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Abstract

The semilinear Cauchy problem $$ u'(t) = A u(t) + G (u(t)),\quad u(0)=x \in \overline{D(A)}, \tag*{(1)}$$ with a Hille-Yosida operator $A$ and a nonlinear operator $G : D(A) \to X$ is considered under the assumption that $$ \| G(x) - G(y) \| \le \|B (x -y ) \| \quad \forall x, y \in D(A) $$ with some linear $B : D(A) \to X$, $$ B(\lambda - A)^{-1}x = \lambda \int_0^\infty e^{-\lambda t} V(s) x d s, $$ where $V$ is of suitable small strong variation on some interval $ [0, \varepsilon) $. We will prove the existence of a semiflow on $[0,\infty) \times \overline{D(A)} $ that provides Friedrichs solutions in $L_1$ for (1). If $X$ is a Banach lattice, we replace the condition above by $$ | G(x) - G(y) | \le B v \quad \hbox{ whenever } x,y,v \in D(A),\, |x-y|\le v, $$ with $B$ being positive. We illustrate our results by applications to age-structured population models.