A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems
DOI:
Keywords:
Differential system, boundary-value problem, continuity in parameterAbstract
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.References
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
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I: General theory, Interscience, New York, 1958. MathSciNet
I. I. Gihman, Concerning a theorem of N. N. Bogolyubov, Ukr. Mat. Zh. 4 (1952), 215-219 (Russian). MathSciNet
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
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
A. S. Goryunov and V. A. Mikhailets, Resolvent convergence of SturmâLiouville operators with singular potentials, Math. Notes 87 (2010), no. 1-2, 287-292. MathSciNet CrossRef
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
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).,
Lars Hormander, The analysis of linear partial differential operators. III: Pseudo-differential operators., Springer-Verlag, Berlin, 1985.
I. T. Kiguradze, Some singular boundary value problems for ordinary differential equations, Tbilisi University, Tbilisi, 1975 (Russian). MathSciNet
I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations, J. Soviet Math. 43 (1988), no. 2, 2259-2339. MathSciNet
I. T. Kiguradze, On boundary value problems for linear differential systems with singularities, Differential Equations 39 (2003), no. 2, 212-225. MathSciNet CrossRef
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).
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
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
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
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
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
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).
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
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.
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
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
Zdzislaw Opial, Continuous parameter dependence in linear systems of differential equations, J. Differential Equations 3 (1967), 571-579. MathSciNet
William T. Reid, Some limit theorems for ordinary differential systems, J. Differential Equations 3 (1967), 423-439. MathSciNet
Frigyes Riesz and Bela Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. MathSciNet
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
