Existence of the mild solution for fractional semilinear initial value problems

In this paper we will study the existence and uniqueness of mild solution for the semilinear initial value problem of non-integer order:u(α)(t)=Au(t)+f(t,u(t),Gu(t),Su(t)),u(α)(t)=Au(t)+f(t,u(t),Gu(t),Su(t)), where, α∈(0,1]α∈(0,1] and f(t,u(t),Gu(t),Su(t))f(t,u(t),Gu(t),Su(t)) is a given function.