Summary: We study the existence of mild solutions for the following system in a general Banach space (with norm ):
Here is the infinitesimal generator of a compact analytic semigroup , , is a multi-valued map and , where and represent the left and right limits of and , respectively. Let , and and are given functions.
By using a fixed point theorem for multi-valued maps due to Dhage, a main existence theorem is established. Finally, we present an example to illustrate this main theorem.