On the minimal number of closed geodesics on positively curved Finsler spheres

In this paper, we proved that for every Finsler metric on $S^n$ $(n\ge 4)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{2n-3}{n-1}{)}^2(\frac{\lambda }{\lambda +1}{)}^2<K\le 1$ and $\lambda <\frac{n-1}{n-2}$, there exist at least $n$ prime closed geodesics on $(S^n,F)$, which solved a conjecture of Katok and Anosov for such positivley curved spheres when $n$ is even. Furthermore, if the number of closed geodesics on such positively curved Finsler $S^n$ is finite, then there exist at least $2\left[\frac{n}{2}\right]-1$ non-hyperbolic closed geodesics.