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Questions tagged [fixed-point-theorems]

Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling $F(x)=x$, under some conditions on the function $F$. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene) etc.

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Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert H_{k}(...
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Fixed points in computability and logic

I asked this question on CS.SE, too: https://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...
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A variation of the square peg problem

This question spawned from this recent thread. The notorious square peg problem states that any continuous, simple and closed curve $\gamma$ in the plane contains the vertices of some square. It has ...
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does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?

I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (...
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Fixed point of a polynomial mapping - what's the relation between the two views

Let $\sigma : \Bbb{C}^3 \to \Bbb{C}^3$ be a polynomial mapping. Let $P:= \Bbb{C}[x,y,z]$ denote the space of polynomial in 3 variables. Then $\sigma$ induces a (linear) mapping $\tilde{\sigma} : P\...
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Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
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Every $f : [a, b] → [a, b]$ has a fixed point where $f$ is continuous. Deduce the intermediate value theorem.

Every $f : [a, b] → [a, b]$ has a fixed point and $f$ is continuous (on $[a,b]$). Deduce the intermediate value theorem. I managed to show the other way, now I'm here. I know that $f(c)=c$ for some $...
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Limit of a Discrete Dynamical System, Part 2

In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered: $$\left[\begin{array}{c}x_{t+1}\\y_{t+1}\end{array}\right]=\left[\begin{array}{c}y_{t}/b\\x_{t}-1+...
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Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
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Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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Connection between codata and greatest fixed points

It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, with constructors are $0 : \mathbb N$ and $s : \mathbb N \to \mathbb ...
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Proving Talagrand's contraction lemma for Gaussian processes with the Banach fixed-point theorem

I've done the standard proof of Talagrand's contraction lemma for Gaussian processes (see Exercise 7.2.13 in Vershynin's High-Dimensional Probability) using the Sudakov-Fernique inequality as ...
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Reference request: equivalence between formulas in fixed point and first-order logic

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in some sort of fixed-point logic to have an equivalent first-...
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Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
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Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function $f:...
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Fixed points of an endomorphism of a ring

Let $R$ be a commutative $k$-algebra, where $k$ is a field of characteristic zero. Let $f$ be a $k$-algebra endomorphism of $R$. ($f$ is not assumed to be either injective nor surjective). Are ...
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Godel's Diagonalization Lemma As Application Of Lawvere's Fixed Point Theorem

I've read through this paper with applications of Lawvere's fixed point theorem. On the diagonalization lemma, they say the following: For one thing, how can $f$ and $\Phi_{\cal E}$ be well ...
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On the in-comparability of the Gauss-Seidel and Jacobi iteration schemes.

Given the system $x_1 + x_2 = 2$ $-x_1 + x_2=0$ $x_1 + 2x_2 - 3x_3=0$ the Jacobi iteration converges and Gauss-Seidel iteration diverges. Is there a way we can derive these two facts using the ...
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Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
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Edelstein Fixed Point Theorem

Let $(M,d)$ be a metric space, $M$ compact. If $f:M \to M$ is continuous and weakly contractive (i.e. $d(f(x), f(y)) < d(x,y) , \forall x,y \in M$), then $\exists x_0 \in M $ unique s.t $f(x_0)=...
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A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
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Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i \operatorname{tr}({\...
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algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
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Point-set topological proof of Brouwer's fixed point theorem

I have tried to understand the point-set topological proof of Brouwer's fixed point theorem presented in Cou11. But I couldn't clarify some parts. Here are the theorem and its proof. Theorem: There ...
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How many fixed points can a function have?

For dimension one, it is easy to think in samples of continuous functions $f:[a,b]\rightarrow [a,b]$ with one, two, three,... fixed points. Or even, infinitely fixed points (take the idendity map). ...
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System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }...
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Fixed point method for non-homogeneous ode and pde

During my course in mathematical methods for physics, my professor introduced us to the "method of fixed point" for studying non-homogeneous systems of odes, odes and pdes. Although I think I ...
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Existence and Uniqueness of Equilibrium Points in Non-Linear Dynamical Systems

Let $\dot{X} = F(X)$ be a non-linear dynamical system. I'm interested in knowing if there are any existence theorems for an equilibrium point, that is, an $X^*$ that satisfies: $$ F(X^*) = 0 $$ I ...
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Equivalence between Brouwer fixed-point theorem and Borsuk-Ulam theorem. Is there a simple proof of equivalence between them?

