An inverse function theorem for Frechet spaces

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue’s dominated convergence theorem and Ekeland’s variational principle. As a consequence, the assumptions are substantially weakened: th...

An Inverse Function Theorem for Metrically Regular Mappings

We prove that if a mapping F : X → → Y , where X and Y are Banach spaces, is metrically regular at x̄ for ȳ and its inverse F−1 is convex and closed valued locally around (x̄, ȳ), then for any function G : X → Y with lipG(x̄) · regF (x̄ | ȳ)) < 1, the mapping (F + G)−1 has a continuous local selection x(·) around (x̄, ȳ + G(x̄)) which is also calm.

On Consistency in Parameter Spaces of Expanding Dimension: an Application of the Inverse Function Theorem

Foutz (1977) uses the Inverse Function Theorem to prove the existence of a unique and consistent solution to the likelihood equations. This note extends his results in three useful directions. The first is to remark that with minor modification the same proof may be used to show that the solution to the likelihood equations converges asymptotically to the least-false parameter (Hjort (1986, 199...

On the Inverse Function Theorem

Theorem for an Inverse Problem

Let VZu+k2u+k2a~(z)u+V.(a~(~)Vu) = -6(cc-y) inR3, where al(z) EL’(D), a&) E H’(D), D E R3_ = {z : 13 < 0) is a finite region, aj(Z) = 0 outside of D, j = 1,2, 1 + 02 > 0, al = El. Assume that thedatau(l,y,k),V~,yEP={I:13=0}8ndallkE(O,ko),ko>O is an arbitrary small number, are given. THEOREM. The above data determine aj(z), j = 1,2, uniquely.