Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 535849, 11 pages
doi:10.1155/2009/535849
Research Article
Advanced Discrete Halanay-Type Inequalities:
Stability of Difference Equations
Ravi P. Agarwal,1 Young-Ho Kim,2 and S. K. Sen1
1
Department of Mathematical Sciences, Florida Institute of Technology,
150 West University Boulevard, Melbourne, FL 32901-6975, USA
2
Department of Applied Mathematics, Changwon National University,
Changwon, Kyeong-Nam 641-773, South Korea
Correspondence should be addressed to Ravi P. Agarwal, agarwal@fit.edu
Received 7 December 2008; Accepted 21 January 2009
Recommended by Martin J. Bohner
We derive new nonlinear discrete analogue of the continuous Halanay-type inequality. These
inequalities can be used as basic tools in the study of the global asymptotic stability of the
equilibrium of certain generalized difference equations.
Copyright q 2009 Ravi P. Agarwal et al. 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
The investigation of stability of nonlinear difference equations with delays has attracted a
lot of attention from many researchers such as Agarwal et al. 1–3, Baı̆nov and Simeonov
4, Bay and Phat 5, Cooke and Ivanov 6, Gopalsamy 7, Liz et al. 8–10, Niamsup et
al. 11, 12, Mohamad and Gopalsamy 13, Pinto and Trofimchuk 14, and references sited
therein. In 15, Halanay proved an asymptotic formula for the solutions of a differential
inequality involving the “maximum” functional and applied it in the stability theory of linear
systems with delay. Such an inequality was called Halanay inequality in several works. Some
generalizations as well as new applications can be found, for instance, in Agarwal et al. 2,
Gopalsamy 7, Liz et al. 8–10, Niamsup et al. 11, 12, Mohamad and Gopalsamy 13,
and Pinto and Trofimchuk 14. In particular, in 2, 6, 10, 12, 13, the authors considered
discrete Halanay-type inequalities to study some discrete version of functional differential
equations.
In the following results of Liz et al. 10, authors showed that some discrete versions
of these maximum inequalities can be applied to study the global asymptotic stability of a
family of difference equations.
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Journal of Inequalities and Applications
Theorem A. Assume that u, v satisfies the system of inequalities
n Cvn Dvn ,
Δun ≤ −Aun Bu
vn ≤ Eun F u
n ,
n ≥ 0,
n ≥ 0,
1.1
where Δun un1 − un , u
n max{un , . . . , un−r }, vn max{vn−1 , . . . , vn−r }, and r > 0 is a natural
number. If B, C, D, E, F ≥ 0, FD B > 0, E F > 0 and
B E FC D < A ≤ 1,
1.2
then there exist constants K1 ≥ 0, K2 ≥ 0, and λ0 ∈ 0, 1 such that
un ≤ K1 λn0 ,
vn ≤ K2 λn0 ,
n ≥ 0.
1.3
Moreover, λ0 can be chosen as the smallest root in the interval 0, 1 of equation hλ 0, where
hλ λ2r1 − 1 − A CEλ2r − B FC EDλr − FD.
1.4
By a simple use of Theorem A, authors also demonstrated the validity of the following
statement, namely, Theorem B.
Theorem B. Assume that f satisfies the following inequalities:
f n, xn , . . . , xn−r ≤ xn , . . . , xn−r , ∀ xn , . . . , xn−r ∈ Rr1 ,
∞
f n, xn , . . . , xn−r − xn ≤ r Δxn−1 , . . . , Δxn−r , ∀ xn , . . . , xn−r ∈ Rr1 .
∞
1.5
If either
a 0 ≤ a ≤ 1 − b, and 0 < br < 1, or
b a < 0, and 0 < br < a b−a b−1
holds, then there exist K > 0 and λ0 ∈ 0, 1 such that for every solution {xn } of
Δxn −axn − bf n, xn , xn−1 , . . . , xn−r ,
a > 0,
1.6
one has
xn ≤ max xi λn ,
0
n ≥ 0,
1.7
where λ0 can be calculated in the form established in Theorem A. As a consequence, the trivial solution
of 1.6 is globally asymptotically stable.
The main aim of the present paper is to establish some new nonlinear retarded
Halanay-type inequalities, which extend Theorem A, along with the derivation of new global
stability conditions for nonlinear difference equations.
