Almost automorphic solutions to a class of semilinear fractional differential equations

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

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