In this paper an H-generalized Cauchy equation S(t + s)C = H(S(s), S(t))is considered, where {S(t)} t≥0 is a one parameter family of bounded linear operators and H :Also a similar equations are studied on C-cosine families and integrated C-semigroups. for x ∈ D(A) is called the infinitesimal generator of T(t). With C = I, the identity operator on X, the C-semigroup {T(t)} t≥0 is said to be a C 0 -semigroup. Regularized semigroups and their connection with the ACP(A; x) have been studied, e.g., in [2,11,13,16].