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On solution manifolds of some differential equations with more general state-dependent delay

Hans-Otto Walther

Abstract

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} in the Banach space $C^1_n=C^1([-h,0],\mathbb{R}^n)$. For a prototypic example we develop a new proof that its solution manifold is diffeomorphic to an open subset of the subspace given by $φ'(0)=0$, without recourse to a restrictive hypothesis about the form of delays which is instrumental in earlier work on the nature of solution manifolds. The new proof uses the framework of algebraic-delay systems.

On solution manifolds of some differential equations with more general state-dependent delay

Abstract

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} in the Banach space . For a prototypic example we develop a new proof that its solution manifold is diffeomorphic to an open subset of the subspace given by , without recourse to a restrictive hypothesis about the form of delays which is instrumental in earlier work on the nature of solution manifolds. The new proof uses the framework of algebraic-delay systems.
Paper Structure (5 sections, 7 theorems, 83 equations)

This paper contains 5 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. A diffeomorphism $C^1\times(-h,0)\to C^1\times(-h,0)$
  3. Transversals to $X_0$ which are small in $C$
  4. Proof of Theorem 1
  5. Examples

Key Result

proposition 1

If condition (e,d) holds then the map $F$ is continuously differentiable and has property (e), and $X_F\neq\emptyset$.

Theorems & Definitions (13)

  • proposition 1
  • proof
  • theorem 1
  • proposition 2
  • proof
  • proposition 3
  • proof
  • proposition 4
  • proof
  • proposition 5
  • ...and 3 more