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 L-p) (-1; 1) is studied in this work. Since the space is nonseparable, a subspace suitable for the spectral problem is defined. The subspace G(p)) (-1;1) of L-p) (-1; 1) generated by shift operator is considered. Basicity of the system of eigenfunctions for the space G(p) )(-1;1)circle plus C, 1 < p < +infinity, is proved. It is shown that the system of eigenfunctions of considered problem forms a basis for G(p)()) (-1;1), 1 < p < +infinity, after removal of any of its even-numbered functions.