Closed geodesics on positively curved Finsler spheres

In this paper, we prove that for every Finsler n-sphere (Sn,F) for n⩾3 with reversibility λ and flag curvature K satisfying , 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(πiμ) with an irrational μ. Furthermore, there always exist three prime closed geodesics on any (S3,F) satisfying the above pinching condition.