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

Let a L l o c 1 ( +) and k C ( +) 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 +. 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 A x = lim t ? 0 + (R (t) x - k (t) x / (a k) (t)) defined on the domain D (A): = { x 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. Copyright © 2012 Carlos Lizama and Felipe Poblete.