Positive solutions of singular fourth-order boundary-value problems

Abstract

In this paper, we present necessary and sufficient conditions for the existence of positive C3 [0, 1] ∩C4 (0, 1) solutions for the singular boundary-value problem x′′′′(t) = p(t)ƒ(x(t)), t ∈ (0, 1); x(0) = x(1) = x′(0) = x′(1) = 0, where ƒ(x) is either superlinear or sublinear, p : (0, 1) → [0, +∞) may be singular at both ends t = 0 and t = 1. For this goal, we use fixed-point index results.