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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, ...

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Is it possible that Successive Over Relaxation (SOR) method converges while Gauss-Siedel method does not?

Is it possible that Successive Over Relaxation (SOR) method converges while Gauss-Siedel method does not? Is the following statement correct? In trying to solve a linear system of equations, the ...
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21 views

Show that $T:\,c_0\to c_0\;\;$, $x\mapsto T(x)=(1,x_1,x_2,\cdots),$ has no fixed points

As a follow-up to my previous question Show that $T:\,c_0\to c_0\;\;$, $x\mapsto T(x)=(1,x_1,x_2,\cdots),$ is non-expansive. Let $X$ be a normed linear space and $X=c_0$ (the space of sequences of ...
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35 views

Is there a contraction mapping on the empty set?

If there is, then that would mean that the empty set is a complete set with no fixed points under the contraction map? This came up as Kolmogorov does not require that the set be non-empty in his ...
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How to find g(x) and aux function h(x) when doing fixed point interation?

I'm learning fixed point iteration (first and second form). My teacher said there are two forms: g(x) = x - f(x) g(x) = x - h(x)f(x) where ...
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15 views

Derivative of an infinite composition of functions

Let $f(x)=g(g(g(...g(x))))$, where the function $g$ is applied to $x$ and infinite amount of times. I am assuming that $x$ is real. What is special about points at which $df/dx=0$? Are they related to ...
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70 views

Picard's method does not solve first order differential equation?

I have the following first order differential equation $$x^\prime(t)=-(x(t))^2+2x(t),\quad t\geq 0,\quad x(0)=1$$ Now I want to obtain an approximation of $x(t)$ by using Picard's method. Then the ...
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36 views

If $g:X \to [0,\infty)$ is defined by $g(x)=d(x,f(x))$, prove that $g$ is uniformly continuous on $X$.

Let $f:X \to X$ be a function on a compact metric space $(X,d)$ such that $d(f(x),f(y)) < d(x,y)$ for all $x\neq y$ a) If $g:X \to [0,\infty)$ is defined by $g(x)=d(x,f(x))$, prove that $g$ is ...
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Root of a plynomial in (0,1)

Define $$f_K(x)=\sum_{i=K+1}^{2K} \binom{2K}{i}x^{i-1}(1-x)^{2K-i}.$$ How to show that $qf_K(x)-f_K(1-x)$ has exactly one real root in $(0,1)$ for any $q > 0$ and $K \geq 1$. The proof for $q=1$ ...
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52 views

Proof Review Fundamental Theorem of Algebra using Brouwers F.P.T. [on hold]

11:42am I deleted everything, I just have one question. Is my proof in my reply good. I will reformat this later, I have an oral spanish test soon and I haven't been able to study at all. This problem ...
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1answer
157 views

$f(a)-f(b)$ is rational iff $f(a-b) $ is rational

Prove that the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfying $f\left(x\right)-f\left(y\right) \in\mathbb{Q} \iff f\left(x-y\right) \in \mathbb{Q}$ is of the form $ f\left(x\right)=ax+...
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1answer
219 views

Find good starting candidates for Newton-Raphson knowing one of the solutions of a parametrized system of nonlinear equations

I have a parameterized system of equations describing the crossed ladders problem. $(x, y)$ are the $2$ horizontal distances respectively on the left/right of the junction of the ladders $(a, b, c)$ ...
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209 views

Showing a function is Frechet Differentiable?

I just started learning the Frechet Derivatives. So I have a function $H:\mathbb{R}^{N\times n}\to\mathbb{R}^{N\times n}$, i.e. $U^T\in\mathbb{R}^{N\times n}$ and $$H(U^T)=GW\times (F(U))^T+S\times U^...
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29 views

Is there a holomorphic diffeomorphism of $\mathbb{C}P^{2n+1}$ without fixed point?

Is there a holomorphic diffeomorphism $f:\mathbb{C}P^{2n+1}\to \mathbb{C}P^{2n+1}$ without fixed point?
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1answer
21 views

Reverse fixed point conclusion

If I have a function such as: $$f:M \rightarrow \mathbb{R} $$ where $M$ is any metric space denoted by : $$(M,d)$$ $$f(x) =d(x,y) $$ where $y \in M$ is a fixed point. I am trying to show that ...
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Fixed-point iteration: $\sqrt{\varepsilon} = \sqrt{c - \varepsilon} \tan (a \sqrt{c - \varepsilon})$

Let $a, c > 0$. Use the Banach fixed-point theorem to show that $$\sqrt{\varepsilon} = \sqrt{c - \varepsilon} \tan (a \sqrt{c - \varepsilon})$$ has at least one solution $\varepsilon \in (0, c)$ ...
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50 views

Is Banach fixed point theorem a necessary and sufficient condition for the existence of a fixed point

Banach fixed point theorem requires a contraction mapping from a metric space into itself, but when I was learning some machine learning algorithms, some questions rise above: k-means is an algorithm ...
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150 views

System of equations $a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2)$

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system $$a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t$$ I have a solution Let $f(x)=\frac t{1-x^2}$ ...
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1answer
10 views

Proving the error estimate of a contraction in Banach space via induction

My task is to prove the following statement: if $T:V$—>$V$ is an $\alpha$-contraction and $V$ is a Banach space, then T has a globally attracting fixed point $\bar v$ in $V$, and for any initial point ...
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1answer
41 views

Is the function is contraction? How to find fixed point?

