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Spectral mapping theorem and semilinear Cauchy problems for integrated semigroups. (Spektraler Abbildungssatz und semilineare Cauchy-Probleme für integrierte Halbgruppen.) (German) Zbl 0802.47042

Tübingen: Universität Tübingen, Math. Fakultät, 81 p. (1992).
In this doctoral thesis the author solves mainly two important problems: the first one concerns the spectral mapping theorem, the other one the existence of solutions of the semilinear Cauchy problem \[ u'(t)= Au(t)+ F(t,u(t)), \qquad u(0)= u_ 0, \] where \(A\) is the generator of an integrated semigroup \((S(t))_ t\) of operators \(S(t)\) on the Banach space \(X\).
The first main and surprising result is the following one: If \((S(t))_ t\) is an \(n\)-times integrated semigroup where \(n\geq 1\) then the spectral mapping theorem holds true, i.e. the spectrum of \((S(t))\) can easily be constructed out of the spectrum of \(A\). As is known this is not true for \(n=0\).
Concerning the Cauchy problem the author generalizes results of Ph. Clément, O. Diekmann, M. Gyllenberg, H. J. A. M. Heijmans and H. R. Thieme [Pitman Res. Notes Math. Ser. 190, 67-89 (1989; Zbl 0675.47036)], W. Desch, W. Schappacher and K. P. Zhang [Houston J. Math. 15, No. 4, 527-552 (1989; Zbl 0712.47052)], and H. R. Thieme [Differential and integral equations 3, 1035-1066 (1990; Zbl 0734.34059)].

MSC:

47D06 One-parameter semigroups and linear evolution equations
47A10 Spectrum, resolvent
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