- Research Article
- Open Access
- Published:
Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation
Advances in Difference Equations volume 2010, Article number: 461014 (2010)
Abstract
Sufficient conditions are obtained for the uniform stability and global attractivity of the zero solution of nonlinear scalar delay impulsive difference equation, which extend and improve the known results in the literature. An example is also worked out to verify that the global attractivity condition is a sharp condition.
1. Introduction and Main Results
Let ,
, and
be the sets of real numbers, natural numbers, and integers, respectively. For any
, define
and
when
.
It is well known that the theory of impulsive differential equations is emerging as an important area of investigation, since it is not only richer than the corresponding theory of differential equations without impulse effects but also represents a more natural framework for mathematical modeling of many world phenomena [1]. Moreover, such equations may exhibit several real-world phenomena, such as rhythmical beating, merging of solutions, and noncontinuity of solutions. And hence ordinary differential equations and delay differential equations with impulses have been considered by many authors, and numerous papers have been published on this class of equations and good results were obtained (see, e.g., [1–10] and references therein).
Since the behavior of discrete systems is sometimes sharply different from the behavior of the corresponding continuous systems and discrete analogs of continuous problems may yield interesting dynamical systems in their own right (see [11–13]), many scholars have investigated difference equations independently. However, there are few concerned with the impulsive difference equations or delay impulsive difference equations (see [14–19]). On the other hand, stability is one of the major problems encountered in applications, but, to the best of our knowledge, very little has been done with the stability of impulsive difference equations (see [15, 20]). Motivated by this, the aim of this paper is devoted to studying the uniform stability and global attractivity of the zero solution of the following nonlinear scalar delay impulsive difference equation:

where denotes the forward difference operator defined by
,
,
is the set of all functions
for some
and
is defined by
for
,
and
, with
as
. By a solution of (1.1), we mean a sequence
of real numbers which is defined for all
and satisfies (1.1) for
for some
. It is easy to see that, for any given
and a given initial function
, there is a unique solution of (1.1), denoted by
such that

We assume that and
, so that
is a solution of (1.1), which we call the zero solution.
For , define the norm of
as

and for any , define

Definition 1.1.
The zero solution of (1.1) is stable, if, for any and
, there exists a
such that
implies that
for
. If
is independent of
, we say that the zero solution of (1.1) is uniformly stable.
Definition 1.2.
The zero solution of (1.1) is globally attractive, if every solution of (1.1) tends to zero as .
A simple example of (1.1) is given by

where ,
and
is a sequence of real numbers. In [15], the author studied the stability of the zero solution of (1.5), where
for
and
, and obtained the following result.
Theorem 1.3.
If

then the zero solution of (1.5) is stable.
In this paper, we assume that there exists a positive constant and a sequence
of nonnegative real numbers such that

where . Furthermore, we assume that there is a sequence
of positive numbers with
such that

The main purpose of this paper is to establish the following theorems.
Theorem 1.4.
Assume that (1.7) and (1.8) hold and

Then the zero solution of (1.1) is uniformly stable.
Remark 1.5.
Theorem 1.4 generalizes and improves Theorem 1.3 greatly.
The next theorem provides a sufficient condition for every solution of (1.1) tends to zero as , that is, the zero solution of (1.1) is globally attractive.
Theorem 1.6.
Assume that (1.7) and (1.8) hold and

and assume that, for each bounded solution , either

or

Then every solution of (1.1) tends to zero as .
Remark 1.7.
An example is worked out in Section 3 to verify that the upper bound in (1.10) is best possible, that is, the upper bound in (1.10) cannot be improved.
One special form of (1.1) is when

where , and
for
,  
. Then for any
, (1.7) holds with
. As a consequence of Theorem 1.4, we have the following.
Corollary 1.8.
Assume that (1.8) holds and

Then the zero solution of the equation

is uniformly stable.
For the sake of convenience, throughout this paper, we will use the convention