Journal of Inequalities and Applications
3
2. Halanay-Type Discrete Inequalities
Let R denote the set of all real numbers, R the set of positive real numbers, R0 the set of
nonnegative real numbers, Z the set of integers, Z the set of positive integers, and Z−r {z ∈
Z : z ≥ −r}. Consider the following nonlinear difference equation:
Δxn f n, xn , xn−1 , . . . , xn−r ,
n ∈ Z ,
2.1
where Δxn xn1 − xn , and f : N × Rr1 → R. Equation 2.1 is a generalized difference
equation see 3, Section 21 and 11. The initial value problem for this equation requires
the knowledge of the initial data {x−r , x−r1 , . . . , x0 }. This vector is called the initial string
in 6. For every initial string, there exists a unique solution {xn }n≥Z−r of 2.1 that can be
calculated using the explicit recurrence formula
xn1 xn f n, xn , xn−1 , . . . , xn−r ,
n ∈ Z0 .
2.2
In this section, we introduce new discrete inequalities which will be used to derive global
stability conditions in the next section.
Theorem 2.1. Let ai , bi , ci , di , ei , fi ∈ R0 , ri0 di e > 0, hi ∈ Z0 i 0, . . . , r, 0 h0 < h1 <
· · · < hr ; hr ∈ Z , and
b c de f < a ≤ 1,
2.3
where a ri0 ai , b ri0 bi , c ri0 ci , d ri0 di , e ri0 ei , and f ri0 fi . Also, let
{un , vn }n∈Z−hr be a sequence of nonnegative real numbers satisfying the system of inequalities
Δun ≤
r
i0
p
− ai un bi un−hi ci vn di vn−hi ,
vn ≤
r e i un i0
p
fi un−hi
n ∈ Z0 ,
2.4
,
n∈Z ,
0
where p ≥ 0 is a constant. Then there exist constants K1 ≥ 0, K2 ≥ 0, and λ0 ∈ 0, 1 such that
un ≤ K1 λn0 ,
vn ≤ K2 λn0 ,
n ∈ Z0 ,
2.5
−nn−hi p
where K1 max0≤i≤r {u−hi , α−1 v−hi }, and K2 αK1 with α e ri0 fi λ0
. Moreover, λ0
can be chosen as the smallest root in the interval 0, 1 of equation gλ 0, where
gλ λ − 1 − a ce −
r
bi cfi λn−hi p−n
i0
−
r
i0
with n ∈ Z0 .
di eλ−hi
r
r
n−hj −hi p−n
di fj λ
−
i0
j0
2.6
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Journal of Inequalities and Applications
Proof. Let {xn , yn }n∈Z−hr be a sequence of nonnegative real numbers satisfying the system of
inequalities
n−1
xn 1 − an x0 1 − an−j−1
j0
×
r
i0
yn r i0
p
− ai xj bi xj−hi ci yj di yj−hi ,
2.7
p
ei xn fi xn−hi ,
where n ∈ Z0 . Since 1 − a ≥ 0, it is easy to prove by induction that if un ≤ xn and vn ≤ yn for
n −hr , . . . , 0, then un ≤ xn and vn ≤ yn for all n ∈ Z0 .
On the other hand, the system 2.7 is equivalent to
Δxn r
i0
yn p
− ai xn bi xn−hi ci yn di yn−hi ,
r ei xn i0
p
fi xn−hi
2.8
,
where n ∈ Z0 . Next we prove, under the assumptions of the theorem, that there exists a
solution {xn , yn }n∈Z−hr to system 2.8 in the form xn λn0 , yn αλn0 with α > 0, λ0 ∈ 0, 1.
Indeed, such {xn , yn }n∈Z−hr is a solution of 2.8 if and only if
1 − aλn0 λn1
0
αλn0
r n−hi p
bi λ0
i0
r ei λn0
i
,
ci αλn0 di αλn−h
0
n−h p
fi λ0 i
n ∈ Z0 ,
2.9
,
n∈Z .
0
i0
This is equivalent to the existence of a solution λ0 ∈ 0, 1 of equation gλ 0, where g is the
polynomial defined by 2.6.
Now, g0 limλ → 0 gλ −∞ < 0 in view of ri0 di e > 0. On the other hand,
g1 a − b − c de f > 0 in view of 2.3. As a consequence, there exists λ0 ∈ 0, 1 such
−nn−hi p
> 0.
that gλ0 0. Hence, λ0 , α is a solution of 2.9 with α e ri0 fi λ0
For this value of λ0 , the pair {Kλn0 , Kαλn0 } is a solution of 2.8 for every K ≥ 0. Thus,
choosing K max0≤i≤r {u−hi , α−1 v−hi }, we have that un ≤ Kλn0 xn , and vn ≤ Kαλn0 yn for all
n −hr , . . . , 0.
