Nonlocal fractional functional differential equations with measure of noncompactness in Banach space

In this paper, we are concerned with the following fractional functional differential equations with nonlocal initial conditions in Banach space Dαx(t)=Ax(t)+f(t,x(t),xt), t∈[0,T], x(0)=ϕ+g(x). By virtue of the theory of measure of noncompactness associated with Darbo’s fixed point theorem, upon making some suitable assumptions, some existence results of mild solutions are obtained. Moreover the results obtained are utilized to study the existence of solutions to fractional parabolic equations as an illustrative example to show the practical usefulness of the analytical results.