On periods of non-constant solutions to functional differential equations

Eugene Bravyi

doi:10.14232/ejqtde.2017.1.14

Abstract

On periods of non-constant solutions to functional differential equations

Highlights

Consider a problem on periodic solutions of the differential equation with deviating argument x(n)(t) = f (x(τ(t)), t ∈ R, (1.1)where x(t) ∈ Rm, f : Rm → Rm is a Lipschitz function, τ : R → R is a measurable function
The estimate (1.2) gives the minimal time required for an object described by a system of ordinary differential equations with the Lipschitz constant L to return to its initial state
For all n, we find a simple representation of the best constants in the estimate for periods of non-constant periodic solutions of some more general equations than (1.1) with Lipschitz nonlinearities

Summary

Introduction

Consider a problem on periodic solutions of the differential equation with deviating argument x(n)(t) = f (x(τ(t)), t ∈ R,. Where x(t) ∈ Rm, f : Rm → Rm is a Lipschitz function, τ : R → R is a measurable function. For periods T of non-constant periodic solutions to (1.1) is obtained in [28] for n = 1 and [16] for n 1 for Lipschitz f in the Euclidian norm, and in [30] for even n and Lipschitz functions f satisfying the condition max i=1,...,m. |xi xi |, x, x ∈ Rm. The estimate (1.2) gives the minimal time required for an object described by a system of ordinary differential equations with the Lipschitz constant L to return to its initial state

Bravyi

Main results

Kn T n

The sharpness of estimates

Conclusion