Advances in Difference Equations
Volume 2010 (2010), Article ID 593834, 20 pages
doi:10.1155/2010/593834
Research Article
Boundary Value Problems for Delay Differential Systems
1Department of Mathematics, Faculty of Science,
University of Žilina, Univerzitná 8215/1, 01026 Žilina, Slovakia
2Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya Str. 3, 01601 Kyiv, Ukraine
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 60200 Brno, Czech Republic
4Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 61600 Brno, Czech Republic
5Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National University of Kyiv, Vladimirskaya Str. 64, 01033 Kyiv, Ukraine
Received 16 January 2010; Revised 27 April 2010; Accepted 12 May 2010
Academic Editor: Ağacik Zafer
Copyright © 2010 A. Boichuk et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Conditions are derived of the existence of solutions of linear Fredholm's boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay, assuming that these solutions satisfy the initial and boundary conditions. Utilizing a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay. As an example of application of the results derived, the problem of bifurcation of solutions of boundary-value problems for systems of ordinary differential equations with a small parameter and with a finite number of measurable delays of argument is considered.