Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 930410, 11 pages
doi:10.1155/2012/930410
Research Article
Global Attractivity of a Higher-Order
Difference Equation
R. Abo-Zeid
Department of Mathematics, College of Science & Arts-Rabigh, King Abdulaziz University,
Rabigh 21911, Saudi Arabia
Correspondence should be addressed to R. Abo-Zeid, abuzead73@yahoo.com
Received 7 May 2012; Accepted 8 July 2012
Academic Editor: M. De la Sen
Copyright q 2012 R. Abo-Zeid. 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.
The aim of this work is to investigate the global stability, periodic nature, oscillation, and
the boundedness of all admissible solutions of the difference equation xn1 Axn−2r−1 /B −
C kil xn−2i , n 0, 1, 2, . . . where A, B, C are positive real numbers and l, r, k are nonnegative
integers, such that l ≤ k.
1. Introduction and Preliminaries
Although some difference equations look very simple, it is extremely difficult to understand
thoroughly the global behaviors of their solutions. One can refer to 1, 2. The study of
nonlinear rational difference equations of higher order is of paramount importance, since we
still know so little about such equations. It is worthwhile to point out that although several
approaches have been developed for finding the global character of difference equations 2–
4, relatively a large number of difference equations have not been thoroughly understood
yet 5–8.
Aloqeili in 9 discussed the stability properties and semicycle behavior of the
solutions of the difference equation:
xn1 xn−1
,
a − xn xn−1
n 0, 1, 2, . . . ,
with real initial conditions and positive real number a.
1.1
2
Discrete Dynamics in Nature and Society
In 10, the authors investigated the global asymptotic stability of the difference
equation:
xn1 Axn−1
,
B C kil xn−2i
n 0, 1, 2, . . . ,
1.2
where A, B, C are nonnegative real numbers and l, k are nonnegative integers, such that l ≤ k.
Also in 11, they discussed the existence of unbounded solutions under certain
conditions of the difference equation:
k
xn1 A
il xn−2i−1
,
k−1
B C il xn−2i
n 0, 1, 2, . . . ,
1.3
where A, B, C are nonnegative real numbers and l, k are nonnegative integers, l < k
In 12, the global asymptotic stability of the difference equation:
xn1 Axn−2r−1
,
B Cxn−2l xn−2k
n 0, 1, 2, . . .
1.4
was discussed, where A, B, C are nonnegative real numbers and r, l, k are nonnegative
integers such that l ≤ k and r ≤ k.
In 13, the global stability and periodic nature of the solutions of the difference
equations:
xn1 xn−2
,
±1 xn xn−1 xn−2
n 0, 1, 2, . . .
1.5
were discussed, where the initial conditions x−2 , x−1 , x0 are real numbers.
In 14, we discussed the oscillation, boundedness, and the global behavior of all
admissible solutions of the difference equation:
xn1 Axn−1
,
B − Cxn xn−2
n 0, 1, 2, . . . ,
1.6
where A, B, C are positive real numbers.
In this paper, we study the global asymptotic stability of the difference equation
xn1 Axn−2r−1
,
B − C kil xn−2i
n 0, 1, 2, . . . ,
1.7
where A, B, C are nonnegative real numbers and l, r, k are nonnegative integers, such that
l ≤ k.
Discrete Dynamics in Nature and Society
3
Consider the difference equation:
xn1 fxn , xn−1 , . . . , xn−k ,
n 0, 1, 2 . . . ,
1.8
where f : Rk1 → R. An equilibrium point for 1.8 is a point x ∈ R such that x fx, x, . . . , x.
1 An equilibrium point x for 1.8 is called locally stable if for every > 0, ∃δ > 0
such that every solution {xn } with initial conditions x−k , x−k1 , . . . , x0 ∈x − δ, x δ
is such that xn ∈x − , x , ∀n ∈ N. Otherwise x is said to be unstable.