Task 1) Show that function $T:C[0, 1] \to C[0, 1]$ is a contraction, then $$T(f)(x)=\int_{0}^x (x-t) f(t) dt,$$ $$x\in [0,1], f\in C[0, 1].$$ 2) Find a fixed point of $T(f)(x)$. My progress 1) I ...
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1answer
16 views

Prove that there exists at least on fixed point of T. Hint consider the map $T_k = (1-\frac{1}{k})T$

Let $\Omega$ = closed ball $B_1(0)$ in $\mathbb{R}^n$ with metric d induced by the Euclidean norm. Suppose the mapping $T: \Omega \to \Omega$ satisfies $d(Tx,Ty) \leq d(x,y)$ for all $x,y \in \Omega$ ...
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66 views

Why is $x=4$ as a fixed point of a map $\sqrt{2}^{x}$ unstable?

My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x \mapsto f(x)$ is given, with $$f(x)= (\sqrt{2})^...
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1answer
22 views

Inverse at a single point using Banach Fixed Point

The Problem: Show that there exists a unique $(x,y)\in \mathbb{R}^2$ so that $\cos(\sin(x))=y$ and $\sin(\cos(y))=x$. I believe you can use the Banach Fixed Point theorem, although I'm sure there is ...
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1answer
32 views

Finding a common Lipschitz constant for a family of contractions

Let $(X,d)$ be a complete metric space and $f:[0,1] \times X \to X$ a family of functions such that $f(t,\cdot)$ is a contraction for every $t \in [0,1]$. Further assume that $f(\cdot,x)$ is ...
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37 views

Iteration of complex rational function

I'm studying the book Iterations of Rational Functions by Alan F. Beardon. Here is the page that all the quotations come from. It is said that: If $z$ is close to the fixed point $\zeta$, then, ...
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1answer
38 views

Contracting mapping theorem - proof

Prove the contracting mapping theorem (the version on $\mathbb R$): Suppose $f : [a,b] \to [a,b]$ is continuous and satisfies $$|f(x) - f(y)| < c|x - y|$$ for some $0 < c < 1$, then $f$ has a ...
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14 views

On a coupled fixed point

Let $X$ be a Banach space. An element $(x^*,y^*)X\times X$ is called a coupled fixed point of a mapping $T : X ×X \rightarrow X$ if $$T(x^*,y^*) = x^*$$ and $$T(y^*,x^*) = y^*$$. While studying ...
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1answer
29 views

Iteration sequence is bounded

Assume we have a continuous map $f:\mathbb{R}^n\to\mathbb{R}^n$, and there exists at least one point $a\in\mathbb{R}^n$ satisfies the sequence $\{a,f(a),f\circ f(a),\cdots\}$ is bounded in $\mathbb{R}^...
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1answer
33 views

$\vert f'(a) \vert < k$,prove that there exists $I$, $a\in I$ that $\vert f'(x)\vert<k$, $f \in C^1$

so here's the question: $\vert f'(a) \vert < k$,prove that there exists $I$, $a\in I$ that $\vert f'(x)\vert<k$, $f \in C^1$, since the first derivative is continuous, I tried using the ...
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40 views

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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On the notion(s) of multiplicity of a fixed point

I am trying to understand the notion of multiplicity of a fixed point of a map $f: M \to M$, say, $M$ being a smooth closed manifold, and $f$ being a smooth diffeomorphism. There is a notion of ...
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24 views

Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions. Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...
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13 views

Bannach Fixed Point Theorem for rotation and scaling the punctured unit disk

I'm trying to show that a weak contraction is not sufficient for the Bannach Fixed point theorem. The best example I could think of is the function defined on the punctured unit disk to itself given ...
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3answers
58 views

Show that $\sqrt{x+2}$ is a contraction

Let $f:X \rightarrow X$ where $X=[0,\infty)$ be defined as $f(x)=\sqrt{x+2}$. I have to show that this mapping is a contraction and find its unique fixed point. The second part is easy: by the CMT, it ...
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27 views

Fixed points of Banach.

