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ON POSITIVITY OF SEVERAL COMPONENTS OF SOLUTION VECTOR FOR SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  04 December 2009

RAVI P. AGARWAL  and
ALEXANDER DOMOSHNITSKY
RAVI P. AGARWAL
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA and Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia e-mail: agarwal@fit.edu
ALEXANDER DOMOSHNITSKY
Affiliation:
Department of Mathematics and Computer Sciences, The Ariel University Center of Samaria, 44837 Ariel, Israel e-mail: adom@ariel.ac.il
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Abstract

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In the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper.

Keywords

34K11
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 115 - 136
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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