Open Access

A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems

Abstract

We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}.$ The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.

Key words: Differential system, boundary-value problem, continuity in parameter.

Article Information

 Title A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 4, 375-386 MathSciNet MR3591086 Milestones Received 15/08/2016 Copyright The Author(s) 2016 (CC BY-SA)

Authors Information

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine; National Technical University of Ukraine ”Kyiv Polytechnic Institute”, 37 Prospect Peremogy, Kyiv, 03056, Ukraine

Aleksandr Murach
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine; Chernihiv National Pedagogical University, 53 Het’mana Polubotka, Chernihiv, 14013, Ukraine

Vitalii Soldatov
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine

Citation Example

Vladimir Mikhailets, Aleksandr Murach, and Vitalii Soldatov, A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems, Methods Funct. Anal. Topology 22 (2016), no. 4, 375-386.

BibTex

@article {MFAT915,
AUTHOR = {Mikhailets, Vladimir and Murach, Aleksandr and Soldatov, Vitalii},
TITLE = {A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {4},
PAGES = {375-386},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=915},
}

References

1. Malkhaz Ashordia, Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations, Czechoslovak Math. J. 46 (1996), no. 3, 385-404.  MathSciNet
2. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I: General theory, Interscience, New York, 1958.  MathSciNet
3. I. I. Gihman, Concerning a theorem of N. N. Bogolyubov, Ukr. Mat. Zh. 4 (1952), 215-219 (Russian).  MathSciNet
4. E. V. Gnyp, T. I. Kodlyuk, and V. A. Mikhailets, Fredholm boundary-value problems with parameter in Sobolev spaces, Ukrainian Math. J. 67 (2015), no. 5, 658-667.  MathSciNet CrossRef
5. Andrii Goriunov, Vladimir Mikhailets, and Konstantin Pankrashkin, Formally self-adjoint quasi-differential operators and boundary-value problems, Electron. J. Differential Equations (2013), no. 101, 1-16.  MathSciNet
6. A. S. Goryunov and V. A. Mikhailets, Resolvent convergence of Sturm—Liouville operators with singular potentials, Math. Notes 87 (2010), no. 1-2, 287-292.  MathSciNet CrossRef
7. A. S. Goryunov and V. A. Mikhailets, Regularization of two-term differential equations with singular coefficients by quasiderivatives, Ukrainian Math. J. 63 (2012), no. 9, 1361-1378.  MathSciNet CrossRef
8. E. Hnyp, V. Mikhailets, and A. Murach, A criterion for continuity in a parameter of solutions to generic boundary-value problems for differential systems in Sobolev spaces (to appear).,
9. Lars Hormander, The analysis of linear partial differential operators. III: Pseudo-differential operators., Springer-Verlag, Berlin, 1985.
10. I. T. Kiguradze, Some singular boundary value problems for ordinary differential equations, Tbilisi University, Tbilisi, 1975 (Russian).  MathSciNet
11. I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations, J. Soviet Math. 43 (1988), no. 2, 2259-2339.  MathSciNet
12. I. T. Kiguradze, On boundary value problems for linear differential systems with singularities, Differential Equations 39 (2003), no. 2, 212-225.  MathSciNet CrossRef
13. T. I. Kodliuk and V. A. Mikhailets, Multipoint boundary-value problems with parameter in Sobolev spaces, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2012), no. 11, 15-19 (Russian).
14. Tatiana I. Kodliuk and Vladimir A. Mikhailets, Solutions of one-dimensional boundary-value problems with a parameter in Sobolev spaces, J. Math. Sci. (N. Y.) 190 (2013), no. 4, 589-599.  MathSciNet CrossRef
15. T. I. Kodlyuk, V. A. Mikhailets, and N. V. Reva, Limit theorems for one-dimensional boundary-value problems, Ukrainian Math. J. 65 (2013), no. 1, 77-90.  MathSciNet CrossRef
16. M. A. Krasnoselskii and S. G. Krein, On the principle of averaging in nonlinear mechanics, Uspehi Mat. Nauk (N.S.) 10 (1955), no. 3, 147-152 (Russian).  MathSciNet
17. Yaroslav Kurcveil and Zdenek Vorel, Continuous dependence of solutions of differential equations on a parameter, Czechoslovak Math. J. 7 (82) (1957), 568-583 (Russian).  MathSciNet
18. A. Ju. Levin, Passage to the limit for nonsingular systems $\dot X=A_n(t)X.$, Dokl. Akad. Nauk SSSR 176 (1967), 774-777 (Russian).  MathSciNet
19. V. A. Mikhailets and G. A. Chekhanova, Fredholm boundary-value problems with parameter on the spaces $C^(n)[a;b]$, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki} YEAR = {2014 no. 7, 24-28 (Russian).
20. V. A. Mikhailets and G. A. Chekhanova, Limit theorems for general one-dimensional boundary-value problems, J. Math. Sci. (N. Y.) 204 (2015), no. 3, 333-342.  MathSciNet CrossRef
21. V. A. Mikhailets, A. A. Murach, and V. Soldatov, Continuity in a parameter of solutions to generic boundary-value problems, Electron. J. Qual. Theory Differ. Equ. (to appear) (2016), no. 87, 1-16.
22. V. A. Mikhailets and N. V. Reva, Passage to the limit in systems of linear differential equations, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2008), no. 8, 28-30 (Russian).  MathSciNet
23. Tkhe Khoan Nguen, Dependence of the solutions of a linear system of differential equations on a parameter, Differential Equations 29 (1993), no. 6, 830-835.  MathSciNet
24. Zdzislaw Opial, Continuous parameter dependence in linear systems of differential equations, J. Differential Equations 3 (1967), 571-579.  MathSciNet
25. William T. Reid, Some limit theorems for ordinary differential systems, J. Differential Equations 3 (1967), 423-439.  MathSciNet
26. Frigyes Riesz and Bela Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955.  MathSciNet
27. V. O. Soldatov, On the continuity in a parameter for the solutions of boundary-value problems total with respect to the spaces $C^ (n+r)[a,b]$, Ukrainian Math. J. 67 (2015), no. 5, 785-794.  MathSciNet CrossRef