Applied Mathematics Letters 38 (2014) 7–13

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

A novel approach to exponential stability of nonlinearnon-autonomous difference equations with variable delaysLe Van Hien ∗

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:Received 26 April 2014Received in revised form 20 June 2014Accepted 23 June 2014Available online 30 June 2014

Keywords:Discrete Halanay inequalityExponential stabilityNon-autonomous systemsTime-varying delays

a b s t r a c t

In this paper, by using a novel approach, we first prove a new generalization of discrete-type Halanay inequality. Based on our new generalized inequality, a novel criterion for theexponential stability of a certain class of nonlinear non-autonomous difference equationsis proposed. Numerical examples are given to illustrate the effectiveness of the obtainedresults.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Time-delay systems naturally arise in modeling a wide range of phenomena in the vivid world [1,2]. These practicalsystems are usually in the form of nonlinear and/or non-autonomous continuous-time systems with time-varying delays.Today, with the dramatically development of computer-based computational techniques, difference equations are foundto be much appropriate mathematical representations for computer simulation, experiment and computation, which playan important role in realistic applications [3]. By a discretization from continuous-time systems, discrete-time systems de-scribed by difference equations inherit the similar dynamical behavior of the continuous ones. Therefore, the problem ofstability analysis for difference equations with delays has received extensive attention from researchers (see, e.g., [4–10]and the references therein).

In most of the reported results concerned with the stability of difference equations with delays, a widely used approachis the Lyapunov–Krasovskii functional method. However, it relies heavily on how to choose an appropriate Lyapunov func-tional candidate and this usually leads to serious difficulties, especially in regard to nonlinear non-autonomous equations.An other effective approach is the use of discrete-type inequalities such as Gronwall inequalities or Halanay inequalities[3,5–7,9].

In [7], Udpin and Niamsup proved the following new discrete Halanay inequality which was shown more general thanthe one given in [5].

Theorem 1.1 ([7]). Let r ∈ Z+, p, qr ∈ R+, qi ∈ R+

0 , hi ∈ Z+, i = 1, . . . , r, where 0 = h0 < h1 < · · · < hr andr

i=0 qi <p ≤ 1. Let (xn)n∈Z−hr be a sequence of real numbers satisfying

∆xn ≤ −pxn +

ri=0

qixn−hi , n ∈ Z0. (1.1)

∗ Tel.: +84 912494391.E-mail address: hienlv@hnue.edu.vn.

http://dx.doi.org/10.1016/j.aml.2014.06.0140893-9659/© 2014 Elsevier Ltd. All rights reserved.

8 L.V. Hien / Applied Mathematics Letters 38 (2014) 7–13

Then there exists λ0 ∈ (0, 1) such that

xn ≤ max{0, x0, x−1, . . . , x−hr }λn0, n ∈ Z0. (1.2)

Based on this new discrete Halanay inequality, new global exponential stability conditions for the following nonlineardifference equation

∆xn = −pxn + f (n, xn, xn−h1 , . . . , xn−hr ), n ∈ Z+, (1.3)

where p > 0, r ∈ Z+, hi ∈ Z+, i = 1, 2, . . . , r , have been derived. However, it should be noted that Theorem 1.1 cannot beapplied to non-autonomous equations. More precisely, even if p = p(n), qi = qi(n) satisfy

ri=0 qi(n) < p(n) ≤ 1 for all n,

the conclusion of Theorem 1.1 is not correct (see the Appendix in this paper). Moreover, (1.1) can be written as follows:

xn+1 ≤ p0xn +

ri=0

qixn−hi , n ∈ Z0, (1.4)

where p0 = 1 − p. Note that, in this case we have δ := p0 +r

i=0 qi ∈ (0, 1).Recently, in [8], a more general class of (1.4) was considered

x(n + 1) ≤ h(n) [P0x(n) + P1x(n − τ1(n)) + · · · + Prx(n − τr(n)) + I] , (1.5)

