Existence results for a fully fourth-order boundary value problem (Q2392174)

Summary: We discuss the existence of solution for the fully fourth-order boundary value problem \[ \begin{aligned} u^{(4)} & = f(t, u, u', u'', u'''), \quad 0 \leq t \leq 1,\\ u(0) &= u(1) = u''(0) = u''(1) = 0. \end{aligned} \] A growth condition on \(f\) guaranteeing the existence of solution is presented. The discussion is based on Fourier analysis and the Leray-Schauder fixed point theorem.