On the average indices of closed geodesics on positively curved Finsler spheres

In this paper, we prove that on every Finsler $n$-sphere $(S^n,F)$ for $n\ge 6$withreversibility$\lambda$andflagcurvature$K$satisfying$(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exist $[\frac{n}{2}]-2$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist $n-3$ closed geodesics possessing irrational average indices provided the number of closed geodesics is finite.