Positive Solutions of the Semipositone Neumann Boundary Value Problem - Henderson, Johnny; Kosmatov, Nickolai (Mathematical Modelling and Analysis, 2015)
In this paper we consider the Neumann boundary value problem at resonance −u’’(t) = f t, u(t) , 0 < t < 1, u’ (0) = u’ (1) = 0. We assume that the nonlinear term satisfies the inequality f(t, z) + α2z + β(t) ≥ 0, t ∈ [0, 1], z ≥ 0, where β : [0, 1] → R+, and α ≠ 0. The problem is transformed into a non-resonant positone problem and positive solutions are obtained by means of a Guo–Krasnoselskii fixed point theorem.