On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem

We consider the nonlocal Cauchy problem for the semilinear functional differential equation with non- integer order: uα(t)=Au(t)+f(t;ut)whereα∈(0;1]andt∈(0;a], u(τk+0)=Qku(τk)≡u(τk)+Iku(τk);k=1;2;...;K, u(t)+(g(ut1,...,utp))(t)=ϕ(t),wheret∈[−r;0].Under suitable conditions we prove the existence and uniqueness of a mild solution to the equation.