ASYMPTOTIC ESTIMATES AND EXPONENTIAL STABILITY FOR HIGHER-ORDER MONOTONE DIFFERENCE EQUATIONS

Asymptotic estimates are established for higher-order scalar di ﬀ erence equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit su ﬃ cient conditions for the global exponential stability of the zero solution are given.


Introduction
Consider the higher-order scalar difference equation x n+1 = f x n ,x n−1 ,...,x n−k , n ∈ N = {0, 1,2,...}, (1.1) where k is a positive integer and f : R k+1 → R. With (1.1), we can associate the discrete dynamical system (T n ) n≥0 on R k+1 , where T : R k+1 → R k+1 is defined by T(x) = f (x),x 0 ,x 1 ,...,x k−1 , x = x 0 ,x 1 ,...,x k ∈ R k+1 . (1.2) As usual, T n denotes the nth iterate of T for n ≥ 1 and T 0 = I, the identity on R k+1 . It follows by easy induction on n that if (x n ) n≥−k is a solution of (1.1), then x n ,x n−1 ,...,x n−k = T n x 0 ,x −1 ,...,x −k , n ≥ 0. (1.3) where µ ≥ 0 is a parameter. In [7], it has been shown that T is monotone (order preserving) under appropriate conditions on f . As a consequence of monotonicity, necessary and sufficient conditions have been given for the boundedness of all solutions and for the local and global stability of an equilibrium of (1.1) (see [7,Section 4]).
In this paper, we give further consequences of the monotonicity of T for (1.1) and for the corresponding difference inequality y n+1 ≤ f y n , y n−1 ,..., y n−k , n ≥ 0, (1.5) under the additional assumption that the nonlinearity f is positively homogeneous (of degree one) on the generating cone C µ , that is, (1.6) An example of (1.1) with property (1.6) is the max type difference equation where k and r are positive integers and the coefficients K i and b are constants. For other examples of higher-order difference equations with a positively homogeneous right-hand side, see, for example, [6].
Using the monotonicity of T and a simple comparison theorem, we give upper exponential estimates for the solutions of (1.5) in terms of the largest positive root of the characteristic equation λ k+1 = f λ k ,λ k−1 ,...,1 . (1.8) As a corollary for the difference inequality y n+1 ≤ k i=0 K i y n−i + b max y n , y n−1 ,..., y n−r , (1.9) we obtain a generalization of earlier results of Ferreiro and the first author [8] on discrete Halanay-type inequalities (see Theorems 1.1 and 3.1). For other related results, see, for example, [1,9,10]. Further, we will show that a mild strengthening of the monotonicity condition in [7] implies that the map T is eventually strongly monotone. As a consequence, a nonlinear version of the Perron-Frobenius theorem [3] applies and we obtain an asymptotic representation of the solutions of (1.1) starting from C µ (see Theorems 1.2 and 3.7). For a similar result, using the standard ordering in R k+1 (µ = 0), see [6].
Finally, we establish an asymptotic exponential estimate for the growth of the solutions of the equation generates a monotone system and the growth of the nonlinearity g : N × R r+1 → R is controlled by a positively homogeneous function which is nondecreasing in each of its variables (see Theorems 1.3 and 3.10). As a corollary, we obtain explicit sufficient conditions for the global exponential stability of the zero solution of (1.10) (see Theorems 1.4 and 3.11).
The following four theorems give a flavor of our more general results presented in Section 3. Without loss of generality, we assume that in all Theorems 1.1, 1.2, 1.3, and 1.4 below, k ≥ r. The first theorem offers an upper estimate for the solutions of inequality (1.9).
The next result shows in case of (1.7) the exponential estimate (1.13) of Theorem 1.1 is sharp. (1.12) holds with a strict inequality for some µ > 0.
where λ 0 has the meaning from Theorem 1.1.
The following theorem provides an estimate for the growth of the solutions of (1.10). where λ 0 has the meaning from Theorem 1.1.
The existence and uniqueness of the solution λ 0 of (1.14) in (µ,∞) is a part of the conclusions of Theorems 1.1, 1.2, and 1.3. This λ 0 is a root of either In the special case K 0 ≥ 0, K i = 0 for i = 1,2,...,k and 0 < b < 1 − K 0 , the conclusion of Theorem 1.1, a discrete analogue of Halanay's inequality, was obtained by Ferreiro and the first author (see [8,Theorem 1]). The same remark holds for Theorem 1.4 (see [8,Theorem 2]).
Under the hypotheses of Theorem 1.4, the global asymptotic stability of the zero solution of (1.10) was established by the second author using a different approach (see [11, Corollary 2 and Remark 2]).
The paper is organized as follows. In Section 2, we discuss the monotonicity properties of the map T defined by (1.2). The main results on the behavior of the solutions of the above higher-order difference equations and inequalities are given in Section 3.

