Advances in Difference Equations Volume 2010, Article ID 461014,17pages doi:10.1155/2010/461014
Research Article
Explicit Conditions for Stability of Nonlinear
Scalar Delay Impulsive Difference Equation
Bo Zheng
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
Correspondence should be addressed to Bo Zheng,[email protected]
Received 20 March 2010; Accepted 2 June 2010
Academic Editor: Leonid Shaikhet
Copyrightq2010 Bo Zheng. 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.
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
LetR,N, andZbe the sets of real numbers, natural numbers, and integers, respectively. For anya, b∈Z, defineZa {a, a1, . . .}andZa, b {a, a1, . . . , b}whena≤b.
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 phenomena1 . 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 obtainedsee, e.g.,1–10 and references therein.
best of our knowledge, very little has been done with the stability of impulsive difference equationssee15,20 . 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:
Δxn fn, xn, n∈Z0, n /nj,
Δxnj
Ij
xnj
, j ∈Z1, 1.1
whereΔdenotes the forward difference operator defined byΔxn xn1−xn,f :
Z0− {nj}×S → R,Sis the set of all functionsφ :Z−k,0 → Rfor somek ∈ N,and
xn∈Sis defined byxnm xnmform∈Z−k,0,Ij :R → R,and 0≤n1< n2 <· · ·< nj< nj1<· · ·, withnj → ∞asj → ∞. By a solution of1.1, we mean a sequence{xn}of
real numbers which is defined for alln∈Zn0−kand satisfies1.1forn∈Zn0for some n0 ∈ Z0. It is easy to see that, for any givenn0 ∈ Z0and a given initial functionφ ∈ S, there is a unique solution of1.1, denoted byxn, n0, φsuch that
xn0m, n0, φ
φm, form∈Z−k,0. 1.2
We assume thatfn,0≡0 andIj0≡0, so thatxn≡0 is a solution of1.1, which we call
the zero solution.
Forφ∈S, define the norm ofφas
φmaxφ
j:j∈Z−k,0, 1.3
and for anyH >0, define
SH
φ∈S:φ< H. 1.4
Definition 1.1. The zero solution of1.1is stable, if, for anyε >0 andn0∈Z0, there exists a δδn0, >0 such thatφ∈Sδimplies that|xn, n0, φ|< εforn∈Zn0. Ifδis independent ofn0, we say that the zero solution of1.1is uniformly stable.
Definition 1.2. The zero solution of1.1is globally attractive, if every solution of1.1tends to zero asn → ∞.
A simple example of1.1is given by
Δxn Cnxn−k 0, n∈Z0, n /nj,
Δxnj
cjx
nj
, j ∈Z1, 1.5
where k ∈ N,C : Z0 → R, and{cj} is a sequence of real numbers. In15 , the author
studied the stability of the zero solution of1.5, whereCn ≥ 0 forn ∈ Z0− {nj}and
Theorem 1.3. If
n
in−k
Ci
nj∈Zi−k,i−1
1cj
−1≤ 3
2, forn∈Zk, 1.6
then the zero solution of 1.5is stable.
In this paper, we assume that there exists a positive constantHand a sequence{Pn} of nonnegative real numbers such that
PnMφ≥ −fn, φ≥ −PnM−φ, forn∈Z0, φ∈SH, 1.7
whereMφ max{0,maxi∈Z−k,0φi}. Furthermore, we assume that there is a sequence
{bj}of positive numbers withbj≤1 such that
bjx2≤x
xIjx
≤x2, forj∈Z1, |x|< H. 1.8
The main purpose of this paper is to establish the following theorems.
Theorem 1.4. Assume that1.7and1.8hold and
n
in−k
Pi
nj∈Zi−k,i−1
b−j1≤ 3 2
1
2k1, forn∈Zk. 1.9
Then the zero solution of 1.1is uniformly stable.
Remark 1.5. Theorem 1.4generalizes and improvesTheorem 1.3greatly.
The next theorem provides a sufficient condition for every solution of1.1tends to zero as n → ∞, that is, the zero solution of1.1is globally attractive.
Theorem 1.6. Assume that1.7and1.8hold and
n
in−k
Pi
nj∈Zi−k,i−1
b−j1≤α < 3 2
1
2k1, forn∈Zk, 1.10
and assume that, for each bounded solution{xn}, either
lim
n→ ∞xn>0, ∞
n0
fn, xn −∞, forn∈Zk, 1.11
or
lim
n→ ∞xn<0, ∞
n0
fn, xn ∞, forn∈Zk. 1.12
Remark 1.7. An example is worked out in Section 3 to verify that the upper bound 3/2
Corollary 1.8. Assume that1.8holds and
n
Then the zero solution of the equation
Δxn
For the sake of convenience, throughout this paper, we will use the convention
j
forj∈Z1. Then 1≤gjx≤1/bjforj ∈Z1if|x|< H. Equation1.1can be rewritten as
Δxn fn, xn, n∈Z0, n /nj,
xnj1
gj−1
xnj
xnj
, j∈Z1.
