Let C : X → X be a bounded linear operator on a Banach space X over the field F(=R or C), and K : [0, T 0) → F a locally integrable function for some 0 < T 0 ≤ ∞. Under some suitable assumptions, we deduce some relationship between the generation of a local (or an exponentially bounded) K-convoluted C 0 0 C-semigroup on X × X with subgenerator (resp., the generator) 0 I B A and one of the following cases: (i) the well-posedness of a complete second-order abstract Cauchy problem ACP(A, B, f, x, y): w (t) = Aw (t) + Bw(t) + f (t) for a.e. t ∈ (0, T 0) with w(0) = x and w (0) = y; (ii) a Miyadera-Feller-Phillips-Hille-Yosida type condition; (iii) B is a subgenerator (resp., the generator) of a locally Lipschitz continuous local α-times integrated C-cosine function on X for which A may not be bounded; (iv) A is a subgenerator (resp., the generator) of a local α-times integrated C-semigroup on X for which B may not be bounded.