where function h(.) : Z0→ R andmatrices Pi ∈ Rm×m

+ , i = 0, 1, 2, . . . , r , satisfy 0 < h(n) ≤ 1 for all n and ρr

i=0 Pi

< 1(ρ(A) denotes the spectral radius of matrix A ∈ Rm×m). An improved time-varying Halanay-type inequality has been provedand sufficient conditions on global exponential stability, global attracting set and ultimately bounded for (1.5) have beenproposed. However, for more convenience in comparing stability conditions, let m = 1, I = 0 then (1.5) is reduced tothe form of (1.4) with variable coefficients which satisfy h(n)

ri=0 Pi ≤

ri=0 Pi =: δ < 1 for all n. It can be seen in the

aforementioned works, a strictly restriction that is required is that the sum of coefficients is uniformly less than one. Thismakes the obtained results conservative. So far, the problem of stability analysis for nonlinear non-autonomous differenceequations without this restriction has not fully been investigated which motivates the present study.

In this paper, by using a novel approach, we first prove a new generalization of discrete-type Halanay inequality of theform

xn+1 ≤ p0(n)xn +

rk=1

pk(n)xn−τk(n), n ∈ Z0, (1.6)

where r ∈ Z+ is given, pk(n) ≥ 0, n ∈ Z0, k = 0, 1, . . . , r , are variable coefficients, 0 ≤ τk(n) ≤ τk ∈ Z+ are variabledelays. We then derive a new criterion for the exponential stability of a more general class of (1.3). Furthermore, the restric-tion that the sum of coefficients is uniformly less than one will be removed. The rest of this paper is organized as follows.Section 2 presents a few preliminaries involved in this paper. Ourmain results are presented in Section 3. Section 4 providessome numerical examples to illustrate the effectiveness of the obtained results. The paper ends with a conclusion and citedreferences.

2. Preliminaries

Throughout this paper, we let Z and Z+ denote the set of integers and positive integers, respectively. For r ∈ Z, wedenote Zr

= {m ∈ Z : m ≥ r} and for r1, r2 ∈ Z, r1 < r2, we denote Z[r1, r2] = {r1, r1 + 1, . . . , r2}.Besides (1.6), we also consider the following general non-autonomous difference equations:

xn+1 = fn(xn, xn−τ1(n), . . . , xn−τr (n)), n ∈ Z0, (2.1)

where fn : Rr+1−→ R is continuous function for each integer n ≥ 0. Assume thatfn(u0, u1, . . . , ur)

≤

rk=0

pk(n)|uk|, (u0, . . . , uk) ∈ Rr+1. (2.2)

For each initial string (xn)n∈Z[−τ ,0], τ = maxk∈Z[1,r] τk, Eq. (2.1) has a unique solution (xn)n∈Z−τ .

Definition 2.1 ([7,9]). Eq. (2.1) is said to be globally exponentially stable (GES) if there exist positive constants δ, λ such thatevery solution (xn)n∈Z−τ of (2.1) satisfies

|xn| ≤ δ maxj∈Z[−τ ,0]

|xj|e−λn, ∀n ∈ Z0.

In Definition 2.1, if the decay function λn is replaced by a general function σ(n), then we say (2.1) is globally generalizedexponentially stable.

L.V. Hien / Applied Mathematics Letters 38 (2014) 7–13 9

Definition 2.2. Eq. (2.1) is said to be globally generalized exponentially stable (GGES) if there exist δ > 0 and a decayfunction, namely σ(n), n ∈ Z0, such that limn→∞ σ(n) = ∞ and every solution (xn)n∈Z−τ of (2.1) satisfies

|xn| ≤ δ maxj∈Z[−τ ,0]

|xj|e−σ(n), ∀n ∈ Z0.

It is easy to see that, if (2.1) is (GES), then it also is (GGES) with decay function σ(n) = λn. However, the inversion is notcorrect as shown in the following example.

