# Conditional Stability and Asymptotic Behavior of Solutions of Weakly Delayed Linear Discrete Systems inR2

Authors: Josef Diblík; Hana Halfarová; Jan Šafařík
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Two-dimensional linear discrete systemsx(k+1)=Ax(k)+∑l=1nBlxl(k-ml),  k≥0, are analyzed, where m1,m2,…,mn are constant integer delays, 0<m1<m2<⋯<mn, A, B1,…,Bn are constant 2×2 matrices, A=(aij), Bl=(bijl),  i,j=1,2,  l=1,2,…,n, and x:{-mn,-mn+1,…}→R2. Under the assumption that the system is weakly delayed, the asymptotic behavior of its solutions is studied and asymptotic formulas are derived.


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## Body

## 1. Introduction and Preliminaries

Throughout the paper, we use the notations:Θ is zero 2×2 matrix, I is the unit 2×2 matrix, θ is zero 2×1 vector, and Zsq={s,s+1,…,q}, where s and q are integers, s≤q. Similarly, a set Zs∞ is defined.In the paper, we investigate discrete planar systems with delays(1)xk+1=Axk+∑l=1nBlxlk-ml,where m1,m2,…,mn are integers such that 0<m1<m2<⋯<mn, k∈Z0∞, A, B1,…,Bn are constant 2×2 matrices, A=(aij), Bl=(bijl),  i,j=1,2,  l=1,2,…,n,(2)Bl≠Θ,l=1,2,…,n,and x:Z-mn∞→R2. An initial problem to (1) is(3)xk=φk,where k=-mn,-mn+1,…,0 and φ:Z-mn0→R2. Obviously, the initial problem (1), (3) has a unique solution on Z-mn∞.


### 1.1. Weakly Delayed Systems

It is well-known that the characteristic equation to (1) is(4)Dλ≔det⁡A+∑l=1nλ-mlBl-λI=0,where λ∈C, and the characteristic equation to a system without delays (5)xk+1=Axk is(6)det⁡A-λI=0.The following definition and lemma are taken from [1].Definition 1.
System (1) is called weakly delayed if equations (4) and (6) are equal, that is, if (7)Dλ=det⁡A-λI,∀λ∈C∖0.Lemma 2.
If system (1) is a weakly delayed system, then its arbitrary linear nonsingular transformation x(k)=Sy(k) with 2×2 matrix S is again a weakly delayed system.In [1], the following necessary and sufficient conditions determining weakly delayed systems are also derived.Theorem 3.
System (1) is a weakly delayed system if and only if(8)b11l+b22l=0,(9)b11lb12lb21lb22l=0,(10)a11a12b21lb22l+b11lb12la21a22=0,(11)b11lb12lb21vb22v+b11vb12vb21lb22l=0,where l,v=1,2,…,n and v>l.


### 1.2. Problem under Consideration

In the paper (in Section2), we are concerned with conditional stability of (1) with the results formulated in Theorems 11–18. In Section 3, asymptotic formulas describing the behavior of solutions (for k→∞) of nontrivial solutions of (1) are derived with the results formulated in Theorems 19–24.To prove such results, we use explicit analytic formulas on the representation of solutions, derived by the first two authors in [1] and, for the reader’s convenience, we recall them in the following part.


### 1.3. Representation of Solutions

#### 1.3.1. Jordan Forms

It is known that, for every matrixA, there exists a nonsingular matrix S transforming it to the corresponding Jordan matrix form Λ=S-1AS. In [[1], Theorem 6], formulas (8)–(11) are analyzed and it is concluded that system (1) can be a weakly delayed system only if matrix A has one of the following three Jordan matrix forms (the case of the roots of (6) being complex conjugate is not compatible with (2) and (8)–(11)):(12)Λ1=λ100λ2,if (6) has two real distinct roots λ1, λ2, and either(13)Λ2=λ00λor(14)Λ3=λ10λin the case of one double real root λ1,2=λ. Transforming (1) by y(k)=S-1x(k), we get(15)yk+1=Λyk+∑l=1nBl∗yk-ml,k∈Z0∞,where Bl∗=S-1BlS, Bl∗=(bij∗l),  l=1,…,n, and i,j=1,2. The transformed initial problem for (15) is(16)yk=φ∗k=φ1∗k,φ2∗kT≔S-1φk,k∈Z-mn0.Below, we will use functions Φi:Z-mn0→R2,  i=1,2, defined as (17)Φ1k≔0,φ1∗kT,Φ2k≔φ2∗k,0T,k∈Z-mn0 and functions Φl∗:Z-mn0→R2,  l=1,2,…,n, defined as(18)Φl∗k=Φl1∗k,Φl2∗kT≔b11∗lφ1∗k+b12∗1b11∗1φ2∗k,-b11∗l2b12∗lφ1∗k+b12∗1b11∗1φ2∗kT,where k∈Z-mn0,  l=1,2,…,n.


#### 1.3.2. Explicit Analytic Formulas in Case (12)

In this case,Λ=Λ1, Λ1k=diag⁡(λ1k,λ2k). From conditions (8)–(11) for (15) it follows (we refer to [[1], Part 2.1.1.]) that either(I)
b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n,or(II)
b11∗l=b22∗l=b12∗l=0,  b21∗l≠0,  l=1,2,…,n.The proofs of the below results are given in [[1], Theorem 5].Theorem 4.
Let (1) be a weakly delayed system and (6) have two real distinct roots λ1, λ2. In case (I), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (I1)
y(k)=φ∗(k) if k∈Z-mn0,(I2)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r[∑l=1nb12∗lΦ2(r-ml)], if k∈Z1m1+1,(I3)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r[∑l=s+1nb12∗lΦ2(r-ml)] + ∑l=1sb12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(I4)
y(k)=Λ1kφ∗(0) + ∑l=1nb12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] if k∈Zmn+2∞.Theorem 5.
Let (1) be a weakly delayed system and (6) have two real distinct roots λ1, λ2. In case (II), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (II1)
y(k)=φ∗(k), if k∈Z-mn0,(II2)
y(k)=Λ1kφ∗(0) + ∑r=0k-1λ2k-1-r[∑l=1nb21∗lΦ1(r-ml)], if k∈Z1m1+1,(II3)
y(k)=Λ1kφ∗(0) + ∑r=0k-1λ2k-1-r[∑l=s+1nb21∗lΦ1(r-ml)] + ∑l=1sb21∗l[∑r=0mlλ2k-1-rΦ1(r-ml) + Φ1(0)∑r=ml+1k-1λ1r-mlλ2k-1-r] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(II4)
y(k)=Λ1kφ∗(0) + ∑l=1nb21∗l[∑r=0mlλ2k-1-rΦ1(r-ml) + Φ1(0)∑r=ml+1k-1λ1r-mlλ2k-1-r] if k∈Zmn+2∞.