Hence, using the first part of the proof, we can conclude that un ≤ xn , and vn ≤ yn for
all n ∈ Z0 .
By the similar argument used in Theorem 2.1, we obtain the following result.
Journal of Inequalities and Applications
5
Theorem 2.2. Let a, b, c, d, e, f ∈ R0 , hi ∈ Z0 , i 0, . . . , r, 0 h0 < h1 < · · · < hr ; r ≥ 1, and
b ce f de fr1 < a ≤ 1
2.10
with ce > 0. Also, let {un , vn }n∈Z−hr be a sequence of nonnegative real numbers satisfying the system
of inequalities
Δun ≤ −aun r
bun−hi cvn r
i0
vn ≤ eun dvn−hi ,
n ∈ Z0 ,
i0
r
fun−hi ,
2.11
n∈Z .
0
i0
Then there exist constants K1 ≥ 0, K2 ≥ 0, and λ0 ∈ 0, 1 such that
un ≤ K1 λn0 ,
vn ≤ K2 λn0 ,
n ∈ Z0 ,
2.12
i
where K1 max0≤i≤r {u−hi , ρ−1 v−hi }, and K2 ρK1 with ρ e f ri0 λ−h
0 . Moreover, λ0 can be
chosen as the smallest root in the interval 0, 1 of equation Fλ 0, where
r1 rn−h
Fλ λ − 1 − a ce − b cf d e fλrn−h
λ
with n ∈ Z0 , h r
i0
2.13
hi .
Proof. Let {xn , yn }n∈Z−hr be a sequence of nonnegative real numbers satisfying the system of
inequalities
Δxn −axn b
r
r
xn−hi cyn d yn−hi ,
i0
yn exn f
n ∈ Z0 ,
i0
r
xn−hi ,
2.14
n∈Z .
0
i0
Since 1 − a ≥ 0, it is easy to prove by induction that if un ≤ xn and vn ≤ yn for n −hr , . . . , 0,
then un ≤ xn and vn ≤ yn for all n ∈ Z0 .
Next we prove that, under the assumptions of the theorem, there exists a solution
{xn , yn }n∈Z−hr to system 2.14 in the form xn λn , yn ρλn with ρ > 0, λ ∈ 0, 1. Indeed,
such {xn , yn }n∈Z−hr is a solution of 2.14 if and only if
λn1 1 − aλn b
r
r
λn−hi cρλn d ρλn−hi ,
i0
i0
r
λn−hi ,
ρλn eλn f
i0
n ∈ Z0 ,
2.15
n∈Z .
0
6
Journal of Inequalities and Applications
This is equivalent to the existence of a solution λ ∈ 0, 1 of equation Fλ 0, where F is the
polynomial defined in 2.13.
Now, in view of ce > 0, we have F0 −1 a − ac < 0 in case rn > h, F0 −1 a − ac − b cf de fr1 < 0 in case rn h, and F0 limλ → 0 Fλ −∞ < 0 in
case rn < h.
On the other hand, F1 a − b − ce f − de fr1 > 0 in view of 2.10. As a
consequence, there exists λ0 ∈ 0, 1 such that Fλ0 0. Hence, λ0 , ρ is a solution of 2.15
i
with ρ e f ri0 λ−h
0 > 0.
For this value of λ0 , the pair {Kλn0 , Kρλn0 } is a solution of 2.14 for every K ≥ 0. Thus,
choosing K max0≤i≤r {u−hi , ρ−1 v−hi }, we have un ≤ Kλn0 , and vn ≤ Kρλn0 for all n −hr , . . . , 0.
These imply un ≤ xn , and vn ≤ yn for all n −hr , . . . , 0. Hence, using the first part of the proof,
we can conclude that un ≤ xn , and vn ≤ yn for all n ∈ Z0 .
