This paper investigates the existence and uniqueness of solutions for fractional differential equations with nonlocal conditions. The results are derived using measure theory, Monch's fixed point theorem, and semigroup theory. An illustrative example is also presented, involving a nonlocal fractional evolution problem in $L^2((0,\pi ),\mathbb{R})$. This example demonstrates the effect of the $\psi$-Caputo derivative on the mild solution in comparison with the standard Caputo derivative, highlighting the influence of non-uniform time scaling and memory effects.