Pseudo almost periodic solutions of some delay differential equations (Q1922965)

The theory of exponential dichotomy is used successfully to establish the existence of pseudo almost periodic solutions of delay-differential equations of the form \(x'(t)=L(t)x_t+f(t)\), \(t\geq \sigma\), \(x_\sigma=\varphi\), where \((\sigma,\varphi)\in\mathbb{R}\times C([-r,0],\mathbb{R}^n)\), \(r>0\), which have pseudo almost periodic coefficients. Also the following theorem is proved: If \(\sigma(A)\cap i\mathbb{R}=\phi\) and \(f:\mathbb{R}\to\mathbb{R}^n\) is continuous and pseudo almost periodic, where \(\sigma(A)\) is the spectrum of the infinitesimal generator \(A\), then \(x'(t)=Lx_t+f(t)\) has a unique bounded solution which is also pseudo almost periodic.