The solvability of inhomogeneous boundary-value problems in Sobolev spaces

0583486 - MÚ 2025 RIV PL eng J - Článek v odborném periodiku
Mikhailets, Volodymyr - Atlasiuk, Olena
The solvability of inhomogeneous boundary-value problems in Sobolev spaces.
Banach Journal of Mathematical Analysis. Roč. 18, č. 2 (2024), č. článku 12. ISSN 2662-2033. E-ISSN 1735-8787
Institucionální podpora: RVO:67985840
Klíčová slova: Fredholm numbers of operator * inhomogeneous boundary-value problem * Sobolev space
Obor OECD: Pure mathematics
Impakt faktor: 1.2, rok: 2022
Způsob publikování: Open access
https://doi.org/10.1007/s43037-023-00316-8

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.
Trvalý link: https://hdl.handle.net/11104/0351446