Basisness and completeness of Fucik eigenfunctions for the Neumann Laplacian

TL;DR

This paper studies the basis properties of Fučík eigenfunctions for the Neumann Laplacian, showing their completeness and Riesz basis properties under certain conditions, with explicit biorthogonal systems.

Contribution

It establishes the completeness and Riesz basis properties of Fučík eigenfunctions for the Neumann Laplacian and provides explicit conditions and biorthogonal systems.

Findings

Fučík eigenfunctions form a complete sequence in L^2(0,π).

They constitute a Riesz basis in the zero-mean subspace.

Explicit biorthogonal systems are constructed under certain eigenvalue conditions.

Abstract

We investigate the basis properties of sequences of Fucik eigenfunctions of the one-dimensional Neumann Laplacian. We show that any such sequence is complete in $L^2(0,\pi)$ and a Riesz basis in the subspace of functions with zero mean. Moreover, we provide sufficient assumptions on Fucik eigenvalues which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$ and we explicitly describe the corresponding biorthogonal system.