Advances in Difference Equations
Volume 2009 (2009), Article ID 104310, 13 pages
doi:10.1155/2009/104310
Research Article

Exponential Stability of Difference Equations with Several Delays: Recursive Approach

Leonid Berezansky1 and Elena Braverman2

1Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary AB, Canada T2N 1N4

Received 8 January 2009; Revised 16 April 2009; Accepted 23 April 2009

Academic Editor: Istvan Gyori

Copyright © 2009 Leonid Berezansky and Elena Braverman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain new explicit exponential stability results for difference equations with several variable delays and variable coefficients. Several known results, such as Clark's asymptotic stability criterion, are generalized and extended to a new class of equations.

1. Introduction and Preliminaries

In this paper we study stability of a scalar linear difference equation with several delays, 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ℎ ( 𝑛 ) 𝑥 𝑙 ( 𝑛 ) , ℎ 𝑙 ( 𝑛 ) ≤ 𝑛 , 𝑛 ≥ 𝑛 0 , ( 1 . 1 ) where ℎ 𝑙 ( 𝑛 ) is an integer for any 𝑙 = 1 , … , 𝑚 , 𝑛 is an integer, 𝑛 0 ≥ 0 . Stability of (1.1) and relevant nonlinear equations has been an intensively developed area during the last two decades.

Let us compare stability methods for delay differential equations and delay difference equations. Many of the methods previously used for differential equations have also been applied to difference equations. However, there are at least two methods which are specific for difference equations. The first approach is reducing a solution of a delay difference equation to the values of a solution of a delay differential equation with piecewise constant arguments at integer points. The second method is based on a recursive form of difference equations and is described in detail later. In this paper we obtain new stability results based on the recursive solution representation.

For (1.1) everywhere below we assume that ℎ 𝑙 ( 𝑛 ) ≥ 𝑛 − 𝑇 ,   𝑛 ≥ 𝑛 0 ≥ 0 , that is, the system has a finite memory, and the following initial conditions are defined 𝑥 ( 𝑛 ) = 𝜑 ( 𝑛 ) , 𝑛 0 − 𝑇 ≤ 𝑛 ≤ 𝑛 0 . ( 1 . 2 )

Definition 1.1. Equation (1.1) is exponentially stable if there exist constants 𝑀 > 0 , 𝜆 ∈ ( 0 , 1 ) such that for every solution { 𝑥 ( 𝑛 ) } of (1.1) and (1.2) the inequality | | | | 𝑥 ( 𝑛 ) ≤ 𝑀 𝜆 𝑛 − 𝑛 0 m a x 𝑛 0 − 𝑇 ≤ 𝑘 ≤ 𝑛 0 | | | | 𝜑 ( 𝑘 ) ( 1 . 3 ) holds for all 𝑛 ≥ 𝑛 0 , where 𝑀 , 𝜆 do not depend on 𝑛 0 ≥ 0 .
Equation (1.1) is stable if for any 𝜀 > 0 there exists 𝛿 > 0 such that m a x 𝑛 0 − 𝑇 ≤ 𝑘 ≤ 𝑛 0 { | 𝜑 ( 𝑘 ) | } < 𝛿 implies | 𝑥 ( 𝑛 ) | < 𝜀 , 𝑛 ≥ 𝑛 0 ; if 𝛿 does not depend on 𝑛 0 , then (1.1) is uniformly stable.
Equation (1.1) is attractive if for any 𝜑 ( 𝑘 ) a solution tends to zero l i m 𝑛 → ∞ 𝑥 ( 𝑛 ) = 0 . It is asymptotically stable if it is both stable and attractive.

One of the methods to establish stability of difference equations is based on a recursive form of these equations; see the monographs [1, 2]. The following result was also obtained by this method.

Lemma 1.2 ([3, 4]). Consider the nonlinear delay difference equation 𝑥 𝑛 + 1 = 𝑓 𝑛 , 𝑥 𝑛 , … , 𝑥 𝑛 − 𝑇 , 𝑛 ≥ 0 . ( 1 . 4 ) Assume that 𝑓 ∶ ℕ × ℝ 𝑇 + 1 → ℝ satisfies | | 𝑓 𝑛 , 𝑢 0 , … , 𝑢 𝑇 | | | | 𝑢 ≤ 𝑏 m a x 0 | | | | 𝑢 , … , 𝑇 | | , ( 1 . 5 ) for some constant 𝑏 < 1 , and for all ( 𝑛 , 𝑢 0 , … , 𝑢 𝑇 ) ∈ ℕ × ℝ 𝑇 + 1 . Then | | 𝑥 𝑛 | | ≤ 𝑏 𝑛 / ( 𝑇 + 1 ) 𝑀 0 , 𝑛 ≥ 0 , ( 1 . 6 ) for every solution { 𝑥 𝑛 } of (1.4), where 𝑀 0 = m a x − 𝑇 ≤ 𝑖 ≤ 0 { | 𝑥 𝑖 | } . In particular, the zero solution of (1.4) is globally exponentially stable.

