Perron-Type Criterion for Linear Difference Equations with Distributed Delay

The characteristics of many real dynamical models are described by delay differential equations, see the list of recent papers [1–6]. However, delay differential equations are hard to manage analytically and thus many articles have examined the models by using the corresponding delay difference equations. In practice, one can easily formulate a discrete model directly by experiments or observations. For simulation purposes, nevertheless, it is important that a discrete analog faithfully inherits the characteristics of the continuous time parent system. In the recent years, the stability of solutions of linear delay difference equations has been extensively studied in the literature. Many authors have addressed this problem by using various methods and applying different techniques, see for instance the papers [7–13] and references therein. It is well known in the theory of ordinary differential equations (see, e.g., [14, page 120]) that if for every continuous function f (t) bounded on [0,∞), the solution of the equation


Introduction and preliminaries
The characteristics of many real dynamical models are described by delay differential equations, see the list of recent papers [1][2][3][4][5][6].However, delay differential equations are hard to manage analytically and thus many articles have examined the models by using the corresponding delay difference equations.In practice, one can easily formulate a discrete model directly by experiments or observations.For simulation purposes, nevertheless, it is important that a discrete analog faithfully inherits the characteristics of the continuous time parent system.In the recent years, the stability of solutions of linear delay difference equations has been extensively studied in the literature.Many authors have addressed this problem by using various methods and applying different techniques, see for instance the papers [7][8][9][10][11][12][13] and references therein.
It is well known in the theory of ordinary differential equations (see, e.g., [14, page 120]) that if for every continuous function f (t) bounded on [0,∞), the solution of the equation satisfying x(0) = 0 is bounded on [0,∞), then the trivial solution of the corresponding homogeneous equation is uniformly asymptotically stable.This result is referred as Perron's criterion [15].Later, Perron's criterion has been extended by Halanay [14,page 371] to differential equations with distributed delay.Indeed, it was shown that if for every continuous function f (t) bounded on [0,∞), the solution of the equation ), then the trivial solution of the equation is uniformly asymptotically stable.Recently, the above result has been carried out for linear impulsive delay differential equations [16] and for linear impulsive differential equations with distributed delay [17].See also the papers [18][19][20][21] for related works.The paper [22] deserves more attention as it discusses a closed result.For a given differential equation, a difference equation approximation would be most acceptable if the solution of the difference equation is the same as the differential equation at the discrete points.But unless we can explicitly solve both equations, it is impossible to satisfy this requirement.Most of the time, it is desirable that a difference equation, when derived from a differential equation, preserves the dynamical features of the corresponding continuous time model.If such discrete models can be derived from continuous time delay models, then the discrete time models can be used without any loss of functional similarities of continuous models.In this paper, we will consider a discrete time analog of the continuous-time Perron criterion.We will show that a discrete time analog will preserve the stability condition of its continuous time counterpart.Our approach is based on the technique followed in [14] and totally different from the one used in [22].
There are several ways of deriving discrete time version of dynamical systems corresponding to continuous time formulations.One of the ways is based on appropriate modifications of the models.For this technique, differential equations with piecewise constant arguments prove helpful; see [23] for more information.Let us assume that the average growth rate in (1.4) changes at regular intervals of time, then we can incorporate this aspect in (1.4) and obtain the following modified equation: where [t] denotes the integer part of t for t ≥ 0. This equation occupies a position midway between differential and difference equations.Integrating (1.5) on any interval of the form [n,n + 1), n = 0,1,2,..., we obtain where Δx(n Let N and Z denote, as usual, the sets of natural and integer numbers, respectively.Let ζ : N × Z → R m×m be a kernel function satisfying the following conditions: , where d > 3 is a positive integer; (ii) there exists a positive real number γ such that sup t≥0 It turns out that the discrete analog of (1.7) should have the form where by For the sake of convenience, however, we will consider an equation of the form For any a,b ∈ N, define N(a) = {a, a + 1,...} and N(a, b) = {a, a + 1,...,b}, where a ≤ b.By a solution of (1.9), we mean a sequence x(n) of elements in R m which is defined for all n ∈ N(n 0 − d + 1) and satisfies (1.9) for n ∈ N(n 0 ) for some n 0 ∈ N. It is easy to see that for any given n 0 ∈ N and initial conditions of the form Equation (1.9) has a unique solution x(n) which is defined for n ∈ N(n 0 − d + 1) and satisfies the initial conditions (1.10).To emphasize the dependence of the solution on the initial point n 0 and the initial functions φ, we may use the notation x(n) = x(n;n 0 ,φ).Consider the nonhomogeneous equation where f is a sequence with values in R m .Perron's condition for (1.9) is formulated as follows.
Definition 1.1.Equation (1.9) is said to verify Perron's condition if for every bounded sequence f (n) for n ≥ 1, the solution of (1.11) with φ(n The main result of this paper is stated in the following theorem. Theorem 1.2.If (1.9) verifies Perron's condition, then its trivial solution is uniformly asymptotically stable.
The proof is provided in the last section.

