Positive solutions for boundary value problems of second order difference equations and their computation

We consider the following two classes of second order boundary value problems for difference equation: Δ ( r i − 1 Δ y i − 1 ) − b i y i + λ a i y i = 0 , 1 ⩽ i ⩽ n , y 0 − τ y 1 = y n + 1 − δ y n = 0 with δ , τ ∈ [ 0 , 1 ] and Δ ( r i − 1 Δ y i − 1 ) − b i y i + λ a i y i = 0 , 1 ⩽ i ⩽ n , y 0 = α y n , y n + 1 = β y 1 with α , β ∈ [ 0 , 1 ] . We establish the existence of positive solutions to both problems. A solver with linear computational complexity for almost tridiagonal linear systems is developed by exploring the special structure of linear system of equations. Based on fast solvers for linear systems, effective algorithms for the computation of positive solutions will be proposed.