NOTE ON LOCAL INTEGRATED $\text{C}$-COSINE FUNCTIONS AND ABSTRACT CAUCHY PROBLEMS

Abstract

Let $\alpha$ be a nonnegative number, and $\text{C}: \text{X} \to \text{X}$ a bounded linear operator on a Banach space $\text{X}$. In this paper, we shall deduce some basic properties of a nondegenerate local $\alpha$-times integrated $\text{C}$-cosine function on $\text{X}$ and some generation theorems of local $\alpha$-times integrated $\text{C}$-cosine functions on $\text{X}$ with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local $\alpha$-times integrated $\text{C}$-cosine function on $\text{X}$ with generator $\text{A}$ and the unique existence of solutions of the abstract Cauchy problem: $$\textrm{ACP}(A,f,x,y) \qquad \begin{cases} u''(t) = Au(t) + f(t) \quad \textrm{for } t \in (0,T_0), \\ u(0) = x, u'(0) = y, \end{cases}$$ just as the case of $\alpha$-times integrated $\text{C}$-cosine function when $\text{C} :\text{X}\to\text{X}$ is injective and $\text{A}:\text{D}(\text{A})\subset\text{X}\to\text{X}$ a closed linear operator in $\text{X}$ such that $\text{C}\text{A}\subset \text{A}\text{C}$. Here $0 \lt T_0 \leq \infty$, $x,y \in X$, and $f$ is an $X$-valued function defined on a subset of $\mathbb{R}$ containing $(0,T_0)$.