Implicit Functions and Problems at Resonance
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) (x − x0) = y −Fx0, 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.
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