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In this article, we give a growth rate about the number of periodic orbits inthe Franks type theorem obtained by the authors \cite{LWY}. As applications, weprove the following two results: there exist infinitely many distinctnon-contractible closed geodesics on $\mathbb{R}P^2$ endowed with a Riemannianmetric such that its Gaussian curvature is positive, moreover, the number ofnon-contractible closed geodesics of length $\leq l$ grows at least like $l^2$;and there exist either two or infinitely many distinct non-contractible closedgeodesics on Finsler $\mathbb{R}P^2$ with reversibility $\lambda$ and flagcurvature $K$ satisfying $\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$,furthermore, if the second case happens, then the number of non-contractibleclosed geodesics of length $\leq l$ grows at least like $l^2$.

The growth of the number of periodic orbits for annulus homeomorphisms  and non-contractible closed geodesics on Riemannian or Finsler  $\mathbb{R}P^2$