Existence results for an evolution problem with fractional nonlocal conditions

A second-order abstract problem involving derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is considered. This problem covers many situations arising in real world phenomena and extends the case of integer order to the case of non-integer order. We establish the existence of mild solutions. Under extra appropriate sufficient assumptions on the nonlinearity and on the initial data these mild solutions are shown to be classical.