On a Functional Equation Associated with ( )-Regularized Resolvent Families

Let and be given. In this paper, we study the functional equation , for bounded operator valued functions defined on the positive real line . We show that, under some natural assumptions on and , every solution of the above mentioned functional equation gives rise to a commutative -resolvent family generated by defined on the domain exists in and, conversely, that each -resolvent family satisfy the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems.