Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 506373, 12 pages
doi:10.1155/2011/506373
Research Article
Global Attractivity of a Family of Max-Type
Difference Equations
Xiaofan Yang,1 Wanping Liu,1 and Jiming Liu2
1
2
College of Computer Science, Chongqing University, Chongqing 400044, China
Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Correspondence should be addressed to Wanping Liu, wanping liu@yahoo.cn
Received 20 October 2010; Accepted 13 January 2011
Academic Editor: Ibrahim Yalcinkaya
Copyright q 2011 Xiaofan Yang 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.
We propose to study a generalized family of max-type difference equations and then prove the
global attractivity of a particular case of it under some parameter conditions. Through some
numerical results of other cases, we finally pose a generic conjecture.
1. Introduction
The study of max-type difference equations is a hotspot in the area of discrete dynamics
because such equations are often closely related to automatic control theory and competitive
dynamics. For recent advances in this direction see 1–8 and the references therein.
Motivated by 9, Liu et al. 10 studied the following nonautonomous max-type
difference equation:
yn p ryn−s
,
q φn yn−1 , . . . , yn−m yn−s
n ∈ N0 ,
1.1
where p ≥ 0, r, q > 0, s, m ∈ N, and φn : R m → R , n ∈ N0 are mappings satisfying
the condition β min{x1 , . . . , xm } ≤ φn x1 , x2 , . . . , xm ≤ β max{x1 , . . . , xm }, for some fixed β ∈
0, ∞. When p 0, β ∈ 0, 1, they proved that every positive solution to 1.1 converges to
zero if r ≤ q, while r
− q/1 β if r > q. If p > 0 and rq ≥ p, then each positive solution
to 1.1 converges to q − r2 4p1 β − q − r/21 β, for some β ∈ 0, ∞, except
for the case q < r, β ∈ β0 , ∞, where β0 4p/q − r2 1. Note that the behavior of positive
solutions to 1.1 for the case q < r, β ∈ β0 , ∞, is still an unsolved open problem as was
mentioned in 10.
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Discrete Dynamics in Nature and Society
Here, we propose to investigate the asymptotic behavior of positive solutions to the
generalized family of max-type difference equations
pi ri xn−s
,
xn max
1≤i≤k qi xn−s fi xn−1 , . . . , xn−m n ∈ N0 ,
1.2
where pi ≥ 0, ri , qi > 0, s, m, k ∈ N, k ≥ 2 and the functions fi : 0, ∞m → 0, ∞,
i 1, 2, . . . , k satisfy the condition
β min{u1 , . . . , um } ≤ fi u1 , u2 , . . . , um ≤ β max{u1 , . . . , um },
1.3
for some fixed β ∈ 0, 1.
In this paper, we mainly consider the particular case that all pi are zero, and then
obviously 1.2 reduces to the following form:
xn xn−s × max
1≤i≤k
ri
,
qi xn−s fi xn−1 , . . . , xn−m n ∈ N0 .
1.4
Let x∗ be a nonnegative equilibrium point of 1.4, then we have
ri
.
x x × max
1≤i≤k qi 1 β x∗
∗
∗
1.5
It follows directly from 1.5 that if 0 < ri ≤ qi for all i 1, 2, . . . , k, then 1.4 has the unique
nonnegative equilibrium x∗ 0, while if there exists at least one j ∈ {1, 2, . . . , k} such that
rj > qj , then 1.4 has a zero equilibrium x∗ 0 and a unique positive equilibrium x∗ max1≤i≤k {ri − qi }/1 β.
Finally, the following two beautiful theorems are derived.
Theorem 1.1. Consider 1.4 with condition 1.3. If 0 < ri ≤ qi for all i 1, 2, . . . , k, then every
positive solution to 1.4 converges to the unique nonnegative equilibrium zero.
Theorem 1.2. Consider 1.4 with positive initial values and positive ri and qi . Let fi : 0, ∞m →
0, ∞ be functions such that for some fixed β ∈ 0, 1, there hold
β min{u1 , . . . , um } ≤ fi u1 , . . . , um ≤ β max{u1 , . . . , um },
i 1, 2, . . . , k.
