On positiveness and contractiveness of the integral operator arising from the beam deflection problem on elastic foundation

We provide a complete proof that there are no nontrivial eigenvalues of the integral operator $\mathcal{K}_l$ outside the interval $(0,1/k)$. $\mathcal{K}_l$ arises naturally from the deflection problem of a beam with length $l$ resting horizontally on an elastic foundation with spring constant $k$, while some vertical load is applied to the beam.