Resonance identity and multiplicity of non-contractible closed geodesics on Finsler RPn

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler n-dimensional real projective space (RPn,F) when there exist only finitely many distinct non-contractible closed geodesics on (RPn,F), where the integer n≥2. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on RPn with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in [14] and [39], it yields that every RPn with a bumpy Finsler metric possesses at least two distinct non-contractible closed geodesics.