Semilinear fractional integro-differential equations with compact semigroup (Q1044509)

The authors give sufficient conditions for the global and local existence, respectively, of mild continuous solutions to the equation \[ u^{(\alpha)}(t) + A u(t) = f(t, u(t)) + \int_0^t q(t-s) g(s, u(s))\,ds,\quad t > 0, \] with some \(\alpha \in (0,1]\). Here apparently \(u^{(\alpha)}\) is meant to be a Riemann-Liouville derivative of order \(\alpha\), although it is unclear whether the starting point of this derivative is supposed to be at \(0\) or somewhere else. The above integro-differential equation is considered in a general Banach space, and \(-A\) is supposed to be an infinitesimal generator of a compact semigroup. For the investigations, the equation is considered in combination with the initial condition \(u(0) = u_0\) which is rather unnatural in the context of Riemann-Liouville derivatives.