Solvability of the Cauchy problem for infinite delay equations (Q1879764)

By using topological methods, expressed in terms of the Kuratowski measure of noncompactness, the authors prove several existence results on mild solutions for the infinite delay functional-integral equation \[ u(t)=g(t)+\int_{\sigma}^tf(t,s,u(s),u_s)\,ds,\quad \sigma\leq t\leq T, \qquad u_\sigma=\varphi, \] and semilinear functional-differential equations of the form \[ u'(t)=Au(t)+f(t,u(t),u_t),\quad 0\leq t\leq T, \qquad u_0=\varphi, \] with \(\varphi\in {\mathcal B}\). Here, \(\mathcal{B}\) is an admissible function space, i.e., a suitable chosen subspace of functions from \((-\infty,\sigma\,]\) to \(X\), with \(X\) a Banach space, \(\varphi\in \mathcal{B}\), \(x_t(s)=x(t+s)\), \(f\) is either in \(C([\,\sigma,T\,]\times [\,\sigma,T\,]\times X\times\mathcal{B};X)\) or in \(C([\,\sigma,T\,]\times X\times\mathcal{B};X)\) and \(A:D(A)\subset X\to X\) is a linear operator. An extension to the case when \(A\) may depend on \(t\) as well is considered, and an application to a functional integro-differential equation of Schrödinger type is included.