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 {equation*} d/dt({mẋ}{√{1-|ẋ|²/c²}}) = -α{x}{|x|³} + ε \, ∇ₓ U(t,x), x ∈ R^d\{0\}, {equation*} with $d=2$ or $d=3$, bifurcating, for ε 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.