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

dc.contributor.authorDiblík, Josefcs
dc.contributor.authorBoháčková, Hanacs
dc.coverage.issue1cs
dc.coverage.volume2013cs
dc.date.issued2013-03-06cs
dc.description.abstractIn 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.en
dc.description.abstractIn 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.en
dc.formattextcs
dc.format.extent1-29cs
dc.format.mimetypeapplication/pdfcs
dc.identifier.citationAdvances in Difference Equations. 2013, vol. 2013, issue 1, p. 1-29.en
dc.identifier.doi10.1186/1687-1847-2013-50cs
dc.identifier.issn1687-1847cs
dc.identifier.orcid0000-0001-5009-316Xcs
dc.identifier.orcid0000-0002-1244-2733cs
dc.identifier.other98366cs
dc.identifier.researcheridD-3530-2014cs
dc.identifier.scopus6701633618cs
dc.identifier.urihttp://hdl.handle.net/11012/137963
dc.language.isoencs
dc.publisherSpringer Naturecs
dc.relation.ispartofAdvances in Difference Equationscs
dc.relation.urihttps://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-50cs
dc.rightsCreative Commons Attribution 2.0 Genericcs
dc.rights.accessopenAccesscs
dc.rights.sherpahttp://www.sherpa.ac.uk/romeo/issn/1687-1847/cs
dc.rights.urihttp://creativecommons.org/licenses/by/2.0/cs
dc.subjectDiscrete equationen
dc.subjectweak delaysen
dc.subjectexplicit solutionen
dc.subjectdimension of the solutions space.en
dc.subjectDiscrete equation
dc.subjectweak delays
dc.subjectexplicit solution
dc.subjectdimension of the solutions space.
dc.titleExplicit general solution of planar linear discrete systems with constant coefficients and weak delaysen
dc.title.alternativeExplicit general solution of planar linear discrete systems with constant coefficients and weak delaysen
dc.type.driverarticleen
dc.type.statusPeer-revieweden
dc.type.versionpublishedVersionen
sync.item.dbidVAV-98366en
sync.item.dbtypeVAVen
sync.item.insts2025.10.14 14:14:29en
sync.item.modts2025.10.14 10:34:16en
thesis.grantorVysoké učení technické v Brně. Fakulta stavební. Ústav matematiky a deskriptivní geometriecs
thesis.grantorVysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií. Ústav matematikycs
thesis.grantorVysoké učení technické v Brně. Fakulta stavební. AdMaS - dotace 12/2010cs
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