Explicit general solution of planar linear discrete systems with constant coefficients and weak delays
In this paper, planar linear discrete systems with constant coefficients and two delays x ( k + 1 ) = A x ( k ) + B x ( k − m ) + C x ( k − n ) are considered where k ∈ Z 0 ∞ : = { 0 , 1 , … , ∞ } , x : Z 0 ∞ → R 2 , m > n > 0 are fixed integers and A = ( a i j ) , B = ( b i j ) and C = ( c i...
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| Published in: | Advances in difference equations Vol. 2013; no. 1 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
06.03.2013
|
| Subjects: | |
| ISSN: | 1687-1847, 1687-1847 |
| Online Access: | Get full text |
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| Summary: | In this paper, planar linear discrete systems with constant coefficients and two delays
x
(
k
+
1
)
=
A
x
(
k
)
+
B
x
(
k
−
m
)
+
C
x
(
k
−
n
)
are considered where
k
∈
Z
0
∞
:
=
{
0
,
1
,
…
,
∞
}
,
x
:
Z
0
∞
→
R
2
,
m
>
n
>
0
are fixed integers and
A
=
(
a
i
j
)
,
B
=
(
b
i
j
)
and
C
=
(
c
i
j
)
are constant
2
×
2
matrices. It is assumed that the 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.
AMS Subject Classification:
39A06, 39A12. |
|---|---|
| ISSN: | 1687-1847 1687-1847 |
| DOI: | 10.1186/1687-1847-2013-50 |