Stepanov-like almost automorphic mild solutions for semilinear fractional differential equations

Abstract

This work is concerned with the existence and uniqueness of Stepanov-like almost automorphic mild solutions for a class of semilinear fractional differential equations

D_t^α x(t) = Ax(t) + D_t^α-1F(t,x(t)), t ∈ ℝ,

where 1 < α < 2, A is a linear densely defined operator of sectorial type of ω < 0 on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of Stepanov-like almost automorphic mild solutions for a fractional relaxation-oscillation equation.