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

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.

### MSC:

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