Local C-semigroups and complete second order abstract Cauchy problems
Keywords:
Integrated C-semigroups, generator, subgenerator, abstract Cauchy problem.Abstract
Let C : X ! X be an injective bounded linear operator on a Banach space X over the field F(=R or C) and 0 < T0 1. Under some suitable assumptions, we deduce some relationship between the generation of a local (or an exponentially bounded) 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): w00(t) = Aw0(t)+Bw(t)+f(t) for a.e. t 2 (0, T0) with w(0) = x and w0(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 C-cosine function on X for which A may not be bounded; (iv) A is a subgenerator (resp., the generator) of a local C-semigroup on X for which B may not be bounded.
Mathematics Subject Classification (2010): 47D60, 47D62.
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