2 The equilibrium point x of 1.8 is called locally asymptotically stable if it is locally
stable and there exists γ > 0 such that for any initial conditions x−k , x−k1 , . . . , x0 ∈
x − γ, x γ, the corresponding solution {xn } tends to x.
3 An equilibrium point x for 1.8 is called global attractor if every solution {xn }
converges to x as n → ∞.
4 The equilibrium point x for 1.8 is called globally asymptotically stable if it is
locally asymptotically stable and global attractor.
The linearized equation associated with 1.8 is
yn1 k
∂f
x, . . . , xyn−i ,
∂x
n−i
i0
n 0, 1, 2, . . . .
1.9
The characteristic equation associated with 1.9 is
λk1 −
k
∂f
x, . . . , xλk−i 0.
∂x
n−i
i0
1.10
Theorem 1.1 see 2. Assume that f is a C1 function and let x be an equilibrium point of 1.8.
Then the following statements are true.
1 If all roots of 1.10 lie in the open disk |λ| < 1, then x is locally asymptotically stable.
2 If at least one root of 1.10 has absolute value greater than one, then x is unstable.
2. Linearized Stability Analysis
Consider the difference equation:
xn1 Axn−2r−1
,
B − C kil xn−2i
n 0, 1, 2, . . . ,
2.1
where A, B, C are nonnegative real numbers and l, r, k are nonnegative integers, such that
l ≤ k.
4
Discrete Dynamics in Nature and Society
The change of variables xn yn1 k−l1
B/Cyn reduces 1.7 to the difference equation:
γyn−2r−1
,
1 − kil yn−2i
n 0, 1, 2, . . . ,
2.2
where γ A/B.
Now we determine the equilibrium points of 2.2 and discuss their local asymptotic
behavior. It is clear that the values of the equilibrium points depend on whether k − l is even
or odd.
When k − l is odd, we have the equilibrium points y 0 and y ± k−l1 1 − γ if γ < 1
and y 0 only if γ ≥ 1.
When k − l is even, we have the equilibrium points y 0 and y k−l1 1 − γ.
Now assume that K max{2k, 2r 1}.
The linearized equation associated with 2.2 about y is
zn1 −
γ
1−y
γy k−l1
z
−
k−l1 n−2r−1
1 − y k−l1
k
2 zn−2i 0,
n 0, 1, 2, . . . .
il
2.3
The characteristic equation associated with this equation is
λK1 −
γy k−l1
γ
1−y
λK−2r−1 − k−l1
1−y
k−l1
2
k
λK−2i 0.
il
2.4
We summarize the results of this section in the following two theorems.
Theorem 2.1. Assume that K 2k > 2r 1. Then the following statements are true.
1 The zero equilibrium point is locally asymptotically stable if γ < 1 and unstable (saddle
point) if γ > 1.
2 When k − l is even, the equilibrium point y k−l1 1 − γ is unstable if γ < 1 and unstable
(saddle point) if γ > 1.
3 When k − l is odd, the equilibrium points y ± k−l1 1 − γ are unstable.
Proof. 1 The linearized equation 2.3 about y 0 is
zn1 − γzn−2r−1 0,
n 0, 1, 2, . . . .
2.5
The characteristic equation associated with this equation is
λ2k1 − γλ2k−2r−1 0.
√
So λ 0, λ ± 2r2 γ. Therefore the result follows.
2.6
Discrete Dynamics in Nature and Society
5
2 Suppose that k − l is even. The linearized equation 2.3 about y zn1 − zn−2r−1 −
k
1
zn−2i 0,
1−γ
γ
il
k−l1
n 0, 1, 2, . . . .
1 − γ is
2.7
The associated characteristic equation 2.4 becomes
k
1
λ2k−2i 0.
1−γ
γ
il
λ2k1 − λ2k−2r−1 −
2.8
Let
fλ λ2k1 − λ2k−2r−1 −
k
1
λ2k−2i 0.
1−γ
γ
il
2.9
We can see that fλ has a real root in 1, ∞ if γ < 1 and when γ > 1, fλ has a root in
1, ∞ and some roots with |λ| < 1. Therefore the result follows.
