Department of Mathematics and Computer Science, The Ariel University Center of Samaria, 44837 Ariel, Israel
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 th 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 , where
the system is considered.