Example 2.1. Let p(n) = e√n−

√n+1, q(n) =

e−√n+1

2n+2 , n ∈ Z0. Consider the following equation:

xn+1 = p(n)xn + q(n)xn−1, n ∈ Z0. (2.3)

For given initial string x−1, x0, (2.3) has a unique solution (xn) which can be written as follows:

xn+1 =

ni=0

p(i)x0 + q(n)xn−1 +

n−1i=0

q(i)n

j=i+1

p(j)xi−1, n ≥ 1,

x1 = p(0)x0 + q(0)x−1.

Therefore

xn+1 = e−√n+1

x0 +

14

ni=0

xi−1

2i

, n ≥ 1. (2.4)

It follows from (2.4) that, if |xk| ≤ 2max{|x0|, |x−1|}e−√k, k ∈ Z[0, n] for some n ∈ Z+, then

|xn+1| ≤ e−√n+1

|x0| +

12max{|x0|, |x−1|}

ni=0

12i

≤ 2max{|x0|, |x−1|}e−

√n+1.

By induction we have |xn| ≤ 2max{|x0|, |x−1|}e−√n for all n ∈ Z0 which shows that Eq. (2.3) is (GGES), where the decay

function is given byσ(n) =√n, n ∈ Z0. On the other hand, let x0 > 0, x−1 > 0, it follows from (2.3) that xn ≥

n−1i=0 p(i)x0 =

x0e−√n. Suppose (2.3) is (GES), then there exist positive constants σ , λ such that

x0e−√n

≤ xn ≤ δ max{x0, x−1}e−λn,

and thus,√n ≥ λn + β, ∀n ∈ Z0, where β = ln x0

δ max{x0,x−1}. Consequently, we have 0 < λ ≤

1√n −

β

n → 0 as n → ∞,which yields a contradiction. This shows that (2.3) is not (GES).

3. Main results

In this section, we first derive an exponential estimate for the inequality (1.6). For this, we make the following assump-tions:

A1. pk(n) ≥ 0, k = 0, 1, . . . , r; 0 <r

k=1 pk(n) ≤ 1 − p0(n), ∀n ∈ Z0.A2. lim supn→∞

rk=1

pk(n)1−p0(n)

< 1.

A3. There exists p0 > 0 such that 1−p0(n+1)1−p0(n)

≤ p0, ∀n ∈ Z0.

Remark 3.1. For the equations considered in [7,8], Assumptions A1–A3 are obviously satisfied. Thus, the proposed condi-tions in this paper are less conservative than those in [5,7,8].

Theorem 3.1. Let Assumptions A1–A3 hold and (xn)n∈Z−τ be a sequence of real numbers satisfying the inequality (1.6). Thenthere exist positive constants δ, λ∗ such that the following inequality holds:

xn ≤ δ[x]+τ exp

−λ∗

nj=0

p0(j), n ∈ Z0, (3.1)

wherep0(n) = 1 − p0(n), n ∈ Z0, and [x]+τ = maxj∈Z[−τ ,0]{0, xj}.

Proof. By induction, it can be seen that xn ≤ [x]+τ , ∀n ∈ Z0. Let γ0 = lim supn→∞

rk=1

pk(n)1−p0(n)

. Then, by A1 and A2, γ0 ∈

[0, 1) and there exists n∗ > τ such that supn≥n∗

rk=1

pk(n)1−p0(n)

≤1+γ0

2 < 1.