This result was applied to a more general than (1.1) nonlinear difference equation 𝑥 𝑛 + 1 − 𝑥 𝑛 = − 𝑁 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( 𝑛 ) 𝑥 𝑛 − 𝑘 + 𝑓 𝑛 , 𝑥 𝑛 , … , 𝑥 𝑛 − 𝑇 . ( 1 . 7 ) Without loss of generality, we can suppose that 𝑁 ≤ 𝑇 . We assume that there exist constants 𝑏 𝑛 ≥ 0 such that | | 𝑓 𝑛 , 𝑢 0 , … , 𝑢 𝑇 | | ≤ 𝑏 𝑛 | | 𝑢 m a x 0 | | | | 𝑢 , … , 𝑇 | | , ( 1 . 8 ) for all 𝑛 ≥ 0 and ( 𝑢 0 , … , 𝑢 𝑇 ) ∈ ℝ 𝑇 + 1 .

Similar argument leads to the following result.

Lemma 1.3 ([4]). Assume that for n large enough inequality (1.8) holds and there exists a constant 𝛾 ∈ ( 0 , 1 ) such that 𝑐 𝑛 | | | | | ∶ = 1 − 𝑁 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 | | | | | + ( 𝑛 ) 𝑁 𝑘 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ( 𝑛 ) 𝑛 − 1 𝑚 = 𝑛 − 𝑘 𝑏 0 𝑥 0 2 0 0 𝑑 𝑚 + 𝑁 𝑘 = 0 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ( 𝑚 ) + 𝑏 𝑛 ≤ 𝛾 . ( 1 . 9 ) Then the zero solution of (1.7) is globally exponentially stable. Moreover, if (1.8) and (1.9) hold for 𝑛 ≥ 0 , then | | 𝑥 𝑛 | | ≤ 𝛾 𝑛 / ( 𝑁 + 𝑇 + 1 ) | | 𝑥 m a x 𝑁 | | | | 𝑥 , … , − 𝑇 | | , 𝑛 ≥ 𝑁 , ( 1 . 1 0 ) for every solution { 𝑥 𝑛 } of (1.7).

Recently several results on exponential stability of high-order difference equations appeared where the results are based on the recursive representations; see, for example, [5, 6]. In particular, in [6, Corollary 7 ] contains the following statement.

Lemma 1.4 ([6]). If there exists 𝜆 ∈ ( 0 , 1 ) such that Λ 𝑛 = | | | | | 𝑁 𝑗 = 0 | | | | | + 0 𝑥 0 2 0 0 𝑑 𝑎 ( 𝑛 − 𝑗 ) − 𝑐 ( 𝑛 ) 𝑁 𝑠 = 1 | | | | | 0 𝑥 0 2 0 0 𝑑 𝑠 − 1 𝑗 = 0 | | | | | | | | | 0 𝑥 0 2 0 0 𝑑 𝑎 ( 𝑛 − 𝑗 ) 𝑐 ( 𝑛 − 𝑠 ) ≤ 𝜆 , ( 1 . 1 1 ) for large 𝑛 , then the zero solution of the equation 𝑥 ( 𝑛 + 1 ) = 𝑎 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑐 ( 𝑛 ) 𝑥 ( 𝑛 − 𝑁 ) ( 1 . 1 2 ) is globally exponentially stable.

In the present paper we obtain some new stability results for (1.1) with several variable delays. In contrast to many other stability tests, we consider the case when the sum of coefficients ∑ 𝑚 𝑘 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( 𝑛 ) or some of its subsum is allowed to be in the interval ( 0 , 2 ] , not just ( 0 , 1 ] . We illustrate our results with several examples.

2. Main Results

Now we can proceed to the main results of this paper. Let us note that any sum where the lower index exceeds the upper index is assumed to vanish.

Theorem 2.1. Suppose that there exist a set of indices 𝐼 ⊂ { 1 , 2 , … , 𝑚 } and 𝛾 ∈ ( 0 , 1 ) such that for n sufficiently large 𝑘 ∈ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ( 𝑛 ) 𝑛 − 1 𝑗 = ℎ 𝑘 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + ( 𝑗 ) 𝑙 ∉ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + | | | | | ( 𝑛 ) 1 − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 | | | | | ( 𝑛 ) ≤ 𝛾 . ( 2 . 1 ) Then (1.1) is exponentially stable.

