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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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$.
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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}^{...
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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 ...
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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
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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| := \...
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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. ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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\;...
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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}: ...
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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 \...
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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 ...
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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 ...
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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[...
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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 ...
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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
$...