Global asymptotic stability of a higher order difference equation
The aim of this work is to investigate the global stability, periodic nature, oscillation and boundedness of solutions of the difference equation.
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Інститут прикладної математики і механіки НАН України
2007
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| Цитувати: | Global asymptotic stability of a higher order difference equation / A.E. Hamza, R. Khalaf-Allah // Український математичний вісник. — 2007. — Т. 4, № 3. — С. 370-377. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1245232017-09-30T03:03:16Z Global asymptotic stability of a higher order difference equation Hamza, A.E. Khalaf-Allah, R. The aim of this work is to investigate the global stability, periodic nature, oscillation and boundedness of solutions of the difference equation. 2007 Article Global asymptotic stability of a higher order difference equation / A.E. Hamza, R. Khalaf-Allah // Український математичний вісник. — 2007. — Т. 4, № 3. — С. 370-377. — Бібліогр.: 4 назв. — англ. 1810-3200 2000 MSC. 39A11. http://dspace.nbuv.gov.ua/handle/123456789/124523 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The aim of this work is to investigate the global stability, periodic nature, oscillation and boundedness of solutions of the difference equation. |
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Hamza, A.E. Khalaf-Allah, R. |
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Hamza, A.E. Khalaf-Allah, R. Global asymptotic stability of a higher order difference equation Український математичний вісник |
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Hamza, A.E. Khalaf-Allah, R. |
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Hamza, A.E. |
| title |
Global asymptotic stability of a higher order difference equation |
| title_short |
Global asymptotic stability of a higher order difference equation |
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Global asymptotic stability of a higher order difference equation |
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Global asymptotic stability of a higher order difference equation |
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Global asymptotic stability of a higher order difference equation |
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global asymptotic stability of a higher order difference equation |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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Global asymptotic stability of a higher order difference equation / A.E. Hamza, R. Khalaf-Allah // Український математичний вісник. — 2007. — Т. 4, № 3. — С. 370-377. — Бібліогр.: 4 назв. — англ. |
| series |
Український математичний вісник |
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AT hamzaae globalasymptoticstabilityofahigherorderdifferenceequation AT khalafallahr globalasymptoticstabilityofahigherorderdifferenceequation |
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2025-07-09T01:33:57Z |
| last_indexed |
2025-07-09T01:33:57Z |
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1837131221590605824 |
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Український математичний вiсник
Том 4 (2007), № 3, 370 – 377
Global asymptotic stability of a higher order
difference equation
Alaa E. Hamza, R. Khalaf-Allah
(Presented by A. E. Shishkov)
Abstract. The aim of this work is to investigate the global stability,
periodic nature, oscillation and boundedness of solutions of the difference
equation
xn+1 =
Axn−1
B + Cxn−2lxn−2k
, n = 0, 1, 2, . . .
where A, B, C are nonnegative real numbers and l, k are nonnegative
integers, l ≤ k.
2000 MSC. 39A11.
Key words and phrases. Difference equation, periodic solution, glob-
ally asymptotically stable.
1. Introduction
Difference equations have always played an important role in the
construction and analysis of mathematical models of biology, ecology,
physics, economic processes, etc. [3].
The study of nonlinear rational difference equations of higher order is
of paramount importance, since we still know so little about such equa-
tions. Cinar [1] examined the global asymptotic stability of all positive
solutions of the rational difference equation
xn+1 =
axn−1
1 + bxnxn−1
, n = 0, 1, 2, . . .
Xiaofan yang et all [4] investigated the asymptotic behavior of solutions
of the difference equations
xn+1 =
axn−1 + bxn−1
c+ dxnxn−1
, n = 0, 1, 2, . . .
Received 17.08.2007
ISSN 1810 – 3200. c© Iнститут математики НАН України
A. E. Hamza, R. Khalaf-Allah 371
where a ≥ 0, b, c, d > 0.
In this paper, we study the global asymptotic stability of the difference
equation
xn+1 =
Axn−1
B + Cxn−2lxn−2k
, n = 0, 1, 2, . . . (1.1)
where A,B,C are nonnegative real numbers and l, k are nonnegative
integers, l ≤ k. The following particular cases can be obtained:
(1) When A = 0, equation (1.1) reduces to the equation
xn+1 = 0, n = 0, 1, 2, . . .
