Dynamical systems method and a homeomorphism theorem

Details Dynamical systems method and a homeomorphism theorem Dynamical systems method and a homeomorphism theorem

2004-08-14

arxiv.org/abs/math/0408192v1

Mathematics

Functional Analysis

Scientific paper

Let $F$ be a nonlinear map in a real Hilbert space $H$. Suppose that $\sup_{u\in B(u_0,R)}$ $\|[F'(u)]^{-1}\|\leq m(R)$, where $B(u_0,R)=\{u:\|u-u_0\|\leq R\}$, $R>0$ is arbitrary, $u_0\in H$ is an element. If $\sup_{R>0}\frac{R}{m(R)}=\infty$, then $F$ is surjective. If $\|[F'(u)]^{-1}\|\leq a\|u\|+b$, $a\geq 0$ and $b>0$ are constants independent of $u$, then $F$ is a homeomorphism of $H$ onto $H$. The last result is known as an Hadamard-type theorem, but we give a new simple proof of it based on the DSM (dynamical systems method).

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