1.6
If there exists at least one j ∈ {1, 2, . . . , k} such that rj > qj , then the unique positive equilibrium of
1.4 is a global attractor.
2. Preliminary Lemmas
For the purpose of establishing the main results, some auxiliary lemmas are essential.
Discrete Dynamics in Nature and Society
3
Lemma 2.1. Consider the first-order difference equation
xn xn−1 × max
1≤i≤k
ri
,
qi xn−1
n ∈ N0 ,
2.1
with positive initial value x−1 and positive ri and qi . If there exists at least one j ∈ {1, 2, . . . , k} such
that rj > qj , then
lim xn max ri − qi : i 1, 2, . . . , k .
n→∞
2.2
Proof. Suppose that max{ri − qi : i 1, 2, . . . , k} rτ − qτ , which is positive, for some τ ∈
{1, 2, . . . , k}. By making the variable change xn rτ − qτ yn into 2.1 and then canceling the
positive term rτ − qτ from the resulting equation, we can derive
ri
yn yn−1 × max
,
1≤i≤k qi rτ − qτ yn−1
n ∈ N0 .
2.3
Note that min{a1 /b1 , a2 /b2 } ≤ a1 a2 /b1 b2 ≤ max{a1 /b1 , a2 /b2 } for ai , bi > 0, i 1, 2.
Then it follows from 2.3 that
yn1
qi yn ri − qi yn
qi yn rτ − qτ yn
≤ max
≤ max yn , 1 .
max
1≤i≤k
1≤i≤k
qi rτ − qτ yn
qi rτ − qτ yn
2.4
In addition, the following two inequalities hold:
qτ yn − 1
ri yn
rτ yn
yn1 − 1 max
2.5
−1 ≥
−1
,
1≤i≤k qi rτ − qτ yn
qτ rτ − qτ yn
qτ rτ − qτ yn
rτ − qτ yn 1 − yn
ri yn
rτ yn
.
yn1 − yn max
− yn ≥
− yn 1≤i≤k qi rτ − qτ yn
qτ rτ − qτ yn
qτ rτ − qτ yn
2.6
In the following, we are confronted with three possibilities.
Case 1. If there exists n0 ≥ −1 such that yn0 1, then it follows from 2.4 and 2.5 that yn 1
holds for all n ≥ n0 .
Case 2. If there exists n0 ≥ −1 such that yn0 > 1, then it follows from 2.5 and 2.6 that
yn0 ≥ yn0 1 ≥ yn0 2 ≥ · · · > 1.
2.7
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Discrete Dynamics in Nature and Society
Thus there is a finite limit γ limn → ∞ yn ≥ 1. By taking the limits on both sides of 2.3 and
canceling the positive factor γ from the resulting equation, we obtain
1 max
ri
2.8
,
qi rτ − qτ γ
1≤i≤k
which implies γ 1. Because if γ > 1, then
1 max
ri
qi rτ − qτ γ
1≤i≤k
ri
< max
1≤i≤k qi rτ − qτ
1,
2.9
leading to a contradiction.
Case 3. If yn < 1 for all n ≥ −1, then it follows from 2.5 and 2.6 that
y−1 < y0 < y1 < · · · < yn < · · · < 1.
2.10
Therefore, the limit of yn exists, denoted by 0 < γ limn → ∞ yn ≤ 1. By taking the limits on
both sides of 2.3 and canceling the nonzero factor γ from the resulting equation, there hold
1 max
1≤i≤k
ri
2.11
,
qi rτ − qτ γ
which implies γ 1. Because if 0 < γ < 1, then
1 max
1≤i≤k
ri
qi rτ − qτ γ
ri
> max
1≤i≤k qi rτ − qτ
1,
2.12
which is a contradiction.
In either of the above three cases, we get limn → ∞ yn 1, implying limn → ∞ xn rτ − qτ .
From Lemma 2.1, we have the following result.
