## The Networking of Chaperones by Co-chaperonesThe volume contains the texts of four courses, given by the authors at a summer school that sought to present the state of the art in the growing field of topological methods in the theory of o.d.e. (in finite and infinitedimension), and to provide a forum for discussion of the wide variety of mathematical tools which are involved. The topics covered range from the extensions of the Lefschetz fixed point and the fixed point index on ANR's, to the theory of parity of one-parameter families of Fredholm operators, and from the theory of coincidence degree for mappings on Banach spaces to homotopy methods for continuation principles. CONTENTS: P. Fitzpatrick: The parity as an invariant for detecting bifurcation of the zeroes of one parameter families of nonlinear Fredholm maps.- M. Martelli: Continuation principles and boundary value problems.- J. Mawhin: Topological degree and boundary value problems for nonlinear differential equations.- R.D. Nussbaum: The fixed point index and fixed point theorems. |

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### Contents

P PITZPATRICK The parity as an invariant for detecting | 2 |

2 The Parity and the Structure of 0 XY | 8 |

3 Local Bifurcation for One Parameter Families | 16 |

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Analysis applications Assume assumption Banach space bifurcation Bifurcation Theory boundary value problems bounded bounded set branch called choose closed compact complete Consequently consider constant contains continuous map convex Corollary corresponding crosses defined definition denote describe DIEGO differential equations eigenvalue equivalent example exists extension fact fixed point index forced Fredholm operators function Furi given Hence hold homotopy implies integer invariant invertible Kerl least Lemma Leray-Schauder degree linear Math Mawhin method metric Moreover multiplicity necessary nonlinear Null obtain open neighborhood operators parametrized parity particular path periodic solutions perturbations positive present problem Proof Proposition prove Remark respect result satisfies simple singular stable subset sufficient Suppose T-periodic Theorem theory topological trivial solutions unbounded Univ zero