Abstract and Applied Analysis
Volume 2011 (2011), Article ID 631412, 19 pages
http://dx.doi.org/10.1155/2011/631412
Research Article

Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

A. Boichuk,1,2 J. Diblík,3,4 D. Khusainov,5 and M. Růžičková2

1Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya Street 3, 01601 Kyiv, Ukraine
2Department of Mathematics, University of Žilina, Univerzitná 8215/1, 01026 Žilina, Slovakia
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic
4Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 10, 616 00 Brno, Czech Republic
5Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National University of Kyiv, Vladimirskaya Street 64, 01033 Kyiv, Ukraine

Received 30 January 2011; Accepted 31 March 2011

Academic Editor: Elena Braverman

Copyright © 2011 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 nonlinear boundary-value problems for systems of 𝑛 ordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity: ̇ 𝑧 ( 𝑡 ) = 𝐴 𝑧 ( 𝑡 − 𝜏 ) + 𝑔 ( 𝑡 ) + 𝜀 𝑍 ( 𝑧 ( ℎ 𝑖 ( 𝑡 ) , 𝑡 , 𝜀 ) , 𝑡 ∈ [ 𝑎 , 𝑏 ] , assuming that these solutions satisfy the initial and boundary conditions 𝑧 ( 𝑠 ) ∶ = 𝜓 ( 𝑠 ) i f 𝑠 ∉ [ 𝑎 , 𝑏 ] , ℓ 𝑧 ( ⋅ ) = 𝛼 ∈ ℝ 𝑚 . The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functional ℓ ) does not coincide with the number of unknowns in the differential system with a single delay.