Abstract
We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.
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Notes
For a Hilbert space \(H\) and a bounded, measurable function \(\phi :\mathbb {R}\rightarrow \mathbb {R}\), we denote
$$\begin{aligned} (\phi (m_0)f)(t):=\phi (t)f(t)\quad (t\in \mathbb {R},f\in H_{\varrho ,0}(\mathbb {R})\otimes H). \end{aligned}$$Such type of problems will be discussed later, when we come to initial value problems.
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The authors thank Wolfgang Ruess for stimulating discussions and encouragement
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Kalauch, A., Picard, R., Siegmund, S. et al. A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term. J Dyn Diff Equat 26, 369–399 (2014). https://doi.org/10.1007/s10884-014-9353-6
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DOI: https://doi.org/10.1007/s10884-014-9353-6