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.For more information see https://ejde.math.txstate.edu/Volumes/2020/30/abstr.html