J. Appl. Math. & Computing Vol. 16(2004), No. 1 - 2, pp. 243 - 249
GLOBAL ATTRACTIVITY OF THE
α − βxn−k
RECURSIVE SEQUENCE xn+1 =
γ + xn
H. M. El-Owaidy, A. M. Ahmed∗ and Z. Elsady
Abstract.
Our aim in this paper is to investigate the global attractivity of the
recursive sequence
α − βxn−k
xn+1 =
,
γ + xn
where α, β, γ > 0 and k = 1, 2, ... We show that the positive equilibrium point of
the equation is a global attractor with a basin that depends on certain conditions
posed on the coefficients.
AMS Mathematics Subject Classification: 39A10, 39A99, 34C99.
Key words and phrases : Difference equations, higher order, attractivity, asymptotic stability.
1. Introduction
H. M. El-Owaidy et al. [3] studied the rational recursive sequence
xn+1 =
−αxn−1
,
β ± xn
where the coefficients α, β > 0 and obtained sufficient conditions for the global
attractivity of the zero equilibrium points with basin that depend on certain
conditions posed on the coefficients.
Also, H. M. El-Owaidy et al. [5] studied the rational recursive sequence
xn+1 =
α − βxn−1
,
γ + xn
n = 0, 1, ...,
Received September 6, 2003. Revised December 9, 2003. ∗ Corresponding author.
c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM
243
244
H. M. El-Owaidy, A. M. Ahmed and Z. Elsady
where the coefficients α, β and γ are non-negative real numbers and obtained
sufficient conditions for the global attractivity of the positive equilibrium point.
M. T. Aboutaleb et al. [1] studied the rational recursive sequence
xn+1 =
α − βxn
,
γ + xn−1
n = 0, 1, ...
where the coefficients α, β and γ are non-negative real numbers and obtained
sufficient conditions for the global attractivity of the positive equilibrium point.
Li and Sun [8] extended the above results to the following rational recursive
sequence
α − βxn
, n = 0, 1, ...
xn+1 =
γ + xn−k
For other related results see [2, 4, 6, 7, 9].
Our aim in this paper is to investigate the global attractivity of the recursive
sequence
α − βxn−k
xn+1 =
, n = 0, 1, ...,
(1.1)
γ + xn
where
α, β, γ > 0 and γ ≥ β.
(1.2)
We show that the positive equilibrium point of equation (1.1) is a global attractor
with a basin that depends on certain conditions posed on the coefficients.
The study of these equations is quite challenging and rewarding and is still
in its infancy.
We believe that the nonlinear rational difference equations are paramount
importance in their own right,and furthermore that results about such equations
offer prototypes for the development of the basic theory of the global behavior
of nonlinear difference equations.
We need the following definitions:
Definition 1.1. The equilibrium point x̄ of the equation
xn+1 = F (xn , xn−1 , ..., xn−k ), n = 0, 1, ...
is the point that satisfies the condition x̄ = F (x̄, x̄, ..., x̄).
Definition 1.2. Let I be an interval of real numbers. The equilibrium point x̄
of equation (1.1) is said to be
(i) locally stable if for every > 0 there exists δ > 0 such that for all x−k , ..., x−1 ,
x0 ∈ I with
|x−k − x̄| + ... + |x−1 − x̄| + |x0 − x̄| < δ,
Global attractivity of the recursive sequence
245
we have |xn − x̄| < for all n ≥ −k,
(ii) locally asymptotically stable if it is locally stable, and if there exist γ > 0 such
that for all x−k , ..., x−1 , x0 ∈ I with
|x−k − x̄| + ... + |x−1 − x̄| + |x0 − x̄| < γ,
we have lim xn = x̄.
n→∞
(iii) global attractor if for all x−k , ..., x−1 , x0 ∈ I, we have lim xn = x̄.
n→∞
(iv) globally asymptotically stable if x̄ is locally stable and x̄ is also global attractor.
(v) unstable if x̄ is not locally stable.
2. Main results
Consider the difference equation (1.1) under conditions (1.2). Equation (1.1)
has two equilibrium points
−(β + γ) ±
x̄i =
q
2
(β + γ) + 4α
,
2
i = 1, 2.
