Analyticity of Bounded Solutions of Analytic State-Dependent Delay Differential Equations

TL;DR

This paper investigates the analyticity of bounded solutions to analytic state-dependent delay differential equations by transforming them into an abstract ODE framework and establishing complex extensions.

Contribution

It introduces a novel method to prove the analyticity of solutions by transforming delay equations into an abstract ODE in sequence space.

Findings

Bounded solutions are analytic under certain conditions.

Solutions can be extended into the complex domain.

The method applies to a class of state-dependent delay equations.

Abstract

We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions by transforming the system of state-dependent delay equations into an abstract ordinary differential equation in a subspace of the sequence space $l^{\infty}(\mathbb{R}^{N+1})$ and prove the existence of complex extension of the bounded solutions. An example is given to illustrate the general results.