Monotonicity
Recall the definition of the discrete exponential ordering from [7]. For every µ ≥ 0, the convex closed cone C µ defined by (1.4) has nonempty interior int C µ given by (2.1)

E. Liz and M. Pituk 45
As a cone in R k+1 , each C µ induces a partial order ≤ µ on R k+1 by x ≤ µ y if and only if y − x ∈ C µ . We write x < µ y if x ≤ µ y and x = y. The strong ordering µ is defined by x µ y if and only if y − x ∈ intC µ . The ordering ≤ µ is called the discrete exponential ordering. Note that the restriction µ < 1 in [7] is not needed here.
The following result follows immediately from the definition of the ordering ≤ µ (see also [7,Proposition 1]). It gives a necessary and sufficient condition for the map T defined by (1.2) to be monotone. Recall that T is said to be monotone (increasing, order preserving) A relatively easily verifiable sufficient condition for (2.3) to hold is given below.
If f is differentiable, then the constants L i in (2.4) may be viewed as the infima of the partial derivatives ∂ f /∂x i (x), where the infimum is taken over all x ∈ R k+1 . The next theorem shows that a mild strengthening of the monotonicity condition (2.3) implies that T is eventually strongly monotone.
Then, T k is strongly monotone with respect to ≤ µ , that is, Proof. Let x, y ∈ R k+1 satisfy x < µ y. We must show that T k (y) µ T k (x). In view of the definition of intC µ and the relation the last inequality is equivalent to the system of inequalities Further, by virtue of (2.9) and the definition of T, we have and hence T(y) > µ T(x). Using (2.6) again, we find Thus, (2.10) holds for i = 0. Suppose for induction that (2.10) holds for some i ≥ 0. By monotonicity, T i+2 (y) ≥ µ T i+2 (x). Moreover, in view of (2.10) and the definition of T, we have Consequently, T i+2 (y) > µ T i+2 (x) and therefore (2.6) and (2.10) imply that Thus, (2.10) holds for all i = 0,1,2,.... As noted before, (2.9) and (2.10) imply that The proof of Proposition 2.4 is an obvious modification of the proof of [7, Proposition 2] and thus it is omitted.

E. Liz and M. Pituk 47
In the next theorem, we describe some further properties of T under the additional assumption that f is continuous and positively homogeneous on C µ . In particular, it can be used to ensure the existence of a strongly positive eigenvector of T.  1 and f (1,1,...,1) < 1.
We conclude this section with some corollaries of the previous results for (1.7), a special case of (1.1) when (2.21) As in Section 1, we assume that k ≥ r in (1.7).
Corollary 2.7. Suppose that b ≥ 0 and µ > 0. Then, the following hold. To prove (iii), observe that, in view of (1.12), we have If (a), (b), or (c) holds, then one of the above inequalities is strict and thus (2.16) holds. The last two conclusions of (iii) follow from Theorem 2.5(ii) and Remark 2.6.

