Basisness of Fucik eigenfunctions for the Dirichlet Laplacian

Abstract

We provide improved sufficient assumptions on sequences of Fučík eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fučík eigenfunctions form a Riesz basis in 𝐿 2 (0, 𝜋). For that purpose, we introduce a criterion for a sequence in a Hilbert space to be a Riesz basis.

Basisness and completeness of Fucik eigenfunctions for the Neumann Laplacian

“…Namely, we want to know if such systems of so-called Fučík eigenfunctions are complete in the real Hilbert space 𝐿 2 (0, 𝜋) or even form a Riesz basis in this space. To the best of our knowledge, basisness for systems of Fučík eigenfunctions has been studied only under Dirichlet boundary conditions, see our previous works [3,4]. We emphasize that the considered problem is a modification of the form of the standard eigenvalue problem for the Laplacian which it is not in the scope of the classical spectral theorem.…”