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

Abstract EN

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.