2.2
Set
yn xn
nj∈Z0,n−1
gj
xnj
, n∈Z−k, 2.3
then2.2reduces to
Δyn fn, xn
nj∈Z0,n−1
gj
xnj
, n∈Z0, n /nj,
Δynj
0, j∈Z1.
2.4
And it is easy to see that
|xn| ≤yn≤ |xn|
nj∈Z0,n−1
b−j1, forj∈Z1, n∈Z−k. 2.5
To prove Theorems1.4and1.6, we need the following lemma.
Lemma 2.1. Let{xn},n ∈ Zn0 −k,be a solution of 1.1,{yn}is defined by2.3,n∗ ∈ Zn02k2, andBn∗ max{|yn|:n∈Zn∗−3k−2, n∗−1}< H. If
n
in−k
Pi
nj∈Zi−k,i−1
b−j1≤c k2
2k1, forn∈Zk 2.6
holds for somec∈k2/2k1,1 and either
yn∗≥0, yn∗> yn∗−1 2.7
or
yn∗<0, yn∗< yn∗−1, 2.8
Proof. We assume thatyn∗≥0 andyn∗> yn∗−1. The case whereyn∗<0 andyn∗< yn∗−1is similar and is omitted.
It is easy to see thatLemma 2.1holds foryn∗ 0.
Ifk0, thenc1 andPn≤2 for alln∈Z0by2.6. Thus we only need to prove that|yn∗| ≤Bn∗. Sinceyn∗>0,yn∗> yn∗−1, by1.7we have
−Pn∗−1M−xn∗−1≤ −fn∗−1, xn∗−1<0, 2.9
that is,
−Pn∗−1max{0,−xn∗−1} ≤ −fn∗−1, xn∗−1<0. 2.10
Soxn∗−1<0 which admits thatyn∗−1<0. Hence
yn∗−yn∗−1 fn∗−1, xn∗−1
nj∈Z0,n∗−2
gj
xnj
≤Pn∗−1max{0,−xn∗−1}
nj∈Z0,n∗−2
gj
xnj
−Pn∗−1yn∗−1≤ −2yn∗−1,
2.11
which implies thatyn∗ ≤ −yn∗−1, so|yn∗| ≤ |yn∗−1| ≤ Bn∗by the definition of Bn∗.
Now, assume thatyn∗>0,yn∗> yn∗−1,andk≥1. By1.7, we also have
−Pn∗−1M−xn∗−1≤ −fn∗−1, xn∗−1<0. 2.12
So,
max
0, max
i∈Z−k,0−xn
∗−1i>0. 2.13
Δyn fn, xn of impulsive points, we always have
In fact, for anyl∈Z0, k,
which shows that2.17holds.
Substituting2.17into2.4, it is easy to get
There are two cases to consider.
So, by2.6,2.15,2.19, and2.25, we have
The proof is completed by combining Cases1and2.
Proof ofTheorem 1.4. By Lemma 2.1, 1.9implies that 2.6 holds with c 1. For any ε ∈
To this end, we first prove that
Ifnl1 n0, then|xnl11| |xn0 I0xn0| ≤ |xn0| < δ < . Ifnl1 > n0, then, for n∈Zn0, nl1, we claim that
|xn|< δ1βnl1−n0 < . 2.31
In fact,
|xn01| ≤ |xn0|fn0, xn0< δPn0δ≤δ
1β< ,
|xn02| ≤ |xn01|fn01, xn01
< δ1ββxn01 ≤δ
1ββδ1β
δ1β2< .
2.32
In general, we can obtain2.31by induction. And so|xn| < δ1βnl1−n0< < Hfor
either case withn∈Zn0, nl1. Forn∈Znl11, nl2, we have
|xnl11| ≤ |xnl1||Il1xnl1| ≤ |xnl1|< δ
1βnl1−n0
,
|xnl12| ≤ |xnl11|fnl11, xnl11
< δ1βnl1−n0βx
nl11
≤δ1βnl1−n0βδ1βnl1−n0
δ1βnl1−n01< .
2.33
And so
|xn|< δ1βnl2−n0−1, forn∈Zn
l11, nl2, 2.34
In general, we have
|xn|< δ1βnli1−n0−1, forn∈Zn
li1, nli1, i1,2, . . . , m−1. 2.35
Now forn∈Znlm1, n02k1,
|xnlm1| ≤ |xnlm||Ilmxnlm| ≤ |xnlm|< δ
1βnlm−n0−1,
|xnlm2| ≤ |xnlm1|fnlm1, xnlm1
< δ1βnlm−n0−1βx
nlm1
≤δ1βnlm−n0−1βδ1βnlm−n0−1
δ1βnlm−n0< .
