### Existence, uniqueness and continuous dependence for a hereditary nonlinear functional partial differential equation of the first order

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The object of this paper is the Hausdorff metric topology on partial maps with closed domains. This topological space is denoted by $(\mathcal{P},{H}_{\rho})$. An equivalence of well-posedness of constrained continuous problems is proved. By using the completeness of the Hausdorff metric on the space of usco maps with moving domains, the complete metrizability of $(\mathcal{P},{H}_{\rho})$ is investigated.

This paper completes and improves results of [10]. Let $(X,{d}_{{}_{X}})$, $(Y,{d}_{{}_{Y}})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence ${\tau}_{{}_{K}}$ and the Kuratowski convergence on compacta ${\tau}_{{}_{K}}^{c}$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology ${\tau}_{{}_{AW}}$ (generated by the box metric of ${d}_{{}_{X}}$ and ${d}_{{}_{Y}}$) and ${\tau}_{{}_{K}}^{c}$ convergence on $G$,...

We study, in Carathéodory assumptions, existence, continuation and continuous dependence of extremal solutions for an abstract and rather general class of hereditary differential equations. By some examples we prove that, unlike the nonfunctional case, solved Cauchy problems for hereditary differential equations may not have local extremal solutions.

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