#### 1.3.3. Explicit Analytic Formulas in Case (13)

We haveΛ=Λ2, Λ2k=diag⁡(λk,λk) and, from the necessary and sufficient conditions (8)–(11) for (15), it follows (see [[1], Part 2.1.3.]) that only the following three cases are possible:(a)
b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n,(b)
b11∗l=b22∗l=b12∗l=0,  b21∗l≠0,  l=1,2,…,n,(c)
b12∗lb21∗l≠0,  l=1,2,…,n.The results formulated below are proved in [[1], Theorems 7 and 8],Theorem 6.
Let (1) be a weakly delayed system, (6) have a twofold root λ1,2=λ, and the matrix Λ have the form (13). In case (a), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (a1)
y(k)=φ∗(k), if k∈Z-mn0,(a2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nb12∗lΦ2(r-ml)], if k∈Z1m1+1,(a3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb12∗lΦ2(r-ml)] + ∑l=1sb12∗l[∑r=0mlλk-1-rΦ2(r-ml) + (k-1-ml)λk-1-mlΦ2(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(a4)
y(k)=Λ2kφ∗(0) + ∑l=1nb12∗l[∑r=0mlλk-1-rΦ2(r-ml) + (k-1-ml)λk-1-mlΦ2(0)] if k∈Zmn+2∞.Theorem 7.
Let (1) be a weakly delayed system, (6) have a twofold root λ1,2=λ, and the matrix Λ have the form (13). In case (b), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (b1)
y(k)=φ∗(k), if k∈Z-mn0,(b2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nb21∗lΦ1(r-ml)], if k∈Z1m1+1,(b3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb21∗lΦ1(r-ml)] + ∑l=1sb21∗l[∑r=0mlλk-1-rΦ1(r-ml) + (k-1-ml)λk-1-mlΦ1(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(b4)
y(k)=Λ2kφ∗(0) + ∑l=1nb21∗l[∑r=0mlλk-1-rΦ1(r-ml) + (k-1-ml)λk-1-mlΦ1(0)] if k∈Zmn+2∞.Theorem 8.
Let system (1) be a weakly delayed system, (6) have two repeated roots λ1,2=λ, and the matrix Λ have the form (13). In case (c), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (c1)
y(k)=φ∗(k), if k∈Z-mn0,(c2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nΦl∗(r-ml)], if k∈Z1m1+1,(c3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nΦl∗(r-ml)] + ∑l=1s[∑r=0mlλk-1-rΦl∗(r-ml)+(k-1-ml)λk-1-mlΦl∗(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(c4)
y(k)=Λ2kφ∗(0) + ∑l=1n[∑r=0mlλk-1-rΦl∗(r-ml)+(k-1-ml)λk-1-mlΦl∗(0)] if k∈Zmn+2∞.


#### 1.3.4. Explicit Analytic Formulas in Case (14)

We haveΛ=Λ3. The necessary and sufficient conditions (8)–(11) for (15) are reduced (we refer to [[1], Part 2.1.6.]) to b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n. The following result is proved in [[1], Theorem 9].Theorem 9.
Let (1) be a weakly delayed system, (6) have a double root λ1,2=λ, and the matrix Λ have the form (14). Then, the solution of the initial problem (1), (3) is x(k)=Sy(k)=Sy1k,y2kT,  k∈Z-mn∞, where 1(y1y2)
y1(k)=φ1∗(k), y2(k)=φ2∗(k), if k∈Z-mn0,2(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑r=0k-1λk-1-r[∑l=1nb12∗lφ2∗(r-ml)], if k∈Z1m1+1,3(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb12∗lφ2∗(r-ml)] + ∑l=1sb12∗l[∑r=0mlλk-1-rφ2∗(r-ml) + (k-1-ml)λk-1-mlφ2∗(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,4(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑l=1nb12∗l[∑r=0mlλk-1-rφ2∗(r-ml) + (k-1-ml)λk-1-mlφ2∗(0)] if k∈Zmn+2∞,5(y2)
y2(k)=λkφ2∗(0) if k∈Z1∞.


## 1.1. Weakly Delayed Systems

It is well-known that the characteristic equation to (1) is(4)Dλ≔det⁡A+∑l=1nλ-mlBl-λI=0,where λ∈C, and the characteristic equation to a system without delays (5)xk+1=Axk is(6)det⁡A-λI=0.The following definition and lemma are taken from [1].Definition 1.
System (1) is called weakly delayed if equations (4) and (6) are equal, that is, if (7)Dλ=det⁡A-λI,∀λ∈C∖0.Lemma 2.
If system (1) is a weakly delayed system, then its arbitrary linear nonsingular transformation x(k)=Sy(k) with 2×2 matrix S is again a weakly delayed system.In [1], the following necessary and sufficient conditions determining weakly delayed systems are also derived.Theorem 3.
System (1) is a weakly delayed system if and only if(8)b11l+b22l=0,(9)b11lb12lb21lb22l=0,(10)a11a12b21lb22l+b11lb12la21a22=0,(11)b11lb12lb21vb22v+b11vb12vb21lb22l=0,where l,v=1,2,…,n and v>l.


## 1.2. Problem under Consideration

In the paper (in Section2), we are concerned with conditional stability of (1) with the results formulated in Theorems 11–18. In Section 3, asymptotic formulas describing the behavior of solutions (for k→∞) of nontrivial solutions of (1) are derived with the results formulated in Theorems 19–24.To prove such results, we use explicit analytic formulas on the representation of solutions, derived by the first two authors in [1] and, for the reader’s convenience, we recall them in the following part.


## 1.3. Representation of Solutions

### 1.3.1. Jordan Forms

It is known that, for every matrixA, there exists a nonsingular matrix S transforming it to the corresponding Jordan matrix form Λ=S-1AS. In [[1], Theorem 6], formulas (8)–(11) are analyzed and it is concluded that system (1) can be a weakly delayed system only if matrix A has one of the following three Jordan matrix forms (the case of the roots of (6) being complex conjugate is not compatible with (2) and (8)–(11)):(12)Λ1=λ100λ2,if (6) has two real distinct roots λ1, λ2, and either(13)Λ2=λ00λor(14)Λ3=λ10λin the case of one double real root λ1,2=λ. Transforming (1) by y(k)=S-1x(k), we get(15)yk+1=Λyk+∑l=1nBl∗yk-ml,k∈Z0∞,where Bl∗=S-1BlS, Bl∗=(bij∗l),  l=1,…,n, and i,j=1,2. The transformed initial problem for (15) is(16)yk=φ∗k=φ1∗k,φ2∗kT≔S-1φk,k∈Z-mn0.Below, we will use functions Φi:Z-mn0→R2,  i=1,2, defined as (17)Φ1k≔0,φ1∗kT,Φ2k≔φ2∗k,0T,k∈Z-mn0 and functions Φl∗:Z-mn0→R2,  l=1,2,…,n, defined as(18)Φl∗k=Φl1∗k,Φl2∗kT≔b11∗lφ1∗k+b12∗1b11∗1φ2∗k,-b11∗l2b12∗lφ1∗k+b12∗1b11∗1φ2∗kT,where k∈Z-mn0,  l=1,2,…,n.


