| A Co semigroup {T(t)|t≥ 0} on a Banach space X is said to be eventually norm continuous semigroup if there exists t0 ≥ 0 such that t → T(t) is norm continuous for t > t0, especially, if t0 = 0, this semigroup is called norm continuous, A be the infinitesimal generator of T(t). As shown in [1], an eventually norm continuous semigroup satisfies the spectrum determined growth assumption which is a very important property concerned with the exponential stability of linear dynamical systems.Seeking characterizations of eventually norm continuous semigroups is an old problem studied by many researchers. In 1983, Pazy in [1] pointed out. " that so far there are no known necessary and sufficient conditions in term of A or the resolvent R(λ, 4), which assure the continuity for t > 0 of T(t) in the uniform operator topology." In 1992, P.You in [2] showed that for semigroups in Hilbert spaces the norm continuity for t > 0 is equivalent to the decay to zero of the resolvent of their generators along song imaginary axis. In 1996, Blasco and Martinez in [3] discussed the property of norm continuous semigroup on Banach space and proved the following assertion in a Hilbert space. Let A be the infinitesimal generator of a Co semigroup T(t) with || T(t) ||Me-t on Hilbert space, then T(t) is norm continuous for t > t0 > 0 if and only if there exits C > 0 such that .But this condition is not easilychecked in Banach spaces, moreover, this characteristic condition has not been solved, therefore, people are going on researching the property of eventually norm continuous semigroups to obtain more concise character in Hilbert spaces and break through this problem on Banach spaces.In this paper, we introduce some fundamental conceptions and conclusions about operator semigroup theory in the first section. In the second section we give some new conclusions about eventually norm continuous semigroups. In the final section we have some applications of operator semigroups theory to impulsive equation.
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