Sufficient conditions for the exponential stability of delay difference equations with linear parts defined by permutable matrices

This paper deals with the stability problem of nonlinear delay difference equations with linear parts defined by permutable matrices. Several criteria for exponential stability of systems with different types of nonlinearities are proved. Finally, a stability result for a model of population dynamics is proved by applying one of them.


Introduction
This paper is concerned with the stability of the nonlinear delay difference equations.Throughout the paper we use for zero matrix notation Θ, I represents identity matrix with I = 1.For given integers s, q, such that s < q we denote Z q s := {s, s + 1, . . ., q} the set of integers.First let us recall the result from the paper [2] concerning the representation of a solution of a linear delay difference system which will be starting point in our further study of the stability problems for nonlinear perturbation of this system.Theorem 1.1.Let ϕ : Z 0 −m → R n be a given function, A, B are n × n constant permutable matrices, i.e.AB = BA with det A = 0. Then the trivial solution of the initial-value problem where This theorem is a discrete version of [6,Th. 2.1].Function e Bk m is called the discrete delayed-matrix exponential [3] and is given by This matrix function was used to construct the general solution of planar linear discrete systems with weak delay in [5].Problems of controllability of linear discrete systems with constant coefficients and pure delay are considered in [4].
Applying Theorem 1.1 to the initial-value problem EJQTDE, 2012 No. 22, p. 2 where m ≥ 1 is a constant delay, ϕ : Z 0 −m → R n a given initial function, f (x(k), x(k − m)) a given function and constant n × n matrices A, B are permutable, we obtain the following representation of its solution: Our aim is to find some sufficient conditions for the exponential stability of the trivial solution of a nonlinear delay difference equation with different types of nonlinearities in the sense of the following definition.
Definition 1.1.Let m > 1, and ϕ : The following lemma will be helpful in our estimations.2 Systems with a nonlinearity independent of delay Now we state sufficient conditions for the exponential stability of the trivial solution of the nonlinear equation Some analogical results for the delay differential equations are proved in the paper [7].
Theorem 2.1.Let A, B be n × n permutable matrices, i.e.AB = BA, then the trivial solution of the equation ( 5) is exponentially stable.
Proof.From Lemma 1.1 we can estimate the solution of the equation (5) as follows EJQTDE, 2012 No. 22, p. 4 Denoting we obtain Now by applying the discrete version of the Gronwall's inequality (cf.[1]) we obtain and this yields the inequality Proof.Similarly to the proof of the previous theorem we derive the following estimate for k ∈ Z ∞ 0 , where we have used the notation of ( 6), (7) and the assumption x(j) < δ for each j ∈ Z k−1 0 .Now let c := max{M, ϕ(0) }, λ(j) := P C −α C (α−1)j , ω := u α .
EJQTDE, 2012 No. 22, p. 5 Then It is easy to see that Consequently, applying the discrete version of the Bihari's theorem (cf.[9]) we obtain where Note that the expression W −1 [W (c) + λ ] is surely less than infinity.If K denotes the constant on the right-hand side of (8) and max{K, c} < δ, we get for P sufficiently small.Since B 1 < − ln A , the trivial solution of the equation ( 5) is exponentially stable.

Systems with a nonlinearity depending on delay
In this section, we consider the system where m ≥ 1 is a constant delay, ϕ : Z 0 −m → R n an initial function and matrices A, B are permutable.Here we derive sufficient conditions for the exponential stability of the trivial solution of equation ( 9) with different types of a given function f.
Then using the notation ( 6) and ( 7) we obtain the estimate for the solution x(k) of ( 3) Now denote g(k) the right-hand side of the above inequality.Note that it is a nondecreasing function.Apparently, u(k) ≤ g(k) and from the property of the maximum we have  Using the Gronwall's inequality we obtain Consequently, for the solution x(k) we have the estimate One can see that the trivial solution of the equation ( 9) is exponentially stable whenever c < δ and Theorem 3.2.Let α 1 > 1, α 2 > 1 and matrices A, B and B 1 be as in the Theorem 3.1.Assume that A e B 1 < 1 and f (x, y) = o( x α 1 + y α 2 ).
Then the trivial solution of the equation ( 9) is exponentially stable.
Proof.Similarly to the proof of the previous theorem we estimate the solution of the equation (9).Supposing x(k) < δ for k ∈ Z ∞ −m and using notation ( 6) and ( 7) we get where Using the property of the maximum (11) we have where λ 2 (j) := 2 α 2 λ 2 (j).Now we apply the estimation proposed by Pinto, Medina (cf.[8]).If Thus one can estimate the function g(k) with a constant and denote it K.
Since 1 P λ 1,2 ∈ l 1 , one can find constant P > 0 such small that the following conditions are fulfilled.Apparently, if B 1 < − ln A the trivial solution of the equation ( 9) is exponentially stable whenever max{K, ϕ } < δ.

Lemma 1 . 1 . 4 )
Let m ≥ 1 be a constant delay.Then for any k ∈ Z the following inequality holds true e Bk m ≤ e B (k+m) .(Proof.Using the definition of delayed matrix exponential one can easily prove the statement.EJQTDE, 2012 No. 22, p. 3