### 1.3.2. Explicit Analytic Formulas in Case (12)

In this case,Λ=Λ1, Λ1k=diag⁡(λ1k,λ2k). From conditions (8)–(11) for (15) it follows (we refer to [[1], Part 2.1.1.]) that either(I)
b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n,or(II)
b11∗l=b22∗l=b12∗l=0,  b21∗l≠0,  l=1,2,…,n.The proofs of the below results are given in [[1], Theorem 5].Theorem 4.
Let (1) be a weakly delayed system and (6) have two real distinct roots λ1, λ2. In case (I), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (I1)
y(k)=φ∗(k) if k∈Z-mn0,(I2)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r[∑l=1nb12∗lΦ2(r-ml)], if k∈Z1m1+1,(I3)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r[∑l=s+1nb12∗lΦ2(r-ml)] + ∑l=1sb12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(I4)
y(k)=Λ1kφ∗(0) + ∑l=1nb12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] if k∈Zmn+2∞.Theorem 5.
Let (1) be a weakly delayed system and (6) have two real distinct roots λ1, λ2. In case (II), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (II1)
y(k)=φ∗(k), if k∈Z-mn0,(II2)
y(k)=Λ1kφ∗(0) + ∑r=0k-1λ2k-1-r[∑l=1nb21∗lΦ1(r-ml)], if k∈Z1m1+1,(II3)
y(k)=Λ1kφ∗(0) + ∑r=0k-1λ2k-1-r[∑l=s+1nb21∗lΦ1(r-ml)] + ∑l=1sb21∗l[∑r=0mlλ2k-1-rΦ1(r-ml) + Φ1(0)∑r=ml+1k-1λ1r-mlλ2k-1-r] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(II4)
y(k)=Λ1kφ∗(0) + ∑l=1nb21∗l[∑r=0mlλ2k-1-rΦ1(r-ml) + Φ1(0)∑r=ml+1k-1λ1r-mlλ2k-1-r] if k∈Zmn+2∞.


### 1.3.3. Explicit Analytic Formulas in Case (13)

We haveΛ=Λ2, Λ2k=diag⁡(λk,λk) and, from the necessary and sufficient conditions (8)–(11) for (15), it follows (see [[1], Part 2.1.3.]) that only the following three cases are possible:(a)
b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n,(b)
b11∗l=b22∗l=b12∗l=0,  b21∗l≠0,  l=1,2,…,n,(c)
b12∗lb21∗l≠0,  l=1,2,…,n.The results formulated below are proved in [[1], Theorems 7 and 8],Theorem 6.
Let (1) be a weakly delayed system, (6) have a twofold root λ1,2=λ, and the matrix Λ have the form (13). In case (a), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (a1)
y(k)=φ∗(k), if k∈Z-mn0,(a2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nb12∗lΦ2(r-ml)], if k∈Z1m1+1,(a3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb12∗lΦ2(r-ml)] + ∑l=1sb12∗l[∑r=0mlλk-1-rΦ2(r-ml) + (k-1-ml)λk-1-mlΦ2(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(a4)
y(k)=Λ2kφ∗(0) + ∑l=1nb12∗l[∑r=0mlλk-1-rΦ2(r-ml) + (k-1-ml)λk-1-mlΦ2(0)] if k∈Zmn+2∞.Theorem 7.
Let (1) be a weakly delayed system, (6) have a twofold root λ1,2=λ, and the matrix Λ have the form (13). In case (b), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (b1)
y(k)=φ∗(k), if k∈Z-mn0,(b2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nb21∗lΦ1(r-ml)], if k∈Z1m1+1,(b3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb21∗lΦ1(r-ml)] + ∑l=1sb21∗l[∑r=0mlλk-1-rΦ1(r-ml) + (k-1-ml)λk-1-mlΦ1(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(b4)
y(k)=Λ2kφ∗(0) + ∑l=1nb21∗l[∑r=0mlλk-1-rΦ1(r-ml) + (k-1-ml)λk-1-mlΦ1(0)] if k∈Zmn+2∞.Theorem 8.
Let system (1) be a weakly delayed system, (6) have two repeated roots λ1,2=λ, and the matrix Λ have the form (13). In case (c), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (c1)
y(k)=φ∗(k), if k∈Z-mn0,(c2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nΦl∗(r-ml)], if k∈Z1m1+1,(c3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nΦl∗(r-ml)] + ∑l=1s[∑r=0mlλk-1-rΦl∗(r-ml)+(k-1-ml)λk-1-mlΦl∗(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(c4)
y(k)=Λ2kφ∗(0) + ∑l=1n[∑r=0mlλk-1-rΦl∗(r-ml)+(k-1-ml)λk-1-mlΦl∗(0)] if k∈Zmn+2∞.


### 1.3.4. Explicit Analytic Formulas in Case (14)

We haveΛ=Λ3. The necessary and sufficient conditions (8)–(11) for (15) are reduced (we refer to [[1], Part 2.1.6.]) to b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n. The following result is proved in [[1], Theorem 9].Theorem 9.
Let (1) be a weakly delayed system, (6) have a double root λ1,2=λ, and the matrix Λ have the form (14). Then, the solution of the initial problem (1), (3) is x(k)=Sy(k)=Sy1k,y2kT,  k∈Z-mn∞, where 1(y1y2)
y1(k)=φ1∗(k), y2(k)=φ2∗(k), if k∈Z-mn0,2(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑r=0k-1λk-1-r[∑l=1nb12∗lφ2∗(r-ml)], if k∈Z1m1+1,3(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb12∗lφ2∗(r-ml)] + ∑l=1sb12∗l[∑r=0mlλk-1-rφ2∗(r-ml) + (k-1-ml)λk-1-mlφ2∗(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,4(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑l=1nb12∗l[∑r=0mlλk-1-rφ2∗(r-ml) + (k-1-ml)λk-1-mlφ2∗(0)] if k∈Zmn+2∞,5(y2)
y2(k)=λkφ2∗(0) if k∈Z1∞.


