Existence of almost periodic solution for nonlocal fractional Cauchy problem with integral initial condition

Abstract

In this research, we intend to show that the nonlocal fractional Cauchy problem $D^{\alpha }u(t)=A(t)u(t)+f(t,u(t))$, $t\in J=[0,1]$ with integral initial condition $u(0)={\displaystyle {\int }_0^{1}g(s,u(s))ds}$, in the Banach space $X$, where $A$ is a generator of $\alpha$-resolvent operator function ${\{T(t)\}}_{t\ge 0}$ and $f$, $g$ are given functions satisfying some assumptions, has an almost periodic solution.