  We prove the existence and uniqueness of mild solutions for a specific class
of time-fractional $\psi$-Caputo evolution systems with a derivative order
ranging from 1 to 2 in Banach spaces. By using the properties of cosine and
sine family operators, along with the generalized Laplace transform, we derive
a more concise expression for the mild solution. This expression is formulated
as an integral, incorporating Mainardi's Wright-type function. Furthermore, we
provide various valuable properties associated with the operators present in
the mild solution. Additionally, employing the fixed-point technique and
Gr\"{o}nwall's inequality, we establish the existence and uniqueness of the
mild solution. To illustrate our results, we conclude with an example of a
time-fractional equation, presenting the expression for its corresponding mild
solution.
