Title: On a theorem for linear evolution equations of hyperbolic type
Abstract: In [1] and [2] T. Kato gave some fundamental and important theorems about evolution operator associated with linear evolution equationsof " hyPerbolic type in a Banach space $X$ .Here, $f$ is a given function from $[0, T]$ into $X,$ $A(t)$ is a given linear operator which is a negative generator of a $C_{0}$ -semigroup in $X$ , and the unknown functionThose theorems are useful in applications to symmetric hyperbolic systems of partial differential equations (for example, see [3] and [7]).The proofs were carried out by using a device due to Yosida $[8, 9]$ , and the proof of Theorem 6.1 of [1] was simplified later by Dorroh [4].It is assumed in those articles that $A(t)$ is norm continuous from $[0, T]$ into $B(Y, X)$ , where $Y$ is a Banach space densely and continuously embedded in $X$ .However, we find it useful to strengthen the theorems by replacing the norm continuity of $A(t)$ with strong continuity.The purpose of the present paper is to show that Theorem 6.1 of [1] is still valid if we assume the strong continuity of $A(t)$ instead of the norm continuity of it.In Section 1 our result is stated.In Section 2 we give a proof of it.In this paper we refer to [1] for notations and definitions.