## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 4, Number 3 (1999), 169-194.

### Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

Gabriele Gühring and Frank Räbiger

#### Abstract

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation $\left(d/dt\right)u\left(t\right)=Au\left(t\right)+B\left(t\right)u\left(t\right)+f\left(t\right),t\in \mathbb{R}$, where $\left(A,D\left(A\right)\right)$ is a Hille-Yosida operator on a Banach space $X,B\left(t\right),t\in \mathbb{R}$, is a family of operators in $\mathcal{L}\left(\overline{D\left(A\right)},X\right)$ satisfying certain boundedness and measurability conditions and $f\in {L}_{\text{\hspace{0.17em}}\text{loc}}^{\text{\hspace{0.17em}}1}\left(\mathbb{R},X\right)$. The solutions of the corresponding homogeneous equations are represented by an evolution family ${\left({U}_{B}\left(t,s\right)\right)}_{t\ge s}$. For various function spaces $\mathcal{F}$ we show conditions on ${\left({U}_{B}\left(t,s\right)\right)}_{t\ge s}$ and $f$ which ensure the existence of a unique solution contained in $\mathcal{F}$. In particular, if ${\left({U}_{B}\left(t,s\right)\right)}_{t\ge s}$ is $p$-periodic there exists a unique bounded solution $u$ subject to certain spectral assumptions on ${U}_{B}\left(p,0\right),f$ and $u$. We apply the results to nonautonomous semilinear retarded differential equations. For certain $p$-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of ${\left({U}_{B}\left(t,s\right)\right)}_{t\ge s}$.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 4, Number 3 (1999), 169-194.

**Dates**

First available in Project Euclid: 9 April 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1049907202

**Digital Object Identifier**

doi:10.1155/S1085337599000214

**Mathematical Reviews number (MathSciNet)**

MR1811234

**Zentralblatt MATH identifier**

0987.34062

**Subjects**

Primary: 34C25: Periodic solutions 34C27

Secondary: 34C28 34G10 47D06 47H15

#### Citation

Gühring, Gabriele; Räbiger, Frank. Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations. Abstr. Appl. Anal. 4 (1999), no. 3, 169--194. doi:10.1155/S1085337599000214. https://projecteuclid.org/euclid.aaa/1049907202