Classical Solutions for a Perturbed N-Body System.- Variational Setting for Newton's Equations.- The Kepler Problem Revisited.- The N-Body Problem.- Results form Critical Point Theory.- Classical Periodic Solutions for the Perturbed N-Body System.- Acknowledgments.- References.- Degree Theory: Old and New.- Degree Theory for Maps in the Sobolev Class H1(S2, S2).- Degree Theory for Maps in the Sobolev Class H1(S1, S1).- Degree Theory for Maps in VMO (Sn, Sn).- Further Properties of VMO Maps in Connection with Topology.- Degree Theory for VMO Maps on Domains.- References.- Global Structure for Nonlinear Operators in Differential and Integral Equations I. Folds.- Fréchet Derivatives.- Fredholm Maps.- Local Structure of Folds.- Abstract Global Characterization of the Fold Map.- Ambrosetti-Prodi and Berger-Podolak - Church Fold Maps.- McKean-Scovel Fold Map.- Giannoni-Micheletti Fold Map.- Mandhyan Fold Map.- Oriented Global Fold Maps.- A Second Mandhyan Fold Map.- Jumping Singularities.- References.- Global Structure for Nonlinear Operators in Differential and Integral Equations II. Cusps.- Critical Values of Fredholm Maps.- Applications of Critical Values to Nonlinear Differential Equations.- Factorization of Differentiate Maps.- Local Structure of Cusps.- Some Local Cusp Results.- von Kármán Equations.- Abstract Global Characterization of the Cusp Map.- Mandhyan Integral Operator Cusp Map.- Pseudo-Cusp.- Cafagna and Donati Theorems on Ordinary Differential Equations.- Micheletti Cusp-like Map.- Cafagna Dirichlet Example.- u3 Dirichlet Map - Initial Results.- u3 Dirichlet Map - The Singular Set and its Image.- u3 Dirichlet Map - The Global Result.- Ruf u3 Neumann Cusp Map.- Ruf's Higher Order Singularities.- Damon's Work in Differential Equations.- References.- Degree for Gradient Equivariant Maps and Equivariant Conley Index.- Basic Notions of Equivariant Topology.- Remarks and Examples.- An Analytic Definition of the Gradient Equivariant Degree.- Technicalities.- Equivariant Conley Index.- Box-like Index Pairs.- The torn Dieck Ring.- Bifurcation.- References.- Variations and Irregularities.- Summary.- Generalized Differential Operators.- Irregularities.- Mass, Length, Energy.- Homogeneous Dirichlet Spaces.- Fractals.- References.- Singularity Theory and Bifurcation Phenomena in Differential Equations.- The Normal Forms for f : ?n ? ?m.- The Malgrange Preparation Theorem.- Singularity Theory for Mappings Between Banach Spaces.- Applications to Elliptic Boundary Value Problems.- First Order Differential Equations.- Global Equivalence Theorems.- Problems with Additional Parameters: Unfoldings.- Bifurcation of Minimal Surfaces.- Singularities at Double Eigenvalues.- Multiplicity by combining Local and Global Information.- Some Numerical Results.- References.- Bifurcation from the Essential Spectrum.- General Setting.- Nonlinear Perturbation of a Self-Adjoint Operator.- Bifurcation from the Infimum of the Spectrum.- Bifurcation into Spectral Gaps.- Semilinear Elliptic Equations.- References.- Rotation of Vector Fields: Definition, Basic Properties, and Calculation.- The Brouwer-Hopf Theory of Continuous Vector Fields.- The Leray-Schauder Theory of Completely Continuous Vector Fields.- Vector Fields with Noncompact Operators.- Some Generalizations and Modifications.- References.

The main purpose of the present volume is to give a survey of some of the most significant achievements obtained by topological methods in nonlin­ ear analysis during the last three decades. It is intended, at least partly, as a continuation of Topological Nonlinear Analysis: Degree, Singularity and Varia­ tions, published in 1995. The survey articles presented are concerned with three main streams of research, that is topological degree, singularity theory and variational methods, They reflect the personal taste of the authors, all of them well known and distinguished specialists. A common feature of these articles is to start with a historical introduction and conclude with recent results, giving a dynamic picture of the state of the art on these topics. Let us mention the fact that most of the materials in this book were pre­ sented by the authors at the "Second Topological Analysis Workshop on Degree, Singularity and Variations: Developments of the Last 25 Years," held in June 1995 at Villa Tuscolana, Frascati, near Rome. Michele Matzeu Alfonso Vignoli Editors Topological Nonlinear Analysis II Degree, Singularity and Variations Classical Solutions for a Perturbed N-Body System Gianfausto Dell 'A ntonio O. Introduction In this review I shall consider the perturbed N-body system, i.e., a system composed of N point bodies of masses ml, ... mN, described in cartesian co­ ordinates by the system of equations (0.1) where f) V'k,m == -£l--' m = 1, 2, 3.