Existence and uniqueness of mild solutions for a class of psi-Caputo time-fractional systems of order from one to two

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.