Proof. Since 𝑥 ( 𝑛 + 1 ) = 𝑥 ( 𝑛 ) − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( ℎ 𝑛 ) 𝑥 𝑘 ( − 𝑛 ) 𝑙 ∉ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ( ℎ 𝑛 ) 𝑥 𝑙 ( = 𝑛 ) 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( 𝑛 ) 𝑛 − 1 𝑗 = ℎ 𝑘 ( 𝑛 ) [ ] 0 𝑥 0 2 0 0 𝑑 𝑥 ( 𝑗 + 1 ) − 𝑥 ( 𝑗 ) + 𝑥 ( 𝑛 ) − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑙 ∉ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ( ℎ 𝑛 ) 𝑥 𝑙 ( 𝑛 ) = − 𝑘 ∈ 𝐼 𝑎 𝑘 ( 𝑛 ) 𝑛 − 1 𝑗 = ℎ 𝑘 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ℎ ( 𝑗 ) 𝑥 𝑙 ( 𝑗 ) + 𝑥 ( 𝑛 ) 1 − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 − ( 𝑛 ) 𝑙 ∉ 𝐼 𝑎 𝑙 ℎ ( 𝑛 ) 𝑥 𝑙 , ( 𝑛 ) ( 2 . 2 ) then | | | | ≤ ⎡ ⎢ ⎢ ⎣ 𝑥 ( 𝑛 + 1 ) 𝑘 ∈ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ( 𝑛 ) 𝑛 − 1 𝑗 = ℎ 𝑘 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + ( 𝑗 ) 𝑙 ∉ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + | | | | | ( 𝑛 ) 1 − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 | | | | | ⎤ ⎥ ⎥ ⎦ ( 𝑛 ) m a x 𝑗 ≤ 𝑛 | | | | 𝑥 ( 𝑗 ) ≤ 𝛾 m a x 𝑗 ≤ 𝑛 | | | | . 𝑥 ( 𝑗 ) ( 2 . 3 ) By Lemma 1.2, (1.1) is exponentially stable.

We can reformulate Theorem 2.1 in the following equivalent form.

Theorem 2.2. Suppose that there exist a set of indices 𝐼 ⊂ { 1 , 2 , … , 𝑚 } and 𝜀 ∈ ( 0 , 1 ) such that for n sufficiently large 𝑘 ∈ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ( 𝑛 ) 𝑛 − 1 𝑗 = ℎ 𝑘 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + ( 𝑗 ) 𝑙 ∉ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | ( 𝑛 ) ≤ m i n 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( 𝑛 ) , 2 − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ( 𝑛 ) − 𝜀 . ( 2 . 4 ) Then (1.1) is exponentially stable.

Assuming first 𝐼 = { 1 } and then 𝐼 = { 1 , 2 , … , 𝑚 } , we obtain the following two corollaries for an equation with a nondelay term.

Corollary 2.3. Let 0 < 𝛾 < 1 and ∑ 𝑚 𝑙 = 2 0 𝑥 0 2 0 0 𝑑 | 𝑎 𝑙 ( 𝑛 ) | + | 1 − 𝑎 1 ( 𝑛 ) | ≤ 𝛾 for 𝑛 large enough. Then the equation 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑎 1 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑚 𝑙 = 2 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 𝑔 ( 𝑛 ) 𝑥 𝑙 ( 𝑛 ) ( 2 . 5 ) is exponentially stable.

Example 2.4. Consider the equation 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 1 . 5 𝑥 ( 𝑛 ) − 0 . 3 s i n ( 𝑛 ) 𝑥 ( ℎ ( 𝑛 ) ) , ( 2 . 6 ) with an arbitrary bounded delay ℎ ( 𝑛 ) : 𝑛 − 𝑇 ≤ ℎ ( 𝑛 ) ≤ 𝑛 for some integer 𝑇 > 0 and any 𝑛 . Since 0 . 3 | s i n 𝑛 | + | 1 − 1 . 5 | ≤ 0 . 8 < 1 , then by Corollary 2.3 this equation is exponentially stable. Lemma 1.4 is formulated for a constant delay, some other tests do not apply since | − 1 . 5 | > 1 .

Corollary 2.5. Suppose that for some 𝛾 ∈ ( 0 , 1 ) the following inequality is satisfied for n large enough: 𝑚 𝑘 = 2 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ( 𝑛 ) 𝑛 − 1 𝑗 = 𝑔 𝑘 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + | | | | | ( 𝑗 ) 1 − 𝑚 𝑘 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 | | | | | ( 𝑛 ) ≤ 𝛾 . ( 2 . 7 ) Then (2.5) is exponentially stable.

