On the global asymptotic stability of a system of difference equations with quadratic terms
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On the global asymptotic stability of a system of difference equations with quadratic terms Erkan Ta¸sdemir1 Received: 15 July 2020 / Revised: 29 September 2020 / Accepted: 3 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we study the global asymptotic stability of following system of difference equations with quadratic terms: xn+1 = A + B y 2yn , yn+1 = A + B x x2 n where A and n−1
n−1
B are positive numbers and the initial values are positive numbers. We also investigate the rate of convergence and oscillation behaviour of the solutions of related system. Keywords Difference equations · Dynamical systems · Global stability · Rate of convergence · Boundedness · Oscillation Mathematics Subject Classification 39A10 · 39A23 · 39A30
1 Introduction Over the last two decades, difference equations or their systems have been huge attention among researchers which are mathematicians or not. Moreover difference equations or systems have too many applications between many branches of science. For example, in [13] Khan et al. studied global dynamics of an one-dimensional discrete-time laser model. Further in [7] Din et al. studied stability of a discrete ecological model. There are many examples related to applications of difference equations or systems. Therefore, studies on difference equations are increasing day by day and will continue to increase. There has been a lot of study about the dynamics of solutions of difference equations or systems (for example, see [1,6,9,10,12,14,15,17,18,20– 23,25,26]). Additionally, there are many papers related to our study as follows:
B 1
Erkan Ta¸sdemir [email protected] Pınarhisar Vocational School, Kırklareli University, 39300 Kırklareli, Turkey
123
E. Ta¸sdemir
In [24], Yang et al. investigated the solutions, stability and asymptotic behaviour of the system of the two nonlinear difference equations Axn Byn p , yn+1 = p. 1 + yn 1 + xn
xn+1 =
In [8], Elabbasy et al. studied the global behaviour of following system of difference equations xn+1 =
a1 x n b1 yn , yn+1 = . r a2 + a3 yn b2 + b3 xnr
In [3], Bacani et al. considered solutions of the following two nonlinear difference equations xn+1 =
q q , yn+1 = . v p + xn − p + ynv
In [11], Hadziabdic et al. studied global behaviours of following system of difference equations xn+1 =
b1 xn2 a2 + c2 yn2 , y = . n+1 A1 + yn2 xn2
In [5], Burgic et al. investigated the global stability properties and asymptotic behaviour of solutions for the system of difference equations xn+1 =
xn yn , yn+1 = . 2 a + yn b + xn2
In [4], Beso et al. studied boundedness of solutions of following difference equation xn+1 = γ + δ
xn 2 xn−1
.
They also investigated global asymptotic stability of related difference equation. Motivated by difference equations and their systems, we consider the following system of difference equations xn+1 = A + B
yn 2 yn−1
, yn+1 = A + B
xn 2 xn−1
(1)
where A and B are positive numbers and the initial values are positive numbers. In th
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