Brouwer's approximate fixed-point theorem is equivalent to Brouwer's fan theorem. Veldman, W. In Logicism, intuitionism, and formalism, volume 341, of Synth. Libr., pages 277–299. Springer, Dordrecht, 2009.
Brouwer's approximate fixed-point theorem is equivalent to Brouwer's fan theorem [link]Paper  doi  bibtex   
@INCOLLECTION{Vel09a,
  author = {Veldman, Wim},
  title = {Brouwer's approximate fixed-point theorem is equivalent to {B}rouwer's
	fan theorem},
  booktitle = {Logicism, intuitionism, and formalism},
  publisher = {Springer, Dordrecht},
  year = {2009},
  volume = {341},
  series = {Synth. Libr.},
  pages = {277--299},
  doi = {10.1007/978-1-4020-8926-8_14},
  keywords = {bib, con},
  mrclass = {03F50},
  mrnumber = {2509663},
  url = {http://dx.doi.org/10.1007/978-1-4020-8926-8_14}
}

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