Periodic Solutions of a System of Nonlinear Difference Equations with Periodic Coefficients

Abstract

This paper is dealt with the following system of difference equations$x_{n+1}=\left(a_n/x_n\right)+\left(b_n/y_n\right),y_{n+1}=\left(c_n/x_n\right)+\left(d_n/y_n\right),$where$n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, the initial values$x_0\hspace{0.17em}\text{and}\hspace{0.17em}{y}_0$are the positive real numbers, and the sequences${\left(a_n\right)}_{n\ge 0}$,${\left(b_n\right)}_{n\ge 0}$,${\left(c_n\right)}_{n\ge 0}$, and${\left(d_n\right)}_{n\ge 0}$are two-periodic and positive. The system is an extension of a system where every positive solution is two-periodic or converges to a two-periodic solution. Here, the long-term behavior of positive solutions of the system is examined by using a new method to solve the system.