Completeness of systems of eigenfunctions for the Sturm–Liouville operator with potential depending on the spectral parameter and for one non-linear problem

The eigenvalue problem for the Sturm–Liouville operator on the closed interval $[0,1]$ with potential depending on the spectral parameter and with zero Dirichlet boundary conditions is considered first. It is proved under certain assumptions about the potential that if a system of eigenfunctions of this problem contains a unique function with $n$ zeros in the interval $(0,1)$ for each non-negative integer $n$, then it is complete in the space $L_2(0,1)$ if and only if the functions in this system are linearly independent in $L_2(0,1)$. Next, this result is used in the study of the spectral problem for a certain non-linear operator of Sturm–Liouville type. The completeness in $L_2(0,1)$ of the corresponding eigenfunctions is proved.