Exponential Stability of Difference Equations with Several Delays: Recursive Approach

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doi:10.1155/2009/104310

Research Article

Exponential Stability of Difference Equations with

Several Delays: Recursive Approach

Leonid Berezansky

1

_{and Elena Braverman}

2

1_{Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel} 2_{Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W.,}

Calgary AB, Canada T2N 1N4

Correspondence should be addressed to Elena Braverman,[email protected]

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

Recommended by Istvan Gyori

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.

Copyrightq2009 L. Berezansky and E. 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.

1. Introduction and Preliminaries

In this paper we study stability of a scalar linear diﬀerence equation with several delays,

xn1−xn − m

l1

alnxhln, hln≤n, n≥n0, 1.1

wherehlnis an integer for anyl 1, . . . , m,nis an integer,n0 ≥ 0. Stability of1.1and

relevant nonlinear equations has been an intensively developed area during the last two decades.

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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 thathln ≥ n−T,n ≥ n0 ≥ 0, that is, the

system has a finite memory, and the following initial conditions are defined

xn ϕn, n0−T ≤n≤n0. 1.2

Definition 1.1. Equation1.1isexponentially stableif there exist constantsM > 0,λ ∈ 0,1 such that for every solution{xn}of1.1and1.2the inequality

|xn| ≤Mλn−n0

max

n0−T≤k≤n0

ϕk

1.3

holds for alln≥n0, whereM, λdo not depend onn0≥0.

Equation 1.1 is stable if for any ε > 0 there exists δ > 0 such that maxn0−T≤k≤n0{|ϕk|} < δimplies|xn| < ε,n ≥ n0; ifδdoes not depend onn0, then1.1 isuniformly stable.

Equation1.1isattractiveif for anyϕka solution tends to zero limn→ ∞xn 0. It is asymptotically stableif 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 monographs1,2 . The following result was also obtained by this method.

Lemma 1.23,4 . Consider the nonlinear delay diﬀerence equation

xn1fn, xn, . . . , xn−T, n≥0. 1.4

Assume thatf:N×RT1 → Rsatisfies

fn, u0, . . . , uT≤bmax{|u0|, . . . ,|uT|}, 1.5

for some constantb <1, and for alln, u0, . . . , uT∈N×RT1. Then

|xn| ≤bn/T1M0, n≥0, 1.6

for every solution{xn}of1.4, whereM0 max−T≤i≤0{|xi|}. In particular, the zero solution of1.4 is globally exponentially stable.

This result was applied to a more general than1.1nonlinear diﬀerence equation

xn1−xn− N

k0

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Without loss of generality, we can suppose thatN≤T. We assume that there exist constants

bn≥0 such that

fn, u0, . . . , uT≤bnmax{|u0|, . . . ,|uT|}, 1.8

for alln≥0 andu0, . . . , uT∈RT1.

Similar argument leads to the following result.

Lemma 1.34 . Assume that for n large enough inequality1.8holds and there exists a constant γ∈0,1such that

Then the zero solution of 1.7is globally exponentially stable. Moreover, if 1.8and1.9hold for n≥0, then

|xn| ≤γn/NT1max{|xN|, . . . ,|x−T|}, n≥N, 1.10

for every solution{xn}of1.7.

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, in6, Corollary 7 contains the following statement.

Lemma 1.46 . If there existsλ∈0,1such that

for largen, then the zero solution of the equation

xn1 anxn−cnxn−N 1.12

is globally exponentially stable.

In the present paper we obtain some new stability results for1.1with several variable delays. In contrast to many other stability tests, we consider the case when the sum of coeﬃcientsm_k₁aknor 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

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Theorem 2.1. Suppose that there exist a set of indicesI ⊂ {1,2, . . . , m}andγ ∈0,1such that for n suﬃciently large

Then1.1is exponentially stable.

Proof. Since

ByLemma 1.2,1.1is exponentially stable.

We can reformulateTheorem 2.1in the following equivalent form.

Theorem 2.2. Suppose that there exist a set of indicesI ⊂ {1,2, . . . , m}andε∈0,1such that for n suﬃciently large

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Corollary 2.3. Let0< γ <1andm_l₂|aln||1−a1n| ≤γfornlarge enough. Then the equation

xn1−xn −a1nxn− m

l2 alnx

gln

2.5

is exponentially stable.

Example 2.4. Consider the equation

xn1−xn −1.5xn−0.3 sinnxhn, 2.6

with an arbitrary bounded delayhn:n−T ≤ hn ≤ nfor some integerT >0 and anyn. Since 0.3|sinn||1−1.5| ≤0.8<1, then byCorollary 2.3this equation is exponentially stable.

Lemma 1.4is formulated for a constant delay, some other tests do not apply since| −1.5|>1.

