Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem (Q618822)

In this paper, the existence and uniqueness of a positive solution is established for the fourth-order boundary value problem with Lidstone boundary conditions \[ u^{(4)}(t)=f(t,u(t)), \,\,\, t\in (0,1), \] \[ u(0)=u(1)=0=u''(0)=u''(1), \] where the function \(f:[0,1]\times [0,\infty)\to [0,\infty)\) is continuous and nondecreasing with respect to the second variable. The analysis relies on a fixed point theorem in partially ordered sets.