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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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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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Fixed point for a continuous function on a compact set?

If $f:X \rightarrow X$ is continuous and X is compact, will $f$ have a fixed point? We know that a contraction will have a fixed point but I have not come across an example of a continuous function ...
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Fixed point combinator and functions with no fixed point

In lambda calculus the fixed point combinator is defined as: It is very easy to see how $Yg =g(Yg)$ for any $g$ by using $\beta$-reduction. At the same time I wonder what is the meaning of $Yg$ ...
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Showing a contraction without a fixed point

Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that: \begin{equation} |f(x)-f(y)| < |x-y| \end{equation} except when $x=y$, but $f$...
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Finding the fixed points of a contraction

Banach's fixed point theorem gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction. I'm interested in how to calculate the ...
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Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
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Understanding Baker and Rippon's proof of a result in iteration theory

Let $a \in \mathbb{C}, b = e^a, T(z) = e^{az}$. Define $W_n = T^n(1)$ where $T^n$ is the nth iterate of $T$. The main result that motivated asking this question is Theorem 1 below. Note: in ...
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Contraction and Fixed Point [duplicate]

How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$. I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ ...
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Are the practical analogies of the Brouwer fixed-point theorem meant to be trivially understood?

When reading about the Brouwer fixed-point theorem on Wikipedia there are some "real world illustrations" of what the theorem says, one of them being the following: [T]ake two sheets of graph paper ...
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Does any continuous function from the open unit interval $(0,1)$ to itself has a fixed point?

I know that the result is true for closed interval $[0,1]$ by using intermediate value property. But in the case where we consider open interval $(0,1)$ does the solution change?
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Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
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continuous map on compact ellipse

will there be any fixed point a continuous $f$ from the ellipse $2x^2+3y^2\le 1$ to itself? Well I think yes but in a solution of a problem hint is given that NO. Just asking to assure myself if I am ...
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Does every continuous map $f$ from $D^2$ minus $k$ disjoint open disks to itself have a fixed point?

Let $A_k$ denote $D^2$ with $k\geq 0$ disjoint open disks removed. For $k=0$, the answer is positive by Brouwer's fixed point theorem. For $1\leq k\neq 2$, it's not difficult to see that the answer is ...
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Fixed point theorems and their applications in Measure Theory

We know that there are many versions for the fixed point theorem and they have many applications. I would like to know whether there is one has an application in Measure Theory. And I would be ...
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On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
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Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem

http://en.wikipedia.org/wiki/Diagonal_lemma I am wondering about the proof of the "Fixed-Point Lemma" $\text{Mod } \Sigma$ is the class of all models of $ \Sigma$. $\text{Th Mod } \Sigma$ is the set ...
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fixed point of homeomorphism and compactness of a complete metric space.

I need to know that the following statements if true or false: Every homeomorphism of $S^2\rightarrow S^2$ has a fixed point. Let $X$ be a complete metric space such that distance between any two ...
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When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $...
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Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let $...
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The equation $2 \cosh(3.1786803659501505 z) = z$?

Let $a$ be a positive real number and $z$ a complex number. I was wondering about the equation $2 \cosh(a z) = z$ where we solve for $z$. Clearly if $z$ is a solution than so is its conjugate. It ...
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$x_{n+1}=\frac{2x_n+3f(x_n)}{5}$ showing $f$ has a fixed point

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a differentiable function , and suppose that there is a constant $A<1$ such that $|f'(t)|\le A$ for all real $t$. Define a sequence $\{x_n\}$ by $ $$$x_{n+...
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Brouwer's fixed point theorem in a practical setting

If we assume that a fluid is a continuum then if we have for example a cup of tea and we stir the fluid then there will be a point in the fluid that is on the same location before and after the ...
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How Gelfond find his limit for $\exp(\pi) $? [duplicate]

$$ a_0 = \frac{1}{\sqrt 2} $$ $$ a_{n+1} = \frac{( \sqrt {1 - a_n^2} -1)^2}{a_n^2} $$ $$ \lim_{n \to \infty} \frac{4^{\frac{1}{2^n}}}{a_{n+1}^{\frac{1}{2^n}}} = \exp(\pi) $$ How did Gelfond find ...
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Fixed point of a differentiable function

Consider a differentiable fuction $f: \mathbb{R} \rightarrow \mathbb{R}$ for which $f'(t) \neq 1$ for every $t \in \mathbb{R}$. Is it true that $f$ has exactly one fixed point? It is clear to me why ...
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Show that $f$ has a unique fixed point.

