On maximal regularity and semivariation of α-times resolvent families

Original · EN

Let 1< α<2 and A be the generator of an α-times resolvent family {Sα(t)}ₜ ≥ ₀ on a Banach space X. It is shown that the fractional Cauchy problem Dₜαu(t) = Au(t)+f(t), t ∈ [0,r]; u(0), u'(0) ∈ D(A) has maximal regularity on C([0,r];X) if and only if Sα(·) is of bounded semivariation on [0,r].