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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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To Prove $T $ is a self map and $T$ have no fixed points

Let $K=\{x=(x(n))\in c_0:0\le x(n)\le 1$ for all $n\in \mathbb{N}\}$. Define $T:K\to c_0$ by $T(x)=(1,x(1),x(2),x(3),...).$ Prove : (a) $T$ is a self map on $K$ and $||Tx-Ty||_\infty=||x-y||_\infty $ ...
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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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1answer
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How to show the optimization/ODE fixed point iteration steps converge?

I have $\vec{C} = G(\vec{\beta})$ by solving a system of ODE numerically. Thanks for the help of Robert the ODE can be found in this link: Solving a system of ODE Also $\vec{\beta}$ should satisfy $$...
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1answer
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Fixed point equation to solve Burgers' equation IVP

Using the equation $u \equiv u ( x , t ) = u _ { 0 } ( x - t u ( x , t ) )$ to compute $u \left( T , x _ { j } \right)$ for the Burgers equation, where the Burgers equation is $u _ { t } + \left( \...
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How to prove the existence of a fixed point of this mapping?

Let $X=\{x_i : i\in I\}\subseteq\mathbf{R}^n$, where $I=\{1,\ldots,m\}$. Then for some initialization $\mu^{(t)}$, and $\pi^{(t)}=\{x\in X : \|x-\mu^{(t)}\|\leq r\}$, $r>0$, we want to prove that a ...
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43 views

fixed point function (nonlinear equation)

here's the following problem, I'm trying to find a real root by fixed-point iteration method but I can't find a properly $g(x)$ that meets the condition that $|g'(x_0)|<1$. Well, my nonlinear ...
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1answer
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Show that $\exists u_0 \in C : g(u_0) = u_0$, if $g$ is nonexpansive over a Banach subspace.

Exercise : Let $X$ be a Banach space, $C \subseteq X$ compact and convex and $g : C \to C$ a nonexpansive operator. Show that $\exists u_0 \in C : g(u_0) = u_0$. Thoughts : In a previous exercise, ...
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Is there a way of proving that a function has a particular number of fixed points.

From my understanding, a function is said to have a fixed point if $f(x) = x$. Is there a way for finding how many fixed points a function has?
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1answer
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Question about Banach's fixed point theorem

Let $(x_n) _{n\ge 1}$ be a sequence and $f:\mathbb{R} \to \mathbb{R} $ a contraction. I know that if $x_{n+1} =f(x_n) $ then $(x_n) _{n\ge 1}$ converges to $f$'s unique fixed point by Banach' s fixed ...
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1answer
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Fixed Point Iteration $x^3 - 3 = 0$

I am having trouble solving $x^3 - 3 = 0$ using the fixed point iteration method. It is advised in the problem to put $g(x)$ in a form similar to $g(x) = x + c(x^2 - 5)$ for $x^2 - 5 = 0$ but I am ...
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I am trying to find the maximum learning rate or stepping rate of steepest descent algorithm in 2 dimensions

Let $f(x,y) = (x-y)^4+2x^2+y^2-x+2y$. I am trying to numerically find the miniumum of $f$. We define a fixed-point iteration scheme \begin{align*} g(x, y) = \vec{x}_{n+1} = \vec{x}_{n} - a\nabla f \...
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Right-preconditioning and fixed point linear iterations

Given a linear system $A\textbf{x}=\textbf{b}$, we can express it into the easier-to-solve right-preconditioned form: $$ AM^{-1}\textbf{y}=\textbf{b}, \quad \textbf{y}= M^{-1}\textbf{x} $$ On the ...
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1answer
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Prob. 7 (b), Sec. 28, in Munkres' TOPOLOGY, 2nd ed: A shrinking self-map of a compact metric space has a unique fixed point

Here is Prob. 7, Sec. 28, in the book Topology by James R. Munkres, 2nd edition: Let $(X, d)$ be a metric space. If $f$ satisfies the condition $$ d\big( f(x), f(y) \big) < d(x, y) $$ for ...
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Proving limit using asymptotically unstable fixed point

$$\dot{x}=x(1-x-\frac{3y}{4(x+1)})$$ $$\dot{y}=y(y-1)$$ One of the fixed points of this system is $(1,0)$ (Easily found through putting $\dot{x}$ and $\dot{y}$ equal to zero). This point is an ...
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Richardson's Iteration, Gradient Method and Spectral Radius

