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Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations
Journal of Inequalities and Applications volume 2009, Article number: 535849 (2009)
Abstract
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.
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], BaÄnov 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.
Theorem 1.
Assume that satisfies the system of inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ1_HTML.gif)
where and
is a natural number. If
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ2_HTML.gif)
then there exist constants and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ3_HTML.gif)
Moreover, can be chosen as the smallest root in the interval
of equation
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ4_HTML.gif)
By a simple use of Theorem A, authors also demonstrated the validity of the following statement, namely, Theorem B.
Theorem 1.
Assume that satisfies the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ5_HTML.gif)
If either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_IEq12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_IEq14_HTML.gif)
holds, then there exist and
such that for every solution
of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ6_HTML.gif)
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ7_HTML.gif)
where 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.
2. Halanay-Type Discrete Inequalities
Let denote the set of all real numbers,
the set of positive real numbers,
the set of nonnegative real numbers,
the set of integers,
the set of positive integers, and
. Consider the following nonlinear difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ8_HTML.gif)
where , and
. 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
. This vector is called the initial string in [6]. For every initial string, there exists a unique solution
of (2.1) that can be calculated using the explicit recurrence formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ9_HTML.gif)
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 and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ10_HTML.gif)
where Also, let
be a sequence of nonnegative real numbers satisfying the system of inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ11_HTML.gif)
where is a constant. Then there exist constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ12_HTML.gif)
where , and
with
Moreover,
can be chosen as the smallest root in the interval
of equation
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ13_HTML.gif)
with
Proof.
Let be a sequence of nonnegative real numbers satisfying the system of inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ14_HTML.gif)
where Since
it is easy to prove by induction that if
and
for
then
and
for all
On the other hand, the system (2.7) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ15_HTML.gif)
where Next we prove, under the assumptions of the theorem, that there exists a solution
to system (2.8) in the form
with
Indeed, such
is a solution of (2.8) if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ16_HTML.gif)
This is equivalent to the existence of a solution of equation
where
is the polynomial defined by (2.6).
Now, in view of
On the other hand,
in view of (2.3). As a consequence, there exists
such that
Hence,
is a solution of (2.9) with
For this value of , the pair
is a solution of (2.8) for every
Thus, choosing
we have that
, and
for all
Hence, using the first part of the proof, we can conclude that , and
for all
By the similar argument used in Theorem 2.1, we obtain the following result.
Theorem 2.2.
Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ17_HTML.gif)
with Also, let
be a sequence of nonnegative real numbers satisfying the system of inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ18_HTML.gif)
Then there exist constants and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ19_HTML.gif)
where , and
with
Moreover,
can be chosen as the smallest root in the interval
of equation
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ20_HTML.gif)
with
Proof.
Let be a sequence of nonnegative real numbers satisfying the system of inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ21_HTML.gif)
Since it is easy to prove by induction that if
and
for
then
and
for all
Next we prove that, under the assumptions of the theorem, there exists a solution to system (2.14) in the form
with
Indeed, such
is a solution of (2.14) if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ22_HTML.gif)
This is equivalent to the existence of a solution of equation
where
is the polynomial defined in (2.13).
Now, in view of , we have
in case
in case
and
in case
On the other hand, in view of (2.10). As a consequence, there exists
such that
Hence,
is a solution of (2.15) with
For this value of , the pair
is a solution of (2.14) for every
Thus, choosing
we have
, and
for all
These imply
, and
for all
Hence, using the first part of the proof, we can conclude that
, and
for all
Remark 2.3.
In [10], a discrete Halanay-type inequality was given as in Theorem A, where the inequalities were replaced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ23_HTML.gif)
where and
is a natural number. Note that if a sequence
of positive real numbers satisfies (2.16), then it also satisfies (2.4). On the other hand, let
and
Then we might easily show that the sequence
satisfies (2.4) but not (2.16). Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ24_HTML.gif)
with On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ25_HTML.gif)
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).
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ26_HTML.gif)
where
Although, for every initial string the solution
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 assume that
satisfies the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ28_HTML.gif)
where and
with
If either
(a) and
or
(b) and
hold, then there exists a constant for every solution
of (3.1) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ29_HTML.gif)
where and
can be chosen as the smallest root in the interval
of equation
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ30_HTML.gif)
with
As a consequence, the trivial solution of (3.1) is globally asymptotically stable.