3 Whenk − l is odd, fλ has a root in 1, ∞ and some roots with |λ| < 1, if γ < 1.
Therefore y ± 1 − γ are unstable.
Theorem 2.2. Assume that K 2r 1 > 2k. Then the following statements are true.
1 The zero equilibrium point is locally asymptotically stable if γ < 1 and a source if γ > 1.
2 If k − l is even, then the equilibrium point y k−l1 1 − γ is unstable (saddle point).
3 If k − l is odd, then the equilibrium points y ± k−l1 1 − γ are unstable (saddle points).
Proof. It is sufficient to consider the linearized equation about y:
zn1 −
γy k−l1
γ
1−y
z
−
k−l1 n−2r−1
1 − yk−l1
k
2 zn−2i 0,
n 0, 1, 2, . . .
il
2.10
and its associated characteristic equation:
λ2r2 −
γ
1 − y k−l1
γy k−l1
−
1 − y k−l1
2
k
λ2r1−2i 0.
il
2.11
3. Oscillation
Let t be the largest nonnegative integer such that 0 < 2t 1 ≤ K and let s be the largest
nonnegative integer such that 0 ≤ 2s ≤ K.
6
Discrete Dynamics in Nature and Society
Theorem 3.1. Assume that γ < 1. Then the interval − k−l1 1 − γ, k−l1 1 − γ is an invariant interval
for 2.2.
Proof. The proofis by induction. Suppose that y−i ∈ − k−l1 1 − γ, k−l1 1 − γ, i 0, 1, . . . , K.
Hence |y−i | < k−l1 1 − γ, i 0, 1, . . . , K.
This implies that | kil y−2i | < 1 − γ. Then
γ y−2r−1 γ y−2r−1 y1 ≤
< y−2r−1 ,
1 − kil y−2i 1 − kil y−2i γ y−2r γ y−2r y2 ≤
< y−2r .
k
k
1 − il y−2i1 1 − il y−2i1 If for a certain n0 ∈ N we have yn0 −K , yn0 −K1 , . . . , yn0 ∈ − k−l1 1 − γ,
k−l1
γ yn− 2r−1 γ yn0 −2r−1 ≤
< yn0 −2r−1 <
0 1
k
k
1 − il y−2i 1 − il y−2i yn
3.1
1 − γ, then
k−l1
1 − γ.
3.2
This completes the proof.
∞
Corollary
of 2.2 such that either y−K , y−K1 , . . . , y−1 , y0 ∈
3.2. Assume
that {yn }n−K be a solution
∞
k−l1
k−l1
1 − γ or−
1 − γ, 0. Then {yn }∞
0,
n−K is positive (or negative). Moreover, {yn }n−K
converges to the zero equilibrium point.
Theorem 3.3. Let {yn }∞
n−K be a nontrivial solution of 2.2 such that either
C1 − k−l1 1 − γ < y−2t−1 , y−2t1 , . . . , y−1 < 0 < y−2s , y−2s2 , . . . , y0 <
k−l1
1−γ
or
C2 − k−l1 1 − γ < y−2s , y−2s2 , . . . , y0 < 0 < y−2t−1 , y−2t1 , . . . , y−1 < k−l1 1 − γ is satisfied.
Then {yn }∞
n−K oscillates about y 0 with semicycles of length one. Moreover y2r1n2j <
or >y2r1n−12j and y2r1n2j1 > or < y2r1n−12j1 , j 1, 2, . . . , r 1 and
n 0, 1, 2, . . ..
Proof. Assume that condition C1 is satisfied. Then we have y1 γy−2r−1 /1 − kil y−2i >
y−2r−1 , and y2 γy−2r /1 − kil y−2i1 < y−2r .
By induction we get 0 > y2r1n2j1 > y2r1n−12j1 , and 0 < y2r1n2j <
y2r1n−12j , n 0, 1, 2, . . ..