Richardson's iteration introduce a scalar $\alpha$ to the update formula: $$ \textbf{x}^{(k+1)} = \textbf{x}^{(k)} + \alpha \textbf{r}^{(k)} $$ And compute $\alpha$ by minimizing the spectral radius:...
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3answers
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Convergence to a fixed point [duplicate]

Let $f : [a,b] \rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $\left\lvert f'(x)\right\rvert \leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ ...
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1answer
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Prob. 7 (a), Sec. 28, in Munkres' TOPOLOGY, 2nd ed: A contraction of a compact metric space has a unique fixed point

Here is Prob. 7, Sec. 28, in the book Topology by James R. Munkres, 2nd edition: Let $(X, d)$ be a metric space. If $f$ satisfies the condition $$ d\big( f(x), f(y) \big) < d(x, y) $$ for ...
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1answer
16 views

Fixed-point iteration: little-oh relation between consecutive pair of elements

Given $x_0 \in [a,b]$, let the sequence $(x_n)$ be defined recursively by: $$ x_n = g( x_{n-1}), n=1,2,... $$ where $g \in C^1 [a,b]$ Assume that $x_n \to c \in [a,b]$, then: $$ c=\lim_{n \to \infty} ...
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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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Contradiction with Banach Fixed Point Theorem

I am trying to find the fixed point of the function $g(x) = e^{-x}$. Wolfram Alpha tells me that this fixed point is approximately $x \approx 0,567$. However, if I apply the Banach fixed point theorem,...
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1answer
36 views

The ordinary iteration method converges faster than any geometric progression.

I have gotten stuck trying to prove that iteration method converges faster than any geometric progression. Background: Assume that the function $g$ is continuously differentiable. Let $x^*$ be the ...
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1answer
39 views

Modifying the “base” of Veblen's hierarchy to exceed $\Gamma_0$

The Veblen hierarchy is usually defined with $\varphi_0(x) = \omega^x$. As a result, we can define the Feferman–Schütte ordinal as the first fixed point of the function $\varphi_\alpha(0) = \alpha$. ...
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1answer
32 views

Schauder fixed point extended

The Schauder fixed point theorem states that if $X$ is a Banach space, $K\subset X$ is a convex, bounded and closed subset and $T:K\rightarrow K$ is compact, then $T$ has, at least, one fixed point in ...
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1answer
35 views

Banach Fixed Point Theorem, System Has Solution

Using Banach's Fixed Point Theorem, show that the following system has at least one solution: $$ x = 0.000001x^2 + 10\sin y + 1 $$ $$y = 0.000001y^3 - 0.01\cos x - 1 $$ Here is what I have tried: ...
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29 views

Prove a Lemma Involving Asympotically Stability

I am trying to prove the following Lemma: Lemma: Suppose that the point $x^*$ is a fixed point of $x(n + 1) = f(x(n))$ (1) while also an asymptotically stable(unstable) fixed point with respect ...
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1answer
35 views

Proving that product of general matrices has small spectral radius

In a Jacobi type of iteration for finding solution to a linear system $Ax=b$, one writes $$x_i^{(k+1)} = Gx_i^{(k)}+c,$$ where $x_i$ is the $i$-th component of vector $x$ and $G=D^{-1}N$, $c = D^{-1}...
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How to show that one not monotonous f doesn't have fixpoints?

I have a question about fixed points If I have one function $f$ (that is not monotonous!) I would like to demostrate that this function hasn't fixed points. I need to find a funciton $f$ for which ...
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1answer
29 views

If the initial points for secant iteration method are sufficiently close to the root, the iteration converges to the root

Well I wish to prove that in case I may choose $x_0,x_1$ close enough to the root $a$ of $f(x)$, then the secant method $x_{n+1} = x_n -\frac{x_n -x_{n-1}}{f(x_n)-f(x_{n-1})}f(x_n)$ converges to the ...
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1answer
35 views

Banach fixed point theorem application

I'm trying to use the Banach fixed point theorem to show that an intergral equation has a unique solution, but can't seem to make my answer work any help would be appreciated. Let $f:[a,b] \...
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2answers
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Exactly one solution to integral equations

Show $\exists$ exactly one solution $U\in C([-1,1])$ to the intergral equation $U(x)=x\int_{0}^{x}t^{2}cos(U(t)) dt $ for $x \in [-1,1]$ Attempt I think I can use the contraction mapping theorem ...
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1answer
45 views

Showing uniqueness of a fixed point on $[0,1]$

Given $g(x)=-x\sin^2(\frac{1}{x})$ for $0<x\leq1$ My attempt: let fixed point given by $g(x)-x=-x\sin^2(\frac{1}{x})-x=0$ $$0=-x\left(\sin^2\left(\frac{1}{x}\right)+1\right) $$ Therefore only for ...
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1answer
47 views

