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How should one begin one's study of fixed point theory, especially of multi-valued maps?

What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have?

Should one begin with metric fixed point theory or topological fixed point theory?

What background is needed for discrete fixed point theory?

Which book(s) should one start with?

Is fixed point theory taught at the undergrad level anywhere in North America, Europe, or Australia?

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  • $\begingroup$ What do you mean by "multi-valued maps"? $\endgroup$ – T_M Jun 3 '18 at 4:10
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For fixed-point type problems, the tools one generally brings to bear depends on the natural structure of the problem you're considering. The main branches of mathematics you'll want to pursue an understanding are:

  • Analysis;
  • General topology;
  • Algebraic topology;
  • Differential topology (depending upon your interests).

A great deal of more sophisticated fixed point theory involves sophisticated technique as well, so the best thing you can do pedagogically (especially as an undergraduate) is to just develop a good core mathematical toolkit, frankly. Learn analysis. Learn topology. But don't try to just skip to fixed point theory. No one likes to hear that, but without a solid at least undergraduate mathematical background, most of the 'real' fixed point theorems are going require tools that will bring a lot more heat than light.

As for books, the classics in each of these fields suffice. For analysis Baby Rudin -> Royden served me well. General topology I prefer Munkres. My personal favorite for algebraic topology is Hatcher, though some people have mixed feelings on it. For the smooth perspective, Guillemin and Pollack is tremendous for an undergraduate, and Hirsch for more advanced interests. Kim Border hasa great book, 'Fixed Point Theorems with Applications to Economics and Game Theory' that is nice, especially if you lack a background in algebraic tools (economics as a field is pathologically averse to algebraic topology). There's also some lovely, in-depth notes here from micro-theorist Andrew McLennan:

Perhaps this is too much of an example, but now below I state a couple famous (and omit many more!) fixed point theorems, and discuss the language one generally needs to be familiar with to prove them, both as specific examples, but also to just illustrate how many paths there are to these results (or, alternatively put, how pervasive they are!).


Consider the following few examples of theorems and proofs of them arising from different fields:

Theorem: (Brouwer) Let $X$ be a compact, convex, finite dimensional topological space and $f:X \to X$ a continuous map. Then $\exists x\in X$ s.t. $f(x)=x$.

You can literally write a book of different ways to prove this theorem. There are 'classical' methods using algebraic topology. For many spaces $X$ one can appeal to Stone-Weierstrass to get a smooth approximation $\bar{f}$ of $f$ and a wonderful proof due to Hirsch gives a purely differential topological proof from there. There's a combinatorial approach that invokes a coloring result known as Sperner's Lemma. Assuming other results (such as Nash's Theorem concerning the existence of equilibria of finite mathematical games, or Gale's theorem that the game of Hex cannot end in a tie, one can also prove Brouwer's fixed point theorem).

Theorem: (Banach) Let $(X,d)$ be a complete metric space, and $f:X\to X$ a contraction. Then $\exists!x\in X$ s.t. $f(x)=x$.

This is completely straightforward to show using purely basic analysis and metric space theory (and is often times very useful to get a criterion for uniqueness in applied work). This is a critical ingredient in showing the existence of solutions to differential equations as well.

Theorem: (Tarski) Let $S$ be a complete lattice and $f:S \to S$ be an order-preserving function. Then the set of fixed points of $f$ are a complete lattice, and hence non-empty.

This is a useful result in my field, economics. This theorem, interestingly, makes zero continuity assumptions of the function $f$ and proceeds purely using order/lattice-theoretic arguments.

Now, switching gears slightly, there is a generalization of the notion of a fixed point called 'coincidence.' Consider two maps $f:X\to Y$ and $g:X\to Y$. These maps are said to be 'coincident' at any point such that $f(x)=g(x)$. Now if $Y=X$ and $g=id$, the coincidence points are nothing more than your traditional fixed points, so this theory subsumes fixed point theory.

Theorem: (Lefschetz) Let $X$ and $Y$ be orientable manifolds such that $\dim X= \dim Y$. Then define the Lefschetz coincidence number of $f$ and $g$ as: $$\Lambda_{f,g}=\sum(-1)^k \textrm{Tr}(D_X \circ g^* \circ D_Y^{-1} \circ f_*),$$

where $f_*$ is the induced map on homology, $g^*$ the induced map on cohomology, and $D_X,D_Y$ are the Poincare-duality isomorphisms for $X$ and $Y$. Then if $\Lambda_{f,g}$ is non-zero, then $f,g$ have a coincidence point.

So ignoring the very technical statement, this is a purely algebraic topological result. But the reason I bring it up (apart from being 'the most general topological fixed point theorem') is that more generally, coincidence theory subsumes the topological study of fixed points of multi-valued maps as well!

Example: Say we have a multi-valued map from a compact, convex topological space $X$ to itself, given by $\phi:X\rightarrow X$. Then we may consider the graph of $\phi$ in $X\times X$, $\Gamma(\phi)$, and two projections from this graph, one to the domain $f$ and to the codomain $g$. Then fixed points of $\phi$ are precisely the coincidence points of $f$ and $g$ and thus we can bring to bear the power of topology to find conditions on $\phi$ that ensure a fixed point, even where it might not appear applicable (for example, the Kakutani/Eilenberg-Montgomery fixed point theorems)!

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    $\begingroup$ Nice answer, but if you are an economist, you should have given a shout-out to Kim Border's book "Fixed point theorems with a pplications to economics and game theory" javascript:OpenURL('www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521388082') $\endgroup$ – Trurl Jun 4 '18 at 18:50
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Andrew McLennan, one of the authors mentioned above, has a new book out called Advanced Fixed Point Theory for Economics. It does not require algebraic topology knowledge but one would want to know some analysis and point-set topology first. It has a lot of material on multivalued maps. Although it is meant primarily for economists, it is a serious math book.

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When it comes to topological fixed point theories, such as Brouwer fixed point theorem, Borsuk-Ulam fixed point theorem and Lefschetz fixed point theory, a great start is Guillemin & Pollack's Differential Topology. It's a wonderful book, and you can read it if you have some background in multivariable calculus or a bit of differential geometry.

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