## 1.3.1. Jordan Forms

It is known that, for every matrixA, there exists a nonsingular matrix S transforming it to the corresponding Jordan matrix form Λ=S-1AS. In [[1], Theorem 6], formulas (8)–(11) are analyzed and it is concluded that system (1) can be a weakly delayed system only if matrix A has one of the following three Jordan matrix forms (the case of the roots of (6) being complex conjugate is not compatible with (2) and (8)–(11)):(12)Λ1=λ100λ2,if (6) has two real distinct roots λ1, λ2, and either(13)Λ2=λ00λor(14)Λ3=λ10λin the case of one double real root λ1,2=λ. Transforming (1) by y(k)=S-1x(k), we get(15)yk+1=Λyk+∑l=1nBl∗yk-ml,k∈Z0∞,where Bl∗=S-1BlS, Bl∗=(bij∗l),  l=1,…,n, and i,j=1,2. The transformed initial problem for (15) is(16)yk=φ∗k=φ1∗k,φ2∗kT≔S-1φk,k∈Z-mn0.Below, we will use functions Φi:Z-mn0→R2,  i=1,2, defined as (17)Φ1k≔0,φ1∗kT,Φ2k≔φ2∗k,0T,k∈Z-mn0 and functions Φl∗:Z-mn0→R2,  l=1,2,…,n, defined as(18)Φl∗k=Φl1∗k,Φl2∗kT≔b11∗lφ1∗k+b12∗1b11∗1φ2∗k,-b11∗l2b12∗lφ1∗k+b12∗1b11∗1φ2∗kT,where k∈Z-mn0,  l=1,2,…,n.


## 1.3.2. Explicit Analytic Formulas in Case (12)

In this case,Λ=Λ1, Λ1k=diag⁡(λ1k,λ2k). From conditions (8)–(11) for (15) it follows (we refer to [[1], Part 2.1.1.]) that either(I)
b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n,or(II)
b11∗l=b22∗l=b12∗l=0,  b21∗l≠0,  l=1,2,…,n.The proofs of the below results are given in [[1], Theorem 5].Theorem 4.
Let (1) be a weakly delayed system and (6) have two real distinct roots λ1, λ2. In case (I), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (I1)
y(k)=φ∗(k) if k∈Z-mn0,(I2)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r[∑l=1nb12∗lΦ2(r-ml)], if k∈Z1m1+1,(I3)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r[∑l=s+1nb12∗lΦ2(r-ml)] + ∑l=1sb12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(I4)
y(k)=Λ1kφ∗(0) + ∑l=1nb12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] if k∈Zmn+2∞.Theorem 5.
Let (1) be a weakly delayed system and (6) have two real distinct roots λ1, λ2. In case (II), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (II1)
y(k)=φ∗(k), if k∈Z-mn0,(II2)
y(k)=Λ1kφ∗(0) + ∑r=0k-1λ2k-1-r[∑l=1nb21∗lΦ1(r-ml)], if k∈Z1m1+1,(II3)
y(k)=Λ1kφ∗(0) + ∑r=0k-1λ2k-1-r[∑l=s+1nb21∗lΦ1(r-ml)] + ∑l=1sb21∗l[∑r=0mlλ2k-1-rΦ1(r-ml) + Φ1(0)∑r=ml+1k-1λ1r-mlλ2k-1-r] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(II4)
y(k)=Λ1kφ∗(0) + ∑l=1nb21∗l[∑r=0mlλ2k-1-rΦ1(r-ml) + Φ1(0)∑r=ml+1k-1λ1r-mlλ2k-1-r] if k∈Zmn+2∞.


## 1.3.3. Explicit Analytic Formulas in Case (13)

We haveΛ=Λ2, Λ2k=diag⁡(λk,λk) and, from the necessary and sufficient conditions (8)–(11) for (15), it follows (see [[1], Part 2.1.3.]) that only the following three cases are possible:(a)
b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n,(b)
b11∗l=b22∗l=b12∗l=0,  b21∗l≠0,  l=1,2,…,n,(c)
b12∗lb21∗l≠0,  l=1,2,…,n.The results formulated below are proved in [[1], Theorems 7 and 8],Theorem 6.
Let (1) be a weakly delayed system, (6) have a twofold root λ1,2=λ, and the matrix Λ have the form (13). In case (a), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (a1)
y(k)=φ∗(k), if k∈Z-mn0,(a2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nb12∗lΦ2(r-ml)], if k∈Z1m1+1,(a3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb12∗lΦ2(r-ml)] + ∑l=1sb12∗l[∑r=0mlλk-1-rΦ2(r-ml) + (k-1-ml)λk-1-mlΦ2(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(a4)
y(k)=Λ2kφ∗(0) + ∑l=1nb12∗l[∑r=0mlλk-1-rΦ2(r-ml) + (k-1-ml)λk-1-mlΦ2(0)] if k∈Zmn+2∞.Theorem 7.
Let (1) be a weakly delayed system, (6) have a twofold root λ1,2=λ, and the matrix Λ have the form (13). In case (b), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (b1)
y(k)=φ∗(k), if k∈Z-mn0,(b2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nb21∗lΦ1(r-ml)], if k∈Z1m1+1,(b3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb21∗lΦ1(r-ml)] + ∑l=1sb21∗l[∑r=0mlλk-1-rΦ1(r-ml) + (k-1-ml)λk-1-mlΦ1(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(b4)
y(k)=Λ2kφ∗(0) + ∑l=1nb21∗l[∑r=0mlλk-1-rΦ1(r-ml) + (k-1-ml)λk-1-mlΦ1(0)] if k∈Zmn+2∞.Theorem 8.
Let system (1) be a weakly delayed system, (6) have two repeated roots λ1,2=λ, and the matrix Λ have the form (13). In case (c), the solution of the initial problem (1), (3) is x(k)=Sy(k),  k∈Z-mn∞, where (c1)
y(k)=φ∗(k), if k∈Z-mn0,(c2)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=1nΦl∗(r-ml)], if k∈Z1m1+1,(c3)
y(k)=Λ2kφ∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nΦl∗(r-ml)] + ∑l=1s[∑r=0mlλk-1-rΦl∗(r-ml)+(k-1-ml)λk-1-mlΦl∗(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,(c4)
y(k)=Λ2kφ∗(0) + ∑l=1n[∑r=0mlλk-1-rΦl∗(r-ml)+(k-1-ml)λk-1-mlΦl∗(0)] if k∈Zmn+2∞.


