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Let Ci (i = 1, 2) be two arbitrary bounded operators on a Banach space. We study (C1,C2)-regularized cosine existence and uniqueness families and their relationship to second order abstract Cauchy problems. We also prove some of their basic properties. In addition, Hille–Yosida type sufficient conditions are given for the exponentially bounded case. Introduction. Let X be a Banach space with norm ‖ · ‖, and B(X) be the set of all bounded operators from X into itself. Consider a well-posed abstract Cauchy problem of the second order (0.1)    d2 dt2 u(t, x, y) = Au(t, x, y), t ∈ R, u(0, x, y) = x, d dt u(0, x, y) = y, or (0.2)    d2 dt2 v(t, x, y) = Av(t, x, y) + x+ ty, t ∈ R, v(0, x, y) = 0, d dt v(0, x, y) = 0, where A is a closed linear operator on X. It is well known that (0.1) or (0.2) is governed by a strongly continuous operator cosine function (see [12, 13]). Cosine families have received less attention than strongly continuous semigroups, partly because (0.1) may be rewritten as a first order abstract 2000 Mathematics Subject Classification: Primary 47D06, 47D09; Secondary 34G10.

Regularized cosine existence and uniqueness families for second order abstract Cauchy problems