Note on some fractional differential equations with nonlocal conditions

Abstract

In this paper, we revisit \cite{mophou2009existence} by using a more suitable definition of mild solution to the Cauchy problem for the fractional differential equation with nolocal conditions $D^{\alpha}x(t)=Ax(t)+t^{n}f(t,Bx(t)),\;\;t\in [0,T],\;\;n\in\mathbb{Z^+},\;\;x(0)+g(x)=x_0$ where $0<\alpha<1$, $A$ is the infinitesimal generator of a $C_0$-semigroup of bounded linear operators on a Banach space $X$. We obtain similar results using classical fixed point theorems along with the H\"older inequality.