Hille–Yosida type theorems for O(ω(t))nO(ω(t))n-times integrated semigroups

For an arbitrary continuous increasing function ω(t):[0,∞)↦Cω(t):[0,∞)↦C, we introduce the concept and characterize the properties of nn-times integrated Laplace transforms of O(ω(t))O(ω(t)) vector-valued functions. Based on these properties, we have established Hille–Yosida type theorems for O(ω(t))nO(ω(t))n-times integrated semigroups. An explanation example is given.