S-asymptotically omega-periodic mild solutions to fractional differential equations

Abstract

This article concerns the existence of mild solutions to the semilinear fractional differential equation $$ D_t^\alpha u(t)=Au(t)+D_t^{\alpha-1} f(t,u(t)),\quad t\geq 0 $$ with nonlocal conditions $u(0)=u_0 + g(u)$ where $D_t^\alpha(\cdot)$ ($1< \alpha < 2$) is the Riemann-Liouville derivative, $A: D(A) \subset X \to X$ is a linear densely defined operator of sectorial type on a complex Banach space $X$, $f:\mathbb{R}^+\times X\to X$ is S-asymptotically $\omega$-periodic with respect to the first variable. We use the Krsnoselskii's theorem to prove our main theorem. The results obtained are new even in the context of asymptotically \omega-periodic functions. An application to fractional relaxation-oscillation equations is given.

Keywords

• s-asymptotically omega-periodic sequence
• fractional semilinear differential equation