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Attouch-Wets convergence and Kuratowski convergence on compact sets

Paolo PiccioneRosella Sampalmieri — 1995

Commentationes Mathematicae Universitatis Carolinae

Let X be a locally connected, b -compact metric space and E a closed subset of X . Let 𝔾 be the space of all continuous real-valued functions defined on some closed subsets of E . We prove the equivalence of the τ a w and τ K c topologies on 𝔾 , where τ a w is the so called topology, defined in terms of uniform convergence of distance functionals, and τ K c is the topology of Kuratowski convergence on compacta.

On the multiplicity of brake orbits and homoclinics in Riemannian manifolds

Roberto GiambòFabio GiannoniPaolo Piccione — 2005

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let M , g be a complete Riemannian manifold, Ω M an open subset whose closure is diffeomorphic to an annulus. If Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω ¯ = Ω Ω starting orthogonally to one connected component of Ω and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating...

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