The existence of affine-periodic solutions for nonlinear impulsive differential equations

In this paper, we study the existence of affine-periodic solutions of nonlinear impulsive differential equations. The affine-periodic solutions have the form x(t+T)=Qx(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x(t+T)=Qx(t)$\end{document} with some nonsingular matrix Q. We give a theorem on the existence of the affine-periodic solutions, respectively, depending on wether det(I−Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{det}(I-Q)$\end{document} (I= identity matrix) is equal to 0 or not.