On a solution to fractional order integrodifferential equations with analytic semigroups (Q1029406)

The authors consider the integrodifferential equation \[ {d^\eta u(t)\over dt^\eta}+ Au(t)= f(t, u(t))+ \int^t_{t_0} a(t- s) g(s, u(s))\,ds, \] for \(t\geq t_0\), in a Banach space \(X\). Here \(-A\) is the generator of an analytic semigroup and \(\eta\in(0, 1]\). The functions \(f\), \(g\) are suitably bounded nonlinear functions and the kernel \(a\in L^1_{\text{loc}}\). Using the contraction mapping principle the existence and regularity of solutions is proved.