Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Abstract

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations ${\mathrm{D}}_t^{\alpha }x\left(t\right)=Ax\left(t\right)+{\mathrm{D}}_t^{\alpha -\mathrm{1}}F\left(t,x\left(t\right),Bx\left(t\right)\right), t\in \mathbb{R},$ where , $A$ is a linear densely defined operator of sectorial type on a complex Banach space $X$ and $B$ is a bounded linear operator defined on $X$, $F$ is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity $F$ is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.