(2.1)
We need the following Theorem:
Theorem A([2]). Assume that a, b ∈ R and k ∈ {1, 2, ...}. Then
|a| + |b| < 1
is a sufficient condition for asymptotic stability of the difference equation
xn+1 − axn + bxn−k = 0,
n = 0, 1, ...
Theorem 2.1. If γ > β, then the positive equilibrium point x̄1 of equation (1.1)
is locally asymptotically stable.
Proof. The linearized equation associated with equation (1.1) about x̄i , i = 1, 2,
has the form
zn+1 +
α − β x̄i
β
zn−k = 0,
zn +
(γ + x̄i )2
(γ + x̄i )
Then the proof follows from Theorem A.
i = 1, 2, n = 0, 1, ...
(2.2)
246
H. M. El-Owaidy, A. M. Ahmed and Z. Elsady
Theorem 2.2. The equation (1.1) has a prime period two solutions iff β =
γ and k is even.
Proof. Let ..., φ, ψ, φ, ψ, ... be a period-two solutions of the equation (1.1).
If k is odd, then γ = −β which is a contradiction.
If k is even, then
α − βψ
φ=
γ+ψ
and
ψ=
α − βφ
.
γ+φ
Thus
(ψ − φ)(γ − β) = 0,
and from which it follows that γ = β.
The period two solution must be of the form
..., u,
α − βu
α − βu
, u,
, ...,
β+u
β+u
where u ∈ R\{−β}. This completes the proof.
In the following results, we assume that
γ > β.
(2.3)
Theorem 2.3([9]). The positive equilibrium point x̄1 of equation (1.1) is a
k+1
.
global attractor with a basin S = [0, α/β)
Lemma 2.4. Assume that condition(2.3) holds. Let {xn } be a solution of
equation (1.1). If xm+i ∈ [0, α/β], i = 0, 1, ..., k for some m ≥ −k, then
xm+i ∈ [C, D],
where
C=
α(γ − β)
α + γ2
∀i ≥ 2k + 2,
and
D=
α
γ
Proof. We can see that xm+k+1 , xm+k+2 , ..., xm+2k+1 ∈ [0, α/γ]. Then
C=
α(γ − β)
α
≤ xm+2k+2 ≤ = D.
α + γ2
γ
(2.4)
Global attractivity of the recursive sequence
247
So, the result follows by induction. Assume that there exists τ ≥ 2 such that the following conditions hold
γ ≥ τ α/β and α ≥ τ β 2 .
(2.5)
Lemma 2.5. Assume that the conditions
(2.5) hold for
some τ ≥ 2. Let {xn } be
α α
a solution of equation (1.1). If xi ∈ −(τ − 1) ,
, i = m, m+1, ..., m+k
β β
for some m ≥ −k, then
xm+i ∈ [C, D],
∀i ≥ 3k + 3.
Proof. We can see that xm+k+1 , xm+k+2 , ..., xm+2k+1 ∈ [0, α/β]. Then the proof
follows immediately from Lemma 2.4 . Lemma 2.6. Assume that the conditions (2.5) hold for some τ ≥ 2. Let {xn }
be a solution of equation (1.1). If xm ∈ [−(τ − 1)α/β, ∞) and xi ∈ [−(τ −
1)α/β, α/β], i = m + 1, m + 2, ..., m + k for some m ≥ −k such that
|xm+k − xm | ≤ 2τ α/β,
then
xm+i ∈ [C, D],
∀i ≥ 3k + 4.
Proof. We can see that
γ + xm+i ≥ α/β, i = 0, 1, ..., k
and
−β(γ + xm+k ) − (τ − 1)α ≤ α − βxm ≤ τ α.
If α − βxm ≥ 0, then
xm+k+1 ∈ [0, α/β].
If α − βxm < 0, then
xm+k+1 ∈ [−α/β, 0].
In both cases Lemma 2.5 yields
xm+i ∈ [C, D],
This completes the proof. ∀i ≥ 3k + 4.
248
H. M. El-Owaidy, A. M. Ahmed and Z. Elsady
Theorem 2.7. If there exists τ ≥ 2 such that conditions (2.5) hold, then the
positive equilibrium point x̄1 of equation (1.1) is a global attractor with a basin
[
k+1
k−1
S = [−(τ − 1)α/β, α/β]
{M } × [−(τ − 1)α/β, α/β]
α/β≤M ≤(2τ +1)α/β
×[M − 2τ α/β, α/β].