2. Proofs of Main Results
Define

for . Then
for
if
. Equation (1.1) can be rewritten as

Set

then (2.2) reduces to

And it is easy to see that

To prove Theorems 1.4 and 1.6, we need the following lemma.
Lemma 2.1.
Let ,
be a solution of (1.1),
is defined by (2.3),
, and
. If

holds for some and either

or

then .
Proof.
We assume that and
. The case where
and
is similar and is omitted.
It is easy to see that Lemma 2.1 holds for .
If , then
and
for all
by (2.6). Thus we only need to prove that
. Since
,
, by (1.7) we have

that is,

So which admits that
. Hence

which implies that , so
by the definition of
.
Now, assume that ,
and
. By (1.7), we also have

So,

Hence, there is such that
and
for all
. So,
and
for all
. For
, by (1.7) and (2.4), one has

provided that contains no impulsive points. And since
if
meets one of impulsive points, we always have

for , where
.
By the choice of , there is a real number
such that

Next, we will show that, for any ,

In fact, for any ,

which shows that (2.17) holds.
Substituting (2.17) into (2.4), it is easy to get

Let

There are two cases to consider.
Case 1 ().
We have by (2.6), (2.16), and (2.19)

Since

we have

Case 2 ().
In this case, there exists such that

Therefore, we may choose a real number such that

Notice that

So, by (2.6), (2.15), (2.19), and (2.25), we have

The proof is completed by combining Cases 1 and 2.
Proof of Theorem 1.4.
By Lemma 2.1, (1.9) implies that (2.6) holds with . For any
, assume that
contains
impulsive points:
. Set

We will prove that implies that

To this end, we first prove that

If , then
. If
, then, for
, we claim that

In fact,

In general, we can obtain (2.31) by induction. And so for either case with
. For
, we have

And so

In general, we have

Now for ,

And so

which has proved that (2.30) holds. Now, we will prove that

Thus we only need to prove that

For any and
we claim that

In fact, we can assume that . The case where
is similar and is omitted. If
, by the definition of
, (2.40) holds. If
, then by Lemma 2.1 we have that (2.40) holds.
Since

by (2.40) we have

By repeatedly using (2.40) we get that (2.39) holds.
Combining (2.30) and (2.39), we find that (2.29) holds and the proof of Theorem 1.4 is complete.
Proof of Theorem 1.6.
In view of Theorem 1.4, we see that the zero solution of (1.1) is uniformly stable. Therefore, for any , there exists
such that
implies that

Next, we will prove that

There are two cases to consider.
Case 1.
is eventually positive, that is, there exists
such that
for all
. Hence, by (1.7) and (1.8), we have
for
. That is,
is eventually nonincreasing and hence
. Thus, by (1.11) we see that (2.44) holds. The case when
is eventually negative is similar and will be omitted.
Case 2.
is oscillatory in the sense that
is neither eventually positive nor eventually negative, so
is also oscillatory. To prove (2.44), we only need to prove that

By the proof of Theorem 1.4, we have

Let

then . It suffices to show that

In fact, if (2.48) does not hold, we assume that and
. The case where
and
is similar and is omitted.
Let , where
is given in Lemma 2.1. Then there exists an integer
such that

Since and
is oscillatory, there must exist an integer
such that

By Lemma 2.1 and (2.49) we have

Equations (2.49) and (2.51) imply that , which contradicts the fact that
; thus (2.48) holds and the proof is complete.
3. Example
In this section we will give a result which guarantees that the upper bound is best possible in Theorem 1.6.
Example 3.1.
Consider the delay impulsive difference equation

where , and
is a sequence of nonnegative real numbers defined by

where and
is an undetermined constant. In view of Theorem 1.6, if

or equivalently

then every solution of (3.1) tends to zero as .
The following theorem shows that if (3.4) does not hold, then there is a solution of (3.1) which does not tend to zero as . This shows that the upper bound
cannot be improved.
Theorem 3.2.
Assume that