(2) When B = 0, equation (1.1) reduces to the equation
xn+1 =
Axn−1
Cxn−2lxn−2k
, n = 0, 1, 2, . . .
This equation can be reduced to the linear difference equation
yn+1 − yn−1 + yn−2l + yn−2k = γ,
by taking
xn = eyn , γ = ln
A
C
.
(3) When C = 0, equation (1.1) reduces to the equation
xn+1 =
A
B
xn−1, n = 0, 1, 2, . . .
which is a linear difference equation.
For various values of l and k, we can get more equation.
2. Preliminaries
Consider the difference equation
xn+1 = f(xn, xn−1, . . . , xn−k), n = 0, 1, . . . (2.1)
where f : Rk+1 → R.
Definition 2.1 ([2]). An equilibrium point for equation (2.1) is a point
x̄ ∈ R such that x̄ = f(x̄, x̄, . . . , x̄).
372 Global Asymptotic stability...
Definition 2.2 ([2]). (1) An equilibrium point x̄ for equation (2.1) is
called locally stable if for every ǫ > 0, ∃ δ > 0 such that every
solution {xn} with initial conditions x−k, x−k+1, . . . , x0 ∈]x̄−δ, x̄+
δ[ is such that xn ∈]x̄ − ǫ, x̄ − ǫ[, ∀n ∈ N. Otherwise x̄ is said to
be unstable.
(2) The equilibrium point x̄ of equation (2.1) is called locally asymp-
totically stable if it is locally stable and there exists γ > 0 such that
for any initial conditions x−k, x−k+1, . . . , x0 ∈]x̄ − γ, x̄ + γ[, the
corresponding solution {xn} tends to x̄.
(3) An equilibrium point x̄ for equation (2.1) is called attractor if every
solution {xn} converges to x̄ as n→∞.
(4) The equilibrium point x̄ for equation (2.1) is called globally asymp-
totically stable if it is locally asymptotically stable and global attrac-
tor.
The linearized equation associated with equation (2.1) is
yn+1 =
k∑
i=0
∂f
∂xn−i
(x̄, . . . x̄)yn−i, n = 0, 1, 2, . . . (2.2)
The characteristic equation associated with equation (2.2) is
λk+1 −
k∑
i=0
∂f
∂xn−i
(x̄, . . . , x̄)λk−i = 0. (2.3)
Theorem 2.1 ([2]). Assume that f is a C1 function and let x̄ be an
equilibrium point of equation (2.1). Then the following statements are
true:
(1) If all roots of equation (2.3) lie in the open disk |λ| < 1, then x̄ is
locally asymptotically stable.
(2) If al least one root of equation (2.3) has absolute value greater than
one, then x̄ is unstable.
The change of variables xn =
√
B
C yn reduces equation (1.1) to the
difference equation
yn+1 =
γyn−1
1 + yn−2lyn−2k
, n = 0, 1, 2, . . . (2.4)
where γ = A
B .
A. E. Hamza, R. Khalaf-Allah 373
3. Linearized stability analysis
In this section we study the asymptotic stability of the nonnegative
equilibrium points of equation (2.4). We can see that equation (2.4) has
two nonnegative equilibrium points ȳ = 0 and ȳ =
√
γ − 1 when γ > 1
and the zero equilibrium only when γ ≤ 1.
The linearized equation associated with equation (2.4) about ȳ is
zn+1 −
γ
1 + ȳ2
zn−1 +
γȳ2
(1 + ȳ2)2
(zn−2l + zn−2k) = 0, n = 0, 1, 2, . . .
(3.1)
The characteristic equation associated with this equation is
λ2k+1 − γ
1 + ȳ2
λ2k−1 +
γȳ2
(1 + ȳ2)2
(λ2k−2l + 1) = 0. (3.2)
We summarize the results of this section in the following theorem.
Theorem 3.1. (1) If γ < 1, then the zero equilibrium point is locally
asymptotically stable.
