Let $K\colon[0,T_0)\to\mathbb F$ be a locally integrable function, and $C\colon X\to X$ a bounded linear operator on a Banach space $X$ over the field $\mathbb F(=\mathbb R\text{ or }\mathbb C)$. In this paper, we will deduce some basic properties of a nondegenerate local $K$-convoluted $C$-cosine function on $X$ and some generation theorems of local $K$-convoluted $C$-cosine functions on $X$ with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local $K$-convoluted $C$-cosine function on $X$ with subgenerator $A$ and the unique existence of solutions of the abstract Cauchy problem: $u''(t)=Au(t)+f(t)$ for a.e. $t\in(0,T_0)$, $u(0)=x$, $u_0(0)=y$ when $K$ is a kernel on $[0,T_0)$, $C\colon X\to X$ an injection, and $A\colon D(A)\subset X\to X$ a closed linear operator in $X$ such that $CA\subset AC$. Here $0<T_0\leq\infty$, $x,y\in X$ and $f\in L^1_{\operatorname{loc}}([0,T_0),X)$.