Abstract
This paper deals with the boundedness, persistence, and global asymptotic stability of positive solution for a system of third-order rational difference equationsxn+1=A+xn/yn-1yn-2,yn+1=A+yn/xn-1xn-2,n=0,1,…, whereA∈(0,∞),x-i∈(0,∞);y-i∈(0,∞),i=0,1,2. Some examples are given to demonstrate the effectiveness of the results obtained.
Highlights
It is known that difference equation appears naturally as discrete analogous and as numerical solutions of differential equation and delay differential equation having many applications in economics, biology, computer science, control engineering, and so forth
The study of discrete dynamical systems described by difference equations has been paid great attention by many mathematical researchers
In 1998, DeVault et al [1] proved every positive solution of the difference equation: xn+1
Summary
It is known that difference equation appears naturally as discrete analogous and as numerical solutions of differential equation and delay differential equation having many applications in economics, biology, computer science, control engineering, and so forth. The persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed in many papers. In 1998, DeVault et al [1] proved every positive solution of the difference equation: xn+1. It is very interesting to investigate the qualitative behavior of the discrete dynamical systems of nonlinear difference equations. Similar results in [12,13,14,15,16,17] have been derived for systems of two nonlinear difference equations. Papaschinopoulos and Schinas [12] investigated the global behavior for a system of the following two nonlinear difference equations: yn , xn−p yn+1. In 2012, Zhang et al [14] investigated the global behavior for a system of the following third-order nonlinear difference equations: xn−2. (iv) If 1 < A < 2/√3, (a1, b1) and (a2, b2) are locally asymptotically stable
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