Background lemmas
This section is devoted to certain auxiliary assertions that will be needed in the proof of Theorem 1.2.Lemma 2.1 which introduces the main result of this section is needed to define an adjoint equation of (1.9).Lemmas 2.3 and 2.6 give representations of solutions of the considered equations.Lemmas 2.9 and 2.10 are concerned with the boundedness of fundamental functions of (1.9).We will construct the adjoint equation of (1.9) which resembles the one constructed by Halanay in [14, page 365].It turns out that the function has the form where by "T we mean the transposition.Consider the equation where 2) is the adjoint equation of (1.9).The following lemma generalizes a fundamental result to linear difference equations with distributed delay.
Lemma 2.1.Let x(n) be any solution of (1.9) and let y(n) be any solution of (2.2), then where , is defined by (2.1).
Proof.Clearly, it suffices to show that Δ x(n), y(n where (2.5) J. O. Alzabut and T. Abdeljawad 5 It is easy to see that By changing the indices and using the properties of ζ, we see that the above equation is equal to zero.The proof is finished.
Lemma 2.6.Let Y (n,α) be a fundamental function of (2.2) and n 0 ∈ N. If y(n) is a solution of (1.10), then For more details on the derivation of solutions representations for similar difference equations, see the papers [24,25].
Remark 2.8.Formulas (2.12) and (2.13) can be derived from function (2.1).Indeed, replacing X by x or Y by y in (2.16), using (2.14) and employing the properties of X and Y , we obtain the desired results.
Let D denote the space of bounded sequences f = { f (n)} with values in R m and equipped with the norm f ∞ = sup n f (n) .Clearly, D is a Banach space.Lemma 2.9.If (1.9) verifies Perron's condition, then there exists a constant C such that n−1 α=0 X(n,α + 1) < C, for ≥ 1. (2.17) Proof.From formula (2.12), the solution x(n) satisfying (1.10) with φ(n) = 0, n ∈ N(−d + 1,1), has the form For each n ∈ N, define a sequence of linear operators By using the estimate it follows that the operators U n are bounded.In virtue of Perron's condition, we deduce that for each f ∈ D, we can find c f > 0 such that sup n U n ( f ) ≤ c f .Hence, by using the Banach-Steinhaus theorem, there exists a constant T > 0 such that For fixed n ≥ 1, let x rk , (1 ≤ r,k ≤ m), be the elements of the matrix X(n,α + 1) where 0 ≤ α ≤ n − 1.Let e p denote the canonical basis having the unity at the pth place and zero otherwise.Let f r α be the element of D with its α-component the vector V r of R m and zeros otherwise, where V r = m k=1 signx rk e k .The vector X f r α (α) will have its rth component equal to m k=1 |x rk |.
From (2.21), we can write where M 2 = T sup r V r .Hence x rk (n,α + 1) ≤ M 2 . (2.23) Since this relation is true for every r, we take the summation m r=1 of both sides to deduce that there exists C such that n−1 α=0 X(n,α + 1) < C for n ≥ 1, (2.24) which completes the proof.
Lemma 2.10.If (1.9) verifies Perron's condition, then there exists a constant M such that X n,n 0 < M, for n ≥ n 0 ≥ 1. (2.25) Proof.Taking into account that Y T (α,t) satisfies (2.2), we take n−1 α=n0 of both sides  (2.29)By using the properties of ζ, it follows that In view of Lemma 2.9, we end up with the desired result.
We are now in a position to give the proof of Theorem 1.2.

Proof of Theorem 1.2
From formula (2.12), the solution of (1.9) has the form Changing the order of summations, we get In virtue of Lemma 2.10, we obtain x n;n 0 ,φ ≤ M 1 φ 0 , ( where x(n) .uniformly with respect to n 0 and φ.For our purpose, let m 0 ≥ n 0 , then the solution has the form x n;n 0 ,φ) = X n,m 0 x m 0 ;n 0 ,φ + (3.10)