Positive Solutions of the Semipositone Neumann Boundary Value Problem

Abstract

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.

Keyword : Neumann boundary condition, resonanc, semipositon