## 1.3.4. Explicit Analytic Formulas in Case (14)

We haveΛ=Λ3. The necessary and sufficient conditions (8)–(11) for (15) are reduced (we refer to [[1], Part 2.1.6.]) to b11∗l=b22∗l=b21∗l=0,  b12∗l≠0,  l=1,2,…,n. The following result is proved in [[1], Theorem 9].Theorem 9.
Let (1) be a weakly delayed system, (6) have a double root λ1,2=λ, and the matrix Λ have the form (14). Then, the solution of the initial problem (1), (3) is x(k)=Sy(k)=Sy1k,y2kT,  k∈Z-mn∞, where 1(y1y2)
y1(k)=φ1∗(k), y2(k)=φ2∗(k), if k∈Z-mn0,2(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑r=0k-1λk-1-r[∑l=1nb12∗lφ2∗(r-ml)], if k∈Z1m1+1,3(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑r=0k-1λk-1-r[∑l=s+1nb12∗lφ2∗(r-ml)] + ∑l=1sb12∗l[∑r=0mlλk-1-rφ2∗(r-ml) + (k-1-ml)λk-1-mlφ2∗(0)] if k∈Zms+2ms+1+1, s=1,2,…,n-1,4(y1)
y1(k)=λkφ1∗(0)+kλk-1φ2∗(0) + ∑l=1nb12∗l[∑r=0mlλk-1-rφ2∗(r-ml) + (k-1-ml)λk-1-mlφ2∗(0)] if k∈Zmn+2∞,5(y2)
y2(k)=λkφ2∗(0) if k∈Z1∞.


## 2. Conditional Stability

In [[1], Theorem 10], it is explained that the space of solutions of a weakly delayed system (1), depending initially on 2(mn+1) parameters (i.e., on the initial data (3)) is reduced (as k≥mn+2) to a space of solutions depending either on mn+1 or even only on 2 parameters. This is also visible from an analysis of formulas describing the behavior of solutions for k≥mn+2 given in Theorems 4–9. In this part, we explain how this property can be used when the stability of a weakly delayed system (1) is considered. Since not all of the initial data have an impact on the behavior of the solution x=x(k) as k≥mn+2, a part of them can be fixed and, under such an assumption, we define a so-called conditional stability below. The fixing of some of the initial data leads to an unexpected phenomenon; the zero solution can be conditionally stable in spite of the fact that it is unstable in terms of the traditional definition of stability.Define a norm of a2×2 matrix A=(Aij),  i,j=1,2, as (19)A≔max⁡A11+A12,A21+A22and, for 2×1 vectors x=x1,x2T, a vector norm(20)x=max⁡x1,x2.For a discrete vector-function ψ:Z-mn0→R2, we define (21)ψmn≔max⁡ψ-mn,ψ-mn+1,…,ψ0. In the following definition, the notion of conditional stability is explained. The two first parts of it are the classical definitions of stability and can be found, for example, in [2, 3].Definition 10.
The zero solutionx(k)=θ, k∈Z-mn∞, of (1) is said to be (a)
stable if, givenε>0 and k0≥0, there exists δ=δ(ε,k0) such that φ(k), k∈Zk0-mnk0, φmn<δ implies x(k,k0,φ)<ε for all k≥k0, uniformly stable if δ may be chosen independently of k0, and unstable if it is not stable;(b)
asymptotically stable if it is stable andlimk→∞⁡x(k)=0;(c)
conditionally stable (conditionally asymptotically stable) if it is unstable (not asymptotically stable), but if it is stable (asymptotically stable) under the condition that there exists a fixed subspaceP⊂R2(mn+1), 1≤dim⁡P<2(mn+1), and the initial data satisfy (22)φ-mn,φ-mn+1,…,φ0∈R2mn+1∖P.Utilizing the formulas on the representation of solutions of (1), we prove results on conditional stability (since the existence of the subspaces P in the proofs is obvious, we do not write them explicitly).Theorem 11.
If the assumptions of Theorem4 hold, |λ1|≤q<1, |λ2|≥1, and φ2∗(0)=0, then the zero solution of (1) is conditionally asymptotically stable.Proof.
In this case,φ∗(0)=φ1∗0,φ2∗0T=φ1∗0,0T and Φ2(0)=φ2∗0,0T=θ. From formula (I4) in Theorem 4, for k∈Zmn+2∞, we get (23)yk≤Λ1kφ∗0+∑l=1nb12∗l∑r=0mlλ1k-1-rΦ2r-ml+Φ20∑r=ml+1k-1λ1k-1-rλ2r-ml≤λ1k00λ2kφ1∗00+∑l=1nb12∗l∑r=0mlλ1k-1-rΦ2r-ml+Φ20∑r=ml+1k-1λ1k-1-rλ2r-ml≤λ1kφ1∗0+∑l=1nb12∗l∑r=0mlλ1k-1-rΦ2r-ml≤qkφ∗mn+∑l=1nb12∗l∑r=0mlqk-1-rφ∗mn≤qk+∑l=1nb12∗l∑r=0mlqk-1-rφ∗mn≤qk+∑l=1nb12∗lqk-1-ml1-qml+11-qφ∗mn=qk1+∑l=1nb12∗lq-ml-1-11-qφ∗mn. Now, it is easy to see that (24)limk→∞⁡yk≤1+∑l=1nb12∗lq-ml-1-11-qφ∗mnlimk→∞⁡qk=0;that is, the zero solution is conditionally asymptotically stable. If φ2∗(0)≠0, then from formula (I4) in Theorem 4, we get (25)y2k=λ2kφ2∗0≠0as k≥mn+2, and the zero solution is unstable if |λ2|>1 and not asymptotically stable if |λ2|=1.Similarly, with the aid of formula(II4), Theorem 5, the following theorem can be proved.Theorem 12.
If the assumptions of Theorem5 hold, |λ2|≤q<1, |λ1|≥1, and φ1∗(0)=0, then the zero solution of (1) is conditionally asymptotically stable.Theorem 13.
If the assumptions of Theorem4 hold, |λ1|=1, |λ2|>1, and φ2∗(0)=0, then the zero solution of (1) is conditionally stable.Proof.
We utilize formula(I4) in Theorem 4 again. Since φ∗(0)=φ1∗0,φ2∗0T=φ1∗0,0T and Φ2(0)=φ2∗0,0T=θ, for k∈Zmn+2∞, we get (26)yk≤Λ1kφ∗0+∑l=1nb12∗l∑r=0mlλ1k-1-rΦ2r-ml+Φ20∑r=ml+1k-1λ1k-1-rλ2r-ml≤λ1k00λ2kφ1∗00+∑l=1nb12∗l∑r=0mlλ1k-1-rΦ2r-ml+Φ20∑r=ml+1k-1λ1k-1-rλ2r-ml≤λ1kφ1∗0+∑l=1nb12∗l∑r=0mlλ1k-1-rΦ2r-ml≤φ∗mn+∑l=1nb12∗l∑r=0mlφ∗mn≤1+∑l=1nb12∗lml+1φ∗mn.We set(27)M≔1+∑l=1nb12∗lml+1,δ≔εM.Then (28)yk≤Mφ∗mn<ε,k∈Zmn+2∞if φ∗mn<δ. As in the proof of Theorem 11, we can show that if φ2∗(0)≠0, the zero solution is unstable.Theorem 14.
If the assumptions of Theorem5 hold, |λ2|=1, |λ1|>1, and φ1∗(0)=0, then the zero solution of (1) is conditionally stable.The proof can be performed similarly to that of Theorem13 with the aid of formula (II4), Theorem 5.Theorem 15.
If the assumptions of Theorem6 hold, |λ|=1 and φ2∗(0)=0, then the zero solution of (1) is conditionally stable.Proof.
We haveφ∗(0)=φ1∗0,φ2∗0T=φ1∗0,0T and Φ2(0)=φ2∗0,0T=θ. For k∈Zmn+2∞, we get by Theorem 6, formula (a4), (29)yk≤Λ2kφ∗0+∑l=1nb12∗l∑r=0mlλk-1-rΦ2r-ml+k-1-mlλk-1-mlΦ20≤λk00λkφ1∗00+∑l=1nb12∗l∑r=0mlλk-1-rΦ2r-ml+k-1-mlλk-1-mlΦ20≤φ1∗0+∑l=1nb12∗l∑r=0mlφ∗mn+k-1-mlΦ20≤1+∑l=1nb12∗lml+1φ∗mn.Let M and δ be given by (27). Then, the last equality implies (30)yk≤Mφ∗mn<ε,k∈Zmn+2∞if φ∗mn<δ. So conditional stability is proved. If φ2∗(0)≠0, then Φ2(0)=φ2∗0,0T≠θ and formula (a4) yields (31)y1k=λkφ1∗0+∑l=1nb12∗l∑r=0mlλk-1-rφ2∗r-ml+k-1-mlλk-1-mlφ2∗0.Then, the zero solution is unstable since, obviously, (32)limsupk→∞⁡y1k=∞.The following two theorems can be proved using a scheme similar to that of the proof of Theorem15 and with the aid of formula (b4), Theorem 7, and formula (c4), Theorem 8, respectively.Theorem 16.
If the assumptions of Theorem7 hold, |λ|=1 and φ1∗(0)=0, then the zero solution of (1) is conditionally stable.Theorem 17.
If the assumptions of Theorem8 hold, |λ|=1 and φ1∗(0)=φ2∗(0)=0, then the zero solution of (1) is conditionally stable.Theorem 18.
If the assumptions of Theorem9 hold, |λ|=1 and φ2∗(0)=0, then the zero solution of (1) is conditionally stable.Proof.
We show that the zero solution of (1) is conditionally stable. For y1(k), k∈Zmn+2∞, we get (by 4(y1), Theorem 9)(33)y1k≤λkφ1∗0+kλk-1φ2∗0+∑l=1nb12∗l∑r=0mlλk-1-rφ2∗r-ml+k-1-mlλk-1-mlφ2∗0≤φ1∗0+∑l=1nb12∗l∑r=0mlφ∗mn≤1+∑l=1nb12∗lml+1φ∗mn,and, for y2(k), k∈Zmn+2∞, we get (by 5(y2), Theorem 9)(34)y2k≤λkφ2∗0=0,k∈Zmn+2∞.From (33), (34), it is easy to see that the zero solution of (1) is conditionally stable. If φ2∗(0)≠0, then (by 5(y2), Theorem 9) (35)limsupk→∞⁡y2k=∞and the zero solution is unstable.