Remark 2.3. In 10, a discrete Halanay-type inequality was given as in Theorem A, where the
inequalities were replaced by
un cvn dvn ,
Δun ≤ −aun b
vn ≤ eun f u
n ,
n ≥ 0,
2.16
n ≥ 0,
where Δun un1 − un , u
n max{un , . . . , un−r }, vn max{vn−1 , . . . , vn−r }, and r ≥ 1 is a
natural number. Note that if a sequence {un }n∈Z−r of positive real numbers satisfies 2.16,
then it also satisfies 2.4. On the other hand, let p r 1, hi i; a 1i0 ai 1, b b0 b1 1/7, c c0 c1 0, d d0 d1 1/7, e e0 e1 1/7, and f f0 f1 1/7. Then
we might easily show that the sequence {1/2n }n∈Z−1 satisfies 2.4 but not 2.16. Indeed,
Δun 1
2n1
−
1
2n
1
,
2n1 1 1 5
1 5
1 1
1
1
<− n 2
7 2n 2n−1
7 2n 7 2n−1 7
−
−
with
1
i0
bi 1
i0 ci
Δun −
1
i0
13 1
,
49 2n
di 1
i0
ei 1
i0
fi <
1
i0
ai 1. On the other hand,
1
2n1
1
1
> − n max
2
7
−
2.17
1
1
, n−1
n
2 2
1
7
1 1
1
max
n−1
72
7
1
2n−1
2.18
31 1
.
49 2n
Therefore, in the case of positive sequences, the discrete inequality 2.4 is less conservative
than the discrete Halanay-type inequality given by 2.16.
Journal of Inequalities and Applications
7
3. Global Stability of Difference Equations
In order to show the applicability of the previous result, in this section we consider the
generalized difference equation
Δxn −axn − bf n, xn , xn−h1 , . . . , xn−hr ,
3.1
where n, hi ∈ Z , i 1, . . . , r, and b > 0.
Although, for every initial string {x−hr , x−hr 1 , . . . , x0 }, the solution {xn } of 3.1 can
be explicitly calculated by a recurrence formula similar to 2.2, it is in general difficult to
investigate the asymptotic behavior of the solutions using that formula. The next result gives
an asymptotic estimate by a simple use of the discrete Halanay inequality.
Theorem 3.1. For all n, xn , xn−h1 , . . . , xn−hr ∈ Z0 × Rr1 , assume that f satisfies the following
inequalities:
r
p
f n, xn , xn−h , . . . , xn−h ≤
βj xn−hj ,
1
r
3.2
j0
r
f n, xn , xn−h , . . . , xn−h − xn ≤
γj Δxn−hj ,
1
r
3.3
j0
where βj , γj , p ∈ R0 ,
· · · < hr . If either
r
i0 γi |a|
> 0, hj ∈ Z0 j 0, . . . , r − 1, and hr ∈ Z with 0 h0 < h1 <
a 0 ≤ a ≤ 1 − b, 0 < bγ < 1, and 0 < β ≤ 1, or
b a < 0 and 0 < bγ < a b−a bβ−1
hold, then there exists a constant λ0 ∈ 0, 1 for every solution {xn } of 3.1 such that
xn ≤
Δxi λn ,
max xi , α−1
0
1
−hr ≤i≤0
n ∈ Z0 ,
3.4
−nn−hi p
where α1 |a| b ri0 βi λ0
, β ri0 βi , γ ri0 γi , and λ0 can be chosen as the smallest
root in the interval 0, 1 of equation g1 λ 0, where
g1 λ λ − 1 − a − b −
r
i0
−hi
b|a|γi λ
−
r
i0
bγi
r
n−hj −hi p−n
bβj λ
3.5
j0
with n ∈ Z0 .
As a consequence, the trivial solution of 3.1 is globally asymptotically stable.
Proof. Let {xn } be a solution of 3.1. Equation 3.1 can be written in the form
Δxn −a bxn − b f n, xn , xn−h1 , . . . , xn−hr − xn .
3.6
8
Journal of Inequalities and Applications
Hence, we know that
n−1
n
n−i−1
−b f i, xi , xi−h1 , . . . , xi−hr − xi ,
xn 1 − a b x0 1 − a b
3.7
i0
where n ∈ Z0 . Thus, using inequality 3.3, we obtain
n−1 r
n−i−1 xn ≤ 1 − a b n x0 bγj Δxi−hj .
1 − a b
3.8
i0 j0
Denote un |xn | for n −hr , . . . , 0, and
n−1 r
n n−i−1 un 1 − a b x0 bγj Δxi−hj 1 − a b
3.9
i0 j0
for n ∈ Z . Then we have |xn | ≤ un and, from inequality 3.9, we obtain
Δun −a bun r
bγj Δxn−hj 3.10
j0
for n ∈ Z . On the other hand, using hypothesis 3.2 in 3.1, we have
r
Δxn ≤ | − a|xn b βj xn−h p
j
j0
≤ |a|un b
r
j0
3.11
p
βj un−hj .
