About Fixed Point Theorems, Wikipedia says:

Results of this kind are amongst the most generally useful in mathematics.

This seems an accurate statement: indeed, there are many journals dedicated exclusively to this topic and many monographs have been published.

  • Could you elaborate on why this kind of theorem is important to the development of current mathematical research? In particular, what is the relationship between fixed point theory and analysis?
  • Also, could you recommend some self-contained but comprehensive monographs?
  • $\begingroup$ The following link could be of your interest karlin.mff.cuni.cz/~prazak/uceni/101/Literatura/… $\endgroup$ – mfl Mar 13 '15 at 23:20
  • $\begingroup$ I believe they occur in mathematical economics. $\endgroup$ – Michael Hardy Mar 13 '15 at 23:33
  • $\begingroup$ The number of papers published reflects the popularity of a mathematical topic, not necessarily its usefulness. But fixed point theorems are useful in economics, among other areas. $\endgroup$ – bubba Mar 14 '15 at 2:19

Many problems can be turned into 'fixed point theorem' problems.

One of the most useful classes of such problems relates to existence/uniqueness theorems for differential equations.

  • $\begingroup$ This was the first main use of the contraction mapping theorem that I encountered in my grad classes. $\endgroup$ – Alan Mar 13 '15 at 23:20
  • $\begingroup$ And if you have a fixed point problem and need solutions to $f(x) = x$, there are lots of numerical methods to solve these kinds of equations. $\endgroup$ – pjs36 Mar 13 '15 at 23:53
  • $\begingroup$ Could you be more specific? And could you address the second point of the question? $\endgroup$ – user62029 Mar 14 '15 at 8:51
  • $\begingroup$ Here's a classic example: en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem There may a book out there on Fixed Point Theorems. But I have never seen one and it could be rather odd, as the context around using an FPTs would be considerable. Better perhaps to study first Metric Spaces, where you prove one version of a FPT or Banach Spaces. Good texts give an example application or two. Then later when you're studying a larger topic and the FPT is deployed, you'll find it straight forward. $\endgroup$ – Simon S Mar 14 '15 at 12:08

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