On Mixed C-Semigroups of Operators on Banach Spaces

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 : B(X) x B(X) -> B(X) is a function. In the special case, when H(S(s), S(t)) = S(s)S(t) + D(S(s) - T(s))(S(t) - T(t)) with D is an element of B(X), solutions of H-generalized Cauchy equation are studied, where {T(t)}(t >= 0) is a C-semigroup of operators. Also a similar equations are studied on C-cosine families and integrated C-semigroups.