S-asymptotically $\omega$-periodic mild solutions to some fractional integro-differential equations with infinite delay

Abstract

Under appropriate conditions and using the Krasnosel'skii's fixed point theorem, we prove that the semilinear fractional integro-differential equation in a Banach space $X$ $u'(t)=\frac{1}{\Gamma(\alpha-1)}\int_{0}^{t}(t-s)^{\alpha-2}Au(s)ds+F(t,u_t),\;\;t\geq 0,$ and $u_0=\phi$, possesses $S$-asymptotically $\omega$-periodic mild solutions where $1<\alpha<2$, $\phi \in \mathcal{B}$ an abstract space, $A:D(A)\subset X \to X$ a closed (not necessarily bounded) linear operator and $F:\mathbb{R}^+\times \mathcal{B}\to X$ a continuous function, $u_t: (-\infty,0]\to X$ with $u_t(\theta)=u(t+\theta)$ is an associated history function to the function $u:\mathbb{R}\to X$.