TY - JOUR
T1 - Differential inequalities for one component of solution vector for systems of linear functional differential equations
AU - Domoshnitsky, Alexander
N1 - Funding Information:
The author thanks the referees for their available remarks. This research was supported by The Israel Science Foundation (Grant no. 828/07).
PY - 2010
Y1 - 2010
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=77955391358&partnerID=8YFLogxK
U2 - 10.1155/2010/478020
DO - 10.1155/2010/478020
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AN - SCOPUS:77955391358
SN - 1687-1839
VL - 2010
JO - Advances in Difference Equations
JF - Advances in Difference Equations
M1 - 478020
ER -