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

Advances in Difference Equations

Josef Diblk¹  Hana Halfarov² 

[1] Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic;Department of Mathematics, Faculty of Electrical Engineering and Communications, Brno University of Technology, Brno, Czech Republic

关键词: discrete equation;    weak delays;    spectrum;    explicit solution;    dimension of the solutions space;   

DOI  :  10.1186/1687-1847-2013-50

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, planar linear discrete systems with constant coefficients and two delaysx(k+1)=Ax(k)+Bx(k−m)+Cx(k−n)are considered wherek∈Z0∞:={0,1,…,∞},x:Z0∞→R2,m>n>0are fixed integers andA=(aij),B=(bij)andC=(cij)are constant2×2matrices. 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 dimension2(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.

【 授权许可】

CC BY