Explicit general solution of planar linear discrete systems with constant coefficients and weak delays

Explicit general solution of planar linear discrete systems with constant coefficients and weak delays

Peer-reviewed article not indexed in WoS or Scopus

In this paper, planar linear discrete systems with constant coefficients and two delays $$ x(k+1)=Ax(k)+Bx(k-m)+Cx(k-n) $$ are considered where $k\in\bZ_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon \bZ_0^{\infty}\to\mathbb{R}^2$, $m>n>0$ are fixed integers and $A=(a_{ij})$, $B=(b_{ij})$ and $C=(c_{ij})$ are constant $2\times 2$ matrices. It is assumed that the system considered system is one with weak delays. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $2(m+1)$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and weak delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

In this paper, planar linear discrete systems with constant coefficients and two delays $$ x(k+1)=Ax(k)+Bx(k-m)+Cx(k-n) $$ are considered where $k\in\bZ_0^{\infty}:=\{0,1,\dots,\infty\}$, $x\colon \bZ_0^{\infty}\to\mathbb{R}^2$, $m>n>0$ are fixed integers and $A=(a_{ij})$, $B=(b_{ij})$ and $C=(c_{ij})$ are constant $2\times 2$ matrices. It is assumed that the system considered system is one with weak delays. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $2(m+1)$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and weak delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

Discrete equation, weak delays, explicit solution, dimension of the solutions space.

Discrete equation, weak delays, explicit solution, dimension of the solutions space.

DIBLÍK, J.; HALFAROVÁ, H.

2014

06.03.2013

Springer Nature

1687-1847

Advances in Difference Equations

2013

1

United States of America

1

29

37

https://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-50

http://hdl.handle.net/11012/137963

1687-1847-2013-50