Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems

The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type ddtmx(center dot)1-|x(center dot)|2/c2=-alpha x|x|3+epsilon del xU(t,x),x is an element of Rd\{0},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{\textrm{d}}{\textrm{d}t}\left( \frac{m\dot{x}}{\sqrt{1-|\dot{x}|<^>{2}/c<^>{2}}}\right) = -\alpha \frac{x}{|x|<^>{3}} + \varepsilon \, \nabla _{x} U(t,x), \qquad x \in \mathbb {R}<^>d\setminus \{0\}, \end{aligned}$$\end{document}with d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} or d=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=3$$\end{document}, bifurcating, for epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that U is T-periodic in time) and the case of the fixed-energy problem (assuming that U is independent of time) are considered.