Asymptotic behavior and periodic nature of two difference equations
We discuss the global asymptotic stability of the solutions of the difference equations $$x_{n+1} = \frac{x_{n−2}}{±1 + x_nx_{n−1}x_{n−2}}, \quad n = 0,1,…,$$ where the initial conditions $x_{−2}, x_{−1}, x_0$ are real numbers.
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| Datum: | 2009 |
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| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509085704650752 |
|---|---|
| author | Khalaf-Allah, R. Халаф-Аллах, Р. |
| author_facet | Khalaf-Allah, R. Халаф-Аллах, Р. |
| author_sort | Khalaf-Allah, R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:24Z |
| description | We discuss the global asymptotic stability of the solutions of the difference equations
$$x_{n+1} = \frac{x_{n−2}}{±1 + x_nx_{n−1}x_{n−2}}, \quad n = 0,1,…,$$
where the initial conditions $x_{−2}, x_{−1}, x_0$ are real numbers. |
| first_indexed | 2026-03-24T02:35:30Z |
| format | Article |
| fulltext |
UDC 517.9
R. Khalaf-Allah (October 6 Univ., Egypt)
ASYMPTOTIC BEHAVIOR AND PERIODIC NATURE
OF TWO DIFFERENCE EQUATIONS
ASYMPTOTYÇNA POVEDINKA TA PERIODYÇNA PRYRODA
DVOX RIZNYCEVYX RIVNQN|
We discuss the global asymptotic stability of the solutions of the difference equations
x
x
x x x
n
n
n n n
+
−
− −
=
± +1
2
1 21
, n = …0 1, , ,
where the initial conditions x−2 , x−1 , x0 are real numbers.
Rozhlqnuto hlobal\nu asymptotyçnu stijkist\ rozv’qzkiv riznycevyx rivnqn\
x
x
x x x
n
n
n n n
+
−
− −
=
± +1
2
1 21
, n = …0 1, , ,
de poçatkovi umovy x−2 , x−1 , x0 [ dijsnymy çyslamy.
1. Introduction and preliminaries. Difference equations, although their forms 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 equ-
ations of higher order is of paramount importance, since we still know so little about
such equations. Cinar [3, 4] examined the global asymptotic stability of all positive so-
lutions of the rational difference equation
xn+1 =
x
x x
n
n n
−
−+
1
11
, n = 0, 1, … .
He also discussed the behavior of the solutions of the difference equation
xn+1 =
x
x x
n
n n
−
−− +
1
11
, n = 0, 1, … .
In this paper, we discuss the global stability and periodic character of all solutions of
the difference equations
xn+1 =
x
x x x
n
n n n
−
− −+
2
1 21
, n = 0, 1, … , (1)
and
xn+1 =
x
x x x
n
n n n
−
− −− +
2
1 21
, n = 0, 1, … . (2)
2. The difference equation x
x
x x xn
n
n n n
++
−−
−− −−
==
++1
2
1 21
. In this section we study
the difference equation
xn+1 =
x
x x x
n
n n n
−
− −+
2
1 21
, n = 0, 1, … .
© R. KHALAF - ALLAH, 2009
834 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6
ASYMPTOTIC BEHAVIOR AND PERIODIC NATURE … 835
Theorem 1. Let x−2 , x−1 and x0 are positive real numbers. Then all solu-
tions of equation (1) are
xn =
x
j
j
n
x
j
n
− =
−
−
+
+ +
= …∏2 0
1 3
1
1 3
1 3 1
1 4 7
1
α
α( )
, , , , ,
( )/
++ +
+ +
= …=
−∏ ( )
( )
, , , , ,
( )/ 3 1
1 3 2
2 5 8
1
0
2 3
0
j
j
n
x
j
n α
α
++ −
+
= …
=∏ ( )
, , , , ,
/ 3 1
1 3
3 6 9
1
3 j
j
n
j
n α
α
(3)
where α = x x x− −2 1 0 .
Proof. Let α = x x x− −2 1 0 . Then
x1 =
x−
+
2
1 α
, x2 = x−
+
+1
1
1 2
α
α
and x3 = x0
1 2
1 3
+
+
α
α
.
Now assume that m ≥ 1. Then we have
x m3 2− = x
j
jj
m
−
=
− +
+ +∏2
0
1 1 3
1 3 1
α
α( )
,
x m3 1− = x
j
jj
m
−
=
− + +
+ +∏1
0
1 1 3 1
1 3 2
( )
( )
α
α
,
x m3 = x
j
jj
m
0
0
1 1 3 2
1 3 3
+ +
+ +=
−
∏ ( )
( )
α
α
.
