Closed geodesics on positively curved spheres $S^n$ with Finsler metric induced by $(\mathbb{R}{P}^n,F)$

It’s well known that the n-sphere $S^n$ is the universal double covering of the $n$-dimensional real projective space $\mathbb{R}{P}^n$ and then any Finsler metric on $\mathbb{R}{P}^n$ induces a Finsler metric of $S^n$. In this paper, we prove that for every Finsler $(S^n,F)$ for $n\ge 3$ whose metric is induced by irreversible Finsler $(\mathbb{R}{P}^n,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda }{\lambda +1}{)}^2<K\le 1$, there exist at least $n-1$ prime closed geodesics on $(S^n,F)$. Furthermore, if there exist finitely many distinct closed geodesics on $(S^n,F)$, then there exist at least $2[\frac{n}{2}]-1$ of them are non-hyperbolic.