Advances in Difference Equations
Volume 2009 (2009), Article ID 784935, 18 pages
doi:10.1155/2009/784935
Research Article

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

J. Diblík,1 D. Ya. Khusainov,2 and Z. Šmarda1

1Brno University of Technology, Brno, Czech Republic
2Kiev University, Kiev, Ukraine

Received 19 January 2009; Accepted 30 March 2009

Academic Editor: Ulrich Krause

Copyright © 2009 J. Diblík et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

1.1. Preliminary Notions and Properties

We use the following notation: for integers 𝑠 , 𝑞 , 𝑠 ≤ 𝑞 , we define ℤ 𝑞 𝑠 ∶ = { 𝑠 , 𝑠 + 1 , … , 𝑞 } where 𝑠 = − ∞ or 𝑞 = ∞ are admitted, too. Throughout this paper, using notation ℤ 𝑞 𝑠 , we always assume 𝑠 ≤ 𝑞 . In this paper we deal with the discrete planar systems 𝑥 ( 𝑘 + 1 ) = 𝐴 𝑥 ( 𝑘 ) + 𝐵 𝑥 ( 𝑘 − 𝑚 ) , ( 1 . 1 ) where 𝑚 ≥ 0 is a fixed integer, 𝑘 ∈ ℤ ∞ 0 , 𝐴 = ( 𝑎 𝑖 𝑗 ) and 𝐵 = ( 𝑏 𝑖 𝑗 ) are constant 2 × 2 matrices, and 𝑥 ∶ ℤ ∞ − 𝑚 → ℝ 2 . Following the terminology (used, e.g., in [1, 2]), (1.1) is referred to as a nondelayed discrete system if 𝑚 = 0 and as a delayed discrete system if 𝑚 > 0 . Together with (1.1), we consider an initial (Cauchy) problem 𝑥 ( 𝑘 ) = 𝜑 ( 𝑘 ) , ( 1 . 2 ) where 𝑘 = − 𝑚 , − 𝑚 + 1 , … , 0 with 𝜑 ∶ ℤ 0 − 𝑚 → ℝ 2 . We will investigate only the case 𝑚 > 0 since the solution of (1.1) for 𝑚 = 0 is given by the known formula 𝑥 ( 𝑘 ) = ( 𝐴 + 𝐵 ) 𝑘 𝜑 ( 0 ) for 𝑘 ∈ ℤ ∞ 1 .

The existence and uniqueness of the solution of the initial problems (1.1) and (1.2) on ℤ ∞ − 𝑚 are obvious. We recall that the solution 𝑥 ∶ ℤ ∞ − 𝑚 → ℝ 2 of (1.1) and (1.2) is defined as an infinite sequence { 𝑥 ( − 𝑚 ) = 𝜑 ( − 𝑚 ) , 𝑥 ( − 𝑚 + 1 ) = 𝜑 ( − 𝑚 + 1 ) , … , 𝑥 ( 0 ) = 𝜑 ( 0 ) , 𝑥 ( 1 ) , 𝑥 ( 2 ) , … , 𝑥 ( 𝑘 ) , … } , ( 1 . 3 ) such that, for any 𝑘 ∈ ℤ ∞ 0 , equality (1.1) holds.

The space of all initial data (1.2) with 𝜑 ∶ ℤ 0 − 𝑚 → ℝ 2 is obviously 2 ( 𝑚 + 1 ) -dimensional. Below we describe the fact that, among the systems (1.1), there are such systems that their space of solutions, being initially 2 ( 𝑚 + 1 ) -dimensional, on a reduced interval turns into a space having dimension less than 2 ( 𝑚 + 1 ) .

1.2. Systems with Weak Delay

We consider the system (1.1) and we look for a solution having the form 𝑥 ( 𝑘 ) = 𝜆 𝑘 , 𝑘 ∈ ℤ ∞ − 𝑚 , 𝜆 = c o n s t with a 𝜆 ≠ 0 . The usual procedure leads to a characteristic equation d e t ( 𝐴 + 𝜆 − 𝑚 𝐵 − 𝜆 𝐼 ) = 0 , ( 1 . 4 ) where 𝐼 is the unit 2 × 2 matrix. Together with (1.1), we consider a system with the terms containing delays omitted 𝑥 ( 𝑘 + 1 ) = 𝐴 𝑥 ( 𝑘 ) , ( 1 . 5 ) and the characteristic equation d e t ( 𝐴 − 𝜆 𝐼 ) = 0 . ( 1 . 6 )

Definition 1.1. The system (1.1) is called a system with weak delay if the characteristic equations (1.4) and (1.6) corresponding to systems (1.1) and (1.5) are equal, that is, if for every 𝜆 ∈ ℂ ⧵ { 0 } d e t ( 𝐴 + 𝜆 − 𝑚 𝐵 − 𝜆 𝐼 ) = d e t ( 𝐴 − 𝜆 𝐼 ) . ( 1 . 7 )