Lemma 2.2. Consider the s-order difference equation
ri
xn xn−s × max
,
1≤i≤k qi xn−s
n ∈ N0 ,
2.13
with positive initial values and ri , qi > 0. If there exists at least one j ∈ {1, 2, . . . , k} such that rj > qj ,
then
lim xn max ri − qi : i 1, 2, . . . , k .
n→∞
2.14
Discrete Dynamics in Nature and Society
5
Proof. Let {xn }n≥−s be an arbitrary positive solution to 2.13. Apparently we know that the
sequence {xn }n≥−s can be divided into s subsequences {xjsk }k≥0 , j −s, −s 1, . . . , −1, which
are, respectively, positive solutions to the first-order equation 2.1 with positive initial values
x−s , x−s1 , . . . , x−1 . According to Lemma 2.1, we derive limk → ∞ xjsk max{ri − qi : i 1, 2, . . . , k} for all j −s, −s 1, . . . , −1, which directly lead to limn → ∞ xn max{ri − qi :
i 1, 2, . . . , k}.
Lemma 2.3. Let a > b > 0, 0 < β < 1, and 0 < < 1 − β/1 βa − b. Define two sequences
{mk } and {Mk } in the following way:
M1 a − b,
, k 1, 2, . . . ,
m k M1 − β M k k
Mk M1 − β mk−1 −
, k 2, 3, . . . .
k − 1
2.15
Then limk → ∞ mk limk → ∞ Mk .
Proof. Observe that
M2 − M1 −β 1 − β a − b − β 1 < 0,
, k 1, 2, . . . ,
mk1 − mk β Mk − Mk1 kk 1
, k 2, 3, . . . .
Mk1 − Mk −β mk − mk−1 kk − 1
2.16
It follows by induction that {mk } is increasing and {Mk } is decreasing. Again by induction
we derive mk < a − b and Mk > 0, k 1, 2, . . . . Hence there are two finite limits ξ limk → ∞ mk
and η limk → ∞ Mk . By taking limits on both sides of 2.15, we derive
ξ a − b − βη,
η a − b − βξ,
2.17
which imply 1 − βξ − η 0. Therefore ξ η a − b/1 β.
3. Proofs of Main Theorems
In this section, we are in a position to prove the main theorems presented in Section 1.
Proof of Theorem 1.1. Note that for the case ri < qi , i 1, 2, . . . , k, the behavior of positive
solutions to 1.4 is quite simple. In this case, we have that
ri
xn ≤ xn−s × max
1≤i≤k qi
μxn−s ,
3.1
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Discrete Dynamics in Nature and Society
where μ max1≤i≤k {ri /qi } < 1. Easily the subsequences {xlsj }l∈N0 , j ∈ {0, 1, . . . , s − 1}
converge to zero, hence the sequence {xn } also converges to zero.
For the case ri ≤ qi , i 1, 2, . . . , k with at least one j ∈ {1, 2, . . . , k} such that rj qj , we
can obtain that
ri
xn ≤ xn−s × max
1≤i≤k qi
xn−s .
3.2
In this case, the subsequences {xlsj }l∈N0 , j 0, 1, . . . , s − 1 are all positive and nonincreasing,
thus they converge, respectively, to some nonnegative limits ψj : liml → ∞ xlsj , j 0, 1, . . . , s−1.
If we replace n in 1.4 by sl j, l ∈ N0 for an arbitrary fixed j ∈ {0, 1, . . . , s − 1} and let
l → ∞, we can get
ri
ψj ψj × max
,
1≤i≤k qi ψj fi ψv1 , . . . , ψvm
3.3
where vi ∈ {0, 1, . . . , s − 1}, i 1, . . . , m. Without loss of generality, assume that ψj /
0, then we
obtain that
1
rτ
,
qτ ψj fτ ψv1 , . . . , ψvm
3.4
with some fixed number τ ∈ {1, 2, . . . , k}. Because rτ ≤ qτ , then it follows from 3.4 that
qτ ψj fτ ψv1 , . . . , ψvm rτ ≤ qτ .