Now
x
x x x
m
m m m
3 2
3 3 1 3 21
−
− −+
=
=
x
j
j
x
j
j
j
m
− =
−
−
+
+ +
+ +
+ +
∏2 0
1
2
1 3
1 3 1
1
1 3
1 3 1
α
α
α
α
( )
( )jj
m
j
m
x
j
j
x
j
=
−
− =
−∏ ∏ + +
+ +
+
0
1
1 0
1
0
1 3 1
1 3 2
1 3( )
( )
(α
α
++
+ +=
−∏ 2
1 3 30
1 )
( )
α
αjj
m
=
=
x
j
j
j
j
j
m
j
− =
−
=
+
+ +
+ +
+ +
∏2 0
1 1 3
1 3 1
1
1 3
1 3 3
α
α
α α
α
( )
( )00
1m−∏
=
x
j
j
m
j
m
− =
− +
+ +
+
+
∏2 0
1 1 3
1 3 1
1
1
1 3
α
α
α
α
( )
=
=
1 3
1 3 1
1 3
1 3 12
0
1+
+ +
+
+ +−
=
−
∏m
m
x
j
jj
mα
α
α
α( ) ( )
= x
j
jj
m
−
=
+
+ +∏2
0
1 3
1 3 1
α
α( )
= x m3 1+ .
This completes the proof.
Remark. If α = x x x− −2 1 0 ≠ – 1/n , for all n ≥ 1, then formula (3) also repre-
sents solutions of equation (1) when x−2 , x−1 and x0 are real numbers.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6
836 R. KHALAF-ALLAH
Theorem 2. Equation (1) has a period-3 solution { , , , , , ,… ϕ ϕ ϕ ϕ ϕ1 2 3 1 2
ϕ3, }… with ϕ ϕ ϕ1 2 3 = α = 0.
Proof. Let α = 0. Using formula (3) it is sufficient to see that
xn =
x n
x n
x n
−
−
= …
= …
= …
2
1
0
1 4 7
2 5 8
3 6 9
, , , , ,
, , , , ,
, , , , ,
therefore, for n = 0, 1, … we have
x m3 = x0 , x m3 1+ = x−1 and x m3 2+ = x−2 .
Now suppose that x−2 = ϕ1 , x−1 = ϕ2 , x0 = ϕ3 . It follows that
{ , , , , , , , }… …ϕ ϕ ϕ ϕ ϕ ϕ1 2 3 1 2 3
is a periodic solution with ϕ ϕ ϕ1 2 3 = α = 0.
This completes the proof.
Theorem 3. The unique equilibrium point x = 0 of equation (1) is nonhyper-
bolic point.
Theorem 4. Assume that α ≠ 0 and α ≠ – 1/n . Then every solution of equa-
tion (1) converges to zero.
Proof. Let { }xn be arbitrary solution of equation (1). We consider only the case
α < 0, the case α > 0 is similar and will be omitted. From formula (3) we have
x m3 1+ = x
j
jj
m
−
=
+
+ +∏2
0
1 3
1 3 1
α
α( )
= x
j
jj
m
−
=
+
+ +∏2
0
1 3
1 3 1
exp ln
( )
α
α
=
= x
j
jj
m
−
=
− + +
+
∏2
0
1 3 1
1 3
exp ln
( ) α
α
= x
jj
m
−
=
− +
+
∑2
0
1
1 3
exp ln
α
α
=
= x c n
j
O
jj n
m
−
=
−
+
+
∑2 0 2
1
1 3
1
0
( ) exp α
α
→ 0, n → ∞ ,
since
1
1 30 +=∑
jj n
m
α
→ – ∞ as n → ∞ and O
jj n
m 1
20
=∑ is convergent.
Here c n( )0 is a positive constant depending on n0 ∈ N .
Similarly x m3 2+ → 0 as n → ∞ and x m3 3+ → 0 as n → ∞ .
This completes the proof.
3. The difference equation x
x
x x xn
n
n n n
++
−−
−− −−
==
−− ++1
2
1 21
. In this section we intro-
duce the following results.
Theorem 5. Let { }xn n = −
∞
2 be a solution of equation (2). Assume that α =
= x x x− −2 1 0 ≠ 1. Then we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6
ASYMPTOTIC BEHAVIOR AND PERIODIC NATURE … 837
x m i3 + =
x i
x
i
x i
m
m
m
−
−
=
=
=
2
1
0
1
2
3
β
β
β
, ,
, ,
, ,
(4)
where
βm =
1
1
1
, ,
, .
m odd
m even
− +
α
Proof. For m = 0 the following results hold
x1 =
x−
− +
2
1 α
, x2 = x− − +1 1( )α and x3 =
x0
1− + α
.
Assume that m > 0. Then if m is even, we have
x
x x x
m
m m m
3 2
3 3 1 3 21
−
− −− +
=
x
x
x
x
m
m
m
m
− −
− −
−
−
−− +
2 1
2 1
1
1
0 11
β
β
β
β
=
=
x m
m
− −
−− +
2 1
11
β
αβ
=
x−
− +
2
1 α
= x m−2β = x m3 1+ .
