Periodicity and attractivity of a nonlinear higher order difference equation (Q1763306)

The authors consider the difference equation of the form \[ x_{n+1}= (a- bx_{n-k})(A+ x_{n-h})^{-1},\quad n\geq 0,\quad n\in\mathbb{Z},\tag{1} \] where \(a\geq 0\), \(A> 0\), \(b> 0\), \((k,h)\in \mathbb{Z}^2\), and study the periodicity, permanence and attractivity of equation (1). If one of the following propositions \(P\), \(Q\), \(R\) is true then the equation (1) has no two-period positive solution: \(P\): \(A> b> 0\), \(Q\): \((A= b)\wedge((k\) is even and \(h\) is odd) \(\vee\) (\(k\) is odd and \(h\) is even) \(\vee\) (\(k\text{ and }h\text{ are odd}))\), \(R\): \((A< b)\wedge((k\text{ and }h\) are odd