Analyticity of Solutions of Differential Equations with a Threshold Delay

Abstract

We consider the differential equationẋ(t) = f (x(t), x(t − r)) where the delay r = r(x(·)) is defined by the threshold condition´t t−r a(x(s),ẋ(s)) ds = ρ with a given ρ > 0. It is shown that if f and a are analytic functions, a is positive, then the globally defined bounded solutions are analytic.Suggested running head: Analyticity of solutions

Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method II: Analytic case

“…The analyticity of periodic solutions is also assumed for a class of differential equations with state dependent delay to prove the global Hopf bifurcation theorem in [HWZ12] Nevertheless, as [MPN14] observes, the time dependent delay can be considered as a linearization version of the theory. Recently, [Kri14] considers a state-dependent delay differential equation in which the delay is defined by a threshold condition. It shows that the globally defined bounded solutions are analytic.…”

Gevrey class regularity for analytic differential-delay equations

“…Mallet-Paret and Nussbaum have constructed a timedependent delay equation in [3] such that a given solution is analytic at certain points of its domain and nonanalytic at others. Krisztin has shown analyticity for a particular class of equations with state-dependent delay in [2]. As far as we know, this is the only positive result in the state-dependent delay case.…”

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

“…We should remark that the work of Mallet-Paret and Nussbaum [6] also presented some examples where bounded solutions are no-longer analytic, while Krisztin [3] showed that globally defined bounded solutions of threshold type delay equations are analytic. Then an important theoretical problem is what would be the most general form of state-dependent delay differential equations for which the conjecture remains true for differential equations with state-dependent delay.…”