Different definition of Veblen functions

Consider the Veblen hierarchy, where $\psi_0(x) = \omega^x$ and $\psi_1(x)$ is the x'th fixed point of $\psi_0$, $\psi_2(x)$ is the x'th fixed point of $\psi_1(x)$, and so on. We eventually get to $\...
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1answer
30 views

Showing a map has a unique fixed point

Show that the function $f:\mathbb{R}^{3} \rightarrow \mathbb{R}^3$ given by $(x,y,z) \mapsto \bigg(\frac{\sin y}{4},\frac{\sin z}{3}+1,\frac{\sin y}{4}+2 \bigg)$ has a unique fixed point. Attempt By ...
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44 views

Fixed point iteration.

I have a general question about fixed point iteration. I have used this method several times in my Numerical Analysis course and sometimes it won't converge to certain root even if the start guess is ...
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2answers
88 views

Fixed points of ordinal exponentiation for bases besides $\omega$

The first fixed point of the map $x \to \omega^x$ is the first epsilon number $\epsilon_0$, which is the supremum of $\omega, \omega^\omega, \omega^{\omega^\omega}, ... = \omega^{\omega^{\omega^{.^{.^{...
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1answer
40 views

Show root using Banach Fixpoint

I'm required to show that: $f(x) = e^x - 4x$ has a root in $(0,1)$ using the Banach Fixpoint theorem. The fact that $f((0,1)) \neq (0,1)$ confuses me. How do I proceed without knowing that $f$ isn't ...
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A Question on Fixed point theorem

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a map such that for $x\in X$ there exists a sequence $(a_n(x))\in [0, \infty)$ such that (A) $\lim _{n\to \infty} a_n(x)=a_{\infty}(x)<1$ ...
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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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2answers
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Let $u'(t)=Au^2-Bu$. Find conditions on A, B to guarantee global solution.

I'm taking a real variable course and we have just covered the Banach Contraction Principle. Our professor sometimes makes problems up on the spot for us to try and figure out together. This is one ...
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Does there exist a triple point map?

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinite double points.also by borsuk-ulam theorem this is true for each map $N:S^n\to \mathbb{R}^n, n\in \mathbb{N}$. A ...
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1answer
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A question about fixed point theorem [closed]

how to prove this theorem how to prove first if the sequence is there that is Cauchy...thanks
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1answer
232 views

Does the sequence $(x_n)$ given by $x_{n+1} = -16+6x_n+\frac{12}{x_n}$ converge?

Question. If $x_0$ is sufficiently close to $2$, then will the sequence obtained as $$x_{n+1} = -16+6x_n+\frac{12}{x_n}$$ converge to 2 ? My attempt : I have shown that if $x_0$ is close to ...
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1answer
82 views

Banach fixed-point theorem : Existence of solution

We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\...
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Looking for an application of DeMarr's point-fixed theorem

I recently discovered the DeMarr theorem: In a vectorial space, if you got two non expensive maps from a convex compact to itself that commutes, they got a common fixed point. I have no example of any ...
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1answer
287 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\...
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2answers
24 views

A question on Contraction and contractive map

how to prove this map is not contraction and have no fixed point and I am proved contractive by using mean value theorem
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18 views

Iterative method with squared function (edited)

For solving a certain equation I've come up with this iterative method $$x^{n+1}=g^2(x^{n}),$$ where $g$ is given by $$g(z)=\frac{1}{2}\left[\sqrt{\left(A\cdot\text{erf}(\sqrt{z}/2)\right)^2+B\cdot\...
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0answers
15 views

Fixed point iteration convergence

Was exploring the following fixed point iteration $$ x \leftarrow 1 + \frac{1}{c*log(1-x) - 1}$$ for $x\in(0,1)$ and $c>1$ and initial guess at $$x_0 = \sqrt{1 - \frac{1}{c}}$$ It seems to ...
2
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1answer
41 views

Kleene's fixed point theorem for cpo's

Let $D$ be a cpo with the Scott topology, then Kleene's fixed point theorem states that every continuous function $f:D\rightarrow D$ has a fixed point: $$ \operatorname{Fix}(f) = \bigsqcup_{n\in\...
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Is this simple demand-based prices game a submodular game?

I have this simple market game: $I=\{1,2...,n\}$ players $S_i$ strategy space of each player $i\in I$ $u_i(s_i,s_{-i})=R_i(s_i)-C(s_i,s_{-i})$ There's only one type of resource. The resource is ...