We consider a linear transformation 𝑥 ( 𝑘 ) = 𝒮 𝑦 ( 𝑘 ) , ( 1 . 8 ) with a nonsingular 2 × 2 matrix 𝒮 . Then the discrete system for 𝑦 is 𝑦 ( 𝑘 + 1 ) = 𝐴 𝒮 𝑦 ( 𝑘 ) + 𝐵 𝒮 𝑦 ( 𝑘 − 𝑚 ) , ( 1 . 9 ) with 𝐴 𝒮 = 𝒮 − 1 𝐴 𝒮 , 𝐵 𝒮 = 𝒮 − 1 𝐵 𝒮 . We show that the property of a system to be the system with weak delay is preserved by every nonsingular linear transformation.

Lemma 1.2. If the system (1.1) is a system with weak delay, then its arbitrary linear nonsingular transformation (1.8) again leads to a system with the weak delay (1.9).

Proof. It is easy to show that 𝐴 d e t 𝒮 + 𝜆 − 𝑚 𝐵 𝒮 𝐴 − 𝜆 𝐼 = d e t 𝒮 − 𝜆 𝐼 ( 1 . 1 0 ) holds since 𝐴 d e t 𝒮 + 𝜆 − 𝑚 𝐵 𝒮 𝒮 − 𝜆 𝐼 = d e t − 1 𝐴 𝒮 + 𝜆 − 𝑚 𝒮 − 1 𝒮 𝐵 𝒮 − 𝜆 𝐼 = d e t − 1 ( 𝐴 + 𝜆 − 𝑚 𝐵 − 𝜆 𝐼 ) 𝒮 = d e t ( 𝐴 + 𝜆 − 𝑚 𝐴 𝐵 − 𝜆 𝐼 ) , d e t 𝒮 𝒮 − 𝜆 𝐼 = d e t − 1 𝒮 𝐴 𝒮 − 𝜆 𝐼 = d e t − 1 ( 𝐴 − 𝜆 𝐼 ) 𝒮 = d e t ( 𝐴 − 𝜆 𝐼 ) , ( 1 . 1 1 ) and the equality d e t ( 𝐴 + 𝜆 − 𝑚 𝐵 − 𝜆 𝐼 ) = d e t ( 𝐴 − 𝜆 𝐼 ) ( 1 . 1 2 ) is assumed.

1.3. Necessary and Sufficient Conditions Determining the Weak Delay

In the forthcoming theorem, we give conditions, in terms of determinants, indicating whether a system is a system with weak delay or not.

Theorem 1.3. System (1.1) is a system with weak delay if and only if the following three conditions hold simultaneously: 𝑏 1 1 + 𝑏 2 2 | | | | 𝑏 = 0 , 1 1 𝑏 1 2 𝑏 2 1 𝑏 2 2 | | | | | | | | 𝑎 = 0 , 1 1 𝑎 1 2 𝑏 2 1 𝑏 2 2 | | | | + | | | | 𝑏 1 1 𝑏 1 2 𝑎 2 1 𝑎 2 2 | | | | = 0 . ( 1 . 1 3 )

Proof. We start with computing the determinant (1.4). We get d e t ( 𝐴 + 𝜆 − 𝑚 | | | | 𝑎 𝐵 − 𝜆 𝐼 ) = 1 1 + 𝑏 1 1 𝜆 − 𝑚 − 𝜆 𝑎 1 2 + 𝑏 1 2 𝜆 − 𝑚 𝑎 2 1 + 𝑏 2 1 𝜆 − 𝑚 𝑎 2 2 + 𝑏 2 2 𝜆 − 𝑚 | | | | = | | | | 𝑎 − 𝜆 1 1 − 𝜆 𝑎 1 2 𝑎 2 1 𝑎 2 2 | | | | − 𝜆 − 𝜆 − 𝑚 + 1 𝑏 1 1 + 𝑏 2 2 + 𝜆 − 𝑚 | | | | 𝑎 1 1 𝑎 1 2 𝑏 2 1 𝑏 2 2 | | | | + | | | | 𝑏 1 1 𝑏 1 2 𝑎 2 1 𝑎 2 2 | | | | + 𝜆 − 2 𝑚 | | | | 𝑏 1 1 𝑏 1 2 𝑏 2 1 𝑏 2 2 | | | | . ( 1 . 1 4 ) Now we see that, for (1.7) to hold, that is, d e t ( 𝐴 + 𝜆 − 𝑚 | | | | 𝑎 𝐵 − 𝜆 𝐼 ) = d e t ( 𝐴 − 𝜆 𝐼 ) = 1 1 − 𝜆 𝑎 1 2 𝑎 2 1 𝑎 2 2 | | | | − 𝜆 , ( 1 . 1 5 ) conditions (1.13) are both necessary and sufficient.

