On Fredholm parameter-dependent boundary-value problems in Sobolev spaces

Atlasiuk, ОМ
Mikhailets, VА
Dopov. Nac. akad. nauk Ukr. 2020, 6:3-6
https://doi.org/10.15407/dopovidi2020.06.003
Section: Mathematics
Language: English
Abstract: 

We consider the most general class of linear inhomogeneous boundary-value problems for systems of r-th order ordinary differential equations whose solutions and right-hand sides belong to appropriate Sobolev spaces. For parameter-dependent problems from this class, we prove a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem.

Keywords: boundary-value problem, continuity in parameter, differential system, Sobolev space
References: 

1. Kodliuk, Т. I. & Mikhailets, V. А. (2013). Solutions of one-dimensional boundary-value problems with a parameter in Sobolev spaces. J. Math. Sci., 190, No. 4, pp. 589-599. https://doi.org/10.1007/s10958-013-1272-2
2. Gnyp, E. V., Kodliuk, Т. I. & Mikhailets, V. A. (2015). Fredholm boundary-value problems with parameter in Sobolev spaces. Ukr. Math. J., 67, No. 5, pp. 658-667. https://doi.org/10.1007/s11253-015-1105-1
3. Hnyp, Y. V., Mikhailets, V. A. & Murach, A. A. (2017). Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces. Electron. J. Different. Equat., No. 81, pp. 1-13.
4. Atlasiuk, O. M. & Mikhailets, V. A. (2019). Fredholm one-dimensional boundary-value problems with parameter in Sobolev spaces. Ukr. Math. J., 70, No. 11, pp. 1677-1687. https://doi.org/10.1007/s11253-019-01599-7
5. Atlasiuk, O. M. & Mikhailets, V. A. (2019). On Solvability of inhomogeneous boundary-value problems in Sobolev spaces. Dopov. Nac. Akad. Nauk Ukr., No. 11, pp. 3-7. https://doi.org/10.15407/dopovidi2019.11.003
6. Dunford, N. & Schwartz, J. T. (1958). Linear operators. I. General theory. New York, London: Interscience Publishers.
7. Maslyuk, H. O. & Mikhailets, V. A. (2018). Continuity in the parameter for the solutions of one-dimensional boundary-value problems for differential systems of higher orders in Slobodetskii spaces. Ukr. Math. J., 70, No. 3, pp. 467-476. https://doi.org/10.1007/s11253-018-1510-3
8. Masliuk, H. & Soldatov, V. (2018). One-dimensional parameter-dependent boundary-value problems in Hölder spaces. Methods Funct. Anal. Topology, 24, No. 2, pp. 143-151.