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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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10k views

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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4answers
300 views

generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
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1answer
549 views

Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question is to check if : $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point. I have first of all considered boundedness. So, $f(x)$ should not have $x$ as ...
4
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1answer
241 views

Fixed point property of Cayley plane

I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed ...
4
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2answers
548 views

Showing that a function $f$ has a unique fixed point in a metric space.

Let $(X, d)$ be a compact metric space, and suppose $f : X → X$ satisfies $$d(f(x), f(y)) < d(x, y)$$ for all $x \neq y \in X$. Show that f has a unique fixed point. All I've gotten it so far ...
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1answer
88 views

Fix point of squaring numbers mod p

Take the set of integers $\{0, 1, .., p-1\}$, square each element, you get the (smaller) set of quadratic residues. Repeat until you get a fix point set. The size of this set is a function of $p$. ...
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1answer
191 views

Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is ...
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1answer
88 views

Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem

Kakutani's fixed-point theorem is quite similar to Brouwer's fixed point theorem - the main difference is that Brouwer speaks about single-valued functions and Brouwer about multi-valued functions. ...
4
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1answer
123 views

Continuous on the unit ball – odd on the unit sphere – does it have a fixed point?

For $n\in\mathbb N$, let \begin{align*} B^n\equiv&\;\{\mathbf x\in\mathbb R^n\,|\,\lVert \mathbf x\rVert\leq 1\}\text{ and}\\ S^{n-1}\equiv&\;\{\mathbf x\in\mathbb R^n\,|\,\lVert \mathbf x\...
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131 views

Counterexample to a “modified” Banach Fixed Point Theorem?

The Banach theorem states that if a (self) map on a complete metric space is Lipschitz with ratio $< 1$, it has a unique fixed point. What about modifying the hypotheses to say that the only ...
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179 views

Continuous mapping and fixed points

Does a continuous mapping $f\colon \mathbb R \to \mathbb R$ which satisfies $f(f(x))=x$ for each $x \in \mathbb R$ necessarily have a fixed point?
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1answer
629 views

Understanding the Banach fixed point theorem

The Banach fixed point theorem is stated in my book (Applied Asymptotic Analysis by Miller) as Let $\mathcal B$ be a Banach space with norm $\|\cdot\|$. Let $X$ be a nonempty bounded subset of $\...
4
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1answer
60 views

Questions about fixed point problem

Consider a homeomorphism $f:[0,1]^n\rightarrow [0,1]^n$, n$\in\mathbb{N}$. Let $S:=\{x\in[0,1]^n:f(x)=x\}$ Can $S$ contain exactly two points? If $S$ is not one point set, then does $S$ include a (...
4
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1answer
772 views

Every 1-Lipschitz function in the closed unit ball has a fixed point

I'm currently trying to solve the following exercise: Let B be the closed unit ball in $\mathbb R^n$ together with the euclidean metric. Show that every 1-Lipschitz function $f:B\to B$ has a fixed ...
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2answers
251 views

Problem in Banach Fixed Point Theorem for a functional equation

I was recently presented this within the context of topological spaces: I am asked to show that there exists a unique continuous function $ f\colon \left[0,\frac{1}{2}\right] \rightarrow \Bbb R $ ...
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1answer
2k views

How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
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314 views

Fixed Point Equivalences of Axiom of Choice

Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular ...
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1answer
116 views

Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
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Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, \frac{1}{3+\...
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1answer
226 views

Unique solution to $x^4 + 7x -1 = 0 $ on $[0,1]$ (Banach's fixed point theorem)

I want to show that $x^4 + 7x -1 = 0 $ has a unique solution on $[0,1]$. The idea is to use Banach's fixed point theorem. However, I see a problem with this as the statement of the theorem says that ...
4
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1answer
658 views

Application of the Banach fixed-point theorem

I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satisfies $$\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$$ for $t\in[0,1]$. The first thing I do is to define a function $A:M\...
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1answer
537 views

Weissinger's Theorem. How to prove?

Theorem (Weissinger). Let $C$ be a (nonempty) closed subset of a Banach space $X$. Suppose $K : C → C$ satisfies $$\|K^nx − K^ny\| ≤ θ_n\|x − y\|, \quad x,y∈ C $$ with $\sum_n θ_n < ∞$. Then $...
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45 views

Proving a function is a fixed point

I'm taking a class in University which involves proving the correctness of computer programs and I'm really bad a proofs, I don't really understand them at all. Can anyone tell me if my proof ...
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1answer
87 views

Regarding a proof of Tarski-Knaster's fixed point theorem

I'm trying to prove the following theorem: let $(X, \leq)$ be a complete lattice. If $\phi:X \to X$ is order preserving, it has a fixed point. So far, I've done the following: I've defined the set $...
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3answers
491 views

Banach fixed point theorem

Given $(X,d)$ complete with $A \subset X$ closed, and $f: A \to A$ satisfying $$ d(f^{n}(x),f^{n}(y)) \leqslant a_{n}d(x,y) \hspace{3mm} \forall x,y \in A \hspace{3mm} n \in \mathbb{N}$$ where $...
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1answer
48 views

Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...
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1answer
194 views

$f:\mathbb R \to \mathbb R$ is a differentiable function such that $f'(x)\le r<1 $ , does $f$ necessarily have a fixed point ? [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a differentiable function . If $\exists r \in \mathbb R $ such that $|f'(x)|\le r<1 , \forall x \in \mathbb R$ then using Lagrange's theorem one can show $f$ is a ...
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2answers
155 views

