# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## Semilinear perturbations of Hille-Yosida operators

### Tom 63 / 2003

Banach Center Publications 63 (2003), 87-122 MSC: Primary 47J35; Secondary 92D25 DOI: 10.4064/bc63-0-3

#### Streszczenie

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.

#### Autorzy

• Horst R. ThiemeDepartment of Mathematics and Statistics
Arizona State University
Tempe, AZ 85287-1804, U.S.A.
e-mail
• Hauke VosselerDepartment of Mathematics and Statistics
Arizona State University
Tempe, AZ 85287-1804, U.S.A.

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