MATHEMATICAL MODELLING OF POPULATION DYNAMICS BANACH CENTER PUBLICATIONS, VOLUME 63 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2004 SEMILINEAR PERTURBATIONS OF HILLE-YOSIDA OPERATORS HORST R. THIEME and HAUKE VOSSELER Department of Mathematics and Statistics, Arizona State University Tempe, AZ 85287-1804, U.S.A. E-mail: h.thieme@asu.edu Abstract. The semilinear Cauchy problem (1) u 0 (t)= Au(t)+ G(u(t)), u(0) = x ∈ D(A), with a Hille-Yosida operator A and a nonlinear operator G : D(A) → X is considered under the assumption that kG(x) - G(y)k≤kB(x - y)k ∀x, y ∈ D(A) with some linear B : D(A) → X, B(λ - A) -1 x = λ Z ∞ 0 e -λt V (s)xds, where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on [0, ∞) × D(A) that provides Friedrichs solutions in L 1 for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)|≤ Bv whenever x, y, v ∈ D(A), |x - y|≤ v, with B being positive. We illustrate our results by applications to age-structured population models. 1. Introduction. Let A be a Hille-Yosida operator in a Banach space X, i.e., after equivalent renormalization and an identity shift, an m-dissipative linear operator. Equiv- alently it is the generator of a locally Lipschitz continuous integrated semigroup S. The part A 0 of A in D(A) generates a C 0 -semigroup ˙ S which is the strong derivative of S. We will consider perturbations A + G with nonlinear operators G that are Lipschitz continuous with respect to A in various ways which we will make precise later, and show that the solutions of the associated Cauchy problem induce a continuous semiflow 2000 Mathematics Subject Classification : Primary 47J35; Secondary 92D25. Research of the first author partially supported by NSF grant DMS-9706787. Research of the second author supported by Studienstiftung des deutschen Volkes. The paper is in final form and no version of it will be published elsewhere. [87]