Remark 1.4. It is easy to see that conditions (1.13) are equivalent to t r 𝐵 = d e t 𝐵 = 0 , d e t ( 𝐴 + 𝐵 ) = d e t 𝐴 . ( 1 . 1 6 )

1.4. Problem under Consideration

The aim of this paper is to show that the dimension of the space of all solutions, being initially equal to the dimension 2 ( 𝑚 + 1 ) of the space of initial data (1.2) generated by discrete functions 𝜑 , is, after several steps, reduced (on an interval of the form ℤ ∞ 𝑠 with an 𝑠 > 0 ) to a dimension less than the initial one. In other words, we will show that the 2 ( 𝑚 + 1 ) -dimensional space of all solutions of (1.1) is reduced to a less-dimensional space of solutions on ℤ ∞ 𝑠 . This problem is solved directly by explicitly computing the corresponding solutions of the Cauchy problems with each of the cases arising being considered. The underlying idea for such investigation is simple. If (1.1) is a system with weak delay, then the corresponding characteristic equation has only two eigenvalues instead of 2 ( 𝑚 + 1 ) eigenvalues in the case of systems with nonweak delay. This explains why the dimension of the space of solutions becomes less than the initial one. The final results (Theorems 2.5–2.8) provide the dimension of the space of solutions.

1.5. Auxiliary Formula

For the reader's convenience we recall one explicit formula (see, e.g., [3]) for the solutions of linear scalar discrete nondelayed equations used in this paper. We consider the first-order linear discrete nonhomogeneous equation 𝑤 𝑘 ( 𝑘 + 1 ) = 𝑎 𝑤 ( 𝑘 ) + 𝑔 ( 𝑘 ) , 𝑤 0 = 𝑤 0 , 𝑘 ∈ ℤ ∞ 𝑘 0 , ( 1 . 1 7 ) with 𝑎 ∈ ℂ and 𝑔 ∶ ℤ ∞ 𝑘 0 → ℂ . Then it is easy to verify that 𝑤 ( 𝑘 ) = 𝑎 𝑘 − 𝑘 0 𝑤 0 + 𝑘 − 1 𝑟 = 𝑘 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 − 1 − 𝑟 𝑔 ( 𝑟 ) , 𝑘 ∈ ℤ ∞ 𝑘 0 + 1 . ( 1 . 1 8 ) Throughout the paper, we adopt the customary notation for the sum: ∑ ℓ 𝑖 = ℓ + 𝑠 0 𝑥 0 2 0 0 𝑑 ℱ ( 𝑖 ) = 0 where ℓ is an integer, 𝑠 is a positive integer and, “ ℱ " denotes the function considered independently of whether it is defined for indicated arguments or not.

2. Results

If (1.7) holds, then (1.4) and (1.6) have only two (and the same) roots simultaneously. In order to prove the properties of the family of solutions of (1.1) formulated in Section 1.4, we will separately discuss all the possible combinations of roots, that is, the cases of two real and distinct roots, a couple of complex conjugate roots, and, finally, a two-fold real root.

2.1. Jordan Forms of Matrix 𝐀 and Corresponding Solutions of The Problem (1.1), (1.2)

It is known that, for every matrix 𝐴 , there exists a nonsingular matrix 𝑆 transforming it to the corresponding Jordan matrix form Λ . This means that Λ = 𝑆 − 1 𝐴 𝑆 , ( 2 . 1 ) where Λ has the following possible forms, depending on the roots of the characteristic equation (1.6), that is, on the roots of 𝜆 2 − 𝑎 1 1 + 𝑎 2 2 𝑎 𝜆 + 1 1 𝑎 2 2 − 𝑎 1 2 𝑎 2 1 = 0 . ( 2 . 2 ) If (2.2) has two real distinct roots 𝜆 1 , 𝜆 2 , then 𝜆 Λ = 1 0 0 𝜆 2 , ( 2 . 3 ) if the roots are complex conjugate, that is, 𝜆 1 , 2 = 𝑝 ± 𝑖 𝑞 with 𝑞 ≠ 0 , then Λ = 𝑝 𝑞 − 𝑞 𝑝 , ( 2 . 4 ) and, finally, in the case of one two-fold real root 𝜆 1 , 2 = 𝜆 , we have either Λ = 𝜆 0 0 𝜆 ( 2 . 5 ) or Λ = 𝜆 1 0 𝜆 . ( 2 . 6 ) The transformation 𝑦 ( 𝑘 ) = 𝑆 − 1 𝑥 ( 𝑘 ) transforms (1.1) into a system 𝑦 ( 𝑘 + 1 ) = Λ 𝑦 ( 𝑘 ) + 𝐵 ∗ 𝑦 ( 𝑘 − 𝑚 ) , 𝑘 ∈ ℤ ∞ 0 , ( 2 . 7 ) with 𝐵 ∗ = 𝑆 − 1 𝐵 𝑆 , 𝐵 ∗ = ( 𝑏 ∗ 𝑖 𝑗 ) , 𝑖 , 𝑗 = 1 , 2 . Together with (2.7), we consider an initial problem 𝑦 ( 𝑘 ) = 𝜑 ∗ ( 𝑘 ) , ( 2 . 8 ) 𝑘 ∈ ℤ 0 − 𝑚 with 𝜑 ∗ ∶ ℤ 0 − 𝑚 → ℝ 2 where 𝜑 ∗ ( 𝑘 ) = 𝑆 − 1 𝜑 ( 𝑘 ) is the initial function corresponding to the initial function 𝜑 in (1.2).

