Existence of positive solutions for an elastic beam equation with nonlinear boundary conditions (Q1714823)

Summary: We study the existence and nonexistence of positive solutions for the following fourth-order two-point boundary value problem subject to nonlinear boundary conditions \(u^{\prime\prime\prime\prime}(t)=\lambda f(t,u(t)), t\in(0,1), u(0)=0,u^\prime(0)=\mu h(u(0)), u^{\prime\prime}(1)=0,u^{\prime\prime\prime}(1)=\mu g(u(1))\), where \(\lambda>0,\mu\geq 0\) are parameters, and \(f:[0,1]\times[0,+\infty)\to(0,+\infty),h:[0,+\infty)\to[0,+\infty)\), and \(g:[0,+\infty)\to(-\infty,0]\) are continuous. By using the fixed-point index theory, we prove that the problem has at least one positive solution for \(\lambda,\mu\) sufficiently small and has no positive solution for \(\lambda\) large enough.