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

Let a∈Lloc1(ℝ+) and k ∈ 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∈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.