If condition C2 is satisfied, the result is similar and will be omitted.
Discrete Dynamics in Nature and Society
7
4. Global Behavior of 2.2
Theorem 4.1. The following statements are true.
1 If γ < 1, then
the zero equilibrium point is a global attractor with basin
− k−l1 1 − γ, k−l1 1 − γK1 .
2 If γ 1, then 2.2 has prime period two solutions of the form . . . , 0, ϕ, 0, ϕ, 0, . . ., where
ϕ ∈ R.
3 If γ > 1, then there exist solutions which are neither bounded nor persist.
, . . . , y−1 , y
∈ − k−l1 1 − γ, k−l1 1 − γ. Then using
Proof. 1 Suppose that y−K , y−K1
0
Theorem 3.1, we have that yn ∈ − k−l1 1 − γ, k−l1 1 − γ, n ≥ 1.
Moreover, we have |yn1 | < |yn−2r−1 |, n 0, 1, 2, . . ..
That is, the subsequences {|y2r1nj |}∞
n−1 , j 1, 2, . . . , 2r 2 are decreasing.
From 2.2 we have
γ y2r1n−1j γ y2r1n−1j y2r1nj .
≤
1 − kil y2r1nj−2i−1 1 − kil y2r1nj−2i−1 4.1
Now suppose that |y2r1nj | → Lj as n → ∞, j 1, 2, . . . , 2r 2. Then the last
inequality implies that
γLj
,
Lj ≤ 1 − kil Lj−2i−1 j 1, 2, . . . , 2r 2.
If for a certain j ∈ {1, 2, . . . , 2r 2} we have Lj / 0, then |1 −
1γ ≥
k
Lj−2i−1 ≥ 1 − γ.
k
il Lj−2i−1 |
4.2
≤ γ. This implies that
4.3
il
This is a contradiction as the the subsequences {|y2r1nj |}∞
n−1 , j 1, 2, . . . , 2r 2 are
decreasing. Therefore, Lj 0, j 1, 2, . . . , 2r 2, and {yn }∞
n−K converges to zero.
2 Clear!
with initialconditions, |y−i | < k−l1 γ − 1 >
3 Let {yn }∞
n−K be a solution of 2.2
k−l1
γ 1, i 2s, 2s − 2, . . . , 2, 0, and |y−i | >k−l1 γ 1 < k−l1 γ − 1, i 2t − 1, 2t − 3, . .
. , 1.
We consider only the case |y−i | < k−l1 γ − 1, i 2s, 2s − 2, . . . 2, 0, and |y−i | > k−l1 γ 1,
i 2t − 1, 2t − 3, . . . , 1.
It follows that | kil y−2i | |y−2k ||y−2k2 | · · · |y0 | < γ − 1.
That is,
−γ 1 <
k
il
y−2i < γ − 1.
4.4
8
Discrete Dynamics in Nature and Society
This implies that −γ 2 < 1 −
k
il y−2i
< γ and so |1 −
k
il y−2i |
γy−2r−1 γ y−2r−1 y1 >
y−2r−1 >
γ
1 − kil y−2i k−l1
< γ. Hence we have
γ 1.
4.5
Also | kil y−2i1 | > γ 1 implies that γ 2 < 1− kil y−2i1 < −γ and so |1− kil y−2i1 | > γ.
Hence we have
γy−2r γ y−2r y2 <
y−2r <
γ
1 − kil y−2i1 γ − 1.
k−l1
4.6
By induction we get
y2r1n2j1 > y2r1n−12j1 >
y2r1n2j < y2r1n−12j <
γ 1,
k−l1
γ − 1,
4.7
k−l1
n ≥ 0 and j 0, 1, . . . , r. Now suppose that
y2r1n−12j1 −→ L2j1 ∈
γ 1, ∞ ,
k−l1
y2r1n−12j −→ L2j ∈ 0,
γ −1 ,
4.8
k−l1
as n → ∞, j 0, 1, . . . , r.