(2) If γ > 1, then the equilibrium points ȳ = 0 and ȳ =
√
γ − 1 are
unstable (saddle points).
Proof. The linearized equation associated with equation (2.4) about ȳ =
0 is
zn+1 − γzn−1 = 0, n = 0, 1, 2, . . . 1
The characteristic equation associated with this equation is
λ2k+1 − γλ2k−1 = 0.
So λ = 0,±√γ.
(1) If γ < 1, then |λ| < 1 for all roots and ȳ = 0 is locally asymptoti-
cally stable.
(2) If γ > 1, it follows that ȳ = 0 is unstable (saddle point).
The linearized equation (3.1) about ȳ =
√
γ − 1 becomes
zn+1 − zn−1 +
(
1− 1
γ
)
(zn−2l + zn−2k) = 0, n = 0, 1, 2, . . .
The associated characteristic equation is
λ2k+1 − λ2k−1 +
(
1− 1
γ
)
(λ2k−2l + 1) = 0.
Let f(λ) = λ2k+1 − λ2k−1 + (1 − 1
γ )(λ2k−2l + 1). We can see that f(λ)
has a root in (−∞,−1). Then the point ȳ =
√
γ − 1 is unstable (saddle
point).
374 Global Asymptotic stability...
4. Global behavior of equation (2.4)
Theorem 4.1. If γ < 1, then the zero equilibrium point is globally
asymptotically stable.
Proof. Let {yn} be a solution of equation (2.4). Hence
yn+1 =
γyn−1
1 + yn−2lyn−2k
< γyn−1, n = 0, 1, 2, . . .
Then limn→∞ yn = 0. In view of Theorem 3.1, ȳ = 0 is globally asymp-
totically stable.
5. Existence of prime period two solutions
This section is devoted to discuss the condition under which there
exist prime period two solutions.
Theorem 5.1. A necessary and sufficient condition for equation (2.4)
to have a prime period two solution is that γ = 1. In this case the
prime period two solution is of the form . . . , 0, ϕ, 0, ϕ, 0, . . . where ϕ > 0.
Furthermore, every solution converges to a period two solution.
Proof. Sufficiency: let γ = 1, then for every ϕ > 0 we have . . . , 0, ϕ, 0, ϕ,
0, . . . is a prime period two solution.
Necessity: assume that equation (2.4) has a prime period two solution
. . . , ψ, ϕ, ψ, ϕ, ψ, . . . . Then
ϕ =
γϕ
1 + ψ2
, ψ =
γψ
1 + ϕ2
.
Hence
(ϕ− ψ) + ϕψ(ψ − ϕ) = γ(ϕ− ψ),
implies
ϕψ = 1− γ. (5.1)
So γ ≤ 1. Similarly,
ϕψ = γ − 1. (5.2)
So γ ≥ 1. Then ϕψ = 0 and the solution is of the form
. . . , 0, ϕ, 0, ϕ, 0, . . . with ϕ > 0.
Now let {yn}∞n=−2k be a solution of equation (2.4) with γ = 1. Then
yn+1 =
γyn−1
1 + yn−2lyn−2k
≤ yn−1, n = 0, 1, 2, . . .
A. E. Hamza, R. Khalaf-Allah 375
and so the even terms {y2n}∞n=0 decreases to a limit ϕ and the odd terms
{y2n+1}∞n=0 decreases to a limit ψ, where ϕ = ϕ
1+ψ2 , ψ = ψ
1+ϕ2 .
Then ϕψ2 = 0 and ψϕ2 = 0. Therefore, {yn}∞n=−2k converges to the
periodic solution . . . , 0, ϕ, 0, ϕ, 0, . . . with ϕ > 0.
6. Semicycle analysis
Here we discuss the existence of semicycles. We need the following
theorem to obtain the main result of this section.
Theorem 6.1. Assume that f ∈ C([0,∞[2k+1, [0,∞[) is increasing in
the even arguments and decreasing in the others. Let ȳ be an equilibrium
point for the difference equation
yn+1 = f(yn, yn−1, . . . , yn−2k), n = 0, 1, 2, . . . (6.1)
Let {yn}∞n=−2k be a solution of equation (6.1) such that either,
(C1) y−2k, y−2k+2, . . . , y0 > ȳ and y−2k+1, y−2k+3, . . . , y−1 < ȳ
or
(C2) y−2k, y−2k+2, . . . , y0 < ȳ and y−2k+1, y−2k+3, . . . , y−1 > ȳ
is satisfied, then {yn}∞n=−2k oscillates about ȳ with semicycles of length
one.
