Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations

Alexander Domoshnitsky

doi:10.1155/2010/478020

Abstract

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.

Highlights

Consider the following system of functional differential equations n Mix t ≡ xi tBij xj t fi t, t ∈ 0, ω, i 1, . . . , n, j1 where x col x1, . . . , xn, Bij : C 0,ω → L 0,ω, i, j 1, . . . , n, are linear continuous operators, C 0,ω and L 0,ω are the spaces of continuous and summable functions y : 0, ω → R1, respectively.Advances in Difference EquationsLet l : Cn0,ω → Rn be a linear bounded functional
The following property is the basis of the approximate integration method by Tchaplygin 2 : from the conditions
Mix t ≥ Miy t, t ∈ 0, ω, i 1, . . . , n, lx ly, 1.4 it follows that xi t ≥ yi t, t ∈ 0, ω, i 1, . . . , n

Summary

Introduction

Bij xj t fi t , t ∈ 0, ω , i 1, . . . , n, j1 where x col x1, . . . , xn , Bij : C 0,ω → L 0,ω , i, j 1, . . . , n, are linear continuous operators, C 0,ω and L 0,ω are the spaces of continuous and summable functions y : 0, ω → R1, respectively. The known book by Krasnosel’skii et al 5 was devoted to approximate methods for operator equations These ideas have been developing in scores of books on the monotone technique for approximate solution of boundary value problems for systems of differential equations. The main idea of our approach is to construct a corresponding scalar functional differential equation of the first order xn t Bxn t f ∗ t , t ∈ 0, ω , 1.11 for nth component of a solution vector, where B : C 0,ω → L 0,ω is a linear continuous operator, f ∗ ∈ L 0,ω. The technique of analysis of the first-order scalar functional differential equations, developed, for example, in the works 10–12 , is used In this paper we consider other boundary conditions that imply that the operator B : C 0,ω → L 0,ω is not a Volterra one even in the case when all Bij : C 0,ω → L 0,ω , i, j 1, . . . , n, are Volterra operators

Construction of Equation for nth Component of Solution Vector

Positivity of the Elements in the Fixed nth Row of Green’s Matrices