Corollary 2.5. Suppose that for someγ∈0,1the following inequality is satisfied for n large enough:

m

k2 |akn|

n−1

jgkn m

l1

al

j1−

m

k1 akn

≤γ. 2.7

Example 2.6. ByCorollary 2.5the equation

xn1−xn −0.50.1 sinnxn−0.6−0.1 sinnxn−1 2.8

is exponentially stable, since|0.50.1 sinn||0.6−0.1 sinn|1.1 and 0.7·1.1|1−1.1|0.87<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 proportionalaln Alrn,l1, . . . , m,there exist r0>0,ε >0and a set of indicesI⊂ {1,2, . . . , m}, such thatrn≥r0>0and

l∈I |Al|

n−1

khln

rk≤ min{rn

l∈IAl,2−rn

l∈IAl} −rn

l /∈I|Al| −ε rnm_l₁|Al|

. 2.9

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Corollary 2.8. Suppose that all coeﬃcients are constantsaln≡aland

lim sup

n→ ∞ m

l1 |al|

m

l1

|al|n−hln<min

_m

l1 al,2−

m

l1 al

. 2.10

Remark 2.9. Corollary 2.8for the case 0<m_l₁al<1 was obtained in Proposition 4.1 of7 .

Now let us consider the equation with one nondelay and one delay terms

xn1−xn −anxn−bnxhn. 2.11

ChoosingI{1},I{1,2}, we obtain Parts1and2ofCorollary 2.10, respectively.

Corollary 2.10. Suppose that there existsγ∈0,1such that at least one of the following conditions holds fornsuﬃciently large:

1|bn||1−an| ≤γ;

2|bn|n_k−1_h_n|ak||bk| |1−an−bn| ≤γ. Then2.11is exponentially stable.

Let us now proceed to equations with three terms in the right-hand side

xn1−xn −anxn−bnxhn−cnxgn. 2.12

ForI{1},{1,2,3},{1,2}and{1,3}we obtain Parts1,2,3,and4, respectively.

Corollary 2.11. Suppose that there existsγ∈0,1such that at least one of the following conditions holds fornsuﬃciently large:

1|bn||cn||1−an| ≤γ;

2|bn|_kn−1_h_n|ak||bk||ck| |cn|n_k−_g1_n|ak||bk||ck| |1−an− bn−cn| ≤γ;

3|bn|n_k−_h1_n|ak||bk||ck| |cn||1−an−bn| ≤γ;

4|cn|n_m−1_g_n|ak||bk||ck| |bn||1−an−cn| ≤γ. Then2.12is exponentially stable.

Theorem 2.1and 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

xn1−xn −a1xn− m

l2

alxn−hl, 2.13

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Corollary 2.12. Letm_l₂|al|<min{a1,2−a1}. Then2.13is exponentially stable.

Remark 2.13. This result is well known; see, for example,8 form2 as well as some results for autonomous equations below. We presented it just to illustrate our method.

Further, Theorem 2.1and Corollaries 2.8and 2.11 can be reformulated for2.13 as follows.

Corollary 2.14. Suppose that there exists a set of indicesI ⊂ {1,2, . . . , m}, with1∈I,such that

m

l1 |al|

k∈I

|ak|hk

l /∈I |al|

1−

k∈I ak

<1. 2.14

Corollary 2.15. Ifm_l₁|al|mk2|ak|hk < min{mk1ak,2−km1ak}, then2.13is exponentially stable.

Consider now an autonomous equation with two delays:

xn1−xn −a0xn−a1xn−h1−a2xn−h2, 2.15

whereh1>0, h2>0.

Corollary 2.16. Suppose that at least one of the following conditions holds

1|a1||a2||1−a0|<1;

2 |a0||a1||a2||a1|h1|a2|h2 |1−a0−a1−a2|<1;

3|a1|h1|a0||a1||a2| |1−a0−a1|<1;

4|a2|h2|a0||a1||a2| |1−a0−a2|<1. Then2.15is 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

xn1

m

l1

alnxhln, 2.16

which is a diﬀerent form of1.1.

We recall that we assumen−hln≤Tfor all delayshlnin this paper.

Theorem 2.17. Suppose that there existsλ∈0,1such that

lim sup

n→ ∞ m

l1 |aln|

m

j1

ajhln−1≤λ. 2.17

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Proof. Without loss of generality we can assume that the expression under lim sup in2.17

Proof. Without loss of generality we can assume that the expression under lim sup in2.21

does not exceed someλ <1 forn≥n0. Since

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3. Discussion and Examples

Let us comment thatTheorem 2.18see alsoCorollary 2.12generalizes the result of Clark8

that|p||q|<1 is a suﬃcient condition for the asymptotic stability of the diﬀerence equation

xn1 pxn qxn−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 2.18gives the same stability condition for the equation with variable delays

xn1 pxhn qxgn0 3.2

once the delays are bounded:n−hn≤T1,n−gn≤T2.