Example 2.6. By Corollary 2.5 the equation 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − ( 0 . 5 + 0 . 1 s i n 𝑛 ) 𝑥 ( 𝑛 ) − ( 0 . 6 − 0 . 1 s i n 𝑛 ) 𝑥 ( 𝑛 − 1 ) ( 2 . 8 ) is exponentially stable, since | 0 . 5 + 0 . 1 s i n 𝑛 | + | 0 . 6 − 0 . 1 s i n 𝑛 | = 1 . 1 and 0 . 7 ⋅ 1 . 1 + | 1 − 1 . 1 | = 0 . 8 7 < 1 .

Now let us assume that all coefficients are proportional. Such equations arise as linear approximations of nonlinear difference equations in mathematical biology. Then a straightforward computation leads to the following result.

Corollary 2.7. Suppose that all coefficients are proportional 𝑎 𝑙 ( 𝑛 ) = 𝐴 𝑙 𝑟 ( 𝑛 ) , 𝑙 = 1 , … , 𝑚 , there exist 𝑟 0 > 0 , 𝜀 > 0 and a set of indices 𝐼 ⊂ { 1 , 2 , … , 𝑚 } , such that 𝑟 ( 𝑛 ) ≥ 𝑟 0 > 0 and 𝑙 ∈ 𝐼 | | 𝐴 0 𝑥 0 2 0 0 𝑑 𝑙 | | 𝑛 − 1 𝑘 = ℎ 𝑙 ( 𝑛 ) ∑ 0 𝑥 0 2 0 0 𝑑 𝑟 ( 𝑘 ) ≤ m i n 𝑟 ( 𝑛 ) 𝑙 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝐴 𝑙 ∑ , 2 − 𝑟 ( 𝑛 ) 𝑙 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝐴 𝑙 ∑ − 𝑟 ( 𝑛 ) 𝑙 ∉ 𝐼 | | 𝐴 0 𝑥 0 2 0 0 𝑑 𝑙 | | − 𝜀 ∑ 𝑟 ( 𝑛 ) 𝑚 𝑙 = 1 | | 𝐴 0 𝑥 0 2 0 0 𝑑 𝑙 | | . ( 2 . 9 ) Then (1.1) is exponentially stable.

Assuming constant coefficients and 𝐼 = { 1 , 2 , … , 𝑚 } we obtain the following corollary.

Corollary 2.8. Suppose that all coefficients are constants 𝑎 𝑙 ( 𝑛 ) ≡ 𝑎 𝑙 and l i m s u p 𝑚 𝑛 → ∞ 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | 𝑛 − ℎ 𝑙 ( 𝑛 ) < m i n 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 , 2 − 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 . ( 2 . 1 0 ) Then (1.1) is exponentially stable.

Remark 2.9. Corollary 2.8 for the case ∑ 0 < 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 < 1 was obtained in Proposition 4 . 1 of [7].

Now let us consider the equation with one nondelay and one delay terms 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑎 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑏 ( 𝑛 ) 𝑥 ( ℎ ( 𝑛 ) ) . ( 2 . 1 1 ) Choosing 𝐼 = { 1 } , 𝐼 = { 1 , 2 } , we obtain Parts ( 1 ) and ( 2 ) of Corollary 2.10, respectively.

Corollary 2.10. Suppose that there exists 𝛾 ∈ ( 0 , 1 ) such that at least one of the following conditions holds for 𝑛 sufficiently large: (1) | 𝑏 ( 𝑛 ) | + | 1 − 𝑎 ( 𝑛 ) | ≤ 𝛾 ;(2) ∑ | 𝑏 ( 𝑛 ) | 𝑛 − 1 𝑘 = ℎ ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 [ | 𝑎 ( 𝑘 ) | + | 𝑏 ( 𝑘 ) | ] + | 1 − 𝑎 ( 𝑛 ) − 𝑏 ( 𝑛 ) | ≤ 𝛾 . Then (2.11) is exponentially stable.

Let us now proceed to equations with three terms in the right-hand side 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑎 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑏 ( 𝑛 ) 𝑥 ( ℎ ( 𝑛 ) ) − 𝑐 ( 𝑛 ) 𝑥 ( 𝑔 ( 𝑛 ) ) . ( 2 . 1 2 ) For 𝐼 = { 1 } , { 1 , 2 , 3 } , { 1 , 2 } and { 1 , 3 } we obtain Parts ( 1 ) , ( 2 ) , ( 3 ) , and ( 4 ) , respectively.