## 3. Asymptotic Formulas

Carefully analyzing the analytical formulas for the solutions of system (1) given in Theorems 4–9, it is possible (under various assumptions for the roots of (6) and for the initial data) to derive asymptotic formulas for the solutions of system (1) or simplify the formulas for the exact solutions. Below, we do such investigation. Set (36)J1≔∑l=1nb12∗l∑r=0mlλ1-rφ2∗r-ml,J2≔∑l=1nb21∗l∑r=0mlλ2-rφ1∗r-ml.The symbol ~ used below stands for what is called the asymptotic equivalence. By definition, two nonzero real functions z1(k) and z2(k) defined for k→∞ are asymptotically equivalent if limk→∞z1(k)/z2(k)=1.Theorem 19.
If the assumptions of Theorem4 hold, then the solution of the initial problem (1), (3) is (37)xk=Syk=Sy1k,y2kT,k∈Z-mn∞,where y2(k)=λ2kφ2∗(0), k∈Z1∞, and (1)
y1(k)~φ1∗(0)+J1λ1-1-φ2∗(0)λ2/λ2-λ1∑l=1nb12∗lλ1-1-mlλ1k if k→∞, |λ1|>|λ2| and φ1∗(0) + J1λ1-1-φ2∗(0)λ2/λ2-λ1∑l=1nb12∗lλ1-1-ml≠0,(2)
y1(k)=0, if k∈Zmn+2∞ and φ1∗(0)=J1=φ2∗(0)=0,(3)
y1(k)=φ2∗(0)λ2/λ2-λ1λ2k∑l=1nb12∗lλ2-1-ml if k∈Zmn+2∞, λ1≠0 and φ1∗(0) + J1λ1-1-φ2∗(0)λ2/λ2-λ1∑l=1nb12∗lλ1-1-ml=0,(4)
y1(k)~φ2∗(0)λ2/λ2-λ1λ2k∑l=1nb12∗lλ2-1-ml if k→∞ and |λ1|<|λ2| and φ2∗(0)∑l=1nb12∗lλ2-1-ml≠0,(5)
y1(k)=λ1kφ1∗(0)+J1λ1k-1 if k∈Zmn+2∞ and φ2∗(0)=0,(6)
y1(k)=λ1kφ1∗(0)+J1λ1k-1 + φ2∗(0)/2∑l=1nb12∗l(-1)k-1-ml-1λ1k-1-ml if k∈Zmn+2∞ and λ2=-λ1.Proof.
Letλ1≠0. Then (38)∑r=ml+1k-1λ1k-1-rλ2r-ml=λ2λ2-λ1λ2k-1-ml-λ1k-1-mland formula (I4), Theorem 4, for solution y(k), k∈Zmn+2∞, can be written as(39)yk=Λ1kφ∗0+J1,0Tλ1k-1+Φ20λ2λ2-λ1∑l=1nb12∗lλ2k-1-ml-λ1k-1-ml.From (39), we get(40)y1k=λ1kφ∗0+J1λ1k-1+φ2∗0λ2λ2-λ1∑l=1nb12∗lλ2k-1-ml-λ1k-1-ml.Then (40), if |λ1|>|λ2|, implies the statement (1). The statement (2) is an obvious consequence of (40). For λ1=0, we get the statement (2) immediately from (I4). Statement (3) is also a straightforward consequence of (40). Let |λ1|<|λ2|. Then from (40) we deduce (41)y1k~φ2∗0λ2λ2-λ1∑l=1nb12∗lλ2k-1-ml=φ2∗0λ2λ2-λ1∑l=1nb12∗lλ2-1-mlλ2kif φ2∗(0)λ2/λ2-λ1∑l=1nb12∗lλ2-1-ml≠0 and the statement (4) is valid. If λ1=0, then (I4) implies (42)y1k=φ2∗0λ2λ2-λ1∑l=1nb12∗lλ2k-1-mland the statement (4) remains true. The statement (5) follows from (40) if λ1≠0. From this representation, we derive the above asymptotic formulas if λ1≠0. If λ1=0, then, by (I4), y1(k)=0 and the statement (5) remains valid. The last statement (6) is a consequence of (40) and remains valid also if λ1=0.Theorem 20.
If the assumptions of Theorem5 hold, then the solution of the initial problem (1), (3) is (43)xk=Syk=Sy1k,y2kT,k∈Z-mn∞,where y1(k)=λ1kφ1∗(0), k∈Z1∞, and (1)
y2(k)~φ2∗(0)+J2λ2-1-φ1∗(0)λ1/λ1-λ2∑l=1nb21∗lλ2-1-mlλ2k if k→∞, |λ2|>|λ1| and φ2∗(0) + J2λ2-1-φ1∗(0)λ1/λ1-λ2∑l=1nb21∗lλ2-1-ml≠0,(2)
y2(k)=0 if k∈Zmn+2 and φ2∗(0)=J2=φ1∗(0)=0,(3)
y2(k)=φ1∗(0)λ1/λ1-λ2λ1k∑l=1nb21∗lλ1-1-ml if k∈Zmn+2∞, λ2≠0, and φ2∗(0) + J2λ2-1-φ1∗(0)λ1/λ1-λ2∑l=1nb21∗lλ2-1-ml=0,(4)
