Let alpha be a nonnegative number, and C : X --> X a bounded linear operator on a Banach space X. In this paper, we shall deduce some basic properties of a nondegenerate local alpha-times integrated C-cosine function on X and some generation theorems of local alpha-times integrated 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 alpha-times integrated C-cosine function on X with generator A and the unique existence of solutions of the abstract Cauchy problem: ACP(A, f, x, y) {u ''(t) = Au(t) + f(t) for t is an element of (0, T-0) u(0) = x,u'(0) = y, just as the case of alpha-times integrated C-cosine function when C : X --> X is injective and A : D(A) subset of X --> X a closed linear operator in X such that CA subset of AC. Here 0 < T-0 <= infinity, x, y is an element of X, and f is an X-valued function defined on a subset of R containing (0, T-0).