Main results
In the theorems below, we assume that f is positively homogeneous and satisfies either the monotonicity condition (  The existence of a root λ 0 of (1.8) in (µ,∞) can be guaranteed by Theorem 2.5(ii). We have the following corollary of Theorems 2.5 and 3.1. Before we present the proof of Theorem 3.1, we establish a comparison theorem which is interesting in its own right. Note that in this theorem we merely assume the monotonicity condition (2.3).
Theorem 3.5. Suppose (2.3) holds for some µ ≥ 0. Let (x n ) n≥−k and (y n ) n≥−k be solutions of (1.1) and (1.5), respectively, such that Then, for all n ≥ 0, In particular, y n ≤ x n , n ≥ −k. We are in a position to give a proof of Theorem 3.1.
Our next aim is to show that for the nontrivial solutions (x n ) n≥−k of (1.1) starting from C µ , the exponential estimate (3.1) of Theorem 3.1 can be replaced with the more precise limit relation lim n→∞ λ −n 0 x n = L, (3.12) where L is a positive constant depending on the initial data. Note that if f in Theorem 3.7 is linear, then the value of the limit (3.12) can be given explicitly in terms of the initial data (x 0 ,x −1 ,...,x −k ) (see [2] or [4] for details).
The proof of Theorem 3.7 will be based on a nonlinear version of the Perron-Frobenius theorem due to Kloeden and Rubinov [3] adapted to our situation. For further related results, see [5].
Theorem 3.8. Let µ ≥ 0. Suppose that T : C µ → R k+1 is a continuous, positively homogeneous map with the following properties: there exist λ > 0 and u µ 0 such that T(u) = λu, (iii) T is monotone on C µ , that is, (iv) some iterate T s (s ≥ 1) of T is strongly monotone on C µ , that is, T s (y) µ T s (x) whenever x, y ∈ C µ satisfy x < µ y, (3.14) Then, for every x ∈ C µ \ {0}, there exists a positive constant K = K(x) such that for some K > 0. By virtue of (1.3), the last limit relation is equivalent to (3.12) with L = Kλ k 0 . Remark 3.9. Theorem 1.2 in Section 1 is a consequence of Theorem 3.7 and Corollary 2.7. Now, we present a theorem concerning the behavior of the solutions of (1.10). We will assume that the linear part of (1.10) generates a monotone system with respect to the ordering ≤ µ and we use the variation-of-constants formula to obtain an exponential estimate for the growth of the solutions. As in Section 1, we assume that k ≥ r in (1.10).
Proof. First, we show that (3.22) has a unique root in (µ,∞). We will apply Theorem 2.5(ii) to the equation (3.25) Since any of the conditions (a), (b), or (c) implies that one of the last two inequalities is strict, (2.16) holds. The existence and uniqueness of λ 0 now follows from Theorem 2.5(ii). Now, we prove (3.21). Let (x n ) n≥−k be an arbitrary solution of (1.10). Consider the solution (y n ) n≥−k of the linear equation (1.11) with the same initial data, (y 0 , y −1 ,..., where (w n ) n≥−k is the solution of (1.11) with initial data (w 0 ,w −1 ,...,w −k ) = (1,λ −1 0 ,..., λ −k 0 ). The same argument applied to the solution (−y n ) n≥−k of (1.11) yields the existence of M 2 > 0 such that −y n ≤ M 2 w n , n ≥ −k. (3.29) Consequently, Writing the variation-of-constants formula for the solution (λ n 0 ) n≥−k of (3.23), we obtain for n ≥ 0, where w n and v n are the solutions of (1.11) defined as before. This and the positive homogeneity of h imply that the right-hand side of (3.35) is equal to Mλ n 0 . Thus, we have shown that (3.33) implies that |x n | ≤ Mλ n 0 . The same argument as in Remark 2.6 shows that the constants M 1 and M 2 in the previous proof and hence M in (3.21) can be written in the form (3.11) (with y replaced with x). Consequently, Theorem 3.10 combined with Remark 2.6 yields the following stability criterion.