A singular third-order 3-point boundary-value problem with nonpositive Green's function

Abstract

We find a Green's function for the singular third-order three-point BVP u‴(t) = -α(t)ƒ(t, u(t)), u(0) = u′(1) = u″(η) = 0 where 0 ≤ η < 1/2. Then we apply the classical Krasnosel'skii's fixed point theorem for finding solutions in a cone. Although this problem Green's function is not positive, the obtained solution is still positive and increasing. Our techniques rely on a combination of a fixed point theorem and the properties of the corresponding vector field.