For systems of linear differential equations of order r ∈ ℕ, we study the most general class of inhomogeneous boundary-value problems whose solutions belong to the Sobolev space W n + r p ([a, b],ℂ m), where m, n + 1 ∈ ℕ and p ∈ [1,∞). We show that these problems are Fredholm problems and establish the conditions under which these problems have unique solutions continuous with respect to the parameter in the norm of this Sobolev space.
Similar content being viewed by others
References
T. I. Kodlyuk, V. A. Mikhailets, and N. V. Reva, “Limit theorems for one-dimensional boundary-value problems,” Ukr. Math. J., 65, No. 1, 77–90 (2013).
I. T. Kiguradze, “Boundary-value problems for systems of ordinary differential equations,” in: VINITI Series in Contemporary Problems of Mathematics (Latest Achievements) [in Russian], 30, VINITI, Moscow (1987), pp. 3–103.
I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilisi University, Tbilisi (1975).
I. T. Kiguradze, “On boundary-value problems for linear differential systems with singularities,” Differents. Uravn., 39, No. 2, 198–209 (2003).
V. A. Mikhailets and N. V. Reva, “Limit transition in systems of linear differential equations,” Dop. Nats. Akad. Nauk Ukr., No. 8, 28–30 (2008).
V. A. Mikhailets and N. V. Reva, “Generalizations of the Kiguradze theorem on well-posedness of linear boundary-value problems,” Dop. Nats. Akad. Nauk Ukr., No. 9, 23–27 (2008).
V. A. Mikhailets and G. A. Chekhanova, “Limit theorems for general one-dimensional boundary-value problems,” J. Math. Sci., 204, No. 3, 333–342 (2015).
V. A. Mikhailets and G. A. Chekhanova, “Fredholm boundary-value problems with parameter in the spaces C (n)[a, b],” Dopov. Nats. Akad. Nauk Ukr., No. 7, 24–28 (2014).
T. I. Kodlyuk and V. A. Mikhailets, “Solutions of one-dimensional boundary-value problems with a parameter in Sobolev spaces,” J. Math. Sci., 190, No. 4, 589–599 (2013).
A. S. Goriunov and V. A. Mikhailets, “Resolvent convergence of Sturm–Liouville operators with singular potentials,” Math. Notes, 87, No. 1–2, 287–292 (2010).
A. S. Goriunov and V. A. Mikhailets, “Regularization of singular Sturm–Liouville equations,” Meth. Funct. Anal. Topol., 16, No. 2, 120–130 (2010).
A. S. Goriunov, V. A. Mikhailets, and K. Pankrashkin, “Formally self-adjoint quasidifferential operators and boundary-value problems,” Electron. J. Different. Equat., No. 101, 1–16 (2013).
N. Dunford and J. T. Schwartz, Linear Operators. Part 2. Spectral Theory. Self-Adjoint Operators in Hilbert Space, Interscience Publ., New York (1963).
V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1989).
H. Triebel, Theory of Function Spaces, Akadem. Verlags. Geest & Portig K.-G., Leipzig (1983).
V. A. Mikhailets and A. A. Murach, H¨ormander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin (2014).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 5, pp. 584–591, May, 2015.
Rights and permissions
About this article
Cite this article
Gnyp, E.V., Kodlyuk, T.I. & Mikhailets, V.A. Fredholm Boundary-Value Problems with Parameter in Sobolev Spaces. Ukr Math J 67, 658–667 (2015). https://doi.org/10.1007/s11253-015-1105-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1105-1