Fréchet differentiability in Fréchet spaces, and differential equations with unbounded variable delay

2018/01/28 by Hans‐Otto Walther, Walther, Hans-Otto
Computer Science · Mathematics · #34 K 05 #37 L 05 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1801.09213

openalex publication_date 2018/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce and discuss Fréchet differentiability for maps between Fréchet spaces. For delay differential equations x'(t)=f(x_t) we construct a continuous semiflow of continuously differentiable solution operators x₀↦ x_t, t≥0, on submanifolds of the Fréchet space C¹((-∞,0],ℝⁿ), and establish local invariant manifolds at stationary points by means of transversality and embedding properties. The results apply to examples with unbounded but locally bounded delay.

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