Below we consider all four possible cases (2.3)–(2.6) separately.

We define Φ 1 ( 𝑘 ) ∶ = 0 , 𝜑 ∗ 1 ( 𝑘 ) 𝑇 , Φ 2 𝜑 ( 𝑘 ) ∶ = ∗ 2 ( 𝑘 ) , 0 𝑇 , 𝑘 ∈ ℤ 0 − 𝑚 . ( 2 . 9 ) Assuming that the system (1.1) is a system with weak delay, the system (2.7), due to Lemma 1.2, is a system with weak delay again.

2.1.1. The Case (2.3) of Two Real Distinct Roots

In this case, we have Λ 𝑘 = d i a g ( 𝜆 𝑘 1 , 𝜆 𝑘 2 ) . The necessary and sufficient conditions (1.13) for (2.7) turn into 𝑏 ∗ 1 1 + 𝑏 ∗ 2 2 | | | | 𝑏 = 0 , ( 2 . 1 0 ) ∗ 1 1 𝑏 ∗ 1 2 𝑏 ∗ 2 1 𝑏 ∗ 2 2 | | | | = 𝑏 ∗ 1 1 𝑏 ∗ 2 2 − 𝑏 ∗ 1 2 𝑏 ∗ 2 1 | | | | 𝜆 = 0 , ( 2 . 1 1 ) 1 0 𝑏 ∗ 2 1 𝑏 ∗ 2 2 | | | | + | | | | 𝑏 ∗ 1 1 𝑏 ∗ 1 2 0 𝜆 2 | | | | = 𝜆 1 𝑏 ∗ 2 2 + 𝜆 2 𝑏 ∗ 1 1 = 0 . ( 2 . 1 2 ) Since 𝜆 1 ≠ 𝜆 2 , (2.10), (2.12) yield 𝑏 ∗ 1 1 = 𝑏 ∗ 2 2 = 0 . Then, from (2.11), we get 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = 0 , so either 𝑏 ∗ 2 1 = 0 or 𝑏 ∗ 1 2 = 0 .

Theorem 2.1. Let (1.1) be a system with weak delay and (2.2) admit two real distinct roots 𝜆 1 , 𝜆 2 . Then 𝑏 ∗ 1 1 = 𝑏 ∗ 2 2 = 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = 0 . The solution of the initial problems (1.1) and (1.2) is 𝑥 ( 𝑘 ) = 𝑆 𝑦 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ − 𝑚 where 𝑦 ( 𝑘 ) has, in the case 𝑏 ∗ 2 1 = 0 , the form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜑 𝑦 ( 𝑘 ) = ∗ ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 1 2 𝑘 − 1 𝑟 = 0 𝜆 1 𝑘 − 1 − 𝑟 Φ 2 ( 𝑟 − 𝑚 ) , i f 𝑘 ∈ ℤ 1 𝑚 + 1 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 1 2 𝑚 𝑟 = 0 𝜆 1 𝑘 − 1 − 𝑟 Φ 2 ( 𝑟 − 𝑚 ) + Φ 2 ( 0 ) 𝑘 − 1 𝑟 = 𝑚 + 1 𝜆 1 𝑘 − 1 − 𝑟 𝜆 2 𝑟 − 𝑚 , i f 𝑘 ∈ ℤ ∞ 𝑚 + 2 , ( 2 . 1 3 ) and, in the case 𝑏 ∗ 1 2 = 0 , the form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜑 𝑦 ( 𝑘 ) = ∗ ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 2 1 𝑘 − 1 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 2 𝑘 − 1 − 𝑟 Φ 1 ( 𝑟 − 𝑚 ) , i f 𝑘 ∈ ℤ 1 𝑚 + 1 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 2 1 𝑚 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 2 𝑘 − 1 − 𝑟 Φ 1 ( 𝑟 − 𝑚 ) + Φ 1 ( 0 ) 𝑘 − 1 𝑟 = 𝑚 + 1 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑟 − 𝑚 𝜆 2 𝑘 − 1 − 𝑟 , i f 𝑘 ∈ ℤ ∞ 𝑚 + 2 . ( 2 . 1 4 )

