Existence of \(n\) solutions and/or positive solutions to a semipositone elastic beam equation (Q860726)

This paper considers nonlinear fourth order boundary value problems of the form \[ u^{(4)}(t)=f(t,u(t),u''(t)), \] \[ u(0)=u(1)=u''(0)=u''(1)=0. \] Under certain conditions on the nonlinearity \(f\), it is proven that the problem has at least \(n\) solutions, where \(n\) is any positive integer, under additional conditions these solutions are positive. The proofs of these results are based on the Krasnosel'skii fixed point theorem for compact mappings in subsets of cones.