Local existence of some nonlinear Volterra integrodifferential equations with infinite delay
The authors prove local existence and uniqueness of solutions to the Volterra integrodifferential equation \[ \begin{cases} u'(t) = Au(t)+\int_0^tk(t,\theta ,u(\theta))\,d\theta +F(t,u_t), & 0\leq t\leq T, \\ u_0 = \varphi \in \mathcal{P}.\end{cases} \] Here, \(A:D(A)\to X\) is a nondensely defined Hille-Yosida operator, the phase space \(\mathcal{P}\) is an abstract admissible linear space of functions mapping \((-\infty ,0]\) into \(X\) and \(u_t\in \mathcal{P}\). Some properties of the solutions are also given.
scientific article; zbMATH DE number 5707735