The infinite generator of α-times Integrated C semigroups and the properties of resolvent are given. At the same time, we discuss the relationship between the existence of strong solution of a class of nonhomogeneous abstract Cauchy problem and α-times integrated C semigroups, and a sufficient and necessary condition is obtained.
 H. R. Thieme, “‘Integrated Semigroup’ and Integrated Solution to Abstract Cauchy Problem,” Journal of Mathematical Analysis and Applications, Vol. 152, No. 2, 1990, pp. 416-447. doi:10.1016/0022-247X(90)90074-P