Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order

In this paper, we prove some existence and uniqueness of mild solutions for a semilinear integrodifferential equation of fractional order with nonlocal initial conditions in αα-norm. We assume that the linear part generates a noncompact analytic semigroup. Our results cover the cases that the nonlinearity FF takes values in different spaces such as X,XαX,Xα and XβXβ, where α≤β∈(0,1)α≤β∈(0,1). Finally, some practical consequences are also obtained.