And so
|xn|< δ1β2k−1, forn∈Znlm1, n02k1 2.37
which has proved that2.30holds. Now, we will prove that
|xn|< , forn∈Zn02k2. 2.38
Thus we only need to prove that
yn< , forn∈Zn02k2. 2.39
For anyn∗∈Zn02k2andBn∗< H,we claim that
yn∗≤Bn∗. 2.40
In fact, we can assume thatyn∗ ≥ 0. The case whereyn∗ < 0 is similar and is omitted. Ifyn∗ ≤ yn∗−1, by the definition ofBn∗,2.40holds. Ifyn∗ > yn∗ −1, then by Lemma 2.1we have that2.40holds.
Since
Bn02k2 maxyn:n∈Z−k, n02k1
< ε < H, 2.41
by2.40we have
yn02k2< ε. 2.42
By repeatedly using2.40we get that2.39holds.
Combining2.30and2.39, we find that2.29holds and the proof ofTheorem 1.4is complete.
Proof ofTheorem 1.6. In view ofTheorem 1.4, we see that the zero solution of1.1is uniformly
stable. Therefore, for anyn0∈Z0, there existsδ >0 such thatφ∈Sδimplies that
|xn|xn, n0, φ< 1
2H, forn∈Zn0. 2.43
Next, we will prove that
lim
n→ ∞xn 0. 2.44
Case 1. {xn}is eventually positive, that is, there existsn1 ∈Zn0such thatxn>0 for all n ∈ Zn1. Hence, by1.7and1.8, we haveΔxn ≤ 0 forn ∈Zn1k. That is,{xn} is eventually nonincreasing and hence limn→ ∞xn ≥ 0. Thus, by1.11we see that 2.44
holds. The case when{xn}is eventually negative is similar and will be omitted.
Case 2. {xn} is oscillatory in the sense that {xn} is neither eventually positive nor
eventually negative, so{yn}is also oscillatory. To prove2.44, we only need to prove that
lim
n→ ∞yn 0. 2.45
By the proof ofTheorem 1.4, we have
yn< 1
2H, forn∈Zn0. 2.46
Let
lim
n→ ∞supyn p, nlim→ ∞infyn q, 2.47
thenq≤0≤p. It suffices to show that
pq0. 2.48
In fact, if2.48does not hold, we assume thatp≥ −qandp >0. The case wherep <−q andq <0 is similar and is omitted.
Let 0 < ε0 < 1−c/2p, where c < 1 is given inLemma 2.1. Then there exists an integerm0∈Z2ksuch that
q−ε0 ≤yn≤pε0, forn∈Zm0. 2.49
Since limn→ ∞supyn p > 0 and{yn}is oscillatory, there must exist an integer m1∈Zm03k2such that
ym1> p−ε0, ym1> ym1−1. 2.50
ByLemma 2.1and2.49we have
p−ε0< ym1 ym1≤cBm1≤c
pε0
, forn∈Zm0. 2.51
3. Example
In this section we will give a result which guarantees that the upper bound 3/21/2k1 is best possible inTheorem 1.6.
Example 3.1. Consider the delay impulsive difference equation
Δxn Pnxn−k 0, n∈Z0, n / 6k1i,
Proof. Let{xn}be a solution of3.1with initial condition of the form
xn , forn∈Z−k,0, 3.6
where >0 is a given constant. Then by3.1and the definition of{Pn}, we have
xn 1b0ε, forn∈Z1, k1,
Δxn − 1
k11b0, forn∈Zk1,2k.
3.7
And hence
xn 1b0ε
!
2− n k1
"
, forn∈Zk1,2k1, 3.8
x2k2 x2k1−cxk1 1b0ε
!
1 k1 −c
"
. 3.9
By virtue of3.1and3.8we get
Δxn −1b0ε 1 k1
!
2−n−k k1
"
, forn∈Z2k2,3k1. 3.10
Summing up from 2k2 to 3k1 and using3.9, we have
x3k2 x2k2− 1b0 k1
3k1
n2k2
!
2−n−k k1
"
1b0
! 1
k1 −c
"
−1b0 k 2k1
−1b0
!
c k−2 2k1
"
def
≡ 1b01.
3.11
Furthermore, we have, by the definition of{Pn},
xn 1b01, forn∈Z3k2,6k1. 3.12
Define the sequence{i}as follows:
0, i1−i !
c k−2 2k1
"
Then we have by3.5
|i1||i|
c k−2
2k1
≥ |i|, fori∈Z0, 3.14
which implies that{|i|}is a non-decreasing sequence and does not tend to zero asn → ∞.
Repeating the above argument, we find that, forn∈Z3k2i6k1, i16k 1, i∈Z0,
xn 1b01b1· · ·1bii1. 3.15
By the definition of{bi}, we know thatni01bi0 asn → ∞, soxn0 asn → ∞.
The proof is complete.
Acknowledgments
The author would like to express her thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project2009.
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