## The Topology of Function Spaces and the Calculus of Variations in the Large |

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### Contents

Merging of critical values | 4 |

APPLICATION TO NONLINEAR INTEGRAL EQUATIONS | 14 |

CRITICAL ssrs OF GEODESIC CURVES | 28 |

THE FUNCTIONAL SPACE R 0F cunves wrrn | 49 |

Acycles in R | 57 |

Products of Vcycles in R The product T X T | 63 |

Some homotopy questions in R | 69 |

The ndimensional case | 76 |

CLOSED GEODESICS ON MANIFOLDS HOMEOMORPHIC | 79 |

95 | |

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### Common terms and phrases

admissible permutations arbitrarily close arbitrary barycentric coordinates calculus of variations choose closed curves closed geodesics closed set coefﬁcients coincides common value compact set Consider contains continuous deformation corresponding critical set critical value curve q cycles V-homologous cyclic coordinate deﬁned deﬁnition deformation D2 degree of covering denote dimensional distance eigenelements eigenvalues elements equal to zero equation f g c ﬁlm ﬁnd ﬁnite ﬁrst ﬁxed follows function f functional spaces geodesic arcs geodesics of length goes homeomorphic homologous to zero i-interval intersection index K-class K-critical K-deformation K-set La,a LEMMA level surface lies Ljusternik lying manifold homeomorphic meridians metric modulo n-dimensional A-cycle n-dimensional sphere neighborhood non-selﬁntersecting curves nonempty normal deformation one-dimensional V-cycle operation orthogonal plane polygon polymeridians projective space proved region Riemannian metric satisﬁes segment set-theoretical intersection simplex Snirel’man sphere 82 stationary points subordinated sufﬁciently close sufﬁciently large sufﬁciently small theorem tion topology two-dimensional vertices