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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)

MSC:

47D06 One-parameter semigroups and linear evolution equations
92D25 Population dynamics (general)

Citations:

Zbl 0727.00007
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