On basicity of eigenfunctions of one discontinuous differential operator in Banach function spaces

This article deals with the basicity properties of the eigenfunctions system of a second-order discontinuous differential operator containing spectral parameters at boundary conditions in Banach function spaces. Investigations are divided into two groups depending on being a rearrangement-invariant space. By imposing some conditions on the Boyd indices of rearrangement-invariant Banach function spaces, we prove some important properties of the basicity of the eigenfunctions system of the spectral problem in suitable separable subspaces of these spaces. These properties are valid in Lebesgue, grand-Lebesgue, Orlics, and Marcinkiewicz. Also, results regarding the basicity of the system of eigenfunctions in non-rearrangement invariant spaces that are a direct sum of rearrangement-invariant spaces with different finite Boyd indices have been obtained. The discontinuity of the differential operator makes it possible to examine the basicity properties of the system of eigenfunctions in the direct sum space of spaces with finite Boyd indices.