Existence of positive solutions for certain partial difference equations (Q2491478)

This paper deals with homogeneous linear partial difference equations \[ aA_{m+1,n+1}+bA_{m,n+1}+cA_{m+1,n} -dA_{m,n}+ P_{m,n}A_{m-k,n-l}=0 \] for positive reals \(a,b,c,d,P_{m,n}\) on a lattice \(\{(m,n)\in\mathbb Z^2: m\geq m_0, n\geq n_0\}\); the delays \(k,l\) are nonnegative integers. Using two different approaches the authors deduce a variety of sufficient criteria for eventual positivity of solutions \(A_{m,n}\). The first approach relies on an order theoretical result (Knaster's fixed point theorem), while the second method is more constructive and based on a representation of the solutions. Both methods are illustrated using an example.