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Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated solution manifold in the Banach space \(C_{n}^{1} = C^{1} \left( {\left[ { - h,0} \right],{\mathbb{R}}^{n} } \right)\) . For a prototypic example we develop a new proof that its solution manifold is diffeomorphic to an open subset of the subspace given by \(\phi^{\prime } \left( 0 \right) = 0 \) , 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.

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On Solution Manifolds of Some Differential Equations with more General State-dependent delay

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  • 发表日期：

    2025年

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  • Hans-Otto Walther

摘要

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated solution manifold in the Banach space \(C_{n}^{1} = C^{1} \left( {\left[ { - h,0} \right],{\mathbb{R}}^{n} } \right)\) . For a prototypic example we develop a new proof that its solution manifold is diffeomorphic to an open subset of the subspace given by \(\phi^{\prime } \left( 0 \right) = 0 \) , 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.