Proof.
Let be a solution of (3.1). Equation (3.1) can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ31_HTML.gif)
Hence, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ32_HTML.gif)
where Thus, using inequality (3.3), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ33_HTML.gif)
Denote for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ34_HTML.gif)
for Then we have
and, from inequality (3.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ35_HTML.gif)
for On the other hand, using hypothesis (3.2) in (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ36_HTML.gif)
Denote We can apply Theorem 2.1 to the system of inequalities (3.10) and (3.11) with
and
Consequently, Theorem 2.1 ensures the validity of the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ37_HTML.gif)
where and
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).
Corollary 3.2.
For all assume that
satisfies inequality (3.2) and the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ38_HTML.gif)
where and
with
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ39_HTML.gif)
holds, where then there exists a constant
for every solution
of (3.1) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ40_HTML.gif)
where , and
can be chosen as the smallest root in the interval
of equation
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ41_HTML.gif)
with
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 assume that
satisfies the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ42_HTML.gif)
where and
with
If
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ43_HTML.gif)
then there exists a constant for every solution
of (3.1) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ44_HTML.gif)
where , and
can be chosen as the smallest root in the interval
of equation
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ45_HTML.gif)
with
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F535849/MediaObjects/13660_2008_Article_1975_Equ46_HTML.gif)
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 and
that ensure the global asymptotic stability of the zero solution. Moreover, from Theorem 3.1 we know that if there exists
such that
for all
and if either
(a) and
or
(b) and
hold, then all solutions of (3.21) converge to zero.
References
Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications. Applied Mathematics and Computation 2005,165(3):599–612. 10.1016/j.amc.2004.04.067
Agarwal RP, Kim Y-H, Sen SK: New discrete Halanay inequalities: stability of difference equations. Communications in Applied Analysis 2008,12(1):83–90.
Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:viii+507.
BaÄnov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.
Bay NS, Phat VN: Stability analysis of nonlinear retarded difference equations in Banach spaces. Computers & Mathematics with Applications 2003,45(6–9):951–960.
Cooke KL, Ivanov AF: On the discretization of a delay differential equation. Journal of Difference Equations and Applications 2000,6(1):105–119. 10.1080/10236190008808216
Gopalsamy K: Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications. Volume 74. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+501.
Liz E, Trofimchuk S: Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality. Journal of Mathematical Analysis and Applications 2000,248(2):625–644. 10.1006/jmaa.2000.6947
Liz E, Ferreiro JB: A note on the global stability of generalized difference equations. Applied Mathematics Letters 2002,15(6):655–659. 10.1016/S0893-9659(02)00024-1
Liz E, Ivanov AF, Ferreiro JB: Discrete Halanay-type inequalities and applications. Nonlinear Analysis: Theory, Methods & Applications 2003,55(6):669–678. 10.1016/j.na.2003.07.013
Niamsup P, Phat VN: Asymptotic stability of nonlinear control systems described by difference equations with multiple delays. Electronic Journal of Differential Equations 2000,2000(11):1–17.
Udpin S, Niamsup P: New discrete type inequalities and global stability of nonlinear difference equations. to appear in Applied Mathematics Letters to appear in Applied Mathematics Letters
Mohamad S, Gopalsamy K: Continuous and discrete Halanay-type inequalities. Bulletin of the Australian Mathematical Society 2000,61(3):371–385. 10.1017/S0004972700022413
Pinto M, Trofimchuk S: Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima. Proceedings of the Royal Society of Edinburgh. Section A 2000,130(5):1103–1118. 10.1017/S0308210500000597
Halanay A: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York, NY, USA; 1966:xii+528.
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The authors thank the referees of this paper for their careful and insightful critique.
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Agarwal, R.P., Kim, YH. & Sen, S.K. Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations. J Inequal Appl 2009, 535849 (2009). https://doi.org/10.1155/2009/535849
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DOI: https://doi.org/10.1155/2009/535849