  In this paper, we first generalize the common index jump theorem for
symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an
enhanced version of it. As its applications, we further prove that for a
compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler
metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge
2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct
non-hyperbolic closed geodesics with odd Morse indices, provided the number of
distinct prime closed geodesics is finite and every prime closed geodesic
satisfies $i(c)>0$. Note that the last non-zero index condition is satisfied if
the flag curvature $K$ satisfies $K\ge 0$. For an odd-dimensional bumpy Finsler
sphere $(S^d,F)$, there exist at least $(d+1)$ distinct prime closed geodesics
with even Morse indices, and at least $(d-1)$ of which are non-hyperbolic,
provided the number of distinct prime closed geodesics is finite and every
prime closed geodesic $c$ satisfies $i(c)\ge 2$. Note that the last index
condition $i(c)\ge 2$ is satisfied if the reversibility $\lambda$ and the flag
curvature $K$ of $(M,F)$ satisfy $\frac{\lambda^2}{(1+\lambda)^2}<K\le 1$. Note
that the first two in the above three lower bound estimates are sharp due to
Katok's examples. In addition, we also prove that either there exists at least
one non-hyperbolic closed geodesic, or there exist infinitely many distinct
closed geodesics on a compact simply connected bumpy Finsler $(M,F)$ satisfying
the above cohomological condition with some even integer $d\ge 2$ and integer
$n\ge 1$.
