Existence of anti-periodic mild solutions for a class of semilinear fractional differential equations

This work is concerned with the existence of anti-periodic mild solutions for a class of semilinear fractional differential equations x ( t ) = Ax ( t ) + D t α - 1 F ( t , x ( t ) ) , t ∈ R , 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 of anti-periodic mild solutions to a fractional relaxation-oscillation equation.