The so-called Homeomorphic mapping theorem is that
If $H(z):\mathbb{R}^n\to\mathbb{R}^n$ is a continuous map and satisfies the
following conditions:
(i) $H(z)$ is injective on $\mathbb{R}^n$,
(ii) $\lim_{\|z\|\to\infty}\|H(z)\|=\infty$,
then $H(z)$ is a homeomorphism of $\mathbb{R}^n$ onto itself.
How to prove the theorem? Please give the proof or recommend a book about the theorem. Thank you very much.
Dear Prof Chen, I suggest you to take a look at this contribution where you can find some variants and extensions of the result you have mentioned.
Sincerely yours
Mauro
G. De Marco, G. Gorni, and G. Zampieri, “Global inversion of functions:
an introduction,” Nonlinear Differential Equations and Applications
NoDEA, vol. 1, no. 3, pp. 229–248, 1994.
Dear Prof. Forti,
Thanks very much for your pertinent suggestion. I believe I will learn more and get the answer from the paper. Moreover, previously, I had read some of your paper on discontinuous neural networks. From these perfect papers, I learn more about discontinuous dynamical system. So thank you very much!
Best regards,
Xiaofeng
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