Some considerations for linear integrodifferential equations (Q1077668)

Consider the equations \[ (1)\quad \dot x(t)=Ax(t)+\int^{t}_{0}B(t- s)x(s)ds+f(t), \] \(x(0)=x_ 0\), \(t\geq 0\) and \(\dot y(\)t)\(=Ay(t)\), \(y(0)=x_ 0\), \(t\geq 0\), defined in a Banach space. A number of relations between the solutions x(\(\cdot)\) and y(\(\cdot)\) are established. Representation formulae for the semigroup generated by A are also derived as a limit of products involving the resolvent of (1).