Continuous mapping on unit ball such that $T(\Bbb x_0)=0$

got a question from a course in functional analysis. " Let $T:\{\Bbb x\in\Bbb R: ||\Bbb x||\leq 1\}\to\Bbb R^n$ be a continuous mapping. Moreover assume that $\langle T(\Bbb x),\Bbb x\rangle>0$ ...
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2answers
152 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
4
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2answers
682 views

Generalization of Banach's fixed point theorem

I wanted to show that if $f:X\to X$ is a function from a complete metric space to itself and if $f^k$ is a contraction, then $f$ has a unique fixed point (say $p$) and for any $x$ in $X$ $f^n(x)\...
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2answers
808 views

Question regarding upper bound of fixed-point function

The problem is to estimate the value of $\sqrt[3]{25}$ using fixed-point iteration. Since $\sqrt[3]{25} = 2.924017738$, I start with $p_0 = 2.5$. A sloppy C++ program yield an approximation to within $...
4
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1answer
85 views

every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point?

can someone please help me to show for every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point? i think since the fuction is not surjective it doesn't contain at least ...
4
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1answer
390 views

If $|\,f'(p)|<1$, prove that $p$ is an attracting fixed point for $f$

Suppose that $f:(a,b) \to (a,b)$ has a fixed point $p$ in $(a, b)$ and that $f$ is differentiable at $p$. Furthermore, assume that $|\,f'(p)|<1$. Question: How do I prove that $p$ is an attracting ...
4
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1answer
161 views

Tarski’s fixed-point theorem and the existence of minimal fixed points

Let $(X,\geq)$ be a partially ordered set. Suppose that any non-empty subset of $X$ has a supremum in $X$ with respect to the order $\geq$; $X$ has a maximum element: $\exists \overline x\in X:\...
4
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1answer
182 views

Algorithm faster than Newton's for calculating $\sqrt{2}$ [duplicate]

It is well known that the iteration scheme $$ x_{n+1} = \frac{1}{2} (x_n + \frac{2}{x_n })$$ converges to $\sqrt{2}$ very fast. More precisely, it converges quadratically. The problem is, is there ...
4
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1answer
176 views

Bourbaki-Witt to Tarski-Knaster Fixed Point Theorem

I was looking at the Bourbaki-Witt Fixed Point Theorem which states that If $X$ is a non-empty, chain complete poset and $f: X \to X$ s.t. $f(x) \geq x$ for all $x$, then $f$ has a fixed point. I ...
4
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2answers
122 views

Breaking Banach's Fixed Point Theorem

In trying to see how Banach's fixed point theorem would break down in an incomplete space, I tried to come up with an example of a function: $f: \mathbb{Q} \longrightarrow \mathbb{Q} \ \ $such that $ ...
4
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1answer
214 views

Fixed points of polynomial ring homomorphism

$S=\mathbb R[x+y+z, xy+yz+zx, xyz]$ is the ring of the symmetric polynomials in $\mathbb R[x,y,z]$. Let $\psi\colon S \to R[x,y,z]$ be a ring homomorphism such that \begin{align} x &\mapsto -x,\\...
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1answer
186 views

Fixed points and infinite series

Consider the formula $1 + \frac{y}2$. This has a fixed point at $y = 2$. And if we use the equation $y = 1 + \frac{y}2$ to substitute for $y$ in our formula, we get $1 + \frac{1 + \frac{y}2}2$, or $1 +...
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1answer
120 views

A technical step in the proof of Atiyah-Bott fixed point formula

From John Roe, Elliptic operators, topology and asymptotic methods, page 135. Here Roe tried to prove that $$ \DeclareMathOperator{\Tr}{Tr} \sum_{q}(-1)^{q} \Tr(Fe^{-t\bigtriangleup})=L(\zeta,\...
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2answers
1k views

Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some $0<\...
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1answer
324 views

A condition implying that a holomorphic function is the identity map

Let $\Omega$ be an open, connected and bounded subset of $\mathbb{C}$, and $\varphi : \Omega \rightarrow \Omega$ a holomorphic function. If there exists a $z_{0} \in \Omega$, such that $$ \varphi(z_0)...
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3answers
311 views

Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
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1answer
2k views

How to prove that the following iteration process converges?

I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$. Q1: How to prove that this iteration process converges for every number $...
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1answer
216 views

Iteration of $x/\log x$

Consider $f(x)= \frac{x}{\log x}$ iterated n times for given x in $Z^+.$ $f^2 = f \circ f.$ Let $x_1 = x^2.$ What I would like to show (or disprove) : $\exists ~\alpha = x_n - e > 0 $ such ...
4
votes
1answer
43 views

Summing tangent slopes for trig function fixed points

Let's consider the values of $x$ for which the tangent line to the graphs of $\sin x$, $\cos x$, $\csc x$, and $\sec x$ passes through the origin. These values of $x$ satisfy $x=\tan x$, $-x=\cot x$, $...
4
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2answers
208 views

Proof (or hints towards proof) for asymptotic shape of orbit $0 \to 1 \to b \to b^b \to \cdots$ with certain class of $b$?

In a paper by Baker & Rippon (1983) the property of being convergent or divergent for iterated exponentials $z_{h+1} \to b^{z_h}$ with $b$ complex and $z_0=1, z_1=b, z_2=b^b, \cdots$ for classes ...
4
votes
1answer
309 views

Prove that there is a unique continuous solution to the following integral equation.

I am trying to prove that there is a unique continuous solution to the integral equation $$F(\alpha) = \int_{0}^{\alpha}F\left(\frac{t}{1-t}\right)\frac{dt}{t}; \qquad F(\alpha)=1 \text{ for } \alpha\...
4
votes
2answers
119 views

Least square circle via fixed point iteration

You have a collection of 2d points that you want to fit to a circle. Form the sum of the squares of the distances from the points to a generic circle. The variables are the $x,y$ coordinates of the ...