Integrodifferential equations with analytic semigroups (Q1412382)

The paper is concerned with the existence and the uniqueness of the local and global mild and classical solutions to the following integrodifferential equation in a Banach space \(X\) \[ u'(t)+Au(t)=f(t,u(t))+K(u)(t), \quad t>t_0, \] \[ u(t_0)=u_0, \] where \[ K(u)(t)=\int_{t_0}^t a(t-s)g(s,u(s))\,ds, \] and \(-A\) generates an analytic semigroup \(S(t), t\geq 0\), on \(X\). The main tool of the proof is the contraction principle combined with the semigroup theory. An example illustrating the abstract theory is presented.