Existence Results for Nonautonomous Impulsive Fractional Evolution Equations

Abstract

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.

Keywords

• nonlocal condition
• fractional nonautonomous differential equations
• measure of noncompactness
• mild solution
• impulsive fractional evolution integrodifferential equations