On solvability of inhomogeneous boundary-value problems in Sobolev - Slobodetskiy spaces

1Mikhailets, VA
Skorobohach, TB
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2020, 4:10-14
https://doi.org/10.15407/dopovidi2020.04.010
Section: Mathematics
Language: English
Abstract: 

We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev—Slobodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boun dary- value problem. The sufficient conditions for the sequence of the characteristic matrices of a specified bounda ryvalue problems to converge are found.

Keywords: Fredholm operator, index of operator, inhomogeneous boundary-value problem, Sobolev—Slobodetskiy 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. (N.Y.), 190, No. 4, pp. 589-599. Doi: 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. Doi: https://doi.org/10.1007/s11253-015-1105-1
3. Hnyp, Y., Mikhailets, V. & Murach, A. (2017). Parameter-depent one-dimensional boundary-value problems in Sobolev spaces. Electron. J. Differ. Equat., No. 81, 13 pp.
4. Atlasiuk, O. M. & Mikhailets, V. A. (2019). Fredholm one-dimensional boundary-value problems in Sobolev spaces. Ukr. Math. J., 70, No. 10, pp. 1526-1537. Doi: https://doi.org/10.1007/s11253-019-01588-w
5. Atlasiuk, O. M. & Mikhailets, V. A. (2019). Fredholm one-dimensional boundary-value problems with a parameter in Sobolev spaces. Ukr. Math. J., 70, No. 11, pp. 1677-1687. Doi: https://doi.org/10.1007/s11253-019-01599-7
6. Atlasiuk, O. M. & Mikhailets, V. A. (2019). On the solvability of inhomogeneous boundary-value problems in Sobolev spaces. Dopov. Nac. akad. nauk Ukr., No. 11, pp. 3-7. Doi: https://doi.org/10.15407/dopovidi2019.11.003
7. Hörmander, L. (1985). The analysis of linear partial differential operators. III: Pseudo-differential operators. Berlin: Springer.