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)⊕C , 1 < p < +∞ , is proved. It is shown that the system of eigenfunctions of considered problem forms a basis for G p) (−1; 1) , 1 < p < +∞ , after removal of any of its even-numbered functions.
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.
The question of the basis property of a system of eigenfunctions of one
spectral problem for a discontinuous second-order differential operator with
a spectral parameter under discontinuity conditions is considered in the
weighted grand-Lebesgue spaces Lp),?(0, 1), 1 < p < +?, with a general
weight ?(?). These spaces are non-separable and therefore it is necessary to
define its subspace associated with differential equation. In this paper,
using the shift operator, a subspace Gp),?(0, 1) is considered, in which the
basis property of exponentials and trigonometric systems of sines and
cosines is established when the weight function ?(?) satisfies the
Muckenhoupt condition. It is proved that the system of eigenfunctions and
associated functions of the discontinuous differential operator
corresponding to the given problem forms a basis in the weighted space
Gp),?(0, 1) ? C,1 < p < +? with the weight ?(?) satisfying the Muckenhoupt
condition. The question of the defect basis property of the system of
eigenfunctions and associated functions of the given problem in the weighted
spaces Gp),?(0, 1),1 < p < +?, is considered.
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