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On stability of delay difference equations with variable coefficients: successive products tests

Abstract

In this paper, we report an error in the paper of the first author in Advances in Difference Equations, 2009, article 104310, present the revised versions of the theorem with several examples, and outline the cases when the previous result is valid.

1 Introduction

The purpose of this short note is to indicate an error in the previous paper [1] published in ‘Advances in Difference Equations’ and an inaccuracy in the recent paper [2]; to present a corrected result for [1] and clarification for [2]; and to outline the cases when the analogue of the result of [1] is still correct.

Consider the equation

x(n+1)= = 1 m a (n)x ( h ( n ) ) for n n 0 ,
(1)

where { a (n)} are sequences of real numbers, and { h (n)} are sequences of integers such that there exists a nonnegative integer τ satisfying nτ h (n)n for all n n 0 and =1,2,,m.

Theorem A Suppose that n h (n)<d for some dN, =1,2,,m, and there exists rN such that

lim sup n j = 0 r = 1 m | a ( n j ) | <1.
(2)

Then (1) is exponentially stable.

Example 1 (Counterexample to Theorem A)

Consider the delay difference equation

x(n+1)=a(n)x ( h ( n ) ) for n0,
(3)

where

a(n)={ p , n = 2 l , q , n = 2 l + 1 andh(n)={ n , n = 2 l , n 1 , n = 2 l + 1
(4)

for some p,qR. Simple computation gives us that the solution of (3) is

x(n)={ q l x ( 0 ) , n = 2 l , p q l x ( 0 ) , n = 2 l + 1 ,
(5)

which is stable if and only if |q|<1. More precisely, we have lim n |x(n)|= for any |q|>1 provided that x(0)0. If we compute (2) with r=1 for (3), we get

lim sup n | a ( n ) a ( n 1 ) | =|pq|
(6)

showing that the assumption of Theorem A is fulfilled if |pq|<1. However, we can find p,qR such that |q|1 and |pq|<1, for instance, p=1/2 and q=6/5. In this case, the right-hand side in (6) is 3/5<1, but by (5) x(2l)= 1.2 l x(0), x(2l+1)=0.5 1.2 l , which is a divergent sequence, the solution is unstable.

Hence, in general, Theorem A is incorrect.

Let us note that in [2] and further in this paper, we apply the idea of reduction of higher (but bounded) order equations to first-order matrix equations. This method was widely used in [35] and in the earlier paper [6].

Also, in the discussion section of [2], the inequality

ρ( A k ( n ) A k ( n ) 1 A n )λ
(7)

is considered as a sufficient asymptotic stability condition for the trivial solution of the first-order matrix equation

X n + 1 = A n X n .
(8)

Here A n are d×d matrices, ρ(A) is the spectral radius of the matrix A, λ(0,1), n 0 N, and k(n)n is a certain number which exists for any n n 0 . Similarly, the condition

lim sup n ρ( A n + k 1 A n + k 2 A n )<1
(9)

is treated as a sufficient exponential stability condition for the trivial solution. This is not true, as the example from [[7], Example 4.17, pp.190-191] illustrates (here k=1, k(n)=n); see also the recent review [8] and Example 2 below.

Example 2 Equation (8), with

A 2 m =( 0 1.2 0.6 0 ), A 2 m + 1 =( 0 0.6 1.2 0 ),m=0,1,2,
(10)

satisfies ρ( A n )=0.6 2 <1 since both A 2 m and A 2 m + 1 have eigenvalues ±0.6 2 . However, if we assume X 0 = ( 0 , 1 ) T , then simple calculations lead to X 2 m = ( 0 , 1.44 m ) T , thus the system is unstable.

On the other hand, if we use in Example 2 the norm

A= sup X = 1 AX,

where is the Euclidean vector norm, instead of the spectral radius, then A m n = A 2 m + 1 =1.2>1 since A 2 m ( 0 , 1 ) T = A 2 m + 1 ( 1 , 0 ) T =1.2, where B T is the transpose of B.

We will use the following result in the recovery of Theorem A, which was obtained in [9]; see also [10, 11].

Theorem B ([[9], Theorem 2])

Let mN and f:Z× R m R. If there exists λ(0,1) such that

| f ( n , u 1 , u 2 , , u m ) | λ max 1 j m { | u j | } for all n n 0 ,

then

x n + 1 =f(n, x n , x n 1 ,, x n m + 1 ) for n n 0

is globally exponentially stable. More precisely, any solution satisfies

| x n | λ ( n n 0 ) / m max n 0 m + 1 j n 0 { | x j | } for all n n 0 .

2 Main results

For kN, define a sequence

b k (n):={ 1 , k = 0 , = 1 m | a ( n ) | b k 1 ( h ( n ) 1 ) , k 1
(11)

for n n 0 +k(τ+1).

