Existence and uniqueness families for the abstract Cauchy problem. (English) Zbl 0766.47011

deLaubenfels, Ralph

J. Lond. Math. Soc., II. Ser. 44, No. 2, 310-338 (1991).

Summary: Suppose that \(C\) is an arbitrary bounded operator on a Banach space. We define a pair of families of operators, one of which yields uniqueness and one of which yields existence, of solutions of the abstract Cauchy problem, for all initial data in the image of \(C\). For exponentially bounded solutions, Hille-Yosida type sufficient conditions are given. We also give a perturbation theory. We apply our results to matrices of operators, acting on the product of (possibly different) Banach spaces.

MSC:

47D06	One-parameter semigroups and linear evolution equations
47E05	General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34G10	Linear differential equations in abstract spaces
47A55	Perturbation theory of linear operators

Keywords:

bounded operator on a Banach space; uniqueness; existence; solution of the abstract Cauchy problem; exponentially bounded solutions; Hille- Yosida type sufficient conditions; perturbation theory; matrices of operators