Abstract and Applied Analysis
Volume 2003 (2003), Issue 15, Pages 843-864

Perturbed Fredholm boundary value problems for delay differential systems

Alexander A. Boichuk1,2 and Myron K. Grammatikopoulos3

1Institute of Mathematics, The National Academy of Science of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv 016 01, Ukraine
2Department of Applied Mathematics, Faculty of Natural Science, University of Zilina, J. M. Hurbana 15, Zilina 01026, Slovakia
3Department of Mathematics, University of Ioannina, Hellas, Ioannina 451 10, Greece

Received 3 March 2003

Copyright © 2003 Alexander A. Boichuk and Myron K. Grammatikopoulos. 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.


Boundary value problems for systems of ordinary differential equations with a small parameter ε and with a finite number of measurable delays of the argument are considered. Under the assumption that the number m of boundary conditions does not exceed the dimension n of the differential system, it is proved that the point ε=0 generates ρ-parametric families (where ρ=nm) of solutions of the initial problem. Bifurcation conditions of such solutions are established. Also, it is shown that the index of the operator, which is determined by the initial boundary value problem, is equal to ρ and coincides with the index of the unperturbed problem. Finally, an algorithm for construction of solutions (in the form of Laurent series with a finite number terms of negative power of ε) of the boundary value problem under consideration is suggested.