Semigroups of operators and abstract dynamic equations on time scales

In this paper we develop the theory of strongly continuous semigroups (C0-semigroups) of bounded linear operators from a Banach space X into itself. Many properties of a C0-semigroup {T(t):t∈T}{T(t):t∈T} and its generator A   are established. Here T⊆R≥0T⊆R≥0 is a time scale endowed with an additive semigroup structure. We also establish necessary and sufficient conditions for the dynamic initial value problem {xΔ(t)=Ax(t),t∈Tx(0)=x0∈D(A),0∈Tto have a unique solution, where D(A) is the domain of A. Finally, we unify the continuous Hille–Yosida–Phillips Theorem and the discrete Gibson Theorem.