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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Vydáno v:	Advances in difference equations Ročník 2013; číslo 1
Hlavní autoři:	Diblík, Josef, Halfarová, Hana
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Cham Springer International Publishing 06.03.2013
Témata:	Analysis Difference and Functional Equations Functional Analysis Mathematics Mathematics and Statistics Ordinary Differential Equations Partial Differential Equations Progress in Functional Differential and Difference Equations explicit solution weak delays discrete equation spectrum dimension of the solutions space
ISSN:	1687-1847, 1687-1847
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Popis
Shrnutí:	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