On basicity of the system of eigenfunctions of one discontinuous spectral problem for second order differential equation for grand-Lebesgue space

Abstract

Basicity of the system of eigenfunctions of some discontinuous spectral problem for a second order differential equation with spectral parameter in boundary condition for grand-Lebesgue space Lp)(−1; 1) is studied in this work. Since the space is nonseparable, a subspace suitable for the spectral problem is defined. The subspace Gp)(−1; 1) of Lp)(−1; 1) generated by shift operator is considered. Basicity of the system of eigenfunctions for the space Gp)(−1; 1)⊕C , 1 < p < +∞, is proved. It is shown that the system of eigenfunctions of considered problem forms a basis for Gp)(−1; 1), 1 < p < +∞, after removal of any of its even-numbered functions.