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

Abstract

Leta∈Lloc1(ℝ+)andk∈C(ℝ+)be given. In this paper, we study the functional equationR(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s), for bounded operator valued functionsR(t)defined on the positive real lineℝ+. We show that, under some natural assumptions ona(·)andk(·), every solution of the above mentioned functional equation gives rise to a commutative(a,k)-resolvent familyR(t)generated byAx=lim t→0+(R(t)x-k(t)x/(a*k)(t))defined on the domainD(A):={x∈X:lim t→0+(R(t)x-k(t)x/(a*k)(t))exists inX}and, conversely, that each(a,k)-resolvent familyR(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.