Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 238068, 6 pages
doi:10.1155/2008/238068
Research Article
Stability of Solutions for a Family of
Nonlinear Difference Equations
Taixiang Sun,1 Hongjian Xi,1, 2 and Caihong Han1
1
College of Mathematics and Information Science, Guangxi University, Nanning,
Guangxi 530004, China
2
Department of Mathematics, Guangxi College of Finance and Economics, Nanning,
Guangxi 530003, China
Correspondence should be addressed to Taixiang Sun, stx1963@163.com
Received 8 September 2007; Accepted 31 January 2008
Recommended by Mariella Cecchi
We consider the family of nonlinear difference equations: xn1 3i1 fi xn , . . . , xn−k f4 xn , . . . ,
5
xn−k f5 xn , . . . , xn−k /f1 xn , . . . , xn−k f2 xn , . . . , xn−k i3 fi xn , . . . , xn−k , n 0, 1, . . . , where
fi ∈ C0, ∞k1 , 0, ∞, for i ∈ {1, 2, 4, 5}, f3 ∈ C0, ∞k1 , 0, ∞, k ∈ {1, 2, . . . } and the
initial values x−k , x−k1 , . . . , x0 ∈ 0, ∞. We give sufficient conditions under which the unique
equilibrium x 1 of these equations is globally asymptotically stable, which extends and includes
corresponding results obtained in the cited references.
Copyright q 2008 Taixiang Sun 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
In 1, Papaschinopoulos and Schinas investigated the global asymptotic stability of the following nonlinear difference equation:
xn1 x
i∈ Zk −{j−1,j} n−i
i∈ Zk
xn−j xn−j1 1
xn−i
,
n 0, 1, . . . ,
1.1
where k ∈ {1, 2, 3, . . .}, {j, j − 1} ⊂ Zk ≡ {0, 1, . . . , k}, and the initial values x−k , x−k1 , . . . , x0 ∈
R ≡ 0, ∞.
Moreover, Kruse and Nesemann 2 studied the global asymptotic stability of the unique
equilibrium of a discrete dynamical system, and as a special result they proved that the unique
equilibrium x 1 of the Putnam difference equation
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Advances in Difference Equations
xn1 xn xn−1 xn−2 xn−3
,
xn xn−1 xn−2 xn−3
n 0, 1, . . . ,
1.2
is globally asymptotically stable, where the initial values x−3 , x−2 , x−1 , x0 ∈ R .
In 3, Çinar et al. investigated the global asymptotic stability of the following nonlinear
difference equation:
xn1 k
xn−i 1
,
xn xn−1 xn ki2 xn−i
xn
i1
n 0, 1, . . . ,
1.3
where k ∈ {1, 2, 3, . . .} and the initial values x−k , x−k1 , . . . , x0 ∈ R . For closely related results,
see 4–10.
In this paper, we consider the family of nonlinear difference equations:
3
xn1 i1 fi xn , . . . , xn−k f4 xn , . . . , xn−k f5 xn , . . . , xn−k ,
f1 xn , . . . , xn−k f2 xn , . . . , xn−k 5i3 fi xn , . . . , xn−k n 0, 1, . . . ,
1.4
where fi ∈ C0, ∞k1 , 0, ∞, for i ∈ {1, 2, 4, 5}, f3 ∈ C0, ∞k1 , 0, ∞, k ∈ {1, 2, . . .},
and the initial values x−k , x−k1 , . . . , x0 ∈ 0, ∞. Our main result is the following theorem.
Theorem 1.1. Let u∗ max{u, 1/u}, for any u ∈ R . If fi u0 , u1 , . . . , uk ∗ ≤ max{u∗0 , u∗1 , . . . ,
u∗k }, for i 1, 2, 4, 5, then x 1 is the unique positive equilibrium of 1.4 which is globally asymptotically stable.
2. The proof of Theorem 1.1
In this section, we will prove Theorem 1.1. To do this, we need the following lemma.
Lemma 2.1. Let a, b, c, d ∈ R4 − {1, 1, 1, 1}, e ∈ 0, ∞, and α max{a∗ , b∗ , c∗ , d∗ }. Then,
1 c d e ab
<
< α.
α cd e a b
2.1
Proof. Since a, b, c, d ∈ R4 − {1, 1, 1, 1}, e ∈ 0, ∞, and α max{a∗ , b∗ , c∗ , d∗ }, we have
α > 1 and either α ≥ β > 1/α or α > β ≥ 1/α, for every β ∈ {a, b, c, d}. If c < 1 or d < 1, then
αcd αa αb αe > ab c d e.
2.2
c d e ab
< α.
cd e a b
2.3
It follows that
If c ≥ 1 and d ≥ 1, then α ≥ c > 1 or α > c ≥ 1 and α ≥ d > 1 or α > d ≥ 1. Thus, we have
the following inequalities:
αa b ≥ 2ab,
αcd αa ≥ αc 1 > 2c,
αcd αb ≥ αd 1 > 2d.
2.4
Taixiang Sun et al.
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It follows from 2.4 that
αcd αa αb αe > ab c d e,
2.5
c d e ab
< α.
cd e a b
2.6
which implies
By the symmetry, we have also that
1 c d e ab
<
.
α cd e a b
2.7
This completes the proof.
Proof of Theorem 1.1. Let {xn }∞
n−k be a positive solution of 1.4 with the initial values x−k ,
x−k1 , . . . , x0 ∈ R . For any n > 0, write
∗
∗
, . . . , xn−k
}.
pn max{xn∗ , xn−1
2.8
From Lemma 2.1, it follows that for any n ≥ 0,
fi xn , . . . , xn−k f4 xn , . . . , xn−k f5 xn , . . . , xn−k
xn1 f1 xn , . . . , xn−k f2 xn , . . . , xn−k 5i3 fi xn , . . . , xn−k
∗
: i 1, 2, 4, 5
≤ max fi xn , . . . , xn−k
∗
≤ max xn−i
: 0 ≤ i ≤ k pn ,
3 i1 fi xn , . . . , xn−k f4 xn , . . . , xn−k f5 xn , . . . , xn−k
xn 1 f1 xn , . . . , xn−k f2 xn , . . . , xn−k 5i3 fi xn , . . . , xn−k
3
i1
≥
≥
max
2.9
1
∗
fi xn , . . . , xn−k
: i 1, 2, 4, 5
1
1
∗
.
p
max xn−i : 0 ≤ i ≤ k
n
By 2.9, we have that for any n ≥ 0,
∗
≤ pn ,
1 ≤ xn1
pn1 ≤ pn .
2.10
From 2.10, we may assume that
lim pn M ≥ 1.
2.11
1
≤ lim infxn ≤ lim supxn ≤ M.
n→∞
M n→∞
2.12
n→∞
Then,
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Advances in Difference Equations
∗
∗
Since pn max{xn∗ , xn−1
, . . . , xn−k
}, there exists a sequence ls → ∞ such that
lim xls M
2.13
1
.
M
2.14
s→∞
or
lim xls s→∞
We may suppose by taking a subsequence that for 1 ≤ i ≤ k 1,
lim xls −i Mi .
s→∞
2.15
From 2.12, it follows that 1/M ≤ Mi ≤ M.
We claim that M 1. Indeed, if M > 1, then fi M1 , . . . , Mk1 /
1, for some i ∈ {1, 2, 4, 5}.
If lims→∞ xls M, then it follows from Lemma 2.1 and 1.4 that
fi M1 , . . . , Mk1 f4 M1 , . . . , Mk1 f5 M1 , . . . , Mk1
M f1 M1 , . . . , Mk1 f2 M1 , . . . , Mk1 5i3 fi M1 , . . . , Mk1
∗
: i 1, 2, 4, 5
< max fi M1 , . . . , Mk1
≤ max Mi : 1 ≤ i ≤ k 1 ≤ M,
3
i1
2.16
which is a contradiction.
If lims→∞ xls 1/M, then it follows from Lemma 2.1 and 1.4 that
3
fi M1 , . . . , Mk1 f4 M1 , . . . , Mk1 f5 M1 , . . . , Mk1
1
i1
M f1 M1 , . . . , Mk1 f2 M1 , . . . , Mk1 5i3 fi M1 , . . . , Mk1
>
≥
max
1
∗
fi M1 , . . . , Mk1
: i 1, 2, 4, 5
2.17
1
1
,
≥
M
max Mi : 1 ≤ i ≤ k 1
which is a contradiction. This completes the proof of the claim.
