D-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations

Michal Fečkan,Natalia Dilna,Mykola Solovyov

doi:10.3390/sym12111761

Abstract

This paper presents a method of establishing the D-stability terms of the symmetric solution of scalar symmetric linear and nonlinear functional differential equations. We determine the general conditions of the unique solvability of the initial value problem for symmetric functional differential equations. Here, we show the conditions of the symmetric property of the unique solution of symmetric functional differential equations. Furthermore, in this paper, an illustration of a particular symmetric equation is presented. In this example, all theoretical investigations referred to earlier are demonstrated. In addition, we graphically demonstrate two possible linear functions with the required symmetry properties.

Highlights

Time-reversal symmetry is one of the fundamental symmetries discussed in natural science.it arises in many physically motivated dynamical systems, in particular, in classical and quantum mechanics [1]
There are various approaches which have attracted our interest that utilize the modeling of reversible dynamical systems, such as the study of time-reversal symmetry in nonequilibrium statistical mechanics [2], the time reversibility investigation in nonequilibrium thermodynamics [3], and the evaluation of certain properties of the Liapunov spectrum for the driven
Using the results from [5,6,11,12,13,14], we investigate the D-stability of the symmetric solutions for functional differential equations

Summary

Introduction

Time-reversal symmetry is one of the fundamental symmetries discussed in natural science. Using the results from [5,6,11,12,13,14], we investigate the D-stability of the symmetric solutions for functional differential equations. The main focus of our research was to obtain the conditions of D-stability of functional differential equations. It allows one to obtain unique solvability conditions and a representation of the unique solution in the form of a functional series. The present paper was mainly motivated by several papers that deal with the conditions of stability and solvability of differential equations with delay. The unique solvability conditions of linear functional differential equations were established in [13,14,20], which represent the unique solution in view of the functional series

Problem Formulation

Notation

Symmetric Properties

Definitions of the D-Stability

Unique Solvability

Section 6.1.

Symmetric Solution

D-Stability of Nonlinear Functional Differential Equations

Application

Conclusions