Proof. Let {xn } be a solution of equation (1.1) with initial conditions
x−k , x−k+1 , ..., x−1 , x0 ∈ S.
Then by Lemma 2.4 and Lemma 2.5,
xm ∈ [0, α/β],
∀m ≥ 2.
By Lemma 2.6,
xm ∈ [C, D],
∀m ≥ 2k + 4,
where C and D are defined in (2.4).
Set
λ = lim inf xn and Λ = lim sup xn .
n→∞
n→∞
Let > 0 be such that < min{(α/β) − Λ, λ}. Then there exists n0 ∈ N such
that
λ − < xn < Λ + , ∀n ≥ n0 .
Hence
α − β(Λ + )
α − β(λ − )
< xn+1 <
,
γ + (Λ + )
γ + (λ − )
Then we get the inequality
∀n ≥ n0 + k.
α − β(λ − )
α − β(Λ + )
<λ<Λ<
.
γ + (Λ + )
γ + (λ − )
This inequality yields
α − βΛ
α − βλ
≤λ≤Λ≤
,
γ+Λ
γ+λ
which implies that
α − βΛ − γλ ≤ α − βλ − γΛ.
Therefore Λ ≤ λ. Hence
λ = Λ = x̄1 .
Then, the proof is complete. Remark 2.1. The results due to equation (1.1) when α = 0 was investigated
by W. S. He et-al. [9].
Global attractivity of the recursive sequence
249
References
1. M. T. Aboutaleb, M. A. E-Sayed and A. E. Hamza, Stability of the recursive sequence
α−βxn
xn+1 = γ+x
, J. Math. Anal. Appl. 261 (2001), 126-133.
n−1
2. R. Devault, W. Kosmala, G. Ladas and S. W. Schultz, Global behavior of yn+1 = (p +
yn−k )/(qyn + yn−k ), Nonlinear Analysis 47 (2001), 4743-4751.
3. H. M. El-Owaidy, A. M. Ahmed and M. S. Mousa, On the recursive sequences xn+1 =
−αxn−1
, J. Appl. Math. Comp. (to appear).
β±xn
4. H. M. El-Owaidy, A. M. Ahmed and Z. Elsady, On the recursive sequences xn+1 =
−αxn−2
, J. Appl. Math. Comp. (to appear).
β±xn
5. H. M. El-Owaidy, A. M. Ahmed and Z. Elsady, Global attractivity of the recursive sequence
α−βx
xn+1 = γ+xn−1 , J. Appl. Math. Computing (to appear).
n
6. V. L. Kocic, and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher
Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
7. M. R. S. Kulenović, G. Ladas and N. R. Prokup, On a rational difference equation, Comp.
Math. Appl. 41 (2001), 671-678.
8. W. T. Li and H. R. Sun, Global attractivity in a rational recursive sequence, Dynamic
Systems and Applications 11 (2002), 339-346.
9. W. S. He and W. T. Li, Attractivity in a a nonlinear delay difference equations, Appl.
Math. E- Notes, (in press).
10. Z. Zhang, B. Ping and W. Dong, Oscillatory of unstable type second-order neutral difference equations, J. Appl. Math. & Computing 9(1) (2002), 87-100.
11. Z. Zhou, J. Yu and G. Lei, Oscillations for even-order neutral difference equations, J.
Appl. Math. & Computing 7(3) (2000), 601-610.
H. M. El-Owaidy received his B. Sc. Engin., B. Sc. Mathematics, Ph. D at Hungarian
Academy of Sc. Budapest Hungary under the direction of Prof. Farkas Miklos. Since 1973
has been at Al-Azhar University, Cairo. He obtained the rank professor in 1983. His research
interests in difference equations and bio-mathematics.
A. M. Ahmed obtained his B. Sc. (1997) and M. Sc. (2000) from Al-Azhar University.
Now he is doing research in difference equations and its applications which leads to obtain
Ph. D.
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City (11884),
Cairo, Egypt
e-mail: [email protected]