The following example outlines the sharpness of the condition that the delays are bounded in Theorems2.1,2.2,2.17, and2.18.

Example 3.1. The equation with constant coeﬃcients

xn1 0.4xn 0.1x0, n0,1, . . . 3.3

satisfies all assumptions of Theorems2.1,2.2,2.17, and2.18but the boundedness of the delay. Since the solutionxnwithx0 6 tends to 1 asn → ∞, then the zero solution of3.3is neither asymptotically nor exponentially stable. Here even the condition limn→ ∞hln ∞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 ofTheorem 2.18are satisfied, this does not imply exponential stability. The equation

xn 0.5xn

2

, n1,2, . . . , 3.4

wherex is the integer part ofx, is asymptotically, but not exponentially stable. Really, its solution

x0,1

2x0, 1 4x0,

1 4x0,

1 8x0,

1

16x0, . . . 3.5

is nonincreasing by the absolute value and

xn x1

n , 3.6

for anyn2k_,_k ₀_,₁_,₂_{, . . .}_{, so lim}

n→0xn 0 for anyx1; the equation is asymptotically

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Next, let us compareCorollary 2.10and Theorem 4 of6 which is alsoLemma 1.4of the present paper.

Example 3.3. Consider1.12, where

an

⎧ ⎨ ⎩

0.8, ifnis even,

0.9, ifnis odd,

cn

⎧ ⎨ ⎩

0.2, ifnis even,

0.8, ifnis odd.

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Then,1.12is exponentially stable for anyNbyCorollary 2.10, Part1; hereλ0.9,γ8/9. If, for example, we assumeN1, then1.11has the form

|anan−1−cn||an| |cn−1| ≤λ <1, 3.8

which is not satisfied for even n, so Lemma 1.4cannot 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

an

⎧ ⎨ ⎩

1

25N1, if nis even, 2, if nis odd,

cn

⎧ ⎨ ⎩

1

25N1, if nis even,

d, if nis odd.

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Let us compare constantsdsuch that1.12is exponentially stable forN1,

xn1 anxn−cnxn−1. 3.10

ByTheorem 2.18we obtain stability whenever

1 50

2|d|<1, or |d|<23. 3.11

Condition3.8ofLemma 1.4becomes

₂₅1 − 1

50

1

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for evennand

₂₅1 −d2 1

50 <1 3.13

for odd n, which gives the intervals |d| < 49 and −0.92 < d < 1, respectively. Finally,

Lemma 1.4implies exponential stability ford∈−0.92,1⊂−23,23.

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 A9–14 . Suppose that

ajn≥0, 0≤j ≤m, m

j0

ajn>0,

∞

n0 m

j0

ajn ∞,

sup

n≥m n

kn−m m

j0 ajk<

3 2

1 2m1.

3.14

Then the equation with several delays

xn1−xn − m

j0 ajnx

n−j, n0,1,2, . . . 3.15

is globally asymptotically stable.

Theorem B15 . Suppose thataln≡al>0and

m

l1

allim sup n→ ∞

n−hln<1

1

e− m

l1

al. 3.16

Then1.1is asymptotically stable.

Let us note that unlike Theorems A and B we do not assume that coeﬃcients are either nonnegativeas in Theorem Aor constantas in Theorem B. Further, let us compare our stability tests with known results, including Theorems A and B.

Example 3.5. Consider the equation

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where

an

⎧ ⎨ ⎩

2, ifnis even,

0.1, ifnis odd,

cn

⎧ ⎨ ⎩

3, ifnis even,

0.05, ifnis odd.

3.18

Repeating the proof ofTheorem 2.17, we have

xn1 anan−5xn−9 cn−5xn−11

cnan−7xn−11 cn−7xn−13 .

3.19

ByTheorem 2.17,3.17is exponentially stable since

|an||an−5||cn−5| |cn||an−7||cn−7|

|an||cn| |an−5||cn−5|

0.15·50.75<1.

3.20

Lemma 1.4cannot be applied since for evenn

|anan−1−cn||an||cn−1||0.2−3|2·0.05>2.8>1. 3.21

Theorem A fails since

n

jn−6

ajbj> an cn an−1 cn−1 5.15> 3

2 1

4. 3.22

Theorem B is applicable to equations with constant coeﬃcients only.

Acknowledgments

The authors are very grateful to the referee for valuable comments and remarks. This paper is partially supported by Israeli Ministry of Absorption and by the NSERC Research Grant.

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