Multiple positive solutions to third-order three-point singular semipositone boundary value problem

Abstract

By using a specially constructed cone and the fixed point index theory, this paper investigates the existence of multiple positive solutions for the third-order threepoint singular semipositone BVP: $$\left\{ \begin{gathered} x'''(t) - \lambda f(t,x) = 0,t \in (0,1); \hfill \\ x(0) = x'(\eta ) = x''(1) = 0, \hfill \\ \end{gathered} \right.$$ where 1/2 < η < 1, the non-linear term ƒ(t, x): (0, 1) × (0, + ∞) → (-∞, + ∞) is continuous and may be singular att = 0,t = 1, andx = 0, also may be negative for some values oft andx, λ is a positive parameter.