Existence Results for Nonautonomous Impulsive Fractional Evolution Equations

Abstract

\noindent {\bf ABSTRACT}\end{center}\par In this paper, we investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations\begin{align*}\begin{cases}\frac{d^q u(t)}{dt^q} &\quad =~~ -A(t)u(t)+f(t,(K_1 u)(t),(K_2 u)(t),\dots,(K_n u)(t),t \in I=[0,T] \\\Delta y|_{t=t_k} &\quad =~~ I_k(y(t_k^-)),t = t_k, k = 1,2,\dots,m, \\u(0) &\quad =~~ A^{-1}(0)g(u)+u_0\end{cases}\end{align*}in Banach spaces, where $T>0, 0<q<1.$ New results are obtained by using Sadovskii's fixed point theorem and the Banach contraction mapping principle. An example is given to illustrate the theory.