  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 $n\geq3$ whose metric is induced by
irreversible Finsler $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flag
curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\leq 1$, 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.