Let $S,T : M \hookleftarrow$. By definition. Let $M$ a set. A fixed point of an application $A:M\hookleftarrow$ is an element $\xi \in M$ satisfying $A(\xi)=\xi$. If $TS=ST$ (commute), then what ...
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1answer
25 views

Fixed point itteration

Let $x_{n+1}=\sqrt{2+x_n}$ with $x_0=0$ and $y_{n+1}=\sqrt{2+y_n}$ with $y_0=2018$. Do those two sequences have the same limit? I think the answer is "Yes", but I am not really sure. Is my argument ...
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1answer
70 views

Convergence analysis for Solving quadratic matrix equation $\mathbf{XBX=A}+\lambda \mathbf{X}$

I seek for a symmetric positive definite (PD) solution $\mathbf{X}$ for the following quadratic matrix equation: \begin{array}{cc} \mathbf{XBX=A}+\lambda \mathbf{X} \tag{1}\label{eq1} \end{array} ...
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How does PageRank deal with nodes that do not have out-links?

I will use the notation that $A_{ij}=1$ if an arrow exists from $j$ to $i$ and otherwise zero. Just to avoid confusion I use in brackets the standard convention $B_{ij}=1$ when $i$ has a directed ...
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1answer
31 views

Proof of fixed point theorem on $D^1$ using the technique used in Brouwer's fixed point theorem.

Let $f : [-1,1] \longrightarrow [-1,1]$ be a continuous function. Then using IVT I have proved that it has a fixed point. Now my question is "Can I prove this result by the technique used in Brouwer's ...
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2answers
171 views

Existence of infinite iteration of functions $f_\infty$?

Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as $f_n(x)=g(x+f_{n-1}(x))$ $f_n(x)=g(xf_{n-1}(x))$ $f_n(x)=g(x/f_{n-1}(x))$ Where $g:=f_1$ is ...
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1answer
38 views

Application of Banach contraction principle

Define $T:\mathbb R^3→\mathbb R^3$, $(x,y,z)\mapsto\left(\dfrac12\cos y +1,\dfrac23\sin z,\dfrac34x\right)$. I have checked that this example is a contraction and now I am trying to apply Banach ...
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1answer
39 views

Proof of convergence rate for iterative methods

Given the sequence $\{x_n\}$ generated from the iterative function $\Phi(x)$, $\{x_n\}$ converges with order $p$ to the fixed point $\alpha$ if: $$ \exists \lim_{n \to \infty} \frac{|x_{n+1}-\alpha|}{...
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How does one find all the fixed points of the operator defined on the factorial function and how it affects the definition of it?

I was reading these notes and I recently asked: Why does the fixed point theorem justify the existence of the factorial function? that outlined the need of fixed points for justifying the definition ...
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1k views

Why does the fixed point theorem justify the existence of the factorial function?

I was learning about fixed point theorem in the context of programming language semantics. In the notes they have the following excerpt: Many recursive definitions in mathematics and computer ...
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2answers
40 views

Infinite composition of function that has a fixed point

I have faced an interesting question working with functions that have a fixed point(i.e such $f$ that $\exists$ $x: f(x)=x$ ) So I asked myself quite a gerenal question that I did't find easy ...
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42 views

Why does apply F to a fixed point show that its a fixed point?

I was learning fixed points in the context of programming languages and the text wants on page 89 wants to show that $fix(\mathcal F)$ is a fix point by applying $\mathcal F$ to the fix point itself. ...
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1answer
45 views

Fixed point of a function defined in terms of a metric on a compact metric space

Let $X$ be a compact metric space, and assume $f : X → X$ satisfies $$d(f(x), f(y))< d(x, y), \forall x \neq y ∈ X.$$ Define a function $g : X → \mathbb{R}$ by $g(x) = d(x,f(x)), \forall x ∈ X$. ...
3
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2answers
40 views

A lemma for the idea in the modern proof of Banach's Contraction Mapping Theorem

Let $(X,d)$ be a metric space and $T: X \rightarrow X$ a contraction mapping with a fixed point $w$. Suppose that $x_0 \in X$ and we define $x_n$ inductively by $x_{n+1}= Tx_n$. Show that $d(x_n,w) \...
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1answer
61 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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1answer
31 views

Verify that $x$ is a fixed point

Given the function $f(x) = {-x^4 \over 4} + x^3 -4x + 4$ I have graphically localized two roots $\alpha$ and $\beta$ (with $\alpha < \beta$). After analyzing them with Newton's algorithm I'm given ...
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15 views

Properties of Alternating Least Squares: fixed points, monotone convergence

Let $y_i$ a scalar and $W_i \in \mathbb{R}^{n \times n}$ for all $i = 1..N$, and $a,b \in \mathbb{R}^n$ . The following optimization problem is bilinear: $min_{a,b} \sum_{i=1}^N \| y_i -a^TW_ib \|_2^...