10 L.V. Hien / Applied Mathematics Letters 38 (2014) 7–13

Consider the following scalar equation:

H(λ) := p0λ +1 + γ0

2eλτ

− 1 = 0, λ ∈ [0, ∞). (3.2)

The function H(λ) is continuous and strictly increasing on [0, ∞), H(0) =1+γ0

2 − 1 < 0, limλ→∞ H(λ) = ∞. Therefore,(3.2) has a unique positive solution λ∗ and H(λ) ≤ 0 for all λ ∈ (0, λ∗]. Using the facts that

nj=n−τk(n)+1

p0(j) =

nj=n−τk(n)+1

[1 − p0(j)] = τk(n) −

nj=n−τk(n)+1

p0(j) ≤ τ ,

and hence, expλn

j=n−τk(n)+1p0(j) ≤ eλτ , k = 1, 2, . . . , r;

λ1 − p0(n + 1)1 − p0(n)

+ eλτr

k=1

pk(n)1 − p0(n)

≤ H(λ) + 1, ∀n ≥ n∗,

we have the following estimation:

rk=1

pk(n)1 − p0(n)

exp

λ

nj=n−τk(n)+1

p0(j)+ λ1 − p0(n + 1)1 − p0(n)

≤ 1 (3.3)

for all n ≥ n∗ and λ ∈ (0, λ∗].On the other hand, by using the fact e−x > 1 − x for all x > 0 we have

1 +exp

−λp0(n + 1)

− 1

1 − p0(n)≥ 1 − λ

1 − p0(n + 1)1 − p0(n)

. (3.4)

Therefore, from (3.3) and (3.4) we obtain

1 +exp(−λp0(n + 1)) − 1

1 − p0(n)≥

rk=1

pk(n)1 − p0(n)

exp

λ

nj=n−τk(n)+1

p0(j). (3.5)

Let us consider the sequence un = δ[x]+τ exp(−λ∗

nj=0p0(j)), n ∈ Z0, where δ = exp(λ∗

n∗

j=0p0(j)). Note that, for anyn ≥ n∗ and m ∈ Z[n − τk(n), n], we have um = un exp(λ∗

nj=m+1p0(j)) and therefore

rk=1

pk(n)un−τk(n) ≤ un

rk=1

pk(n) exp

λ∗

nj=n−τk(n)+1

p0(j), n ≥ n∗. (3.6)

From (3.5) and (3.6) we readily obtain

un+1 − p0(n)un −

rk=1

pk(n)un−τk(n) ≥ un

exp

−λ∗p0(n + 1)

− p0(n)

−

rk=1

pk(n) exp

λ∗

nj=n−τk(n)+1

p0(j) ≥ 0, ∀n ≥ n∗.

Thus

un+1 ≥ p0(n)un +

rk=1

pk(n)un−τk(n), n ≥ n∗.

Wewill prove that xn ≤ un, ∀n ∈ Z0, which completes the proof of Theorem 3.1. By definition of δ we have xn ≤ un, ∀n ∈

Z[0, n∗]. Assume that xm ≤ um, ∀m ∈ Z[0, n] for some n ≥ n∗. Then we have

xn+1 − un+1 ≤ p0(n)(xn − un) +

rk=1

pk(n)xn−τk(n) − un−τk(n)

≤ 0,

and hence, xn+1 ≤ un+1. By induction we obtain xn ≤ un for all n ∈ Z0. Consequently

xn ≤ δ[x]+τ exp

−λ∗

nj=0

p0(j), n ∈ Z0.

The proof is completed. �

L.V. Hien / Applied Mathematics Letters 38 (2014) 7–13 11

Remark 3.2. To avoid taking limit, in (3.2) we can replace 1+γ02 by η = supn≥n∗

rk=1

pk(n)1−p0(n)

< 1. Thus, the followingprocedure can be used to derive the exponential factor

• Let n∗ = minn ∈ Z+ : n − τk(n) ≥ 0, ∀k, and supm≥n

rk=1

pk(m)

1−p0(m)< 1

.

• The scalar equation λp0 + ηeλτ− 1 = 0, where η = supn≥n∗

rk=1

pk(n)1−p0(n)

, has a unique positive solution λ∗.• The exponential factor is given by

δ = exp

λ∗

n∗j=0

(1 − p0(j))

. (3.7)

It is worth noting that, Theorem 1.1 can be obtained from Theorem 3.1 as the following corollary.

