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 $C_n^1=C^1([-h,0],{\mathbb{R}}^n)$. For a prototypic example we develop a new proof that its solution manifold is diffeomorphic to an open subset of the subspace given by ${\varphi }^{\prime }(0)=0$, 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.

16 pages