Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay

J Diblík,D Ya Khusainov,Z Šmarda

doi:10.1155/2009/784935

Abstract

Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced.

Highlights

In this paper we deal with the discrete planar systems x k 1 Ax k Bx k − m, 1.1 where m ≥ 0 is a fixed integer, k ∈ Z∞0, A aij and B bij are constant 2 × 2 matrices, and x : Z∞−m → R2
We show that the property of a system to be the system with weak delay is preserved by every nonsingular linear transformation
Together with 2.7, we consider an initial problem y k φ∗ k, 2.8 k ∈ Z0−m with φ∗ : Z0−m → R2 where φ∗ k S−1φ k is the initial function corresponding to the initial function φ in 1.2

Summary

Introduction

Preliminary Notions and Properties We use the following notation: for integers s, q, s ≤ q, we define Zqs : {s, s 1, . Q} where s −∞ or q ∞ are admitted, too. Throughout this paper, using notation Zqs, we always assume s ≤ q. In this paper we deal with the discrete planar systems x k 1 Ax k Bx k − m , 1.1 where m ≥ 0 is a fixed integer, k ∈ Z∞0 , A aij and B bij are constant 2 × 2 matrices, and x : Z∞−m → R2. E.g., in 1, 2 , 1.1 is referred to as a Advances in Difference Equations nondelayed discrete system if m 0 and as a delayed discrete system if m > 0.

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Conclusion