Almost automorphic solutions to a class of semilinear fractional differential equations

We study almost automorphic (mild) solutions of the semilinear fractional equation ∂tαu=Au+∂tα−1f(⋅,u),1<α<2, considered in a Banach space XX, where AA is a linear operator of sectorial type ω<0ω<0. We prove the existence and uniqueness of an almost automorphic mild solution assuming that f(t,x)f(t,x) is almost automorphic in tt for each x∈Xx∈X, satisfies some Lipschitz type conditions and takes values on XX.