Corollary 2.11. Suppose that there exists 𝛾 ∈ ( 0 , 1 ) such that at least one of the following conditions holds for 𝑛 sufficiently large: (1) | 𝑏 ( 𝑛 ) | + | 𝑐 ( 𝑛 ) | + | 1 − 𝑎 ( 𝑛 ) | ≤ 𝛾 ;(2) ∑ | 𝑏 ( 𝑛 ) | 𝑛 − 1 𝑘 = ℎ ( 𝑛 ) ∑ 0 𝑥 0 2 0 0 𝑑 [ | 𝑎 ( 𝑘 ) | + | 𝑏 ( 𝑘 ) | + | 𝑐 ( 𝑘 ) | ] + | 𝑐 ( 𝑛 ) | 𝑛 − 1 𝑘 = 𝑔 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 [ | 𝑎 ( 𝑘 ) | + | 𝑏 ( 𝑘 ) | + | 𝑐 ( 𝑘 ) | ] + | 1 − 𝑎 ( 𝑛 ) − 𝑏 ( 𝑛 ) − 𝑐 ( 𝑛 ) | ≤ 𝛾 ;(3) ∑ | 𝑏 ( 𝑛 ) | 𝑛 − 1 𝑘 = ℎ ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 [ | 𝑎 ( 𝑘 ) | + | 𝑏 ( 𝑘 ) | + | 𝑐 ( 𝑘 ) | ] + | 𝑐 ( 𝑛 ) | + | 1 − 𝑎 ( 𝑛 ) − 𝑏 ( 𝑛 ) | ≤ 𝛾 ;(4) ∑ | 𝑐 ( 𝑛 ) | 𝑛 − 1 𝑚 = 𝑔 ( 𝑛 ) 0 𝑥 0 2 0 0 𝑑 [ | 𝑎 ( 𝑘 ) | + | 𝑏 ( 𝑘 ) | + | 𝑐 ( 𝑘 ) | ] + | 𝑏 ( 𝑛 ) | + | 1 − 𝑎 ( 𝑛 ) − 𝑐 ( 𝑛 ) | ≤ 𝛾 .Then (2.12) is exponentially stable.

Theorem 2.1 and its corollaries imply new explicit conditions of exponential stability for autonomous difference equations with several delays, as well as a new justification for known ones.

Consider the autonomous equation 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑎 1 𝑥 ( 𝑛 ) − 𝑚 𝑙 = 2 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 𝑥 𝑛 − ℎ 𝑙 , ( 2 . 1 3 ) where ℎ 𝑙 > 0 . Choosing 𝐼 = { 1 } we immediately obtain the following stability test.

Corollary 2.12. Let ∑ 𝑚 𝑙 = 2 0 𝑥 0 2 0 0 𝑑 | 𝑎 𝑙 | < m i n { 𝑎 1 , 2 − 𝑎 1 } . Then (2.13) is exponentially stable.

Remark 2.13. This result is well known; see, for example, [8] for 𝑚 = 2 as well as some results for autonomous equations below. We presented it just to illustrate our method.

Further, Theorem 2.1 and Corollaries 2.8 and 2.11 can be reformulated for (2.13) as follows.

Corollary 2.14. Suppose that there exists a set of indices 𝐼 ⊂ { 1 , 2 , … , 𝑚 } , with 1 ∈ 𝐼 , such that 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | 𝑘 ∈ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑘 | | ℎ 𝑘 + 𝑙 ∉ 𝐼 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | + | | | | | 1 − 𝑘 ∈ 𝐼 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 | | | | | < 1 . ( 2 . 1 4 ) Then (2.13) is exponentially stable.

Corollary 2.15. If ∑ 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 | 𝑎 𝑙 | ∑ 𝑚 𝑘 = 2 0 𝑥 0 2 0 0 𝑑 | 𝑎 𝑘 | ℎ 𝑘 ∑ < m i n { 𝑚 𝑘 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 ∑ , 2 − 𝑚 𝑘 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑘 } , then (2.13) is exponentially stable.

Consider now an autonomous equation with two delays: 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑎 0 𝑥 ( 𝑛 ) − 𝑎 1 𝑥 𝑛 − ℎ 1 − 𝑎 2 𝑥 𝑛 − ℎ 2 , ( 2 . 1 5 ) where ℎ 1 > 0 , ℎ 2 > 0 .

Corollary 2.16. Suppose that at least one of the following conditions holds (1) | 𝑎 1 | + | 𝑎 2 | + | 1 − 𝑎 0 | < 1 ;(2) ( | 𝑎 0 | + | 𝑎 1 | + | 𝑎 2 | ) ( | 𝑎 1 | ℎ 1 + | 𝑎 2 | ℎ 2 ) + | 1 − 𝑎 0 − 𝑎 1 − 𝑎 2 | < 1 ;(3) | 𝑎 1 | ℎ 1 ( | 𝑎 0 | + | 𝑎 1 | + | 𝑎 2 | ) + | 1 − 𝑎 0 − 𝑎 1 | < 1 ;(4) | 𝑎 2 | ℎ 2 ( | 𝑎 0 | + | 𝑎 1 | + | 𝑎 2 | ) + | 1 − 𝑎 0 − 𝑎 2 | < 1 .Then (2.15) is exponentially stable.

