differential systems, Čas. Pěst. Mat. 110 (1985), 183–192. (1985) MR0796568
  7. Linear Hamiltonian difference systems: transformations, recessive solutions, generalized reciprocity, Dynamical Systems and Applications 8 (1999), 401–420. (1999) MR1722970
  8. Global Properties of Linear Ordinary Differential Equations, Mathematics and Its Applications (East European Series), Kluwer Acad. Publ., Dordrecht, 1991. (1991) Zbl0784.34009MR1192133
  9. On transformation of solutions of the differential equation y ' ' = Q ( t ) y with a complex coefficient of a real variable, Acta Univ. Palack. Olomucensis, F.R.N. 88 Math. 26 (1987), 57–83. (1987) MR1033331
  10. Otakar Borůvka and Differential Equations, PhD. thesis, MU, Brno, 1998. (1998) 

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