A Nash-Moser theorem with near-minimal hypothesis. (English) Zbl 1021.47043

Neuberger, J. W.

Int. J. Pure Appl. Math. 4, No. 3, 269-280 (2003).

This article deals with a new Nash-Moser type inverse function theorem. Let \(H,K,J\) be Banach spaces and \(H\) embedded in \(J\) in the sense that \(H\) forms a linear subspace of \(J\) and every bounded sequence \((x_k)^\infty_{k =1}\) in \(H\) has a subsequence convergent in \(J\) to an element of \(x\in H\) so that \(\|x\|_H\leq \liminf_{k\to\infty} \|x_k\|_H\). Further, let an operator \(F:B_r\to K\) \((B_r\) is the ball in \(H\) with radius \(r\) and center \(0)\), \(F0=0\), be continuous as a function on \(J\cap D(F)\), \(g\in K\), and for each \(\varepsilon>0\) and \(x\) in the interior of \(B_r\) assume that there is a sequence \((x_n)^\infty_{n=1}\) in \(H\) and a sequence \((t_n)^\infty_{n=1}\) of numbers decreasing to 0 such that \[ \left\|{1\over t}(Fx_n-Fx)-g\right \|_K \leq \varepsilon,\quad \text{and} \quad\|x_n-x \|_H\leq Mt_n,\;n=1,2, \dots \] In this case, if \(\lambda\in [0,r/M)\), then there is \(x\in B_{\lambda M}\) such that \(Fx=\lambda g\).
The main difference of this result in comparison with classical Nash-Moser results is related to the method of its proof; the approach of the author is not based on modified (“smooth”) Newton iterations; moreover, the author underlines that the main idea of his approach is near to the method based on the “continuous Newton’s method” that reduces solving a nonlinear operator equation \(Fx=g\) to the study of the nonlocal Cauchy problem \[ z' (t)=F'\bigl(z(t) \bigr)^{-1}g,\;z(0)=0,\;t\in[0,1) \] (the latter approach was directly used in the article by A. Castro and J. W. Neuberger [Nonlinear Anal. 47, 3223-3229 (2001)]. However, the theorem presented above implies a Moser-like inverse function theorem for operators acting in a scale of Hilbert spaces with multiplicative inequalities for norms.

Reviewer: Peter Zabreiko (Minsk)

MSC:

47J05	Equations involving nonlinear operators (general)
58C15	Implicit function theorems; global Newton methods on manifolds

Keywords:

Nash-Moser inverse function theorem; continuous Newton method; operators in Hilbert spaces; nonlocal Cauchy problem