Diekmann, Odo; Gyllenberg, Mats; Thieme, Horst R. Semigroups and renewal equations on dual Banach spaces with applications to population dynamics. (English) Zbl 0799.47022 Delay differential equations and dynamical systems, Proc. Conf., Claremont/CA (USA) 1990, Lect. Notes Math. 1475, 116-129 (1991). [For the entire collection see Zbl 0727.00007.]If \(T_ 0\) is a strongly continuous semigroup of linear operators on the Banach space \(X\), with generator \(A_ 0\), and \(C\) a bounded linear operator from \(X^ \odot\) (sun dual) to \(X^*\) then \(A^*_ 0+C\) generates a \(w^*\)-semigroup \(T^ \times\) on \(X^*\). It is shown how this perturbation problem gives rise to a renewal equation whose solution can be used to obtain the semigroup \(T^ \times\). Conversely, starting from an abstract renewal equation the authors give conditions when the solutions of this equation determine a semigroup on \(X^*\).The results of this paper are motivated by the theory of age structured population dynamics, and the notions are interpreted in terms of this theory. Reviewer: J.Voigt (Dresden) Cited in 2 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 92D25 Population dynamics (general) Keywords:one-parameter semigroup; integrated semigroup; sun dual; perturbation problem; structured population dynamics Citations:Zbl 0727.00007 PDFBibTeX XMLCite \textit{O. Diekmann} et al., Lect. Notes Math. None, 116--129 (1991; Zbl 0799.47022)