Existence of strictly decreasing positive solutions of linear differential equations of neutral type

Josef Diblík; Zdeněk Svoboda

doi:10.3934/dcdss.2020004

Article Contents

2020, Volume 13, Issue 1: 67-84. Doi: 10.3934/dcdss.2020004

This issue Previous Article A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback Next Article Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

Existence of strictly decreasing positive solutions of linear differential equations of neutral type

Josef Diblík^, and
Zdeněk Svoboda

CEITEC - Central European Institute of Technology, Brno University of Technology, Brno, Czech Republic

^* Corresponding author: Josef Diblík
^* Corresponding author: Josef Diblík

Received: November 30, 2016

Revised: March 31, 2017

Published: January 2020

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Abstract

The paper is concerned with a linear neutral differential equation

$ \dot y(t) = -c(t)y(t-\tau(t))+d(t)\dot y(t-\delta(t)) $

where $ c\colon [t_0,\infty)\to (0,\infty) $, $ d\colon [t_0,\infty)\to [0,\infty) $, $ t_0\in {\Bbb{R}} $ and $ \tau, \delta \colon [t_0,\infty)\to (0,r] $, $ r\in{\mathbb{R}} $, $ r>0 $ are continuous functions. A new criterion is given for the existence of positive strictly decreasing solutions. The proof is based on the Rybakowski variant of a topological Ważewski principle suitable for differential equations of the delayed type. Unlike in the previous investigations known, this time the progress is achieved by using a special system of initial functions satisfying a so-called sewing condition. The result obtained is extended to more general equations. Comparisons with known results are given as well.

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Mathematics Subject Classification: Primary: 34K40, 34K25; Secondary: 34K12.

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