Denote vn |Δxn |. We can apply Theorem 2.1 to the system of inequalities 3.10 and 3.11
with ri0 ai a b, bi 0, ci 0, di bγj , ri0 ei |a|, and fi bβj . Consequently,
Theorem 2.1 ensures the validity of the following inequality:
xn ≤
Δxi λn ,
max xi , α−1
0
1
−hr ≤i≤0
n ∈ Z0 ,
3.12
where λ0 and α1 are chosen as in Theorem 3.1. This completes the proof of the theorem.
Next, we obtain new conditions for the asymptotic stability of 3.1 using inequality
3.13 instead of 3.3.
Journal of Inequalities and Applications
9
Corollary 3.2. For all n, xn , xn−h1 , . . . , xn−hr ∈ Z0 × Rr1 , assume that f satisfies inequality 3.2
and the following condition:
r
p
f n, xn , xn−h , . . . , xn−h − xn ≤
γj xn−hj δj Δxn ηj Δxn−hj ,
1
r
3.13
j0
where γj , δj , ηj , p ∈ R0 ,
· · · < hr . If
r
j0
ηj |a| > 0, hj ∈ Z0 j 0, . . . , r − 1, and hr ∈ Z with 0 h0 < h1 <
bγ bδ η|a| bβ < a b ≤ 1
3.14
holds, where β ri0 βi , γ ri0 γi , δ ri0 δi , and η ri0 ηi , then there exists a constant
λ0 ∈ 0, 1 for every solution {xn } of 3.1 such that
xn ≤
Δxi λn ,
max xi , α−1
0
1
n ∈ Z0 ,
−hr ≤i≤0
3.15
−nn−hi p
, and λ0 can be chosen as the smallest root in the interval 0, 1 of
where α1 |a| b ri0 βi λ0
equation g2 λ 0, where
r
g2 λ λ − 1 − a − b b|a|δ −
b γi bδβi λn−hi p−n
i0
−
r
i0
−hi
b|a|ηi λ
−
r
i0
bηi
r
3.16
n−hj −hi p−n
bβj λ
j0
with n ∈ Z0 .
As a consequence, the trivial solution of 3.1 is globally asymptotically stable.
Similarly, using Theorem 2.2 instead of Theorem 2.1, we obtain the following result.
Theorem 3.3. For all n, xn , xn−h1 , . . . , xn−hr ∈ Z0 × Rr1 , assume that f satisfies the following
inequalities:
r f n, xn , xn−h , . . . , xn−h ≤ β xn−h ,
1
r
j
j0
r r f n, xn , xn−h , . . . , xn−h − xn ≤ γ
xn−h δΔxn η Δxn−h ,
1
r
j
j
j0
3.17
j0
where β, γ, δ, η ∈ R0 , hj ∈ Z0 , j 0, . . . , r − 1, and hr ∈ Z with 0 h0 < h1 < · · · < hr . If |a|δ > 0
and
r1
bγ bδ |a| bβ bη |a| bβ
< a b ≤ 1,
3.18
10
Journal of Inequalities and Applications
then there exists a constant λ0 ∈ 0, 1 for every solution {xn } of 3.1 such that
xn ≤
where ρ1 |a| bβ
F1 λ 0, where
r
hi
i0 λ0 ,
max {xi , ρ1−1 Δxi } λn0 ,
−hr ≤i≤0
n ∈ Z0 ,
3.19
and λ0 can be chosen as the smallest root in the interval 0, 1 of equation
r1 rn−h
λ
F1 λ λ − 1 − a b |a|bδ − b γ bβδ η |a| bβλrn−h
3.20
with n ∈ Z0 , h ri0 hi .
As a consequence, the trivial solution of 3.1 is globally asymptotically stable.
Remark 3.4. Equation 3.1 covers a variety of difference equations. For instance, we can
consider the following difference equation:
Δxn −axn − bf xn−k ,
b > 0.
3.21
Next, we study the asymptotic behavior of the solutions of 3.21. We can apply Theorem 3.1,
Corollary 3.2, or Theorem 3.3 to obtain some relations between coefficients a and b that ensure
the global asymptotic stability of the zero solution. Moreover, from Theorem 3.1 we know that
if there exists β, γ ∈ R such that |fx| ≤ β|x|p , |fx − x| ≤ γ|Δx| for all x, and if either
a 0 < a ≤ 1 − b, 0 < bγ < 1, and 0 < β ≤ 1, or
b a < 0 and 0 < bγ < a b−a bβ−1
hold, then all solutions of 3.21 converge to zero.
Acknowledgment
The authors thank the referees of this paper for their careful and insightful critique.
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