Affiliation:
1. Department of Mathematics and Computer Science , Abdelhafid Boussouf University , , Mila Algeria
2. LMAM Laboratory , Mohamed Seddik Ben Yahia University , , Jijel Algeria
Abstract
Abstract
In this paper, we study the periodicity, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of p nonlinear difference equations
x
n
+
1
(
1
)
=
A
+
x
n
−
1
(
1
)
x
n
(
p
)
,
x
n
+
1
(
2
)
=
A
+
x
n
−
1
(
2
)
x
n
(
p
)
,
…
,
x
n
+
1
(
p
−
1
)
=
A
+
x
n
−
1
(
p
−
1
)
x
n
(
p
)
,
x
n
+
1
(
p
)
=
A
+
x
n
−
1
(
p
)
x
n
(
p
−
1
)
$$\begin{equation*}x^{(1)}_{n+1}=A+\dfrac{x^{(1)}_{n-1}}{x^{(p)}_{n}},\quad x^{(2)}_{n+1}=A+\dfrac{x^{(2)}_{n-1}}{x^{(p)}_{n}},\quad\ldots,\quad x^{(p-1)}_{n+1}=A+\dfrac{x^{(p-1)}_{n-1}}{x^{(p)}_{n}},\quad x^{(p)}_{n+1}=A+\dfrac{x^{(p)}_{n-1}}{x^{(p-1)}_{n}}
\end{equation*}
$$
where n ∈ ℕ0, p ≥ 3 is an integer, A ∈ (0, +∞) and the initial conditions
x
−
1
(
j
)
$x_{-1}^{(j)}$
,
x
0
(
j
)
$x_{0}^{(j)}$
, j = 1, 2, …, p are positive numbers.
Reference38 articles.
1. Akrour, Y.—Touafek, N.—Halim, Y.: On a system of difference equations of second order solved in closed form, Miskolc Math. Notes 20 (2019), 701–717.
2. Amleh, A. M.—Grove, E. A.—Ladas—Georgiou, D. A.: On the recursive sequencexn+1=A+xn−1xn$x_{n+1}=A+\dfrac{x_{n-1}}{x_{n}}$, J. Math. Anal. Appl. 233 (1999), 790–798.
3. Elaydi, S.: An Introduction to Difference Equations, Springer-Verlag New York, 1995.
4. Belhannache, F.—Touafek, N.—Abo-Zeid, R.: Dynamics of a third-order rational difference equation, Bull. Math. Soc. Sci. Math. Roum. Nouv. Ser. 107 (2016), 13–22.
5. Elsayed, E. M.: On a system of two nonlinear difference equations of order two, Proceedings Jagiellonian Mathematics Society 18 (2015), 353–369.
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1 articles.
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