Positive periodic solutions for higher order functional difference equations (Q602836)

The paper is concerned with existence of positive periodic solutions for \[ x(n+m-k) - ax(n+m) - bx(n+k) + abx(n) = f(n,x(n-\tau(n)) \] where \(a\neq 1\), \(b\neq 1\) are positive constants, \(\tau : \mathbb Z\to\mathbb Z\) is \(\omega\)-periodic, \(f(\cdot,u)\) is \(\omega\)-periodic for any \(u\in \mathbb R\), \(\omega,n,k\in\mathbb N\), motivated by discrete time mathematical models of biology. The results are based on a fixed point theorem on Banach spaces endowed with a cone. Existence of at least one positive \(\omega\)-periodic solution is proved and, further, existence of at least \(N\) positive \(\omega\)-periodic solutions with \[ p_i\leq \|x_k\|\leq p_{k+1},\quad k=1,2,\dots,N \] where \(0<p_1<p_2<\dots< p_N<p_{N+1}\) are some constants that occur in the assumptions of the Theorem. Some examples are discussed.