y2(k)~φ1∗(0)λ1/λ1-λ2λ1k∑l=1nb21∗lλ1-1-ml if k→∞, |λ2|<|λ1|, and φ1∗(0)∑l=1nb21∗lλ1-1-ml≠0,(5)
y2(k)=λ2kφ2∗(0)+J2λ2k-1 if k∈Zmn+2∞ and φ1∗(0)=0,(6)
y2(k)=λ2kφ2∗(0)+J2λ2k-1+φ1∗(0)/2∑l=1nb21∗l(-1)k-1-ml-1λ2k-1-ml if k∈Zmn+2∞ and λ1=-λ2.Proof.
Letλ2≠0. Then, (44)∑r=ml+1k-1λ1r-mlλ2k-1-r=λ1λ1-λ2λ1k-1-ml-λ2k-1-ml,and formula (II4) of Theorem 5 for the solution y(k), k∈Zmn+2∞, can be written as(45)yk=Λ1kφ∗0+0,J2Tλ2k-1+Φ10λ1λ1-λ2∑l=1nb21∗lλ1k-1-ml-λ2k-1-ml.From (45), we get (46)y2k=λ2kφ2∗0+J2λ2k-1+φ1∗0λ1λ1-λ2∑l=1nb21∗lλ1k-1-ml-λ2k-1-ml.Now, to prove statements (1)–(6) of the theorem, we can proceed in much the same way as in the proof of Theorem 19.Theorem 21.
If the assumptions of Theorem6 hold, then the solution of the initial problem (1), (3) is (47)xk=Syk=Sy1k,y2kT,k∈Z-mn∞,where y2(k)=λkφ2∗(0), k∈Z1∞, and(1)
y1(k)~φ2∗(0)∑l=1nb12∗l(k-1-ml)λ-1-mlλk if k→∞, φ2∗(0)≠0 and λ≠0,(2)
y1(k)=λkφ1∗(0)+J1λk-1 if k∈Zmn+2∞ and φ2∗(0)=0.Proof.
Formula(a4), Theorem 6, can be written as (48)yk=Λ2kφ∗0+J1,0Tλk-1+∑l=1nb12∗lk-1-mlλk-1-mlΦ20,y1k=λkφ1∗0+J1λk-1+∑l=1nb12∗lk-1-mlλk-1-mlφ2∗0.If φ2∗(0)≠0 and λ≠0, then the last addition term is the leading term and formula (1) is proved. If φ2∗(0)=0, we get formula (2).Theorem 22.
If the assumptions of Theorem7 hold, then the solution of the initial problem (1), (3) is (49)xk=Syk=Sy1k,y2kT,k∈Z-mn∞,where y1(k)=λkφ1∗(0), k∈Z1∞, and (1)
y2(k)~φ1∗(0)∑l=1nb21∗l(k-1-ml)λ-1-mlλk if k→∞, φ1∗(0)≠0, and λ≠0,(2)
y2(k)=λkφ2∗(0)+J2λk-1 if k∈Zmn+2∞ and φ1∗(0)=0.Proof.
Formula(b4), Theorem 7, can be written as (50)yk=Λ2kφ∗0+0,J2Tλk-1+∑l=1nb21∗lk-1-mlλk-1-mlΦ10,y2k=λkφ2∗0+J2λk-1+∑l=1nb21∗lk-1-mlλk-1-mlφ1∗0.Further, we can proceed as in the proof of Theorem (34).The following theorem uses the vector norm, given by (20).Theorem 23.
If the assumptions of Theorem8 hold and λ≠0, then the solution of the initial problem (1), (3) is (51)xk=Syk=Sy1k,y2kT,k∈Z-mn∞,where (1)
y(k)~λk-1∑l=1n(k-1-ml)λ-mlΦl∗(0) if k→∞, ∑l=1nΦl∗(0)≠0,(2)
y(k)=λkφ∗(0) + λk-1∑l=1n∑r=0ml-1λ-rΦl∗(r-ml), if k∈Zmn+2∞, ∑l=1nΦl∗(0)=0.Proof.
Formula(c4), Theorem 8, can be written as (52)yk=Λ2kφ∗0+λk-1∑l=1n∑r=0mlλ-rΦl∗r-ml+∑l=1nk-1-mlλk-1-mlΦl∗0if k∈Zmn+2∞. If ∑l=1nΦl∗(0)≠0, then the second sum is a leading term in this formula. If ∑l=1nΦl∗(0)=0, the first two terms are leading.Theorem 24.
If the assumptions of Theorem9 hold, then the solution of the initial problem (1), (3) is (53)xk=Syk=Sy1k,y2kT,k∈Z-mn∞,where y2(k)=λkφ2∗(0) if k∈Z1∞ and (1)
y1(k)~kλk-1φ2∗(0) + ∑l=1nb12∗l(k-1-ml)λk-1-mlφ2∗(0) if k→∞, φ2∗(0)≠0 and λ≠0,(2)
y1(k)=λkφ1∗(0)+J1λk-1 if k∈Zmn+2∞ and φ2∗(0)=0.Proof.
The formula4(y1), Theorem 9, can be written as (54)y1k=λkφ1∗0+kλk-1φ2∗0+J1λk-1+∑l=1nb12∗lk-1-mlλk-1-mlφ2∗0if k∈Zmn+2∞. The second and fourth terms are leading terms if φ2∗(0)≠0 and λ≠0. If φ2∗(0)=0, then the first and third terms are leading terms.


