Abstract
We study the boundary-value problem for a system of partial differential equations with constant coefficients with conditions nonlocal in time. By using a metric approach, we prove the well-posedness of the problem in the scale of Sobolev spaces of functions periodic in space variables. By using matrix calculus, we construct an explicit representation of a solution.
Similar content being viewed by others
References
A. A. Dezin, General Questions of the Theory of Boundary-Value Problems [in Russian], Nauka, Moscow 1980.
V. K. Romanko, “Solvability of boundary-value problems for operator-differential equations of higher order,” Differents. Uravn., 14, No. 6, 1081–1092 (1978).
G. B. Savchenko, “On the well-posedness of one boundary-value problem for systems of linear differential equations,” Differents. Uravru, 14, No. 11, 2082–2085 (1978).
V. M. Borok and 1.1. Antypko, “A criterion of unconditional well-posedness of a boundary-value problem in a layer,” Teor. Funkts, Funkts. Anal Prilozhen., Issue 26, 3–9 (1976).
A. A. Makarov, “Existence of a well-posed two-point boundary-value problem for systems of pseudodifferential equations in a layer,” Differents. Uravn., 30, No. 6, 144–150 (1994).
B. I. Ptashnyk, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev 1984.
V. S. Il’kiv and B. I. Ptashnyk, “A problem with nonlocal boundary conditions for systems of partial differential equations with constant coefficients,” Differents. Uravn., 20, No. 6, 1012–1023 (1984).
V. S. Il’kiv, A Nonlocal Problem for Operator-Differential Equations. I, II, Deposited in VINITI, No. 3942–82, Lvov (1982), pp. 8–12, 13–18.
V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Operator-Differential Equations [in Russian], Naukova Dumka, Kiev 1984. English translation: Kluwer, Dordrecht (1991).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow 1981. English translation: Chelsea, New York (1990).
P. S. Kazimirskii, Factorization of Matrix Polynomials [in Ukrainian], Naukova Dumka, Kiev 1981.
P. Lancaster, Theory of Matrices, Academic Press, New York 1969.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
H’kiv, V.S., Ptashnyk, B.I. Representation and investigation of solutions of a nonlocal boundary-value problem for a system of partial differential equations. Ukr Math J 48, 207–219 (1996). https://doi.org/10.1007/BF02372046
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02372046