This paper is devoted to the study existence of locally/globally mild solutions for fractional differential equations with $\psi$-Caputo derivative with a nonlocal initial condition. We firstly establish the local existence by making use of usual fixed point arguments, where computations and estimates are essentially based on continuous and bounded properties of the Mittag-Leffler functions. Secondly, we establish the called $\psi$-Hölder continuity of solutions, which shows how $|u(t’) – u(t)|$ tends to zero with respect to a small difference $|\psi(t’) – \psi(t)|^{\beta}$, $\beta \in (0,1)$. Finally, by using contradiction arguments, we discuss the existence of a global solution or maximal mild solution with blowup at finite time.