Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem

By using Krasnoselskii’s fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem Δ2u t − 1 a t f u t 0, t ∈ {1, 2, . . . , T}, u 0 β ∑η s 1 u s , u T 1 α ∑η s 1 u s , where f is continuous, T ≥ 3 is a fixed positive integer, η ∈ {1, 2, ..., T − 1}, 0 < α < 2T 2 /η η 1 , 0 < β < 2T 2 − αη η 1 /η 2T − η 1 , and Δu t − 1 u t − u t − 1 . We show the existence of at least one positive solution if f is either superlinear or sublinear.