Proof. In the case considered we have 𝑏 ∗ 1 1 = 𝑏 ∗ 2 2 = 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = 0 and the transformed system (2.7) takes either the form 𝑦 1 ( 𝑘 + 1 ) = 𝜆 1 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 2 𝑦 2 ( 𝑘 − 𝑚 ) , 𝑘 ∈ ℤ ∞ 0 , 𝑦 ( 2 . 1 5 ) 2 ( 𝑘 + 1 ) = 𝜆 2 𝑦 2 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ 0 , ( 2 . 1 6 ) if 𝑏 ∗ 2 1 = 0 or the form 𝑦 1 ( 𝑘 + 1 ) = 𝜆 1 𝑦 1 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ 0 𝑦 , ( 2 . 1 7 ) 2 ( 𝑘 + 1 ) = 𝜆 2 𝑦 2 ( 𝑘 ) + 𝑏 ∗ 2 1 𝑦 1 ( 𝑘 − 𝑚 ) , 𝑘 ∈ ℤ ∞ 0 , ( 2 . 1 8 ) if 𝑏 ∗ 1 2 = 0 . We investigate only the initial problem (2.15), (2.16), (2.8) since the initial problem (2.17), (2.18), (2.8) can be examined in a similar way. From (2.16) and (2.8), we get 𝑦 2 𝜑 ( 𝑘 ) = ∗ 2 ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , 𝜆 𝑘 2 𝜑 ∗ 2 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 1 . ( 2 . 1 9 ) Then (2.15) becomes 𝑦 1 𝜆 ( 𝑘 + 1 ) = 1 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 2 𝜑 ∗ 2 ( 𝑘 − 𝑚 ) , i f 𝑘 ∈ ℤ 𝑚 0 , 𝜆 1 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 2 𝜆 2 𝑘 − 𝑚 𝜑 ∗ 2 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 𝑚 + 1 . ( 2 . 2 0 ) First we solve this equation for 𝑘 ∈ ℤ 𝑚 0 . This means that we consider the problem 𝑦 1 ( 𝑘 + 1 ) = 𝜆 1 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 2 𝜑 ∗ 2 ( 𝑘 − 𝑚 ) , 𝑘 ∈ ℤ ∞ 0 , 𝑦 1 ( 0 ) = 𝜑 ∗ 1 ( 0 ) . ( 2 . 2 1 ) With the aid of formula (1.18), we get 𝑦 1 ( 𝑘 ) = 𝜆 𝑘 1 𝜑 ∗ 1 ( 0 ) + 𝑏 ∗ 1 2 𝑘 − 1 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜑 ∗ 2 ( 𝑟 − 𝑚 ) , 𝑘 ∈ ℤ 1 𝑚 + 1 . ( 2 . 2 2 ) Now we solve (2.20) for 𝑘 ∈ ℤ ∞ 𝑚 + 1 , that is, we consider the problem (with initial data deduced from (2.22) 𝑦 1 ( 𝑘 + 1 ) = 𝜆 1 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 2 𝜆 2 𝑘 − 𝑚 𝜑 ∗ 2 ( 0 ) , 𝑘 ∈ ℤ ∞ 𝑚 + 1 , 𝑦 1 ( 𝑚 + 1 ) = 𝜆 1 𝑚 + 1 𝜑 ∗ 1 ( 0 ) + 𝑏 ∗ 𝑚 1 2 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑚 − 𝑟 𝜑 ∗ 2 ( 𝑟 − 𝑚 ) . ( 2 . 2 3 ) Applying formula (1.18) yields (for 𝑘 ∈ ℤ ∞ 𝑚 + 2 ) 𝑦 1 ( 𝑘 ) = 𝜆 1 𝑘 − ( 𝑚 + 1 ) 𝑦 1 ( 𝑚 + 1 ) + 𝑏 ∗ 1 2 𝜑 ∗ 2 ( 0 ) 𝑘 − 1 𝑟 = 𝑚 + 1 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜆 2 𝑟 − 𝑚 = 𝜆 1 𝑘 − ( 𝑚 + 1 ) 𝜆 1 𝑚 + 1 𝜑 ∗ 1 ( 0 ) + 𝑏 ∗ 𝑚 1 2 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑚 − 𝑟 𝜑 ∗ 2 ( 𝑟 − 𝑚 ) + 𝑏 ∗ 1 2 𝜑 ∗ 2 ( 0 ) 𝑘 − 1 𝑟 = 𝑚 + 1 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜆 2 𝑟 − 𝑚 = 𝜆 𝑘 1 𝜑 ∗ 1 ( 0 ) + 𝑏 ∗ 𝑚 1 2 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜑 ∗ 2 ( 𝑟 − 𝑚 ) + 𝑏 ∗ 1 2 𝜑 ∗ 2 ( 0 ) 𝑘 − 1 𝑟 = 𝑚 + 1 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜆 2 𝑟 − 𝑚 . ( 2 . 2 4 ) Picking up all particular cases (2.8), (2.22), and (2.24), we have 𝑦 1 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜑 ( 𝑘 ) = ∗ 1 ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , 𝜆 𝑘 1 𝜑 ∗ 1 ( 0 ) + 𝑏 ∗ 1 2 𝑘 − 1 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜑 ∗ 2 ( 𝑟 − 𝑚 ) , i f 𝑘 ∈ ℤ 1 𝑚 + 1 , 𝜆 𝑘 1 𝜑 ∗ 1 ( 0 ) + 𝑏 ∗ 1 2 𝑚 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜑 ∗ 2 ( 𝑟 − 𝑚 ) + 𝜑 ∗ 2 ( 0 ) 𝑘 − 1 𝑟 = 𝑚 + 1 0 𝑥 0 2 0 0 𝑑 𝜆 1 𝑘 − 1 − 𝑟 𝜆 2 𝑟 − 𝑚 , i f 𝑘 ∈ ℤ ∞ 𝑚 + 2 . ( 2 . 2 5 ) Now, taking into account (2.9), the formula (2.13) is a consequence of (2.19) and (2.25). The formula (2.14) can be proved in a similar way.
Finally, we note that both formulas (2.13) and (2.14) remain valid for 𝑏 ∗ 1 2 = 𝑏 ∗ 2 1 = 0 as well. In this case, the transformed system (2.7) reduces to a system without delay.

