ON LOCAL INTEGRATED $C$-COSINE FUNCTION AND WEAK SOLUTION OF SECOND ORDER ABSTRACT CAUCHY PROBLEM

Abstract

Let $\alpha$ be a nonnegative number, $C : X \to X$ a bounded linear injection on a Banach space $X$ and $A : D(A) \subset X \to X$ a closed linear operator in $X$ which satisfies $C^{−1}AC = A$ and may not be densely defined. We prove some equivalence relations between the generation of a local $\alpha$-times integrated $C$-cosine function on $X$ with generator $A$ and the uniqueness existence of weak solutions of the abstract Cauchy problem: \[ \operatorname{ACP}_{2}(A,f,x,y) \qquad \begin{cases} u''(t) = Au(t) + f(t) \quad \textrm{for } t \in (0,T_0), \\ u(0) = x, \; u'(0) = y, \end{cases} \] where $x,y \in X$ are given and $f$ is an $X$-valued function defined on a subset of $\mathbb{R}$.