Let us present two more results which can be easily deduced from the recursive representation of solutions. To this end we consider the equation 𝑥 ( 𝑛 + 1 ) = 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ℎ ( 𝑛 ) 𝑥 𝑙 ( 𝑛 ) , ( 2 . 1 6 ) which is a different form of (1.1).

We recall that we assume 𝑛 − ℎ 𝑙 ( 𝑛 ) ≤ 𝑇 for all delays ℎ 𝑙 ( 𝑛 ) in this paper.

Theorem 2.17. Suppose that there exists 𝜆 ∈ ( 0 , 1 ) such that l i m s u p 𝑚 𝑛 → ∞ 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | ( 𝑛 ) 𝑚 𝑗 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑗 ℎ 𝑙 | | ( 𝑛 ) − 1 ≤ 𝜆 . ( 2 . 1 7 ) Then (2.16) is exponentially stable.

Proof. Without loss of generality we can assume that the expression under l i m s u p in (2.17) does not exceed some 𝜆 < 1 for 𝑛 ≥ 𝑛 0 . Since 𝑥 ℎ 𝑙 = ( 𝑛 ) 𝑚 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 ℎ 𝑙 𝑥 ℎ ( 𝑛 ) − 1 𝑗 ℎ 𝑙 ( 𝑛 ) − 1 , ( 2 . 1 8 ) then 𝑥 ( 𝑛 + 1 ) = 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ℎ ( 𝑛 ) 𝑥 𝑙 = ( 𝑛 ) 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 ( 𝑛 ) 𝑚 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 ℎ 𝑙 𝑥 ℎ ( 𝑛 ) − 1 𝑗 ℎ 𝑙 . ( 𝑛 ) − 1 ( 2 . 1 9 ) Hence for 𝑛 ≥ 𝑛 0 + 2 𝑇 + 1 we have | | | | ≤ 𝑥 ( 𝑛 + 1 ) 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | ( 𝑛 ) 𝑚 𝑗 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑗 ℎ 𝑙 | | ( 𝑛 ) − 1 m a x 𝑛 − 2 𝑇 − 1 ≤ 𝑘 ≤ 𝑛 | | | | 𝑥 ( 𝑘 ) ≤ 𝜆 m a x 𝑛 − 2 𝑇 − 1 ≤ 𝑘 ≤ 𝑛 | | 𝑥 | | . ( 𝑘 ) ( 2 . 2 0 ) Thus by Lemma 1.2 | 𝑥 ( 𝑛 ) | ≤ 𝑀 0 𝜇 𝑛 − 𝑛 0 , where 𝜇 = 𝜆 1 / ( 2 𝑇 + 2 ) , 𝑀 0 = 𝜆 − 2 𝑇 − 1 m a x − 𝑇 ≤ 𝑘 ≤ 𝑛 0 | 𝑥 ( 𝑘 ) | , for 𝑛 ≥ 𝑛 0 , so (2.16) is exponentially stable.

Theorem 2.18. Suppose that there exists 𝜆 ∈ ( 0 , 1 ) , 𝑁 ∈ ℕ such that l i m s u p 𝑁 𝑛 → ∞ 𝑗 = 0 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | ( 𝑛 − 𝑗 ) ≤ 𝜆 . ( 2 . 2 1 ) Then (2.16) is exponentially stable.

Proof. Without loss of generality we can assume that the expression under l i m s u p in (2.21) does not exceed some 𝜆 < 1 for 𝑛 ≥ 𝑛 0 . Since | | | | ≤ 𝑥 ( 𝑛 + 1 ) 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | ( 𝑛 ) m a x 𝑛 − 𝑇 ≤ 𝑘 ≤ 𝑛 | | | | ≤ 𝑥 ( 𝑘 ) 𝑁 𝑗 = 0 0 𝑥 0 2 0 0 𝑑 𝑚 𝑙 = 1 | | 𝑎 0 𝑥 0 2 0 0 𝑑 𝑙 | | ( 𝑛 − 𝑗 ) m a x 𝑛 − 𝑁 𝑇 ≤ 𝑘 ≤ 𝑛 | | 𝑥 | | ( 𝑘 ) ≤ 𝜆 m a x 𝑛 − 𝑁 𝑇 ≤ 𝑘 ≤ 𝑛 | | | | , 𝑥 ( 𝑘 ) ( 2 . 2 2 ) then the reference to Lemma 1.2 completes the proof.

