Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions

Abstract A second-order abstract problem of neutral type with derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions. AMS Subject Classification 26A33, 34K40, 35L90, 35L70, 35L15, 35L07