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

Abstract

AbstractIn 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 compareallthe components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity ofallelements 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 thenth row of the Green's matrices of the initial and two other problems, non-oscillation of thenth 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 thenth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for thenth 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.