2.1.2. The Case (2.4) of Two Complex Conjugate Roots

The necessary and sufficient conditions (1.13) for (2.7) take the forms (2.10) and (2.11) and | | | | 𝑏 𝑝 𝑞 ∗ 2 1 𝑏 ∗ 2 2 | | | | + | | | | 𝑏 ∗ 1 1 𝑏 ∗ 1 2 | | | | 𝑏 − 𝑞 𝑝 = 𝑝 ∗ 1 1 + 𝑏 ∗ 2 2 𝑏 + 𝑞 ∗ 1 2 − 𝑏 ∗ 2 1 = 0 . ( 2 . 2 6 ) The system of conditions (2.10), (2.11), and (2.26) gives 𝑏 ∗ 1 2 = 𝑏 ∗ 2 1 , ( 𝑏 ∗ 1 1 ) 2 = − ( 𝑏 ∗ 1 2 ) 2 and admits only one possibility, namely, 𝑏 ∗ 1 1 = 𝑏 ∗ 2 2 = 𝑏 ∗ 1 2 = 𝑏 ∗ 2 1 = 0 . ( 2 . 2 7 ) Consequently, 𝐵 ∗ = 0 and 𝐵 = 0 as well. The initial problems (1.1) and (1.2) reduces to a problem without delay 𝑥 ( 𝑘 + 1 ) = 𝐴 𝑥 ( 𝑘 ) , 𝑘 ∈ ℤ 0 − 𝑚 , 𝑥 ( 𝑘 ) = 𝜑 ( 𝑘 ) , 𝑘 ∈ ℤ 0 − 𝑚 , ( 2 . 2 8 ) and, obviously, 𝑥 ( 𝑘 ) = 𝜑 ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , 𝐴 𝑘 𝜑 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 1 . ( 2 . 2 9 )

2.1.3. The Case (2.5) of Two-Fold Real Root

We have Λ 𝑘 = d i a g ( 𝜆 𝑘 , 𝜆 𝑘 ) . The necessary and sufficient conditions (1.13) are, for (2.7), reduced to (2.10), (2.11), and | | | | 𝑏 𝜆 0 ∗ 2 1 𝑏 ∗ 2 2 | | | | + | | | | 𝑏 ∗ 1 1 𝑏 ∗ 1 2 | | | | 𝑏 0 𝜆 = 𝜆 ∗ 1 1 + 𝑏 ∗ 2 2 = 0 . ( 2 . 3 0 ) From (2.10), (2.11), and (2.30), we get 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = − ( 𝑏 ∗ 1 1 ) 2 . Now we will analyse the two possible cases: 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = 0 and 𝑏 ∗ 1 2 𝑏 ∗ 2 1 ≠ 0 .

