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 -