Abstract
The purpose of this article is to demonstrate with a simple example the complexities that arise when considering boundary-value problems for differential-difference equations. In this paper we discuss in detail smoothness issues of solutions to the Dirichlet problem for linear differential-difference equation with integer shifts. After analysis of this example we consider more general statement of the problem and certain results concerning the solvability and smoothness of solutions.
Similar content being viewed by others
REFERENCES
A. V. Bitsadze and A. A. Samarskii, ‘‘On some simplest generalizations of linear elliptic boundary-value problems,’’ Sov. Math. Dokl. 10, 398–400 (1969)
G. A. Kamenskii and A. D. Myshkis, ‘‘On the formulation of boundary-value problems for differential equations with deviating argument and several highest terms,’’ Differ. Uravn. 10, 409–418 (1974).
N. N. Krasovskii, Control Theory of Motion (Nauka, Moscow 1968) [in Russian].
D. A. Neverova and A. L. Skubachevskii, ‘‘On the classical and generalized solutions of boundary-value problems for difference-differential equations with variable coefficients,’’ Math. Notes 94, 653–667 (2013).
G. G. Onanov and A. L. Skubachevskii, ‘‘Differential equations with displaced arguments in stationary problems in the mechanics of a deformed body,’’ Sov. Appl. Mech. 15, 39–47 (1979).
G. G. Onanov and A. L. Skubachevskii, ‘‘Nonlocal problems in the mechanics of the three-layer shells,’’ Math. Model. Nat. Phenom. 12, 192–207 (2017).
A. L. Skubachevskii, ‘‘Eigenvalues and eigenfunctions of some nonlocal boundary-value problems,’’ Differ. Equat. 25, 1245–1254 (1989).
A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Vol. 91 of Operator Theory. Advances and Applications (Birkhäuser, Basel, Boston, Berlin, 1997).
Funding
This work is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by A. B. Muravnik)
Rights and permissions
About this article
Cite this article
Neverova, D.A. Model Example for Study Smoothness of Solutions of Boundary-Value Problems for Differential-Difference Equations. Lobachevskii J Math 42, 967–973 (2021). https://doi.org/10.1134/S1995080221050115
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080221050115