The Case 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = 0

Theorem 2.2. Let (1.1) be a system with weak delay, (2.2) admit a two-fold root 𝜆 1 , 2 = 𝜆 , 𝑏 ∗ 1 2 𝑏 ∗ 2 1 = 0 and the matrix Λ has the form (2.5). Then the solution of the initial problems (1.1) and (1.2) is 𝑥 ( 𝑘 ) = 𝑆 𝑦 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ − 𝑚 where 𝑦 ( 𝑘 ) has, in the case 𝑏 ∗ 2 1 = 0 , the form 𝑦 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜑 ( 𝑘 ) = ∗ ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 1 2 𝑘 − 1 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 𝑘 − 1 − 𝑟 Φ 2 ( 𝑟 − 𝑚 ) , i f 𝑘 ∈ ℤ 1 𝑚 + 1 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 1 2 𝑚 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 𝑘 − 1 − 𝑟 Φ 2 ( 𝑟 − 𝑚 ) + ( 𝑘 − 1 − 𝑚 ) 𝜆 𝑘 − 1 − 𝑚 Φ 2 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 𝑚 + 2 , ( 2 . 3 1 ) and, in the case 𝑏 ∗ 1 2 = 0 , the form 𝑦 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜑 ( 𝑘 ) = ∗ ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 2 1 𝑘 − 1 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 𝑘 − 1 − 𝑟 Φ 1 ( 𝑟 − 𝑚 ) , i f 𝑘 ∈ ℤ 1 𝑚 + 1 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑏 ∗ 2 1 𝑚 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 𝑘 − 1 − 𝑟 Φ 1 ( 𝑟 − 𝑚 ) + ( 𝑘 − 1 − 𝑚 ) 𝜆 𝑘 − 1 − 𝑚 Φ 1 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 𝑚 + 2 . ( 2 . 3 2 )

Proof. The assumption 𝑏 ∗ 2 1 = 0 or 𝑏 ∗ 1 2 = 0 leads to 𝑏 ∗ 1 1 = 𝑏 ∗ 2 2 = 0 . Then the following cases arise. Either 𝑏 ∗ 1 2 ≠ 0 , 𝑏 ∗ 2 1 = 0 or 𝑏 ∗ 1 2 = 0 , 𝑏 ∗ 2 1 ≠ 0 or 𝑏 ∗ 1 2 = 𝑏 ∗ 2 1 = 0 . The latter case is covered by the above formulas (2.31) and (2.32) since it can be treated as system (2.28) considered previously (with 𝐴 = Λ ) when 𝐵 ∗ = 𝐵 = 0 , and the corresponding solution is described by the formula (2.29). If 𝑏 ∗ 1 2 ≠ 0 , then (2.7) turns into the system 𝑦 1 ( 𝑘 + 1 ) = 𝜆 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 2 𝑦 2 ( 𝑘 − 𝑚 ) , 𝑘 ∈ ℤ ∞ 0 , 𝑦 2 ( 𝑘 + 1 ) = 𝜆 𝑦 2 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ 0 , ( 2 . 3 3 ) and, if 𝑏 ∗ 2 1 ≠ 0 , then (2.7) turns into the system 𝑦 1 ( 𝑘 + 1 ) = 𝜆 𝑦 1 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ 0 , 𝑦 2 ( 𝑘 + 1 ) = 𝜆 𝑦 2 ( 𝑘 ) + 𝑏 ∗ 2 1 𝑦 1 ( 𝑘 − 𝑚 ) , 𝑘 ∈ ℤ ∞ 0 . ( 2 . 3 4 ) System (2.33) can be solved in much the same way as the systems (2.15) and (2.16) if we put 𝜆 1 = 𝜆 2 = 𝜆 , and the discussion of the system (2.34) copies the discussion of the systems (2.17) and (2.18) with 𝜆 1 = 𝜆 2 = 𝜆 . Formulas (2.31) and (2.32) are consequences of (2.13) and (2.14).

The Case 𝑏 ∗ 1 2 𝑏 ∗ 2 1 ≠ 0
For 𝑘 ∈ ℤ 0 − 𝑚 we define 𝑏 Φ ( 𝑘 ) ∶ = ∗ 1 1 𝜑 ∗ 1 𝑏 ( 𝑘 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 𝑘 ) , − ( 𝑏 ∗ 1 1 ) 2 𝑏 ∗ 1 2 𝜑 ∗ 1 𝑏 ( 𝑘 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 𝑘 ) 𝑇 . ( 2 . 3 5 )

Theorem 2.3. Let the system (1.1) be a system with weak delay, (2.2) admit two repeated roots 𝜆 1 , 2 = 𝜆 , 𝑏 ∗ 1 2 𝑏 ∗ 2 1 ≠ 0 , and the matrix Λ has the form (2.5). Then the solution of the initial problems (1.1) and (1.2) is given by 𝑥 ( 𝑘 ) = 𝑆 𝑦 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ − 𝑚 where 𝑦 ( 𝑘 ) has the form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜑 𝑦 ( 𝑘 ) = ∗ ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑘 − 1 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 𝑘 − 1 − 𝑟 Φ ( 𝑟 − 𝑚 ) , i f 𝑘 ∈ ℤ 1 𝑚 + 1 , Λ 𝑘 𝜑 ∗ ( 0 ) + 𝑚 𝑟 = 0 0 𝑥 0 2 0 0 𝑑 𝜆 𝑘 − 1 − 𝑟 Φ ( 𝑟 − 𝑚 ) + ( 𝑘 − 1 − 𝑚 ) 𝜆 𝑘 − 1 − 𝑚 Φ ( 0 ) , i f 𝑘 ∈ ℤ ∞ 𝑚 + 2 . ( 2 . 3 6 )

