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

Let a epsilon L-loc(1)(R+) and k epsilon C(R+) be given. In this paper, we study the functional equation R (s) (a (*) R)(t) - (a (*) R)(s)R(t) = k(s) (a (*) R)(t) - k(t) (a (*) R) (s), for bounded operator valued functions R(t) defined on the positive real line R+. We show that, under some natural assumptions on a (.) and k (.), every solution of the above mentioned functional equation gives rise to a commutative (a, k)-resolvent family R (t) generated by Ax = lim(t) (,0+) (R(t)x - k (t)x / (a (*) k) (t)) defined on the domain D(A) := {x epsilon X : lim(t -> 0+) (R(t)x - k (t)x/(a (*) k) (t)) exists in X} and, conversely, that each (a, k)-resolvent family R (t) 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.