Let $M$ be complete and let $f:M\to M$ be continuous .If $f^k$ is a strict contraction for some integer $k>1$ ,show that $f$ has a unique fixed point. My try: Let us fix $x_0$.Let $g=f^k$. Put ...
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Find if a fixed-point iteration converges for a certain root

I'm asked to find if the fixed-point iteration $$x_{k+1} = g(x_k)$$ converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$. ...
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When to use Newtons's, bisection, fixed-point iteration and the secant methods?

I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of ...
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Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...
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Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
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Example of a function, so that $g(x)\neq x$

I'm trying to find an example of a function $g:\mathbb{R}\to \mathbb{R}$ (or $g:[1,\infty) \to \mathbb{R}$), so that $$|g(x_1)-g(x_2)|<|x_1-x_2|$$ for all $x_1, x_2\in \mathbb{R}$ ( or $x_1,x_2\in ...
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Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$

My knowledge of the fixed points and iteration equals zero, same for the notation and terminology but I really need to know if this deduction has trivial errors or is really as nice as it seems. I ...
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1answer
240 views

Holomorphic function with a unique fixed point

Let $\omega \subset \mathbb C$ be a simple connected set and $\,f:\omega \to A$ is an analytic function where $A \subset \omega$ is compact. Show that $f$ has an unique fixed point. I think we can ...
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Brouwer's fixed point theorem (for unit balls) and retractions

Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$. I want to prove that the following proposition $B$ is a fixed-point space if ...
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Brouwer's fixed point theorem for infinite dimensional real space in subsystems of second order arithmetic

$\text{WKL}_0$ proves Brouwer's fixed point theorem for continuous functions on $\lbrack 0,1 \rbrack^n$, when $n$ is finite. What subsystem of second order arithmetic is needed to prove Brouwer's ...
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Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
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Prove that $E=\{x=(x_n)\in \ell^\infty(\Bbb N): (x_n)_n~~\text{is periodic}\}$ is not complete.

Let $E=\{x=(x_n)\in \ell^\infty(\Bbb N): (x_n)~~\text{is periodic}\}$ Defintion: $x=(x_n)$ is periodic means there exists $p\in \Bbb N$ such that, $x_{n+p} =x_n ~~~\forall ~~n\in\Bbb N.$ we ...
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1answer
202 views

Convergence of a recursively defined sequence

Let a sequence $(a_n)_{n=0}^\infty$ be defined recursively $a_{n+1} = (1-a_n)^{\frac1p}$, where $p>1$, $0<a_0<(1-a_0)^{\frac1p}$. Let $a$ be the unique real root of $a=(1-a)^{\frac1p}$, $0<...
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What is the value of $z$ for any Julia set? Does it influence the graphical result?

I'm trying to understand how Julia sets iterations are done and how those iterations differ from the ones that generate the Mandelbrot set. Both of them use the following function: $f(z) = z^2 + C$ ...
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Find all $k_0$'s such that $x_{n+1}=f(x_n)$ will remain in $[0,1]$ for the “tent map” of height $3$

Let $f: [0,1]\rightarrow \mathbb{R}$: $f(x)=3x \, \, $ if $0\le x\le \frac{1}{2}$; $f(x)=3-3x \, \, $ if $\frac{1}{2}<x\le 1$. Let a sequence $k_{n+1}=f(k_n)$. Find all possible value of $...
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Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space over ...
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Counterexamples of Brouwer fixed point theorem applied on the close unit ball

Brouwer fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least one fixed point. Brouwer fixed point theorem applies in particular on the ...
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Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, find the first 6 terms in the Taylor expansion solution $y=\varphi (x)$

Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, use the method of reduction to an integral equation and successive approximation to find the first 6 terms in the Taylor expansion ...
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Proof that a continuous function from the unit ball to itself without fixed points implies existence of retract from unit ball to unit sphere

Assume $f:B_{1}\to B_{1}$ (where $B_{1}$ is the closed unit-ball in $\mathbb{R}^{n}$ ) is a continuous function that has no fixed points I need to construct a function $g:B_{1}\to B_{1}$ which ...