I wonder if Brouwer's fixed-point theorem and Borsuk-Ulam's theorem are equivalent. Brouwer's fixed-point theorem (simple form). Let $B_{\mathbb{R}^{n}}[0,1]=\{x\in \mathbb{R}^n: \|x-0\|\leq 1\}$ ...
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Fredholm alternative for elliptic PDE: Is $L_\mu^{-1}$ contractive?

In Evans' PDE, §6.2, in Theorem 4 he proves a Fredholm alternative based dichotomy for second order uniformly elliptic pde. The general idea: For $\mu\geq\gamma\geq0$ with $\gamma\geq0$ from Thm. 2, ...
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differentiable contractive mappings

Some time ago, I managed to show that $sin(x)$ and $cos(x)$ were both contractive almost everywhere and then I noted that $cos$ was the derivative of $sin$ so there might be a deeper connection. Let'...
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What is this numerical method for solving Ax=b called?

Following is the pseudo-code of a simple iterative method of solving $Ax=b$ where $A$ is an $n\times n$ matrix. ...
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Fixed-point iteration and continuity of parameters

Let $X$ a compact set and $A\subseteq \mathbb{R}$. Consider a continuous function $f\colon X\times A\to X$ and construct a fixed-point iteration as follows $$ x_{k+1}=f(x_k,a),\quad x_0\in X, a \in A.\...
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Extension of Kakutani's fixed point theorem.

Can the Kakutani's fixed point theorem's be extended to say that there exists a fixed point inside the set (not on boundary)(I am not sure how to formally state this). For a $n$-dimensional compact, ...
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Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
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Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
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313 views

fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle xt_{ij}\left(1-\frac{g}{r}\...
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On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
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Existence of fixed points for this Markov operator.

Perhaps math overflow is a better place to put this but I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point. In particular ...
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Fixed Point of an infinite-dimensional map

I have the following fixed point equation: $$ F(t)= \displaystyle\sum_{r=1}^M \int_{\mathbb{R}} F ((t-c)r) g(r) h(c) \: \mathrm{d}c$$ where: $F$ is a cumulative density function of a certain random ...
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Convergence of a fixed point iteration with nonlinearity

If the following is too specific, my question can be simplified as fixed point iteration $${\mathbf X}_{k+1}={\mathbf G}f({\mathbf X}_k)+h({\mathbf X}_k)$$ and given that the spectrum radius of $\bf ...
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Problem in proving fixed point theorem

Let $ f(x)=x^r, 1<r<\infty, x\in \mathbb{R^+}=[0,\infty) ~and ~n\in \mathbb{N}.$ Define $$\pi(x)= 1 ~~if~~ x\leq n ~and =0 ~if~ x< (n+1).$$ Then for any $x,y\in\mathbb{R},~~ $ I want to ...
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26 views

Reference Request Fixed Point with correction

Suppose we have a mapping: $$ T: \operatorname{CDF} \rightarrow \mathbb{R}^{[0,\infty)}: F \mapsto F $$ which maps Cumulative Distribution Functions to functions on $[0,\infty)$. I am looking for a ...
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Fixed point theorems, convexity, contractibility

The proof I see for the Schauder fixed point theorem for Banach spaces is basically the following: If $K$ is compact and convex and $A\colon K\to K$ is continuous, for each $\epsilon$ find a finite ...
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Inverse function theorem via degree theory

In this MO question Terrence Tao writes the following about the inverse function theorem. The Brouwer fixed point theorem gives local surjectivity, and degree theory gives local injectivity if $\...
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Is it hard to find out for which $r$ the infinite power tower $r\uparrow r \uparrow r\uparrow \cdots$ converges?

It is well known that the infinite power tower $$r\uparrow r\uparrow r\uparrow\cdots $$ with $r>0$ converges if and only if $e^{-e}\le r\le e^{1/e}$. I tried to prove it and I got stuck in the ...
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Brouwer fixed point theorem and completeness axiom.

I want to show that the completeness axiom of the real numbers is equivalent to the Brouwer fixed point theorem (in $\mathbb{R}^2$), i.e., without loss of generality: For all nonempty set $A\subset (...
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Is there a fixed point theorem I can use here?

Let $A$ be the following set of 2-dimensional functions of one variable with strictly increasing and continuous components: $$A = \{(x_1(\cdot), x_2(\cdot)): x_i: [0,1] \rightarrow \mathbb{R}, x_i(\...