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In this paper, we prove that for every irreversible Finsler $n$-dimensionalreal projective space $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flagcurvature $K$ satisfying$\frac{16}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ with $\lambda<3$,there exist at least $n-1$ non-contractible closed geodesics. In addition, ifthe metric $F$ is bumpy with$\frac{64}{25}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ and$\lambda<\frac{5}{3}$, then there exist at least $2[\frac{n+1}{2}]$non-contractible closed geodesics, which is the optimal lower bound due toKatok's example. The main ingredients of the proofs are the Fadell-Rabinowitzindex theory of non-contractible closed geodesics on $(\mathbb{R}P^n,F)$ andthe $S^1$-equivariant Poincar$\acute{e}$ series of the non-contractiblecomponent of the free loop space on $\mathbb{R}P^n$.

The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$