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

Abstract

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation $\left(d/dt\right)u\left(t\right)=Au\left(t\right)+B\left(t\right)u\left(t\right)+f\left(t\right),t\in ℝ$, where $\left(A,D\left(A\right)\right)$ is a Hille-Yosida operator on a Banach space $X,B\left(t\right),t\in ℝ$, is a family of operators in $ℒ\left(\overline{D\left(A\right)},X\right)$ satisfying certain boundedness and measurability conditions and $f\in L_{\text{loc}}^1\left(ℝ,X\right)$. 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 $ℱ$ 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 $ℱ$. 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 ${\left(U_B\left(t,s\right)\right)}_{t\ge s}$.