On basicity of the system of eigenfunctions of one discontinuous spectral problemfor 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)⊕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.

On basicity of exponential and trigonometric systems in grand Lebesgue spaces

“…Let us extend the function f by zero to the entire axis R , i.e. [34,35]). Next, we need the fact that the singular integral is bounded in grand Lebesgue spaces.…”

“…Questions of solvability of Dirichlet problems for elliptic equations were considered in [7,8]. Korovkin-type theorems, as well as spectral problems with a spectral parameter in boundary conditions in grand Lebesgue spaces, were studied in [34] and [35], respectively. Note that the problem considered in [35] in Morrey-type spaces was studied in [3].…”

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

“…Spectral problems have a pivotal role in many branches of mathematics, mechanics and mathematical physics (for instance with the Fourier method many boundary value problems for partial differential equations (mostly mixed or elliptic types) are reduced to investigating corresponding spectral problems 5 for differential operators regarding space coordinates). It should be noted that many problems encountered in various fields of natural sciences are reduced to the spectral problems regarding discontinuous differential operators (for more information one can see works [1,2,3,4,5,6,7]).…”

“…It should be noted that the spectral problem (1)-( 2) was considered in Morrey type space [10,11], grand-Lebesgue space [7] and weighted Lebesgue spaces [12].…”

“…Lots of basic problems of analysis are required for the theory of differential equations and its union with harmonic analysis in these spaces. Many classical facts of harmonic analysis such as boundedness problem of singular operator with Cauchy kernel, maximal function, Hilbert transform have been extended to these spaces (for more details, see [28,17,29,30,3,31,32,33,34,35,36,37,38,39,40,41,7]). These results stimulate consideration of the problems of the theory of differential equations and the theory of partial differential equations in these spaces.…”