## 4. Example

To illustrate the results derived, consider the following example. The computations below were partly implemented by WolframAlpha software (http://www.wolframalpha.com/). Let system (1) be reduced to (55)x1k+1=-6.5x1k+3.5x2k+4x1k-1-2x2k-1-2x1k-2+x2k-2,x2k+1=-7x1k+4x2k+8x1k-1-4x2k-1-4x1k-2+2x2k-2and let initial problem (3) be(56)xk=φk=1,1T,k∈Z-20.In this case, n=2, m1=1, m2=2, (57)A=-6.53.5-74,B1=4-28-4,B2=-21-42.The eigenvalues of A are λ1=0.5, λ2=-3. System (55) is weakly delayed by Theorem 3. Indeed, (8) holds since (58)b111+b221=4-4=0,b112+b222=-2+2=0,(9) holds since (59)b111b121b211b221=4-28-4=0,b112b122b212b222=-21-42=0,(10) holds since (60)a11a12b211b221+b111b121a21a22=-6.53.58-4+4-2-74=0,a11a12b212b222+b112b122a21a22=-6.53.5-42+-21-74=0,and (11) holds since (61)b111b121b212b222+b112b122b211b221=4-2-42+-218-4=0.A nonsingular matrix S transforming A to the corresponding Jordan form is (62)S=1121,S-1=-112-1.Then, (63)Λ1=S-1AS=-112-1-6.53.5-741121=0.500-3,Λ1k=0.5k00-3k,k∈Z0∞,B1∗=S-1B1S=-112-14-28-41121=0200,B2∗=S-1B2S=-112-1-21-421121=0-100.The transformed initial problem (56) is (64)φ∗k=S-1φk=S-11,1T=0,1T,k∈Z-20.The transformed system (55) with the initial problem is(65)yk+1=Λ1yk+B1∗yk-1+B2∗yk-2,k∈Z0∞,where (66)yk=φ∗k=φ1∗k,φ2∗k=0,1T,k∈Z-20or (67)y1k+1=0.5y1k+2y2k-1-x2k-2,y2k+1=-3y2k,where the initial data are (68)y1k=0,y2k=1,k∈Z-20.The function Φ2:Z-20→R2 is defined as (69)Φ2k≔φ2∗k,0T=1,0T,k∈Z-20.All assumptions of Theorem 4 are true. Therefore, the solution of the initial problem (55), (56) is x(k)=Sy(k), k∈Z-2∞, where (I1∗)
y(k)=(y1(k),y2(k)) = (φ1∗(k),φ2∗(k))=0,1T if k∈Z-20,(I2∗)
y(k)=Λ1kφ∗(0)+∑r=0k-1λ1k-1-r∑l=12b12∗lΦ2(r-ml) = (0.5)k00(-3)k01+10∑r=0k-1(0.5)k-1-r if k∈Z12,(I3∗)
y(3)=Λ13φ∗(0) + ∑r=02λ12-rb12∗2Φ2(r-2) + b12∗1[∑r=01λ12-rΦ2(r-1)+Φ2(0)λ2]  =  (0.5)300(-3)301-10∑r=02(0.5)2-r + 20[∑r=01(0.5)2-r-3](I4∗)
y(k)=Λ1kφ∗(0) + ∑l=12b12∗l[∑r=0mlλ1k-1-rΦ2(r-ml) + Φ2(0)∑r=ml+1k-1λ1k-1-rλ2r-ml] = (0.5)k00(-3)k01+20[∑r=01(0.5)k-1-r + ∑r=2k-1(0.5)k-1-r(-3)r-1] + -10[∑r=02(0.5)k-1-r + ∑r=3k-1(0.5)k-1-r(-3)r-2] if k∈Z4∞.The asymptotic behavior of the solutions of systems satisfying the assumptions of Theorem4 is covered by Theorem 19. In the considered case, the assumptions of formula (4) in Theorem 19 are satisfied. Then, (70)y2k=λ2kφ2∗0=-3k,k∈Z1∞,y1k~φ2∗0λ2λ2-λ1λ2k∑l=1nb12∗lλ2-1-ml=-3-3-0.5-3k2-3-2-1-3-3=2-3k-2ifk⟶∞;that is, (71)y1k~2-3k-2k⟶∞.The last formula can be deduced from (I4∗) as it is done (in the general case) in the proof of Theorem 19.Finally, we select initial data for system (55) to derive a result on conditional asymptotic stability by Theorem 11. Let, for k=0, the initial data be given by the equation (72)x0=φ0=φ10,φ20T=c,2cT,where c is an arbitrary constant. Then, the initial data y(0) for (65), deduced from (16), are (73)y0=φ∗0=φ1∗0,φ2∗0T=S-1φ0=S-1c,2cT=c,0T.Since |λ1|=0.5<1, |λ|2=3>1, and φ2∗(0)=0, all assumptions of Theorem 11 are true. Therefore, the zero solution of system (55) is conditionally stable if the initial data (74)xk=φk=φ1k,φ2kT,k∈Z-20satisfy the equation (φ1(0),φ2(0))=(c,2c).In the proof of Theorem11 this statement is derived from formula (I4) in Theorem 4. In the considered particular case we have (75)Φ2k≔φ2∗k,0T,k∈Z-2-1,Φ20≔φ2∗0,0T=0,0T,yk=Λ1kc0+20,5k-1φ2∗-10-0,5k-1φ2∗-20+0,5k-2φ2∗-10=c0.5k0-0,5k-1φ2∗-20=c-2φ2∗-20.5k0.Then (76)limk→∞⁡yk=θsince, by the above formula, (77)limk→∞⁡y1k=0,y2k=0,k∈Z0∞.


## 5. Conclusions

The paper deals with conditional stability and asymptotic representation of solutions to linear weakly delayed discrete systems (1) if the Jordan form of the matrix A equals (12), (13), or (14), that is, Λ1, Λ2, or Λ3. For these cases, conditional stability is investigated and formulas are deduced expressing the asymptotic behavior of solutions as k→∞. For further results related to weakly delayed systems, representations of solutions of discrete systems, their stability, and asymptotic behavior, we refer to [1–11] and to the references therein. It is an open question whether similar results can be derived in the case of system (1) consisting of three equations.



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*Source: 6028078-2017-06-11.xml*