Proof. In this case, all the entries of 𝐵 ∗ are nonzero and, from (2.10), (2.11), and (2.30), we get 𝐵 ∗ = ⎛ ⎜ ⎜ ⎜ ⎝ 𝑏 ∗ 1 1 𝑏 ∗ 1 2 − 𝑏 ∗ 1 1 2 𝑏 ∗ 1 2 − 𝑏 ∗ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ . ( 2 . 3 7 ) Then the system (2.7) reduces to 𝑦 1 ( 𝑘 + 1 ) = 𝜆 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 1 𝑦 1 ( 𝑘 − 𝑚 ) + 𝑏 ∗ 1 2 𝑦 2 𝑦 ( 𝑘 − 𝑚 ) , ( 2 . 3 8 ) 2 ( 𝑘 + 1 ) = 𝜆 𝑦 2 𝑏 ( 𝑘 ) − ∗ 1 1 2 𝑏 ∗ 1 2 𝑦 1 ( 𝑘 − 𝑚 ) − 𝑏 ∗ 1 1 𝑦 2 ( 𝑘 − 𝑚 ) , ( 2 . 3 9 ) where 𝑘 ∈ ℤ ∞ 0 . It is easy to see (multiplying (2.39) by 𝑏 ∗ 1 2 / 𝑏 ∗ 1 1 and summing both equations) that 𝑦 1 𝑏 ( 𝑘 + 1 ) + ∗ 1 2 𝑏 ∗ 1 1 𝑦 2 𝑦 ( 𝑘 + 1 ) = 𝜆 1 𝑏 ( 𝑘 ) + ∗ 1 2 𝑏 ∗ 1 1 𝑦 2 ( 𝑘 ) , 𝑘 ∈ ℤ ∞ 0 . ( 2 . 4 0 ) We can see (2.40) as a homogeneous equation with respect to the unknown expression 𝑦 1 ( 𝑘 ) + ( 𝑏 ∗ 1 2 / 𝑏 ∗ 1 1 ) 𝑦 2 ( 𝑘 ) . Then, using (1.18), we obtain 𝑦 1 ( 𝑏 𝑘 ) + ∗ 1 2 𝑏 ∗ 1 1 𝑦 2 ( ⎧ ⎪ ⎨ ⎪ ⎩ 𝜑 𝑘 ) = ∗ 1 𝑏 ( 𝑘 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 𝑘 ) , i f 𝑘 ∈ ℤ 0 − 𝑚 , 𝜆 𝑘 𝜑 ∗ 1 𝑏 ( 0 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 1 . ( 2 . 4 1 ) With the aid of (2.41), we rewrite the systems (2.38) and (2.39) as follows: 𝑦 1 ( ⎧ ⎪ ⎨ ⎪ ⎩ 𝑘 + 1 ) = 𝜆 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 1 𝜑 ∗ 1 𝑏 ( 𝑘 − 𝑚 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 𝑘 − 𝑚 ) , i f 𝑘 ∈ ℤ 𝑚 0 , 𝜆 𝑦 1 ( 𝑘 ) + 𝑏 ∗ 1 1 𝜆 𝑘 − 𝑚 𝜑 ∗ 1 𝑏 ( 0 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 𝑚 + 1 , 𝑦 2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( 𝑘 + 1 ) = 𝜆 𝑦 2 𝑏 ( 𝑘 ) − ∗ 1 1 2 𝑏 ∗ 1 2 𝜑 ∗ 1 𝑏 ( 𝑘 − 𝑚 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 𝑘 − 𝑚 ) , i f 𝑘 ∈ ℤ 𝑚 0 , 𝜆 𝑦 2 𝑏 ( 𝑘 ) − ∗ 1 1 2 𝑏 ∗ 1 2 𝜆 𝑘 − 𝑚 𝜑 ∗ 1 𝑏 ( 0 ) + ∗ 1 2 𝑏 ∗ 1 1 𝜑 ∗ 2 ( 0 ) , i f 𝑘 ∈ ℤ ∞ 𝑚 + 1 . ( 2 . 4 2 ) It is easy to see that the system (2.42) is decomposed into two separate equations. Solving each of them in a similar way as in the proof of Theorem 2.1 using (1.18) (details are omitted), we conclude 𝑦 1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜑 ( 𝑘 ) = ∗ 1 (