Existence of Nonnegative Solutions of Linear Autonomous Functional Differential Equations

Mihály Pituk

doi:10.3390/math8071098

Abstract

It is shown that if we exclude the existence of nontrivial small solutions, then a linear autonomous functional differential equation has a nontrivial nonnegative solution if and only if it has a nonnegative eigenfunction.

Highlights

Let Rn be the n-dimensional space of real column vectors with any norm | · |
Before we present the proof of Theorem 2, we recall some facts from the decomposition theory of linear autonomous functional differential equations given in ([1], Chap. 7) and we establish two preliminary results
The basic oscillation theorem for differential equations with constant coefficients and several delays was obtained by Arino and Győri [9]

Summary

Introduction

Let Rn be the n-dimensional space of real column vectors with any norm | · |. A linear autonomous ordinary differential equation cannot have a nontrivial small solution. The existence of nontrivial small solutions of Equation (1) is a consequence of the fact that the phase space C is infinite dimensional. 7, Corollary 8.1), Equation (1) has no nontrivial small solutions if and only if the exponential type of the characteristic function det ∆ is equal to nr, or, equivalently, the system of eigenfunctions and generalized eigenfunctions of the generator of Equation (1) is complete. We will show that if we exclude the existence of nontrivial small solutions, the “natural” sufficient condition ∆(μ)v = 0 for some μ ∈ R and v ∈ Rn+ \ {0} is necessary for the existence of a nontrivial nonnegative solution of Equation (1).

Main Result

Discussion