Advances in Difference Equations
Volume 2010 (2010), Article ID 478020, 14 pages
Research Article

Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations

Department of Mathematics and Computer Science, The Ariel University Center of Samaria, 44837 Ariel, Israel

Received 24 December 2009; Accepted 26 April 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 Alexander Domoshnitsky. 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.


The method to compare only one component of the solution vector of linear functional differential systems, which does not require heavy sign restrictions on their coefficients, is proposed in this paper. Necessary and sufficient conditions of the positivity of elements in a corresponding row of Green's matrix are obtained in the form of theorems about differential inequalities. The main idea of our approach is to construct a first order functional differential equation for the nth component of the solution vector and then to use assertions about positivity of its Green's functions. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. It should be also noted that the sufficient conditions, obtained in this paper, cannot be improved in a corresponding sense and does not require any smallness of the interval [0,ω], where the system is considered.