Theorem 1 (Correction of Theorem A)

Suppose that there exists rN such that

lim sup n b r (n)<1.

Then (1) is exponentially stable.

Proof Let us prove for all kN that

| x ( n + 1 ) | b k (n) max n k ( τ + 1 ) j n { | x ( j ) | } for all n n 0 +k(τ+1).
(12)

We proceed by induction in k. From (1), for k=1, we have

| x ( n + 1 ) | = 1 m | a ( n ) | | x ( h ( n ) ) | = 1 m | a ( n ) | max n ( τ + 1 ) j n { | x ( j ) | } = b 1 ( n ) max n ( τ + 1 ) j n { | x ( j ) | }
(13)

for all n n 0 +τ+1. Thus, the claim is true for k=1. Assume now that the claim is true for some k1. From (12) and (13), for all n n 0 +(k+1)(τ+1), we have

| x ( n + 1 ) | = 1 m | a ( n ) | b k ( h ( n ) 1 ) max h ( n ) 1 k ( τ + 1 ) j h ( n ) 1 { | x ( j ) | } = 1 m | a ( n ) | b k ( h ( n ) 1 ) max n ( k + 1 ) ( τ + 1 ) j n { | x ( j ) | } = b k + 1 ( n ) max n ( k + 1 ) ( τ + 1 ) j n { | x ( j ) | } ,

which shows that (12) is true when k is replaced with (k+1). Using (12) with k=r, we see that the solution is exponentially stable by Theorem B. □

Theorem 1 with r=1 immediately yields the following result.

Corollary 1 Assume that

lim sup n = 1 m | a ( n ) | <1.

Then (1) is exponentially stable.

Remark 1 The claim of Theorem A for r=0 is correct.

Setting r=2 in Theorem 1, we obtain the following corollary, which is also proved in [[1], Theorem 2.17].

Corollary 2 Assume that

lim sup n 1 = 1 m | a 1 ( n ) | 2 = 1 m | a 2 ( h 1 ( n ) 1 ) | <1.

Then (1) is exponentially stable.

Remark 2 Theorem A for the nondelay equation

x(n+1)=a(n)x(n)for n n 0

is correct. Indeed, Theorem 1 reduces to Theorem A since for k1, we get

b k (n)= j = 0 k 1 | a ( n j ) | for n n 0 +k.

Setting r=3 in Theorem 1 gives us the following corollary.

Corollary 3 Assume that

lim sup n 1 = 1 m | a 1 ( n ) | 2 = 1 m | a 2 ( h 1 ( n ) 1 ) | 3 = 1 m | a 3 ( h 2 ( h 1 ( n ) 1 ) 1 ) | <1.

Then (1) is exponentially stable.

Example 3 Consider the delay difference equation (3) with (4), where p,qR, which can be written in the two equivalent forms:

x(n+1)= a 1 (n)x(n) a 2 (n)x(n1)for n0
(14)

and

x(n+1)x(n)= ( a 1 ( n ) + 1 ) x(n) a 2 (n)x(n1)for n0,
(15)

where

a 1 (n)={ p , n = 2 l , 0 , n = 2 l + 1 and a 2 (n)={ 0 , n = 2 l , q , n = 2 l + 1 .

Computing { b k (n)} defined by (11), we see that

b k (n)={ | p q k 1 | , n = 2 l , | q k | , n = 2 l + 1 for kN,n2k.

Equation (3) is exponentially stable by Theorem 1 if |q|<1 because there always exists rN such that |p q r 1 |<1 and | q r |<1. From (5), we see that |q|<1 is the best possible condition for the global exponential stability of (3) with (4).

Application of a recent result [[12], Theorem 6] to (14) gives us (1+|p|)|q|<1, which implies |q|<1.

The so-called ‘3/2-test’ (see [13] and [[1], Theorem A]) can be applied to (15) if p>1 and q>0, and ensures global exponential stability when

p+q+2< 3 2 + 1 2 2 = 7 4 or equivalentlyp+q< 1 4

for which 0<q<3/4 is necessary.

It is obvious that these two results and Corollary 1 cannot deliver any answer for the exponential stability when p=1 and q=1/2.

As mentioned in Remark 2, Theorem A is valid for a nondelay scalar equation. Next, any higher-order (of the order not exceeding d) equation (1), with nd< h (n)n, can be rewritten as the first-order system

X(n+1)=A(n)X(n),n=0,1,2,,
(16)

where X(n) R d , A(n) are d×d matrices. Indeed, denote X(0)= ( x ( d + 1 ) , x ( d + 2 ) , , x ( 0 ) ) T , X(n)= ( x ( n d d + 1 ) , x ( n d d + 2 ) , , x ( n d ) ) T and rewrite (1) as

x(n)= j = 1 d b(n,j)x(nj),nN,
(17)

where

b(n,j)= l { 1 , , m | h l ( n ) = n j } a l (n).

