On a theorem for linear evolution equations of hyperbolic type
Kazuo Kobayasi · Journal of the Mathematical Society of Japan · 1979
In [1] and [2] T. Kato gave some fundamental and important theorems about evolution operator associated with linear evolution equationsThose 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.