3.5
ψj fτ ψv1 , . . . , ψvm 0,
3.6
Therefore we have
leading to ψj 0, which is a contradiction. Hence we have that ψj 0, j 0, 1, . . . , s − 1, and
every positive solution to 1.4 converges to zero, if ri ≤ qi for all i 1, 2, . . . , k.
Proof of Theorem 1.2. Suppose that max{ri − qi : i 1, 2, . . . , k} rτ − qτ > 0 for some τ ∈
{1, 2, . . . , k}. Let be an arbitrary fixed real number with 0 < < 1 − β/1 βrτ − qτ .
Define two sequences {Mk } and {mk } in the way shown in 2.15 with a rτ , b qτ .
Let {xn } be an arbitrary positive solution to 1.4. Next, we proceed by proving two
claims.
Claim 1. There exists N1 ∈ N such that m1 − ≤ xn ≤ M1 for all n ≥ N1 .
Proof of Claim 1. Note that
ri
xn ≤ xn−s × max
,
1≤i≤k qi xn−s
n 0, 1, 2, . . . .
3.7
Discrete Dynamics in Nature and Society
7
Consider the following difference equation:
1
zn
1
zn−s
× max
1≤i≤k
ri
1
qi zn−s
,
n 0, 1, 2, . . . .
1
1
3.8
1
1
Let {zn } be a positive solution to 3.7 with the initial values z−1 x−1 , z−2 x−2 , . . . , z−s x−s .
Note that the mapping hx rx/q x is strictly increasing on the interval 0, ∞.
1
1
It follows by induction that xn ≤ zn for all n ≥ −s. By Lemma 2.2, we have limn → ∞ zn rτ − qτ M1 . Hence there is an integer N1 ∈ N such that xn ≤ M1 for n ≥ N1 .
Let t max{s, m}. Note that
ri
,
xn ≥ xn−s × max
1≤i≤k qi xn−s βM1 n ≥ N1 t.
3.9
Consider the difference equation
1
yn
1
yn−s
1
1
1
1
× max
1≤i≤k
ri
1
qi yn−s βM1 n ≥ N1 t,
,
3.10
1
with yN t−1 xN1 t−1 , yN t−2 xN1 t−2 , . . . , yN xN1 . Note the monotonicity of hx, it
1
1
1
follows by induction that xn ≥ yn for all n ≥ N1 . By Lemma 2.2, we get that limn → ∞ yn m1 .
Thus there exists an integer N1 ≥ N1 such that xn ≥ m1 − for all n ≥ N1 .
Working inductively, we will reach the following claim.
Claim 2. For every k ∈ N, there exists Nk ∈ N such that
mk −
≤ xn ≤ Mk ,
k
k
3.11
for all n ≥ Nk .
Proof of Claim 2. Obviously, the case k 1 follows directly from Claim 1. In the following, we
proceed by induction. Assume that the assertion is true for k ωω ≥ 1. Then it suffices to
prove the assertion is also true for k ω 1.
Note that
xn ≤ xn−s × max
1≤i≤k
ri
,
qi xn−s βmω − /ω
n ≥ Nω t.
3.12
Consider the difference equation
ω1
zn
ω1
zn−s
× max
1≤i≤k
ri
ω1
qi zn−s βmω − /ω
,
n ≥ Nω t,
3.13
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Discrete Dynamics in Nature and Society
ω1
ω1
ω1
with zNω t−1 xNω t−1 , zNω t−2 xNω t−2 , . . . , zNω
ω1
zn
xNω . Note the monotonicity of hx,
it follows by induction that xn ≤
for all n ≥ Nω . By Lemma 2.2, we have that
ω1
limn → ∞ zn
Mω1 . So there is an integer Nω1
∈ N such that xn ≤ Mω1 /ω 1
for all n ≥ Nω1 . Then note that
ri
,
xn ≥ xn−s × max
1≤i≤k qi xn−s βMω1 /ω 1
n ≥ Nω1
t.
