| Abstract: |
| We present results on existence and uniqueness, mild solutions, and a measure-based integration approach for a broad class of differential and pseudo-differential equations with distributional delay on $[-1,0]$. We consider the differential problem\r\n\begin{equation*}\r\nx`(t) =\int_{[-1, 0]} x(t + s) \, d \mu (s; t), \r\n\end{equation*} \r\nor, more generally, \r\n\begin{equation*}\r\n(D x)(t) =\int_{[-1, 0]} x(t + s) \, d \mu(s; t),\r\n\end{equation*}\r\nwith an appropriate initial condition $x{\restriction}_{[-1,0]}$. Here $D$ is the pseudo-differential operator defined for some signed measure $\alpha$ via a generalized Newton-Leibniz identity\r\n\begin{equation*}\r\nx(t) - x(s) =\int_{[s, t]} (D x) (\xi) \, d \alpha(\xi). \r\n\end{equation*}\r\nFor a given measure $\alpha$, we define the function space on which the operator $D$ is well-defined. We also present random versions of these equations, as well as models based on partial differential equations, which appear to be more suitable for applications in biology and mathematical ecology. Such problems provide a starting point for the study of random dynamical systems generated by the associated solution operator |
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