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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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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Is Schauder's conjecture resolved?
Schauder's conjecture: "Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point." [Problem 54 in The ...
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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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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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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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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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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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Sperner Lemma Applications
I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of ...
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Periodic orbit property
A topological space $X$ satisfies the "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $...
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fixed point property for maps of compacts
Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point.
Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
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Does every automorphism of a separably rationally connected variety have a fixed point?
Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ that is separably rationally connected, i.e., there exists a $k$-morphism $u:\mathbb{P}^1_k \to X$ such that ...
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What are the major differences between real and complex Banach space?
Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa.
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Banach and Knaster-Tarski fixed point theorems -- are they related?
It there any known way of obtaining the Banach fixed-point theorem from the Tarski fixed-point theorem or vice-versa?
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How is this fixed point theorem related to the axiom of choice?
I'm hoping the answer to this is well-known.
Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) \...
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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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Converse to Banach's fixed point theorem?
Let $(X,d)$ be a metric space. Banach's fixed point theorem states that if $X$ is complete, then every contraction map $f:X\to X$ has a unique fixed point. A contraction map is a continuous map for ...
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Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?
I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
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Topological Content of the Kakutani Fixed Point Theorem
Reading about general equilibria, the Kakutani fixed point theorem seems to be a central tool. It states (following Wikipedia)
For $S \subset \mathbb{R}^n$, non-empty, compact and convex, and $\phi : ...
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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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Converse of Knaster-Tarski's theorem as choice principle
Knaster-Tarski's theorem States that if $(A,\le)$ is a complete lattice, then every monotone function $f :A \to A$ has a fixed point. The proof is carried out in $\mathsf{ZF}$.
By $\mathsf{KTC}$ we ...
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Fixed point theorem for the uncountable power of an interval
Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa\geq\aleph_1$ ?
That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ ...
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Reference request: an elementary proof of Brouwer fixed-point theorem.
One of the elementary way to prove of the Brouwer fixed-point theorem is, making it follow from the (smooth) Non-Retraction theorem. The latter is then proven by contradiction by means of a simple ...
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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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Connection between codata and greatest fixed points
This is, I'm afraid, another question that MSE couldn't answer.
It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, ...
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Homeo-Fixed point property
Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
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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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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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Is there a least-fixed-point formulation of inaccessible cardinals?
The infinity axiom can be formulated by defining a function $S$ as
$$S(N) = \{0\} \cup \{n+1\\ |\\ n \in N\}$$
(FWIW, I'm assuming the von Neumann ordinals.) The axiom is then
$$\exists I . I = S(I)...
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Does Peano's existence theorem admits a constructive proof?
$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
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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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Fundamental groups of reduced subgroup lattices
Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
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Equivariant Riemann-Hurwitz
The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e_P$ and computes invariants of $X$, such as ...
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Can we internalize topological fixed point theorems in an effective topos?
Reflective oracles are a kind of Turing oracle that give stochastic answers about the outputs of Turing machines. This works in a self-referential way, where they can answer queries about Turing ...
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Knaster Tarski theorem, example needed
http://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem
Let $L$ be a complete lattice and let $f : L \to L$ be an order-preserving function. Then the set of fixed points of $f$ in $L$ is also a ...
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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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Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
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Is $\{x_n\}$ a Cauchy sequence?
Let $(X,d)$ be a complete metric space and $f$ a mapping of $X$ into itself. Let $\{f^n(x)\}=\{x_n\}$ be the sequence of iterated transforms.
Suppose $f$ satisfies that for each $\varepsilon >0$,...
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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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A question on fixed point theory
I asked this question in MSE, but I did not received any answer, so I repeat it here:
https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property
Assume that $0<k<n-1$, ...
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Infinite-dimensional hex
Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...
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Continuity of mapping sending a function to its (brouwer) fixed point
Let $f:[0,1]^n \rightarrow [0,1]^n$ be a continuous mapping. Brouwer's fixed point theorem says that $f$ has a fixed point, i.e., some $x$ such that $f(x) = x$.
Suppose we have a continuous family, i....
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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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algebraic proof of Atiyah-Bott fixed point formula?
Hi,
Atiyah and Bott apparently proved the following theorem:
Let $X$ be a smooth projective complex variety and $L$ a line bundle on $X$.
Let $f:X\to X$ be an automorphism of $(X,L)$ with finitely ...
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A property stronger than the fixed point property
Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...
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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 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 a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?
If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...