Cesaro and Abel ergodic theorems for integrated semigroups

Let {S(t)}(t >= 0) be an integrated semigroup of bounded linear operators on the Banach space X into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesaro mean and the Abel average of S(t) converge uniformly on B(X). More precisely, we show that the Abel average of S(t) converges uniformly if and only if X = R(A) circle plus N(A), if and only if R(A(k)) is closed for some integer k and parallel to lambda(2) R(lambda, A)parallel to -> 0 as lambda -> 0(+), where R(A), N(A) and R(lambda, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t(2)-> 0 as t -> infinity, then the Cesaro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.