Questions tagged [fixed-point-theorems]

A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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5 answers
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Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...
Qiaochu Yuan's user avatar
44 votes
35 answers
9k views

Fixed point theorems

It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and ...
32 votes
4 answers
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Does the Brouwer fixed point theorem admit a constructive proof?

Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
coudy's user avatar
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31 votes
3 answers
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Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
Mostafa Mirabi's user avatar
8 votes
0 answers
270 views

Fixed points and universal maps for posets

In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed ...
Włodzimierz Holsztyński's user avatar
71 votes
3 answers
5k views

Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
Paul Cusson's user avatar
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47 votes
6 answers
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Can we actually find any fixed points with Brouwer's theorem?

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
Vidit Nanda's user avatar
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38 votes
3 answers
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Do finite groups acting on a ball have a fixed point?

Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point? A fixed point for $G$ is a point $p \in B^n$ where for all $g \...
Sam Nead's user avatar
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16 votes
3 answers
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Has the Fundamental Theorem of Algebra been proved using just fixed point theory?

Question: Is there already in the literature a proof of the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem? N.B. The original post contained superfluous ...
David Bradway's user avatar
1 vote
1 answer
203 views

Totally non fixed point property

Edit: According to the comment of Pietro Majer, I revise the question Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?: "Constant ...
Ali Taghavi's user avatar
44 votes
2 answers
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Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...
Martin Brandenburg's user avatar
14 votes
0 answers
625 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
Gil Kalai's user avatar
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13 votes
3 answers
963 views

Closed manifolds with the fixed point property

The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{...
coudy's user avatar
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12 votes
2 answers
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A variation of the Ryll-Nardzewski fixed point theorem

Is there a fixed-point theorem that implies the following result? Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
Anton Petrunin's user avatar
12 votes
1 answer
929 views

Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?

Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2? Thanks
Pedro Perez's user avatar
12 votes
1 answer
886 views

Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \...
James Propp's user avatar
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11 votes
1 answer
288 views

Infinite vertex-transitive graph where every automorphism has a fixed vertex

This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof. Let $G = (V,E)$ be a graph with $V$ infinite. ...
Sam Hopkins's user avatar
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9 votes
2 answers
378 views

Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point. However, these fixed points cannot be chosen ...
Alex Mennen's user avatar
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9 votes
1 answer
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Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?

This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT). Yanofsky [0] has demonstrated several applications of LFPT to ...
jpt4's user avatar
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8 votes
1 answer
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What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
Mikhail Katz's user avatar
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8 votes
0 answers
227 views

Do commuting homeomorphisms of the $2$-disk have a common fixed point?

Problem 2.20 (attributed to Lima) in Kirby's list of unsolved problems in low-dimensional topology asks: Are there commuting homeomorphisms of the $2$-ball $B^2$ without a common fixed point? ...
Dejan Govc's user avatar
7 votes
1 answer
365 views

Fixed points and their continuity

Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the function $x \mapsto f(x,y)$ admits a fixed point. I want to ask whether ...
LSZ's user avatar
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6 votes
2 answers
2k views

Is the Binomial Expectation of Convex Function Convex in p?

Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function. Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
Hugh Medal's user avatar
5 votes
1 answer
362 views

Connected vertex-transitive graph with the fixed-point property

Many connected vertex-transitive graphs $G=(V,E)$ have the property that some of their automorphisms other than the identity have fixed points. To point out two simple examples: If $G = K_3$ then the ...
Dominic van der Zypen's user avatar
4 votes
1 answer
236 views

A closed chain of $2n+1$-gon around $2n+1$-points

I posed a generalization of Theorem 3.2 In my paper Conjecture: Let $P_1, P_2,....,P_{2n+1}$ and $O$ be $2n+2$ points in plane. Construct a chain $2n+1$ regular ${2n+1}$-gons $A_{1\;1}A_{1\;2}...A_{1\;...
Đào Thanh Oai's user avatar
4 votes
0 answers
202 views

Conjecture on convergence of iterated near-matrix square root

Here is a simple problem that has stumped me for some time; sharing with the community, as I suspect it has been solved somewhere, or is immediately implied by the correct theorem. Let $\textbf{diag}: ...
Keith Rush's user avatar
4 votes
0 answers
304 views

Banach's fixed point theorem for quasilinear parabolic PDEs

I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$ \begin{cases} \partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
Peter Koepernik's user avatar
4 votes
2 answers
414 views

An approximate infinite-dimensional fixed point theorem

Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||_{\infty} < \epsilon\right\}\right)$? In ...
Paul Christiano's user avatar
4 votes
1 answer
375 views

Fixed points of finite order isometries of metric spaces

I would like to show the following: Let $X$ be a complete metric space that is uniquely geodesic (i.e. each two distinct points are connected by a unique geodesic segment) and $\phi\colon X\to X$ an ...
Florian's user avatar
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3 votes
1 answer
2k views

leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...
mnmn1993's user avatar
2 votes
2 answers
441 views

A Fixed point Theorem that does not need the convexity of set valued map?

I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued. Something like contractiblity or other properties can be replaced with ...
behrad mahboobi's user avatar
0 votes
0 answers
110 views

Fixed point of a contraction map

This question is a continuation of Is this a contraction mapping for small $T$? Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm $...
GJC20's user avatar
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