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On the behavior of solutions of the system of rational difference equations
Advances in Difference Equations volume 2011, Article number: 40 (2011)
Abstract
In this article, we investigate the solutions of the system of difference equations , , where x 0, x -1, y 0, y -1, z 0, z -1 real numbers such that y 0 x -1 ≠ 1, x 0 y -1 ≠ 1 and y 0 z 0 ≠ 0.
1. Introduction
In [1], Kurbanli et al. studied the behavior of positive solutions of the system of rational difference equations
In [2], Cinar studied the solutions of the systems of difference equations
In [3], Kurbanli, studied the behavior of solutions of the system of rational difference equations
In [4], Papaschinnopoulos and Schinas proved the boundedness, persistence, the oscillatory behavior, and the asymptotic behavior of the positive solutions of the system of difference equations
In [5], Clark and Kulenović investigate the global stability properties and asymptotic behavior of solutions of the system of difference equations
In [6], Camouzis and Papaschinnopoulos studied the global asymptotic behavior of positive solutions of the system of rational difference equations
In [7], Kulenović and Nurkanović studied the global asymptotic behavior of solutions of the system of difference equations
In [8], Özban studied the positive solutions of the system of rational difference equations
In [9], Zhang et al. investigated the behavior of the positive solutions of the system of the difference equations
In [10], Yalcinkaya studied the global asymptotic stability of the system of difference equations
In [11], Irićanin and Stević studied the positive solutions of the system of difference equations
Although difference equations are very simple in form, it is extremely difficult to understand throughly the global behavior of their solutions, for example, see Refs. [12–34].
In this article, we investigate the behavior of the solutions of the difference equation system
where x 0, x -1, y 0, y -1, z 0, z -1 real numbers such that y 0 x -1 ≠ 1, x 0 y -1 ≠ 1 and y 0 z 0 ≠ 0.
2. Main results
Theorem 1. Let y 0 = a, y -1 = b, x 0 = c, x -1 = d, z 0 = e, z -1 = f be real numbers such that y 0 x -1 ≠ 1, x 0 y -1 ≠ 1 and y 0 z 0 ≠ 0. Let {x n , y n , z n } be a solution of the system (1.1). Then all solutions of (1.1) are
Proof. For n = 0, 1, 2, 3, we have
for n = k, assume that
and
are true. Then, for n = k + 1 we will show that (1.2), (1.3), and (1.4) are true. From (1.1), we have
Also, similarly from (1.1), we have
Also, we have
and
□
Corollary 1. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0 and b ≠ 0. Also, if ad, cb ∈ (1, 2) and b > a then we have
and
Proof. From ad, cb ∈ (1, 2) and b > a we have 0 < ad -1 < 1 and 0 < cb - 1 < 1.
Hence, we obtain
and
Similarly, from ad, cb ∈ (1, 2) and b > a, we have 0 < ad - 1 < 1 and 0 < cb - 1 < 1.
Hence, we obtain
and
□
Corollary 2. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0 and b ≠ 0. If a = b and cb = ad = 2 then we have
and
Proof. From a = b and cb = ad = 2 then we have, cb - 1 = ad - 1 = 1. Hence, we have
and
Also, we have
and
Similarly, we have
and
□
Corollary 3. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0 and b ≠ 0. Also, if 0 < a, b, c, d, e, f < 1 then we have
and
Proof. From 0 < a, b, c, d, e, f < 1 we have -1 < ad - 1 < 0 and - 1 < cb - 1 < 0. Hence, we obtain
and
Similarly, we have
and
□
Corollary 4. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0, and b ≠ 0. Also, if 0 < a, b, c, d, e, f < 1 then we have
and
Proof. The proof is clear from Theorem 1. □
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Kurbanli, A.S. On the behavior of solutions of the system of rational difference equations . Adv Differ Equ 2011, 40 (2011). https://doi.org/10.1186/1687-1847-2011-40
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DOI: https://doi.org/10.1186/1687-1847-2011-40