Positive solutions to a generalized second-order difference equation with summation boundary value problem (Q442962)

Summary: By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem \(\Delta^2 u(t - 1) + a(t)f(u(t)) = 0, t \in \{1, 2, \dots, T\}, u(0) = \beta \sum^\eta_{s=1} u(s), u(T + 1) = \alpha \sum^\eta_{s=1} u(s)\), where \(f\) is continuous, \(T \geq 3\) is a fixed positive integer, \(\eta \in \{1, 2, \dots, T - 1\}, 0 < \alpha < (2T + 2)/\eta(\eta + 1), 0 < \beta < (2T + 2 - \alpha \eta(\eta + 1))/\eta(2T - \eta + 1)\), and \(\Delta u(t - 1) = u(t) - u(t - 1)\). We show the existence of at least one positive solution if \(f\) is either superlinear or sublinear.