If m is odd, then
x
x x x
m
m m m
3 2
3 3 1 3 21
−
− −− +
=
x
x
x
x
m
m
m
m
− −
− −
−
−
−− +
2 1
2 1
1
1
0 11
β
β
β
β
=
=
x m
m
− −
−− +
2 1
11
β
αβ
=
x−
−
−
− +
− + − +
2
1
1
1
1 1
( )
( )
α
α α
= x−2 = x m−2β = x m3 1+ .
This completes the proof.
Theorem 6. The equilibrium points x = 0 and x = 23 of equation (2)
are nonhyperbolic points.
Theorem 7. Every solution of equation (2) is periodic with period 6.
Proof. Let { }xn n = −
∞
2 be a solution of equation (2) then we have
x m i( )3 6+ + =
x i
x
i
x i
m
m
m
− +
−
+
+
=
=
=
2 2
1
2
0 2
1
2
3
β
β
β
, ,
, ,
, ,
=
x i
x
i
x i
m
m
m
−
−
=
=
=
2
1
0
1
2
3
β
β
β
, ,
, ,
, ,
= x m i3 + .
This completes the proof.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6
838 R. KHALAF-ALLAH
Corollary 1. Let { }xn n = −
∞
2 be a solution of equation (2) with α = 2. Then
{ }xn n = −
∞
2 is periodic with period 3.
Corollary 2. Let { }xn n = −
∞
2 be a solution of equation (2) where x−2 , x−1
and x0 are positive real numbers such that α = x x x− −2 1 0 > 1. Then the soluti-
on { }xn n = −
∞
2 is positive.
Corollary 3. Let { }xn n = −
∞
2 be a solution of equation (2) where x−2 , x−1
and x0 are negative real numbers. Then the solution { }xn n = −
∞
2 oscillates with
semicycles of length 3.
Acknowledgements. Many thanks to Dr. Alaa E. Hamza for his help and support.
1. Agarwal R. P. Difference equations and inequalities. – First edition. – Marcel Dekker, 1992.
2. Kocic V. L., Ladas G. Global behavior of nonlinear difference equations of higher order with
applications. – Dordrecht: Kluwer Acad., 1993.
3. Cinar C. On the positive solution of the difference equation x
x
x x
n
n
n n
+
−
−
=
+1
1
11
// Appl. Math.
and Comput. – 2004. – 150. – P. 21 – 24.
4. Cinar C. On the positive solution of the difference equation x
x
x x
n
n
n n
+
−
−
=
− +1
1
11
// Ibid. – 2004.
– 158. – P. 816 – 819.
Received 24.12.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 6
|
| id | umjimathkievua-article-3061 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:30Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/87/315a9be93e44ff584198d185a0b7af87.pdf |
| spelling | umjimathkievua-article-30612020-03-18T19:44:24Z Asymptotic behavior and periodic nature of two difference equations Асимптотична поведінка та періодична природа двох різницевих рівнянь Khalaf-Allah, R. Халаф-Аллах, Р. We discuss the global asymptotic stability of the solutions of the difference equations $$x_{n+1} = \frac{x_{n−2}}{±1 + x_nx_{n−1}x_{n−2}}, \quad n = 0,1,…,$$ where the initial conditions $x_{−2}, x_{−1}, x_0$ are real numbers. Розглянуто глобальну асимптотичну стійкість розв'язків різницевих рівнянь $$x_{n+1} = \frac{x_{n−2}}{±1 + x_nx_{n−1}x_{n−2}}, \quad n = 0,1,…,$$ де початкові умови $x_{−2}, x_{−1}, x_0$ є дійсними числами. Institute of Mathematics, NAS of Ukraine 2009-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3061 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 6 (2009); 834-838 Український математичний журнал; Том 61 № 6 (2009); 834-838 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3061/2871 https://umj.imath.kiev.ua/index.php/umj/article/view/3061/2872 Copyright (c) 2009 Khalaf-Allah R. |
| spellingShingle | Khalaf-Allah, R. Халаф-Аллах, Р. Asymptotic behavior and periodic nature of two difference equations |
| title | Asymptotic behavior and periodic nature of two difference equations |
| title_alt | Асимптотична поведінка та періодична природа двох різницевих рівнянь |
| title_full | Asymptotic behavior and periodic nature of two difference equations |
| title_fullStr | Asymptotic behavior and periodic nature of two difference equations |
| title_full_unstemmed | Asymptotic behavior and periodic nature of two difference equations |
| title_short | Asymptotic behavior and periodic nature of two difference equations |
| title_sort | asymptotic behavior and periodic nature of two difference equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3061 |
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