3. Discussion and Examples

Let us comment that Theorem 2.18 (see also Corollary 2.12) generalizes the result of Clark [8] that | 𝑝 | + | 𝑞 | < 1 is a sufficient condition for the asymptotic stability of the difference equation 𝑥 ( 𝑛 + 1 ) + 𝑝 𝑥 ( 𝑛 ) + 𝑞 𝑥 ( 𝑛 − 𝑁 ) = 0 . ( 3 . 1 ) We note that there are not really many publications on difference equations with variable delays, and the present paper partially fills up this gap. In particular, Theorem 2.18 gives the same stability condition for the equation with variable delays 𝑥 ( 𝑛 + 1 ) + 𝑝 𝑥 ( ℎ ( 𝑛 ) ) + 𝑞 𝑥 ( 𝑔 ( 𝑛 ) ) = 0 ( 3 . 2 ) once the delays are bounded: 𝑛 − ℎ ( 𝑛 ) ≤ 𝑇 1 , 𝑛 − 𝑔 ( 𝑛 ) ≤ 𝑇 2 .

The following example outlines the sharpness of the condition that the delays are bounded in Theorems 2.1, 2.2, 2.17, and 2.18.

Example 3.1. The equation with constant coefficients 𝑥 ( 𝑛 + 1 ) = 0 . 4 𝑥 ( 𝑛 ) + 0 . 1 𝑥 ( 0 ) , 𝑛 = 0 , 1 , … ( 3 . 3 ) satisfies all assumptions of Theorems 2.1, 2.2, 2.17, and 2.18 but the boundedness of the delay. Since the solution 𝑥 ( 𝑛 ) with 𝑥 ( 0 ) = 6 tends to 1 as 𝑛 → ∞ , then the zero solution of (3.3) is neither asymptotically nor exponentially stable. Here even the condition l i m 𝑛 → ∞ ℎ 𝑙 ( 𝑛 ) = ∞ is not satisfied.

Example 3.2. Let us demonstrate that in the case when the arguments tend to infinity but the delays are not bounded and all other conditions of Theorem 2.18 are satisfied, this does not imply exponential stability. The equation 𝑛 𝑥 ( 𝑛 ) = 0 . 5 𝑥 2 , 𝑛 = 1 , 2 , … , ( 3 . 4 ) where [ 𝑥 ] is the integer part of 𝑥 , is asymptotically, but not exponentially stable. Really, its solution 1 𝑥 ( 0 ) , 2 1 𝑥 ( 0 ) , 4 1 𝑥 ( 0 ) , 4 1 𝑥 ( 0 ) , 8 1 𝑥 ( 0 ) , 8 1 𝑥 ( 0 ) , 8 1 𝑥 ( 0 ) , 8 1 𝑥 ( 0 ) , 1 6 𝑥 ( 0 ) , … ( 3 . 5 ) is nonincreasing by the absolute value and 𝑥 ( 𝑛 ) = 𝑥 ( 1 ) 𝑛 , ( 3 . 6 ) for any 𝑛 = 2 𝑘 , 𝑘 = 0 , 1 , 2 , … , so l i m 𝑛 → 0 𝑥 ( 𝑛 ) = 0 for any 𝑥 ( 1 ) ; the equation is asymptotically stable. Since the solution decay is not faster than 1 / 𝑛 , then the equation is not exponentially stable.

Next, let us compare Corollary 2.10 and Theorem 4 of [6] which is also Lemma 1.4 of the present paper.

Example 3.3. Consider (1.12), where 𝑎 ( 𝑛 ) = 0 . 8 , i f 𝑛 i s e v e n , 0 . 9 , i f 𝑛 i s o d d , 𝑐 ( 𝑛 ) = 0 . 2 , i f 𝑛 i s e v e n , 0 . 8 , i f 𝑛 i s o d d . ( 3 . 7 ) Then, (1.12) is exponentially stable for any 𝑁 by Corollary 2.10, Part ( 1 ) ; here 𝜆 = 0 . 9 , 𝛾 = 8 / 9 .
If, for example, we assume 𝑁 = 1 , then (1.11) has the form | | | | + | | | | | | | | 𝑎 ( 𝑛 ) 𝑎 ( 𝑛 − 1 ) − 𝑐 ( 𝑛 ) 𝑎 ( 𝑛 ) 𝑐 ( 𝑛 − 1 ) ≤ 𝜆 < 1 , ( 3 . 8 ) which is not satisfied for even 𝑛 , so Lemma 1.4 cannot be applied to deduce exponential stability.

