On the solution for a system of two rational difference equations (Q2794911)

The authors consider the system of two rational difference equations NEWLINE\[NEWLINE \displaystyle{x_{n+1} = {{x_{n-3}}\over{A+x_{n-3}y_{n-1}}}\;,\;y_{n+1} = {{y_{n-3}}\over{B+y_{n-3}x_{n-1}}}}, NEWLINE\]NEWLINE with \(A>0\), \(B>0\) and positive initial conditions \(x_i,y_i\), \(i=-3,-2,-1,0\). The following main results are provedNEWLINENEWLINE\(1^o\) If \(A>1\), \(B>1\), then \((0,0)\) is the unique equilibrium point of the system.NEWLINENEWLINE\(2^o\) For all positive solutions of the system, the authors obtain upper bounds of the solutions which depend on their own initial conditions, e.g. NEWLINE\[NEWLINE 0\leq x_n\leq x_{-3}/A^{k+1}\;,\;n=4k+1\;;\;0\leq y_n\leq y_{-3}/B^{k+1}\;,\;n=4k+1, NEWLINE\]NEWLINE etc.NEWLINENEWLINE\(3^o\) If \(A>1\), \(B>1\), the solutions converge exponentially to the equilibrium point. Also, the equilibrium is globally asymptotically stable.NEWLINENEWLINE\(4^o\) If \(A<1\), \(B<1\), the equilibrium is unstable.NEWLINENEWLINE\(5^o\) If \(A=B=1\), then every solution with positive initial conditions is bounded.NEWLINENEWLINEFinally, the complete expressions of the solutions in the case \(A=B\) are given.