Non-autonomous nonlinear integro-differential equations with infinite delay
The authors prove the local existence and uniqueness of strict solutions of the nonlinear functional integro-differential problem with infinite delay \[ \begin{cases} u'(t)=A(t)u(t)+\int_0^tk(t,s,u(s))\,ds+F(t,u_t),\quad 0\leq t\leq T,\\ u_0=\varphi\in\mathcal P,\end{cases} \] in a Banach space \(E\), where \((A(t))_{0\leq t\leq T}\) is a family of linear operators defined on a non-dense set \(D\) of \(E\), \(k\in C(\Delta(0,T)\times D,E)\), \(F\in C^1([0,T]\times {\mathcal P}, E)\), and the phase space \((\mathcal P,\|\cdot\|_{\mathcal P})\) is the Banach space of functions from \((-\infty,0]\) into \(E\), which satisfy some assumptions. Three other related functional integro-differential problems are also investigated.
nonlinear functional integro-differential equation
scientific article; zbMATH DE number 5532910