Then we can define the matrix A(0)= ( c ( i , j ) ) i = 1 d as follows:

and X(1)=A(0)X(0). Similarly, we construct A(n), nN and obtain system (16). Since X(n)|x(j)|, j=ndd+1,ndd+2,,nd, exponential (asymptotical) stability of (16) implies the relevant stability of (1). We recall that (16) is exponentially stable if there exist n 0 N, L>0, and μ(0,1) such that X(n)L μ n X(0), n n 0 .

Theorem 2 If there exist λ(0,1), M>0, and n 0 N such that A(n)M and j = n n + k 1 A(j)λ for every n n 0 and for some positive integer k, then (16) is exponentially stable.

Proof Without loss of generality, we can assume M>1 and j = 0 n 0 1 A(j)M. Further, for any n n 0 denote m=[ n n 0 1 k ], where [t] is the integer part of t, and obtain the estimate

X ( n ) = A ( n 1 ) A ( 0 ) X ( 0 ) A ( n 1 ) A ( 0 ) X ( 0 ) = j = n 0 + k m n 1 A ( j ) j = n 0 n 0 + k 1 A ( j ) j = 0 n 0 1 A ( j ) X ( 0 ) j = n 0 + k m n 1 A ( j ) j = n 0 n 0 + k 1 A ( j ) j = 0 n 0 1 A ( j ) X ( 0 ) M λ m M k M k + 1 ( 1 λ ) 1 + n 0 / k ( λ 1 / k ) n X ( 0 ) = L μ n X ( 0 )

for n n 0 , where L= M k + 1 λ 1 n 0 / k , μ= λ 1 / k . □

Example 4 (Example 6 in [2])

If in (16)

A 2 m = 1 2 ( 5.1 4.9 4.9 5.1 ), A 2 m + 1 = 1 2 ( 7.1 6.9 6.9 7.1 ),m=0,1,2,

then both A 2 m and A 2 m + 1 have the norms exceeding one (they have eigenvalues of 5 and 7, respectively), but the product

A 2 m A 2 m + 1 = A 2 m + 1 A 2 m =( 0.6 0.1 0.1 0.6 )

has the norm A 2 m A 2 m + 1 = A 2 m + 1 A 2 m =0.7<1, thus (16) is exponentially stable.

Example 5 Consider (16) with a 4-periodic matrix A(n), where A 4 m + j = ( 0 1.1 1.1 0 ) , m=0,1,2, , j=1,2,3, A 4 m = ( 0 0.7 0.7 0 ) . Then A 4 m + j =1.1>1, j=1,2,3, but (16) is exponentially stable since A n + 3 A n + 2 A n + 1 A n = ( 0.9317 0 0 0.9317 ) for any n=0,1,2, , and λ= A n + 3 A n + 2 A n + 1 A n =0.9317<1.

3 Discussion

The dynamics of higher-order difference equations with variable coefficients, as well as of non-autonomous systems of difference equations, is much more complicated than that of the relevant autonomous models; see, for example, [8, 14]. For example, the fact that the spectral radius of each matrix is less than one does not imply exponential stability of the system. On the other hand, as demonstrated in Example 5, non-autonomous systems, where some matrices have norms exceeding one, can still be exponentially stable. The challenge is to extend recursive results to other type of stability, for example, asymptotic and l p stability; see, for example, [3].

Regarding generalizations to some types of nonlinear models, the analogue of Theorem 1 can be found in [2], while Theorem 2 can be reformulated for the nonlinear first-order system

X(n+1)= F n ( X ( n ) ) ,n=0,1,2,,
(18)

in the following way, with the same proof repeated.

Theorem 3 If there exist λ(0,1), kN, M>0, and n 0 N such that F n (X)MX and F n + k 1 ( F n + 1 ( F n (X)))λX for any X R d and n n 0 , then (18) is uniformly exponentially stable.

Again, the case of possible asymptotic stability when

F n + k 1 ( F n + 1 ( F n ( X ) ) ) λ n X

and lim sup n λ n =1 is still to be considered.

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Acknowledgements

The authors are grateful to the anonymous reviewer whose valuable comments greatly contributed to the presentation of the paper. EB was partially supported by the NSERC Research grant. This work was partially completed while BK was on one-year leave at the Department of Mathematics and Statistics, University of Calgary, Canada, in the framework of Doctoral Research Scholarship of the Council of Higher Education of Turkey. The authors are also grateful to the University of Calgary who supported this publication.

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Correspondence to Elena Braverman.

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Braverman, E., Karpuz, B. On stability of delay difference equations with variable coefficients: successive products tests. Adv Differ Equ 2012, 177 (2012). https://doi.org/10.1186/1687-1847-2012-177

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Keywords

  • Difference Equation
  • Asymptotic Stability
  • Variable Coefficient
  • Trivial Solution
  • Spectral Radius