On a conjecture of Anosov

In this paper, we prove that for every bumpy Finsler $n$-sphere $(S^n,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda }{\lambda +1}{)}^2<K\le 1$, there exist $2[\frac{n+1}{2}]$ prime closed geodesics. This gives a confirmed answer to a conjecture of D. V. Anosov \cite{Ano} in 1974 for a generic case.