Example 3.4. We will modify Example 8 in [6] to compare Theorem 2.18 and Lemma 1.4. Consider 1 𝑎 ( 𝑛 ) = 1 2 5 ( 𝑁 + 1 ) , i f 𝑛 i s e v e n , 2 , i f 𝑛 i s o d d , 𝑐 ( 𝑛 ) = 2 5 ( 𝑁 + 1 ) , i f 𝑛 i s e v e n , 𝑑 , i f 𝑛 i s o d d . ( 3 . 9 ) Let us compare constants 𝑑 such that (1.12) is exponentially stable for 𝑁 = 1 , 𝑥 ( 𝑛 + 1 ) = 𝑎 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑐 ( 𝑛 ) 𝑥 ( 𝑛 − 1 ) . ( 3 . 1 0 ) By Theorem 2.18 we obtain stability whenever 1 + 1 5 0 | | 𝑑 | | | | 𝑑 | | 5 0 2 + < 1 , o r < 2 3 . ( 3 . 1 1 ) Condition (3.8) of Lemma 1.4 becomes | | | 1 − 1 2 5 | | | + 1 5 0 | | 𝑑 | | 5 0 < 1 ( 3 . 1 2 ) for even 𝑛 and | | | 1 | | | 1 2 5 − 𝑑 + 2 5 0 < 1 ( 3 . 1 3 ) for odd 𝑛 , which gives the intervals | 𝑑 | < 4 9 and − 0 . 9 2 < 𝑑 < 1 , respectively. Finally, Lemma 1.4 implies exponential stability for 𝑑 ∈ ( − 0 . 9 2 , 1 ) ⊂ ( − 2 3 , 2 3 ) .

In addition to Theorems 2.1, 2.2, 2.17, and 2.18, let us review some other known stability conditions for equations with several delays. For comparison, we will cite the following two results.

Theorem 3 A ([9–14]). Suppose that 𝑎 𝑗 ( 𝑛 ) ≥ 0 , 0 ≤ 𝑗 ≤ 𝑚 , 𝑚 𝑗 = 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 ( 𝑛 ) > 0 , ∞ 𝑛 = 0 0 𝑥 0 2 0 0 𝑑 𝑚 𝑗 = 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 ( 𝑛 ) = ∞ , s u p 𝑛 𝑛 ≥ 𝑚 𝑘 = 𝑛 − 𝑚 0 𝑥 0 2 0 0 𝑑 𝑚 𝑗 = 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 3 ( 𝑘 ) < 2 + 1 2 . ( 𝑚 + 1 ) ( 3 . 1 4 ) Then the equation with several delays 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) = − 𝑚 𝑗 = 0 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 ( 𝑛 ) 𝑥 ( 𝑛 − 𝑗 ) , 𝑛 = 0 , 1 , 2 , … ( 3 . 1 5 ) is globally asymptotically stable.

Theorem 3 B ([15]). Suppose that 𝑎 𝑙 ( 𝑛 ) ≡ 𝑎 𝑙 > 0 and 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 l i m s u p 𝑛 → ∞ 𝑛 − ℎ 𝑙 1 ( 𝑛 ) < 1 + 𝑒 − 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑙 . ( 3 . 1 6 ) Then (1.1) is asymptotically stable.

Let us note that unlike Theorems A and B we do not assume that coefficients are either nonnegative (as in Theorem A) or constant (as in Theorem B). Further, let us compare our stability tests with known results, including Theorems A and B.

Example 3.5. Consider the equation 𝑥 ( 𝑛 + 1 ) = 𝑎 ( 𝑛 ) 𝑥 ( 𝑛 − 4 ) + 𝑐 ( 𝑛 ) 𝑥 ( 𝑛 − 6 ) , ( 3 . 1 7 ) where 𝑎 ( 𝑛 ) = 2 , i f 𝑛 i s e v e n , 0 . 1 , i f 𝑛 i s o d d , 𝑐 ( 𝑛 ) = 3 , i f 𝑛 i s e v e n , 0 . 0 5 , i f 𝑛 i s o d d . ( 3 . 1 8 ) Repeating the proof of Theorem 2.17, we have [ ] [ 𝑎 ] . 𝑥 ( 𝑛 + 1 ) = 𝑎 ( 𝑛 ) 𝑎 ( 𝑛 − 5 ) 𝑥 ( 𝑛 − 9 ) + 𝑐 ( 𝑛 − 5 ) 𝑥 ( 𝑛 − 1 1 ) + 𝑐 ( 𝑛 ) ( 𝑛 − 7 ) 𝑥 ( 𝑛 − 1 1 ) + 𝑐 ( 𝑛 − 7 ) 𝑥 ( 𝑛 − 1 3 ) ( 3 . 1 9 )
By Theorem 2.17, (3.17) is exponentially stable since | | 𝑎 | |