2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157025In this paper, we prove that for every Finsler $n$-sphere $(S^n, F)$ for $n\ge 3$ with reversibility $λ$ and flag curvature $K$ satisfying $(\fracλ{λ+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form $\exp(πi μ)$ with an irrational $μ$. Furthermore, there always exist three prime closed geodesics on any $(S^3, F)$ satisfying the above pinching condition.41 pages. Revised version. To appear in Adv. MathDifferential Geometry53C22, 53C60, 58E10text