The paper deals with the following partial functional integro-differential equation:
where is the infinitesimal generator of a compact, analytic semigroup, , and are some given functions. The equation is considered here together with the conditions:
It is shown that, under suitable conditions on the functions , and for any , the above Cauchy problem has at least one mild solution on . The proof of this result employs the Leray-Schauder alternative. The author also presents an illustrative example at the end of the paper.