3.14
Consider the following difference equation
ω1
yn
ω1
ω1 t−1
with yN ω1
yn−s
× max
1≤i≤k
ri
ω1
qi yn−s
ω1
ω1 t−2
xNω1
t−1 , zN βMω1 /ω 1
ω1
xNω1
t−2 , . . . , zN ω1
ω1
yn
,
n ≥ Nω1
t,
3.15
xNω1
. By the monotonicity of
hx, it follows by induction that xn ≥
for all n ≥ Nω1
. By Lemma 2.2, we have that
ω1
limn → ∞ yn
mω1 . So there is an integer Nω1 ≥ Nω1 such that xn ≥ mω1 − /ω 1 for
all n ≥ Nω1 .
From Claim 2, we derive
≤ lim xn ≤ lim xn ≤ lim Mk lim Mk .
lim mk lim mk −
n→∞
k→∞
k→∞
k→∞
k→∞
k
k
n→∞
3.16
This plus Lemma 2.3 leads to that
lim xn lim mk lim Mk n→∞
k→∞
k→∞
rτ − qτ
.
1β
3.17
4. Simulations and Future Work
In the previous section, we proved the global attractivity of 1.2 when all pi are zero. In this
section, we investigate the dynamic behavior of 1.2 provided that all pi are not zero. First,
it is trivial to confirm that when
all pi are not zero, 1.2 has the following unique positive
equilibrium point x∗ max1≤i≤k { qi − ri 2 4pi 1 β ri − qi }/21 β. In the following,
some numerical results are presented.
Experiment 1. Consider the first-order difference equation
xn max
0.2 0.6xn−1
rxn−1
,
,
0.6 xn−1 0.3xn−1 q xn−1 0.3xn−1
where r, q > 0 and the initial value x0 > 0. See Figures 1 and 2.
n ∈ N,
4.1
Discrete Dynamics in Nature and Society
9
0.7
0.4
0.38
0.65
0.36
0.6
0.34
0.32
0.55
0.3
0.28
0.5
0.26
0.45
0.24
0.4
0.22
0.2
0
10
20
30
40
50
0.35
0
a Initial value x0 0.2
0.9
0.8
0.88
0.75
0.86
0.7
0.84
0.65
0.82
0.6
0.8
0.55
0.78
10
20
30
30
40
50
40
50
√
26/13 ≈ 0.3922.
0.85
0
20
b Initial value x0 0.7
Figure 1: r 1, q 2; x∗ 0.5
10
40
50
0.76
0
a Initial value x0 0.5
10
20
30
b Initial value x0 0.9
Figure 2: r 2, q 1; x∗ 10/13 ≈ 0.7692.
Experiment 2. Consider the second-order difference equation
xn max
0.5 xn−2
0.8 rxn−2
,
,
1 xn−2 0.5xn−1 q xn−2 0.5xn−1
n ≥ 2,
4.2
where r, q > 0 and the initial values x0 , x1 > 0. See Figures 3 and 4.
Experiment 3. Consider the third-order difference equation
xn max
⎧
⎪
⎨
⎫
⎪
⎬
0.5 xn−3
3xn−3
,
,
⎪
⎩ 1 xn−3 0.9 x2 x2 /2 2 xn−3 0.9 x2 x2 /2 ⎪
⎭
n−2
n−2
n−1
n−1
n ≥ 3,
4.3
where the initial values x0 , x1 , x2 > 0. See Figure 5.
10
Discrete Dynamics in Nature and Society
0.65
0.8
0.6
0.75
0.55
0.7
0.5
0.65
0.6
0.45
0.55
0.4
0.5
0.35
0.45
0.3
0.4
0.25
0.35
0.2
0.3
0
10
20
30
40
50
0
a Initial values x0 0.2, x1 0.4
10
20
30
40
50
b Initial values x0 0.8, x1 0.3
Figure 3: r 1, q 2; x∗ √
3/3 ≈ 0.5774.
