GENERALIZED PERIODIC ORBITS IN SOME RESTRICTED THREE-BODY PROBLEMS

The paper proves that, for small mass ratio, the restricted three‑body problem has at least l generalized Tε‑periodic solutions of the first kind by reducing to a forced Kepler equation, invoking KS regularization, and applying a Weinstein-type persistence result (Theorem 2.1) and then specializing to R3BP (Theorem 4.1) . The candidate solution instead treats R3BP (in an inertial frame) as a small Tε‑periodic C∞ perturbation of the autonomous Kepler problem on a collision‑free annulus, selects l resonant Kepler tori whose periods divide Tε, and applies Weinstein’s theorem on periodic manifolds. After minor fixes (restrict the perturbation estimates to a neighborhood of the chosen periodic manifolds away from the singularities, and note the use of q‑fold covered Kepler orbits), the model’s strategy yields the same conclusion—l generalized solutions of the first kind with Zm=∅ as in the paper’s definition . Hence both are correct, via different (but related) routes.