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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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Suppose $k_n\subset[0,\infty)$ with $k_n\to1$ as $n\to\infty$ and $\alpha_n$ is in $[0,1)$ is $\sum\alpha_n(k_n(M+1)-1)<\infty$ where $M>1$

we am working on a mapp to show that the Mann iteration converges. At the end of the computation we came up the series $\sum\alpha_n(k_n(M+1)-1)$, from previous work it is known that the series $\sum\...
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Fixed-point iterations for quadratic function $x\mapsto x^2-2$

Let $f(x)$ be $x^2-x-2$. I want to find the root using FPI in an interval where it will converge. I have chosen $g(x)=x^2-2$ and so $g'(x)=2x$. The convergence condition, $|g'(x)|<1$ is ...
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Fixed point iteration with the same power coefficient

I have a function where $f(x) = x^3cos(x)-x^3/10$, with that said, how do i find the fixed point iteration formula for it. I have tried adding an unknown to it and get the $x$ but it does not converge ...
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Convexity is crucial in Schauder-Tychnonov fixed-point theorem.

The following Theorem is well-known: Schauder-Tychnonov fixed-point theorem: Let $K$ be a compact convex subset of a Banach space, $E$. If $T:K\to K$ is continuous, then $T$ has a fixed point. I'm ...
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The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1],$ unit closed, need not have a fixed point.

The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1]$, need not have a fixed point. Know about the Brouwer fixed point Theorem on $\mathbb{R} ^n$ which states that if $ \...
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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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Sufficient and necessary condition for the global uniqueness of fix-points

https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...
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Uniformly convex implies strictly convex [closed]

How to prove Uniformly convex implies strictly convex
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Variant of the Contraction Mapping Theorem

Let (C, ||.||) be a closed subset of a Banach space, and let f: C -> C be a mapping such that ||f(x) - f(y)|| < ||x - y|| for all x, y in C. Must there exist a fixed point in C that f maps to ...
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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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Rigorous proof of a unique solution using Banach's Fixed Point Theorem

I would like to have feedback on the overall quality of the following proof. Question: Prove that $x^5+7x-1=0$ has a unique solution in $[0,1]$. Proof: Let $f(x)=\frac{1-x^5}{7}$ and note that any ...
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Is the set of nash equilibria/correlated equilibria convex?

I am curious about the geometry of these sets (assuming compact, convex action space and concave utility function, so the nash must exist). Is there any general argument about when will any solution ...
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A question about fixed points and non-expansive map

Let $$K=\{x=(x(n))_n\in l_2(\mathbb{N}):\|x\|_2\le 1\ \text{ and } x(n)\ge 0 \text{ for all } n\in \mathbb{N} \}$$ and define $T:K\to c_0$ by $T(x)=(1-\|x\|_2,x(1),x(2),\ldots)$. Prove : (1) $T$ is ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Locally convergence for Fixed -point iterations

Taking the following iteration $$x_{n+1}=-\ln(x_n),\quad x_0\in]0,+\infty[$$ Study the convergence of this iteration. By applying Fixed-point theorem, I found that the function is not contraction on $...
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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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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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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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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
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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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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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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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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
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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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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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67 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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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 ...