Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation

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:

(1.1)

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

(1.2)

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

For , define the norm of as

(1.3)

and for any , define

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

The main purpose of this paper is to establish the following theorems.

Theorem 1.4.

Assume that (1.7) and (1.8) hold and

(1.9)

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

(1.10)

and assume that, for each bounded solution , either

(1.11)

or

(1.12)

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

(1.13)

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

(1.14)

Then the zero solution of the equation

(1.15)

is uniformly stable.

For the sake of convenience, throughout this paper, we will use the convention

(1.16)

2. Proofs of Main Results

Define

(2.1)

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

(2.2)

Set

(2.3)

then (2.2) reduces to

(2.4)

And it is easy to see that

(2.5)

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

(2.6)

holds for some and either

(2.7)

or

(2.8)

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

(2.9)

that is,

(2.10)

So which admits that . Hence

(2.11)

which implies that , so by the definition of .

Now, assume that , and . By (1.7), we also have

(2.12)

So,

(2.13)

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

(2.14)

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

(2.15)

for , where .

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

(2.16)

Next, we will show that, for any ,

(2.17)

In fact, for any ,

(2.18)

which shows that (2.17) holds.

Substituting (2.17) into (2.4), it is easy to get

(2.19)

Let

(2.20)

There are two cases to consider.

Case 1 ().

We have by (2.6), (2.16), and (2.19)

(2.21)

Since

(2.22)

we have

(2.23)

Case 2 ().

In this case, there exists such that

(2.24)

Therefore, we may choose a real number such that

(2.25)

Notice that

(2.26)

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

(2.27)

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

(2.28)

We will prove that implies that

(2.29)

To this end, we first prove that

(2.30)

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

(2.31)

In fact,

(2.32)

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

(2.33)

And so

(2.34)

In general, we have

(2.35)

Now for ,

(2.36)

And so

(2.37)

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

(2.38)

Thus we only need to prove that

(2.39)

For any and we claim that

(2.40)

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

(2.41)

by (2.40) we have

(2.42)

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

(2.43)

Next, we will prove that

(2.44)

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

(2.45)

By the proof of Theorem 1.4, we have

(2.46)

Let

(2.47)

then . It suffices to show that

(2.48)

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

(2.49)

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

(2.50)

By Lemma 2.1 and (2.49) we have

(2.51)

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

(3.1)

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

(3.2)

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

(3.3)

or equivalently

(3.4)

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

(3.5)

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

(3.6)

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

(3.7)

And hence

(3.8)

(3.9)

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

(3.10)

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

(3.11)

Furthermore, we have, by the definition of ,

(3.12)

Define the sequence as follows:

(3.13)

Then we have by (3.5)

(3.14)

which implies that is a non-decreasing sequence and does not tend to zero as .

Repeating the above argument, we find that, for

(3.15)

By the definition of , we know that as , so as . The proof is complete.