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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 L2(0,1) if and only if the functions in this system are linearly independent in L2(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 L2(0,1) of the corresponding eigenfunctions is proved.