Continuity in a parameter of solutions to linear boundary-value problems in Hölder–Zygmund spaces

1Murach, AA, 1Soldatov, VO
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2016, 10:15-21
https://doi.org/10.15407/dopovidi2016.10.015
Section: Mathematics
Language: Ukrainian
Abstract: 

We introduce and investigate the broadest class of linear boundary-value problems for the systems of first-order ordinary differential equations, whose solutions belong to the complex Hölder–Zygmund space. For these problems, we establish a constructive criterion, under which their solutions are continuous in a parameter in this space.

Keywords: boundary-value problem, continuity in a parameter, Hölder–Zygmund space, system of differential equations
References: 
  1. Gikhman I.I. Ukr. Mat. Zh., 1952, 4, No 2: 215-219 (in Russian).
  2. Krasnosel'skii M.A., Krein S.G. Uspekhi Mat. Nauk, 1955, 10, No 3: 147-153 (in Russian).
  3. Kurzweil J., Vorel Z. Czechoslovak Math. J., 1957, 7, No 4: 568-583 (in Russian).
  4. Samoilenko A.M. Ukr. Mat. Zh., 1962, 14, No 3: 289—298 (in Russian). https://doi.org/10.1007/BF02526637
  5. Kiguradze I.T. Some singular boundary-value problems for ordinary differential equations, Tbilisi: Izd-vo Tbilisi University, 1975 (in Russian).
  6. Kiguradze I.T. J. Soviet Math., 1988, 43, Iss. 2: 2259-2339. https://doi.org/10.1007/BF01100360
  7. Ashordia M. Czechoslovak Math. J., 1996, 46, No 3: 385—404.
  8. Mikhailets V.A., Reva N.V. Dopov. Nac. acad. nauk Ukr. 2008, No 9: 23-27 (in Russian).
  9. Kodlyuk T.I., Mikhailets V.A., Reva N.V. Ukr. Math. J., 2013, 65, No 1: 77-90. https://doi.org/10.1007/s11253-013-0766-x
  10. Mikhailets V.A., Chekhanova G.A. J. Math. Sci., 2015, 204, No 3: 333-342. https://doi.org/10.1007/s10958-014-2205-4
  11. Mikhailets V.A., Reva N.V. Dopov. Nac. akad. nauk Ukr. 2008, No 8: 28-30 (in Russian).
  12. Kodlyuk T.I., Mikhailets V.A. J. Math. Sci., 2013, 190, No 4: 589-599. https://doi.org/10.1007/s10958-013-1272-2
  13. Gnyp E.V., Kodlyuk T.I., Mikhailets V.A. Ukr. Math. J., 2015, 67, No 5: 658-667. https://doi.org/10.1007/s11253-015-1105-1
  14. Mikhailets V.A., Chekhanova G.A. Dopov. Nac. akad. nauk Ukr. 2014, No 7: 24-28 (in Russian).
  15. Triebel H. Interpolation theory, function spaces, differential operators, 2-nd ed., Heidelberg: Johann Ambrosius Barth, 1995.