On upper and lower functions for a fourth-order ordinary differential equation. I (Q2468717)

The author proves the existence of solutions for the problem \[ x''''=f(t,x,x''),\, t \in [0,\tau],\quad x(0)=x''(0)=x(\tau)=x''(\tau)=0 \] with \(f\) a continuous function. By using a more restrictive definition than usual of lower and upper solutions, the existence of at least one solution of the considered problem lying between both functions is deduced. The proofs are based on the properties of the related Green's function.