On Mixed C-Semigroups of Operators on Banach Spaces

Abstract

In this paper a $H$ generalized Cauchy equation
\[S(t+s)C=H(S(s), S(t))\]
will be considered, where $\{S(t)\}_{t\geq 0}$ is a one parameter family of bounded operators and $H:B(X)\times B(X)\to B(X)$ is a function. For the special case, when $H(S(s),S(t))=S(s)S(t)+D(S(s)-T(s))(S(t)-T(t))$ with $D\in B(X)$, solutions of $H$ generalized Cauchy equation will be studied, where $\{T(t)\}_{t\geq 0}$ is a $C$-semigroup of operators. Also a similar equation will be studied on integrated $C$-semigroup.