Positive solutions for semipositone fourth-order two-point boundary value problems

In this paper we investigate the existence of positive solutions of the following nonlinear semipositone fourth-order two-point boundary-value problem with second derivative: $$displaylines{ u^{(4)}(t) = f(t, u(t), u''(t)), quad 0 leq t leq 1, cr u'(1) = u''(1) = u'''(1) = 0, quad k u(0) = u'''(0), }$$ where $-6 < k < 0$, $f geq - M$, and $M$ is a positive constant. Our approach relies on the Krasnosel'skii fixed point theorem