By 1.4 and 2.12, it follows that limn→∞ xn 1 and
3
1
f i 1, . . . , 1 f4 1, . . . , 1f5 1, . . . , 1
.
f1 1, . . . , 1f2 1, . . . , 1 5i3 fi 1, . . . , 1
i1
2.18
Thus, x 1 is the unique positive equilibrium of 1.4.
For any 0 < ε < 1, choose δ ε/ε 1 and let {xn }∞
n−k be a solution of 1.4 with the
initial values x−k , x−k1 , . . . , x0 ∈ 1 − δ, 1 δ. Then, for any −k ≤ i ≤ 0, we have that xi < 1 ε
and 1/xi < 1/1 − δ 1 ε. By 2.9 it follows that for any n ≥ 0,
1−ε <
1
1
≤
≤ xn1 ≤ pn ≤ p0 < 1 ε,
p0 pn
which implies that x 1 is globally asymptotically stable. This completes the proof.
2.19
Taixiang Sun et al.
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3. Example
In this section, we will give an application of Theorem 1.1.
Example 3.1. Consider the following equation:
xn−i xn−j g xn , . . . , xn−k xn−s xn−t
xn1 ,
xn−i xn−j g xn , . . . , xn−k xn−s xn−t
n 0, 1, . . . ,
3.1
where k ∈ {1, 2, . . .}, i, j, s, t ∈ {0, 1 . . . , k}, the initial conditions x−k , x−k1 , . . . , x0 ∈ R , and
g ∈ C0, ∞k1 , 0, ∞. Then, x 1 is the unique positive equilibrium of 3.1 which is
globally asymptotically stable.
Proof. Let f1 u0 , u1 , . . . , uk ui , f2 u0 , u1 , . . . , uk uj , f3 u0 , u1 , . . . , uk gu0 , u1 , . . . , uk ,
f4 u0 , u1 , . . . , uk us , and f5 u0 , u1 , . . . , uk ut . It is easy to verify that fi u0 , u1 , . . . , uk ∗ ≤
max{u∗0 , u∗1 , . . . , u∗k }, for i 1, 2, 4, 5. By Theorem 1.1, we know that x 1 is the unique positive
equilibrium of 3.1 which is globally asymptotically stable.
Remark 3.2. Let k ≥ 3, f1 u0 , u1 , . . . , uk 1, f2 u0 , u1 , . . . , uk ut , for some t ∈ Zk − {j − 1, j},
f3 u0 , u1 , . . . , uk i∈
u , f u , u , . . . , uk uj−1 , and f5 u0 , u1 , . . . , uk uj . Then,
Zk −{j−1,j,t} i 4 0 1
1.4 is 1.1, since fi u0 , u1 , . . . , uk ∗ ≤ max{u∗0 , u∗1 , . . . , u∗k }, for i 1, 2, 4, 5. By Theorem 1.1,
we know that the unique positive equilibrium x 1 of 1.1 is globally asymptotically stable.
Remark 3.3. Let k 3, f1 u0 , u1 , u2 , u3 u0 , f2 u0 , u1 , u2 , u3 u1 , f3 u0 , u1 , u2 , u3 0,
f4 u0 , u1 , u2 , u3 u2 , and f5 u0 , u1 , u2 , u3 u3 . Then, 1.4 is 1.2, since fi u0 , u1 , . . . , uk ∗ ≤
max{u∗0 , u∗1 , . . . , u∗k }, for i 1, 2, 4, 5. By Theorem 1.1, we know that the unique positive equilibrium x 1 of 1.2 is globally asymptotically stable.
Remark 3.4. Let f1 u0 , u1 , . . . , uk 1/u0 , f2 u0 , u1 , . . . , uk u1 , f3 u0 , u1 , . . . , uk u2 · · ·uk−1 ,
f4 u0 , u1 , . . . , uk uk , and f5 u0 , u1 , . . . , uk 1. Then, 1.4 is 1.3, since fi u0 , u1 , . . . , uk ∗
≤ max{u∗0 , u∗1 , · · · , u∗k }, for i 1, 2, 4, 5. By Theorem 1.1, we know that the unique positive equilibrium x 1 of 1.3 is globally asymptotically stable.
Acknowledgements
The project was supported by NSFC 10461001 and NFSGX 0640205, 0728002.
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Advances in Difference Equations
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