But as
γ y2r1n−12j γ y2r1n−12j y2r1n2j ≤
,
1 − kil y2r1n2j−2i−1 1 − kil y2r1nj−2i−1 4.9
then
γL2j
,
L2j ≤ 1 − kil L2j−2i−1 j 0, 1, . . . , r.
4.10
We claim that for each j 0, 1, . . . , r, L2j 0.
For the sake of contradiction suppose that there exists j ∈ {0, 1, . . . , r} with L2j ∈
0, k−l1 γ − 1.
Then 4.10 gives
k
L2j−2i−1 ≤ γ.
1 −
il
4.11
Discrete Dynamics in Nature and Society
9
10
8
6
4
2
0
−2
0
10
20
30
40
50
Figure 1: The difference equation yn1 0.6yn−1 /1 − yn yn−2 .
1
0.5
0
−0.5
−1
0
10
20
30
40
50
Figure 2: The difference equation yn1 0.7yn−3 /1 − yn yn−2 .
This implies that
−γ 1 ≤
k
4.12
L2j−2i−1 ≤ γ 1.
il
As L2j1 ∈ k−l1 γ 1, ∞, j 0, 1, . . . , r, we have a contradiction.
Thus it is true that for each j 0, 1, . . . , r we have L2j 0 and so limn → ∞ y2n 0.
We now claim that for each j 0, 1, . . . , r, L2j ∞.
For the sake of contradiction, suppose that there exists j ∈ {0, 1, . . . , r} with L2j1 ∈
k−l1 γ 1, ∞. Then
L2j1
γlimn → ∞ y2r1n−12j1 lim y2r1n−12j1 n→∞
1 − kil limn → ∞ y2r1n2j−2i γlimn → ∞ y2r1n−12j1 γL2j1
≥
γL2j1 .
k
1
limn → ∞ y2r1n2j−2i
1 k L2j
il
il
4.13
10
Discrete Dynamics in Nature and Society
×105
Global attractivity of a higher
10
8
6
4
2
0
0
10
20
30
40
Figure 3: The difference equation yn1 2yn−1 /1 − yn yn−2 .
2000
1500
1000
500
0
0
10
20
30
40
Figure 4: The difference equation yn1 2yn−3 /1 − yn yn−2 .
This is a contradiction. Therefore for each j 0, 1, . . . , r we have L2j1 ∞ and so
limn → ∞ y2n1 ∞.
The case when |y−i | > k−l1 γ 1, i 2s, 2s − 2, . . . , 2, 0 and |y−i | < k−l1 γ − 1, i 2t −
1, 2t − 3, . . . , 1 is similar and will be omitted.
5. Numerical Examples
Example 5.1. Figure 1 shows that if r 0, l 0, k 1 K max{2k, 2r 1} 2 and γ 0.6,
then the solution {yn }∞
n−2 with initial conditions y−2 −1, y−1 1.3, y0 1.1 converges to
zero.
Example 5.2. Figure 2 shows that if r 1, l 0, k 1 K max{2k, 2r 1} 3 and γ 0.7,
then the solution {yn }∞
n−3 with initial conditions y−3 −1, y−2 −1.2, y−1 1.3, y0 1.1
converges to zero.
Example 5.3. Figure 3 shows that if r 0, l 0, k 1 K max{2k, 2r 1} 2 and γ 2, then
the solution {yn }∞
n−3 with initial conditions y−2 2, y−1 0.4, y0 2.1 is unbounded.
Discrete Dynamics in Nature and Society
11
Example 5.4. Figure 4 shows that if r 1, l 0, k 1 K max{2k, 2r 1} 3 and γ 2,
then the solution {yn }∞
n−3 with initial conditions y−3 0.5, y−2 2, y−1 0.4, y0 2.1 is
unbounded.
Acknowledgment
This paper was funded by the Deanship of the Scientific Research DSR, King Abdulaziz
University, Jeddah, under Grant no. 15-662-D1432. The author, therefore, acknowledge with
thanks DSR technical and financial support.
References
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