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 equationsDtαx(t)=Ax(t)+Dtα-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 of anti-periodic mild solutions to a fractional relaxation-oscillation equation.

► Some sufficient conditions for the existence and uniqueness of anti-periodic mild solutions for a class of semilinear fractional differential equations are given. ► The existence and uniqueness of anti-periodic mild solutions for a fractional relaxation- oscillation equation, which is illustrated by example, are in good agreement with the theoretical analysis. ► The methods used in this paper can also be applied to deal with the existence of periodic mild solutions for the semilinear fractional differential equations.