Proof. Assume that f is increasing in the even arguments and decreasing
in the others. Let f be satisfying condition (C1), we have
y1 = f(y0, y−1, y−2, . . . , y−2k+1, y−2k)
< f(ȳ, y−1, ȳ, . . . , y−2k+1, ȳ)
< f(ȳ, ȳ, ȳ, . . . , ȳ, ȳ, ) = ȳ,
y2 = f(y1, y0, y−1, y−2, . . . , y−2k+2, y−2k+1)
> f(ȳ, y0, ȳ, . . . , y−2k+2, ȳ)
> f(ȳ, ȳ, ȳ, . . . , ȳ, ȳ, ) = ȳ.
by induction we obtain the result.
If f satisfies condition (C2), we can prove the result similarly.
Corollary 6.1. Assume that γ > 1 and let {yn}∞n=−2k be a solution of
equation (2.4) such that either (C1) or (C2) is satisfied. Then {yn}∞n=−2k
oscillates about the positive equilibrium point ȳ =
√
γ − 1 with semicycles
of length one.
Proof. The proof follows directly from the previous theorem.
376 Global Asymptotic stability...
7. Existence of unbounded solutions
Finally we show that, under certain initial condition, unbounded so-
lution will be obtained.
Theorem 7.1. Assume that γ > 1. Let {yn}∞n=−2k be a solution of
equation (2.4) and ȳ =
√
γ − 1, the positive equilibrium point. Then the
following statements are true:
(1) If y−2k, y−2k+2, . . . , y0 > ȳ and y−2k+1, y−2k+3, . . . , y−1 < ȳ, then
{y2n} increases to ∞ and {y2n+1} decreases to 0.
(2) If y−2k, y−2k+2, . . . , y0 < ȳ and y−2k+1, y−2k+3, . . . , y−1 > ȳ, then
{y2n} decreases to 0 and {y2n+1} increases to ∞.
Proof. (1) Let {yn}∞n=−2k be a solution of equation (2.4) with initial con-
ditions y−2k, y−2k+1, . . . , y0 > ȳ and y−2k+1, y−2k+3, . . . , y−1 < ȳ. Then
y2n+2 =
γy2n
1 + y2n−2l+1y2n−2k+1
>
γy2n
1 + ȳ2
= y2n
and
y2n+3 =
γy2n+1
1 + y2n−2l+1y2n−2k+1
<
γy2n+1
1 + ȳ2
= y2n+1
and so {y2n} increases to ∞ and {y2n+1} decreases to 0.
(2) The proof is similar and will be omitted.
Acknowledgment. Many thanks to Dr. Adel El-Tohamy for his help
and support.
References
[1] C Cinar, On the positive solution of the difference equation xn+1 =
ax
n−1
1+bxnx
n−1
//
Appl. Math. and Comp. 156 (2004), 578–590.
[2] V. L. Cocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of
Higher Order with Applications. Kluwer Academic, Dordrecht, 1993.
[3] R. E. Michens, Difference Equations. Theorey and Applications. 2nd Edition. Van
Nostrand Renhold, 1990.
[4] X. Yang, W. Su, B. Chen, G. M. Megson, D. J. Evans, On the recurcive sequence
xn+1 =
axn+bx
n−1
c+dxnx
n−1
// Appl. Math. and Comp. doi:10.1016/j.amc.2004.03.023.
A. E. Hamza, R. Khalaf-Allah 377
Contact information
Alaa E. Hamza Department of Mathematics
Faculty of Science
Cairo University
Giza, 12211
Egypt
E-Mail: hamzaaeg2003@yahoo.com
R. Khalaf-Allah Department of Mathematics
Faculty of Science
Helwan University
Cairo, 11795
Egypt
E-Mail: abuzead73@yahoo.com
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