Corollary 3.1. Let (xn) be a sequence of real numbers satisfying (1.1). Then the following exponential estimate holds

xn ≤ max{0, x−1, . . . , x−hr }e−λ0(n−hr ), n ∈ Z0,

where λ0 is the unique positive solution of the following equation

λ +

ri=1

qi exp

λ

phr

− p = 0. (3.8)

Noticing that, in order to obtain the convergence rate, Udpin and Niamsup in [7] had to solve a polynomial equation oforder hr + 1 to find the unique solution λ0 ∈ (0, 1), in this paper, we obtain the exponential convergence rate by findingthe positive solution of (3.8).

Next, we will derive exponential stability conditions for (2.1) based on our new generalized discrete Halanay inequalityobtained in Theorem 3.1. Let us define the sequence pa0(n) =

1n

ni=0 p0(i), n ∈ Z+.

Theorem 3.2. Under the assumptions A1–A3, the following assertions hold:(i) If lim supn→∞ pa0(n) < 1, then Eq. (2.1) is globally exponentially stable.(ii) If limn→∞ n

1 − pa0(n)

= ∞, then Eq. (2.1) is globally generalized exponentially stable.

Remark 3.3. The stability conditions proposed in [7] obviously satisfy (i). Moreover, in [8], to guarantee the asymptotic sta-bility of the unique equilibrium with p0(n) = κ0h(n), where 0 < κ0 < 1, 0 < h(n) ≤ 1, ∀n, it is required that limn→∞ h(n)= 0. However, in this case, pa0(n) ≤ κ0

1 +

1n

and we have lim supn→∞ pa0(n) ≤ κ0 < 1. Therefore, condition (i) is satisfied

without any requirement on h(n).Proof. Let (xn)n∈Z−τ be a solution of (2.1). We define a sequence (un)n∈Z−τ as un = |xn| for n ∈ Z[−τ , 0], and

un+1 = p0(n)|xn| +

rk=1

pk(n)|xn−τk(n)|, n ∈ Z0. (3.9)

Using (2.2) we have

|xn+1| ≤ p0(n)|xn| +

rk=1

pk(n)|xn−τk(n)|, n ∈ Z0.

Using (3.9) and by induction we have |xn| ≤ un, ∀n ∈ Z−τ , which yields

un+1 ≤ p0(n)un +

rk=1

pk(n)un−τk(n), n ∈ Z0.

By Theorem 3.1, there exist δ > 0, λ∗ > 0 such that

un ≤ δ maxj∈Z[−τ ,0]

uj exp

−λ∗

nj=0

(1 − p0(j))

, n ∈ Z0. (3.10)

Proof. (i) Assume that lim supn→∞ pa0(n) < 1, then there exist 0 ≤ γ1 < 1 and n∗∈ Z+ such that pa0(n) ≤ γ1 for all n ∈ Zn∗

.Therefore, un ≤ δ maxj∈Z[−τ ,0] |xj|e−λ∗(1−γ1)n−λ∗ , n ∈ Zn∗

. In addition, let γ2 = maxj∈Z[0,n∗] p0(j) then γ2 ∈ [0, 1) and, by(3.10), we have un ≤ δ maxj∈Z[−τ ,0] |xj|e−λ∗(1−γ2)(n+1), n ∈ Z[0, n∗]. Let γ = max{γ1, γ2} < 1 and δ1 = δe−λ∗(1−γ ) wereadily obtainxn ≤ un ≤ δ1 max

j∈Z[−τ ,0]|xj|e−λ∗(1−γ )n, ∀n ∈ Z0.

This shows that (2.1) is globally exponentially stable. The proof of (i) is completed.