1.4
1.3
1.35
1.2
1.3
1.1
1.25
1
1.2
0.9
1.15
0.8
1.1
0.7
1.05
0.6
1
0.5
0
10
20
30
40
50
0
a Initial values x0 1.3, x1 1.0
10
20
30
40
50
b Initial values x0 0.5, x1 0.9
√
Figure 4: r 2, q 1; x∗ 5.8 1/3 ≈ 1.1361.
Inspired by this work and the results of 10, here we pose the following conjecture.
Conjecture 4.1. Consider 1.2 with nonnegative pi and positive ri and qi . Let fi : 0, ∞m →
0, ∞, i 1, 2, . . . , k be k functions such that for some fixed β ∈ 0, 1, there hold
β min{u1 , . . . , uk } ≤ fi u1 , . . . , uk ≤ β max{u1 , . . . , uk }.
4.4
If ri qi ≥ pi for all i 1, 2, . . . , k, then every positive solution to 1.2 converges to the equilibrium
point
1
x∗ max
2 1 β 1≤i≤k
qi − ri
2
4pi 1 β ri − qi .
4.5
Discrete Dynamics in Nature and Society
11
0.6
1
0.55
0.9
0.5
0.8
0.45
0.4
0.7
0.35
0.6
0.3
0.5
0.25
0.4
0.2
0.3
0.15
0.1
0.2
0
50
100
0
150
a Initial values x0 0.1, x1 0.3, x2 0.5
0.8
1.1
0.7
1
0.6
0.9
0.5
0.8
0.4
0.7
0.3
0.6
0.2
0.5
0.1
0
50
100
150
c Initial values x0 0.8, x1 0.3, x2 0.1
50
100
150
b Initial values x0 0.3, x1 0.7, x2 1.0
0.4
0
50
100
150
d Initial values x0 1.1, x1 0.8, x2 0.6
Figure 5: x∗ 10/19 ≈ 0.5263.
Acknowledgments
This work was financially supported by the Fundamental Research Funds for the Central
Universities no. CDJXS10181130, the National Natural Science Foundation of China no.
10771227, and the New Century Excellent Talent Project of China no. NCET-05-0759.
References
1 K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max
difference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193–1199,
2006.
2 G. Ladas, “On the recursive sequence xn max{A1 /xn−1 , A2 /xn−2 , . . . , Ap /xn−p },” Journal of Difference
Equations and Applications, vol. 2, no. 3, pp. 339–341, 1996.
3 W. Liu and X. Yang, “Quantitative bounds for positive solutions of a Stević difference equation,”
Discrete Dynamics in Nature and Society, vol. 2010, Article ID 235808, 14 pages, 2010.
p
p
4 S. Stević, “On the recursive sequence xn1 max{c, xn /xn−1 },” Applied Mathematics Letters, vol. 21, no.
8, pp. 791–796, 2008.
5 S. Stević, “Global stability of a difference equation with maximum,” Applied Mathematics and
Computation, vol. 210, no. 2, pp. 525–529, 2009.
6 S. Stević, “Boundedness character of a class of difference equations,” Nonlinear Analysis: Theory,
Methods & Applications, vol. 70, no. 2, pp. 839–848, 2009.
12
Discrete Dynamics in Nature and Society
7 S. Stević, “Global stability of a max-type difference equation,” Applied Mathematics and Computation,
vol. 216, no. 1, pp. 354–356, 2010.
8 I. Szalkai, “On the periodicity of the sequence xn1 max{A0 /xn , A1 /xn−1 , . . . , Ak /xn−k },” Journal of
Difference Equations and Applications, vol. 5, no. 1, pp. 25–29, 1999.
9 B. D. Iričanin, “Dynamics of a class of higher order difference equations,” Discrete Dynamics in Nature
and Society, vol. 2007, Article ID 73849, 6 pages, 2007.
10 W. Liu, X. Yang, and J. Cao, “On global attractivity of a class of nonautonomous difference equations,”
Discrete Dynamics in Nature and Society, vol. 2010, Article ID 364083, 13 pages, 2010.
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Stochastic Analysis
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International Journal of
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Discrete Dynamics in
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Optimization
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