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Implicit Functions and Problems at Resonance

  • Klaus Deimling

Abstract

In the preceding chapters we used global concepts to study existence and uniqueness of solutions to F x = y. This time we start with the description of the local behaviour of the nonlinear map F by means of purely analytical methods. If F is differentiable in a neighbourhood of x0 it is natural to assume something about the ‘first approximation’ F'(x 0 ), i.e. to linearize the nonlinear problem to the linear problem F'(x0) (xx0) = yFx0, and to study the implications for F near x0 of such assumptions about F'(x0). The simplest result of this type is the inverse function theorem, saying that F is a homeomorphism from a small neighbourhood U of x0 onto F(U)if F is C1 near x0 and F'(x0) is a homeomorphism, together with its companion for parameter-dependent F, the classical implicit function theorem.

Keywords

Banach Space Implicit Function Theorem Fredholm Operator Degree Theory Complex Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Klaus Deimling
    • 1
  1. 1.Gesamthochschule PaderbornPaderbornFederal Republic of Germany

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