Then there exists a solution of (3.1) which does not tend to zero as .
Proof.
Let be a solution of (3.1) with initial condition of the form

where is a given constant. Then by (3.1) and the definition of
, we have

And hence


By virtue of (3.1) and (3.8) we get

Summing up from to
and using (3.9), we have

Furthermore, we have, by the definition of ,

Define the sequence as follows:

Then we have by (3.5)

which implies that is a non-decreasing sequence and does not tend to zero as
.
Repeating the above argument, we find that, for

By the definition of , we know that
as
, so
as
. The proof is complete.
References
Lakshmikantham V, BaÄnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.
Hale J: Theory of Functional Differential Equations, Applied Mathematical Sciences. Volume 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365.
Nieto JJ, Tisdell CC: On exact controllability of first-order impulsive differential equations. Advances in Difference Equations 2010, 2010:-9.
Lakshmikantham V, Liu X, Sathananthan S: Impulsive integro-differential equations and extension of Lyapunov's method. Applicable Analysis 1989,32(3-4):203-214.
Xing Y, Wang Q, Chen D: Antiperiodic boundary value problem for second-order impulsive differential equations on time scales. Advances in Difference Equations 2009, 2009:-14.
Yu JS: Stability for nonlinear delay differential equations of unstable type under impulsive perturbations. Applied Mathematics Letters 2001,14(7):849-857. 10.1016/S0893-9659(01)00055-6
Yu JS: Explicit conditions for stability of nonlinear scalar delay differential equations with impulses. Nonlinear Analysis: Theory, Methods & Applications 2001,46(1):53-67. 10.1016/S0362-546X(99)00445-9
Xing Y, Han M: A new approach to stability of impulsive functional differential equations. Applied Mathematics and Computation 2004,151(3):835-847. 10.1016/S0096-3003(03)00540-X
Yan J: Stability for impulsive delay differential equations. Nonlinear Analysis: Theory, Methods & Applications 2005,63(1):66-80. 10.1016/j.na.2005.05.001
Yan J: Global attractivity for impulsive population dynamics with delay arguments. Nonlinear Analysis: Theory, Methods & Applications 2009,71(11):5417-5426. 10.1016/j.na.2009.04.030
May RM: Simple mathematical models with very complicated dynamics. Nature 1976,261(5560):459-467. 10.1038/261459a0
Mohamad S, Gopalsamy K: Dynamics of a class of discrete-time neural networks and their continuous-time counterparts. Mathematics and Computers in Simulation 2000,53(1-2):1-39. 10.1016/S0378-4754(00)00168-3
Mohamad S, Gopalsamy K: Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Applied Mathematics and Computation 2003,135(1):17-38. 10.1016/S0096-3003(01)00299-5
Abdullin RZ: The comparison method in the stability of nonlinear difference equations with impulse actions. Automation and Remote Control 2000,61(11, part 1):1796-1807.
Zhang Q: On a linear delay difference equation with impulses. Annals of Differential Equations 2002,18(2):197-204.
Peng M: Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses. Computers & Mathematics with Applications 2002,44(5-6):741-748. 10.1016/S0898-1221(02)00187-6
He Z, Zhang X: Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions. Applied Mathematics and Computation 2004,156(3):605-620. 10.1016/j.amc.2003.08.013
Geng F, Xu Y, Zhu D: Periodic boundary value problems for first-order impulsive dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):4074-4087. 10.1016/j.na.2007.10.038
Zhang Y, Sun J, Feng G: Impulsive control of discrete systems with time delay. IEEE Transactions on Automatic Control 2009,54(4):871-875.
Zhang QQ, Zhou Z: Stability for difference equations of unstable type under impulsive perturbations. Hunan Daxue Xuebao 2002,29(3):4-9.
Acknowledgments
The author would like to express her thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project (2009).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zheng, B. Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation. Adv Differ Equ 2010, 461014 (2010). https://doi.org/10.1155/2010/461014
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/461014
Keywords
- Ordinary Differential Equation
- Discrete System
- Initial Function
- Discrete Analog
- Explicit Condition