On solutions of some nonlinear recurrences (Q1088984)

Conditions which ensure that there is a unique solution in certain non- linear recurrences in vector lattices are studied. The application to orthogonal polynomials is considered, where the lattice is R. Recurrences of the type \(x_ n=f_ n(x_{n-k_ 1},...,x_{n-k_ m})\) for \(n\geq 1\) are studied in more detail. Let L be an order continuous Banach lattice and \(f_ n:L^ m\to L\) be a sequence of functions and \(k_ i\in Z\setminus \{0\}\) for \(i=1,...,m\). If \(f_ n:L^ n\to L\) is continuous, positive and decreasing on \(L^ m_+\) or if a certain inequality is fulfilled for any two positive solutions then there is a unique positive solution to the above recurrence. for \(\gamma^ 2/a_ n=a_{n+1}+a_ n+a_{n-1}+\beta\) for \(n\geq 1\), \(a_ 0=a\) a positive solution exists and conditions are given for this solution to be unique.