Global dynamics for a higher-order rational difference equation (Q370860)

Consider the rational difference equation of higher order NEWLINE\[NEWLINEx_{n+1}=\frac{px_{n}+x_{n-k}}{r+qx_{n}+x_{n-k}},~\;\;n=0,1,2,\dots,~\tag{\(*\)} NEWLINE\]NEWLINE where the parameters \(p,q\) and \(r\) are nonnegative real numbers, \(k\) is a positive integer, and the initial conditions \(x_{-k},\dots,x_{-1},x_{0}\) are nonnegative real numbers. The authors prove the following theorems:NEWLINENEWLINETheorem 1. The origin of equation (\(*\)) is globally asymptotically stable for \(p+1<r\).NEWLINENEWLINETheorem 2. The positive equilibrium \(x^{\ast }=(p+1-r)/(q+1)\) is locally asymptotically stable, provided one of the following conditions is satisfied: NEWLINENEWLINE(i) \(1\leq q\leq p-r\) or \(q\leq p-r<(3q+1)/(1-q)\); NEWLINENEWLINE(ii) \(q>p-r\) and \(q(1-r)\leq p\) or \(q(1-r)>p>[(1-r)(q-1)]/(q+3)\).NEWLINENEWLINEThe existence of eventual prime period two solutions, the existence and asymptotic behavior of non-oscillatory solutions are investigated.