Fredholm Boundary-Value Problems with Parameter in Sobolev Spaces
Article
First Online:
Received:
- 25 Downloads
- 4 Citations
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 p n + r ([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.
Keywords
Banach Space Cauchy Problem Sobolev Space Vector Function Matrix Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- 1.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).CrossRefMathSciNetMATHGoogle Scholar
- 2.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.Google Scholar
- 3.I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilisi University, Tbilisi (1975).Google Scholar
- 4.I. T. Kiguradze, “On boundary-value problems for linear differential systems with singularities,” Differents. Uravn., 39, No. 2, 198–209 (2003).MathSciNetGoogle Scholar
- 5.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).Google Scholar
- 6.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).Google Scholar
- 7.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).CrossRefMathSciNetGoogle Scholar
- 8.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).Google Scholar
- 9.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).CrossRefMathSciNetGoogle Scholar
- 10.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).CrossRefMathSciNetMATHGoogle Scholar
- 11.A. S. Goriunov and V. A. Mikhailets, “Regularization of singular Sturm–Liouville equations,” Meth. Funct. Anal. Topol., 16, No. 2, 120–130 (2010).MathSciNetMATHGoogle Scholar
- 12.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).Google Scholar
- 13.N. Dunford and J. T. Schwartz, Linear Operators. Part 2. Spectral Theory. Self-Adjoint Operators in Hilbert Space, Interscience Publ., New York (1963).Google Scholar
- 14.V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1989).Google Scholar
- 15.H. Triebel, Theory of Function Spaces, Akadem. Verlags. Geest & Portig K.-G., Leipzig (1983).CrossRefGoogle Scholar
- 16.V. A. Mikhailets and A. A. Murach, H¨ormander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin (2014).Google Scholar
Copyright information
© Springer Science+Business Media New York 2015