12 L.V. Hien / Applied Mathematics Letters 38 (2014) 7–13

Proof. (ii) Note that vn = n1 − pa0(n)

is an increasing sequence and hence the limit limn→∞ vn exists. Assume that

limn→∞ vn = ∞ then

∞

j=0(1 − p0(j)) = ∞. Let us define the function σ(n) = λ∗

nj=0(1 − p0(j)) then σ(n) is a positive

increasing function satisfying limn→∞ σ(n) = ∞. From (3.10) we have

|xn| ≤ δ maxj∈Z[−τ ,0]

|xj|e−σ(n), n ∈ Z0,

which proves that (2.1) is globally generalized exponentially stable. The proof is completed. �

Remark 3.4. It is worth noting that, condition (ii) does not require pa0(n) is uniformly less than one. When p0(n) tends toone with slow enough convergence rate, (2.1) can be globally generalized exponentially stable.

4. Illustrative examples

In this section, we give some examples to illustrate the effectiveness of our obtained results.

Example 4.1. Consider a non-autonomous difference equation of Mackey–Glass type [10]

xn+1 =1

n + 1

12xn +

13

xn−τ

1 + x2n−τ

, n ∈ Z0, (4.1)

where τ ∈ Z+ is known constant delay. Note that (4.1) satisfies (2.2) with p0(n) =1

2n+2 and p1(n) =1

3n+3 . We have,p0(n) + p1(n) ≤

56 , ∀n, limn→∞

p1(n)1−p0(n)

= 0 and 1−p0(n+1)1−p0(n)

≤32 , ∀n. Therefore, Assumptions A1–A3 are satisfied. Moreover,

lim supn→∞ pa0(n) ≤12 . By Theorem 3.2, Eq. (4.1) is globally exponentially stable. For τ = 6, we have λ∗ ≃ 0.0536 and every

solution of (4.1) satisfies

|xn| ≤ maxj∈Z[−6,0]

|xj|e−0.0268n, n ∈ Z0.

Example 4.2. Consider the following equation:

xn+1 =

1 −

1n + 1

xn

1 + x2n+

14ln1 +

12n + 2

4k=1

xn−k, n ∈ Z0. (4.2)

It is easy to verify that (4.2) satisfies Assumptions A1–A3. Therefore, Eq. (4.2) is globally generalized exponentially stable.Moreover, by solving the scalar equation λ +

12 e

4λ− 1 = 0 we obtain λ∗ ≃ 0.0682 and every solution of (4.2) satisfies

|xn| ≤ maxj∈Z[−4,0]

e−σ(n), n ∈ Z0,

where σ(n) = 0.0628n

j=01

j+1 .

5. Conclusion

In this paper, we have proved an extension of discrete type Halanay inequalities for variable coefficients. Based on theobtained inequality, a novel criterion for the generalized exponential stability of a class of nonlinear non-autonomous dif-ference equations has been proposed. Numerical examples have been provided to illustrate the effectiveness of the obtainedresults.

Acknowledgments

The author would like to thank the editor(s) and anonymous reviewers for their constructive comments which helpedto improve the present paper. This work was supported by the National Foundation for Science & Technology Development(101.01-2011.51) and the Ministry of Education and Training of Vietnam (B2013.17.42).

Appendix

Consider the sequence (xn)n∈Z−1 , where xn =n+2

k=11k − ln(n + 2), n ≥ −1. It is well known that (xn) is a decreasing

sequence which converges to Euler’s constant γ . Thus, 0.5 < γ < xn ≤ 1, ∀n ∈ Z−1. Let p(n) =ln1+ 1

n+2

−

1n+3

γ−0.5 and q(n) =

12p(n), n ∈ Z0, then 0 < q(n) < p(n) < 1, ∀n ∈ Z0. Although the sequence (xn) satisfies the following inequality

∆xn ≤ −p(n)xn + q(n)xn−1, n ∈ Z0,

it does not converge to zero and consequently Theorem 1.1, in general, does not hold for non-autonomous differenceequations.

L.V. Hien / Applied Mathematics Letters 38 (2014) 7–13 13

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