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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19 views

Why does Scarf's algorithm only need to examine a small fraction of points in the simplex?

Scarf's algorithm for finding the Brouwer fixed-point searches for the fixed-point in an non-repeating fashion examining a finite number of points. It finds the fixed point however by examining a very ...
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27 views

How to prove the divergence of a fixed point iteration, in the context of the power tower

EDIT 2: I realise now that such conditions are quite context-dependent. To include the original context from which I considered this question, I was researching the convergence of the power tower, ...
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17 views

Limiting points and fixed points of a system of differential equations

Consider a system of differential equations $$ \frac{d}{dt}f(t) = F(t, f(t), g(t)), $$ $$ \frac{d}{dt}g(t) = G(t, f(t), g(t)). $$ Assume $F, G \in C^{\infty}$. What is the necessary and sufficient ...
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59 views

Does there exist an unique continuous bounded $f$ such that $f(x)= \frac{\sin(f(x))}{2+x^2} - \frac{\cos^2(x)}{1+e^x}$?

Does there exist an unique continuous bounded $f$ such that $f(x)= \frac{\sin(f(x))}{2+x^2} - \frac{\cos^2(x)}{1+e^x}$? I wanted to prove this by proving this is a strict contraction and than applying ...
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21 views

Contraction mapping and linear continous operator

I'm working on contraction mapping theorem with parameter, and this leads me to Appendix D of G. Da Prato, Introduction to stochastic analysis and Malliavin calculus. In the book, it says whereas I ...
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35 views

Perron-Frobenius Theorem poof by Brouwer fixed point

Could you suggest me a book where I can find a proof of Perron-Frobenius theorem (especially for nonnegative matrices) based on a Brouwer fixed point theorem?
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48 views

Fixed Point Iterations on $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$

Say I have two nonlinear equations of the form $$ \begin{bmatrix} u \\ v \end{bmatrix} = f(u,v) = \begin{bmatrix} f_1(u,v) \\ f_2(u,v) \end{bmatrix}, \tag*{(1)} $$ where $u,v \in \mathbb{R}$ and I ...
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54 views

Brouwer's fixed-point theorem, permutations and coffee

One of my friends pointed out an interesting application of the Brouwer's fixed-point theorem: You cannot stir a coffee in a mug such that all of the coffee particles have changed their position. ...
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54 views

functional iteration convergence

functional iteration sequence $x_{n+1} = 2 - (1+c)x_n + cx_{n}^3$ will converge for some values of c to $ \alpha = 1$ for what values of c this sequence will converge? My attempt to solve this was ...
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30 views

prove sequence of functional iteration converges

Prove that the sequence generated by the iteration $x_{n+1} = F(x_n)$ will converge if $|F^{'}(x)| \le \lambda < 1$ on the interval $[x_0 -p , x_0 + p]$ where $p = \frac{|F(x_0) - x_0|}{1- \lambda} ...
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Simple Iteration Covergence

A possible simple iteration rearrangement of the equation f (x) = 0 is in the form x = g(x) where g(x) = x − k(f(x)/f'(x)) and k is some real number. Determine the values of the parameter k for which ...
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Using contraction mapping theorem with parameter to prove some lemma of generalised Picard-Lindelöf Theorem

I posted a question about contraction mapping theorem with parameter several hours ago, and I kind of understand why we can't use chain rule directly since we don't know if the function is ...
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39 views

Contraction mapping Theorem with parameter

I encountered contraction mapping theorem in the lecture and by doing some extra reading, I started to consider a parameter-dependent case. It runs as follows: If P is an open set of a complete metric ...
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25 views

Newton's Method Convergence Rate when first derivative is non-zero and second derivative is zero

Using the Taylor expansion it is easy to show that the convergence of Newton's method for a root $\alpha$ is quadratic when $f'(\alpha)\neq0$ and $f''(\alpha)\neq0$. If instead $f'(\alpha)=0$ and $f''(...
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66 views

A variant of Brouwer's fixed point theorem.

In the course on algebraic topology I came across a result called Brouwer's Fixed Point Theorem which states that any map (continuous function) $f : D^n \longrightarrow D^n$ has a fixed point for $n \...
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39 views

Fixed point iteration without contractivity

Let $x_0>0$, $a>0$, $b>0$ be given and define $$x_{n+1}:= a+b x_{n}^{1/4}$$. Question: What is this speed of convergence of $x_n$ to the unique solution $x>0$ of $x=a+bx^{1/4}$? Lacking ...
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22 views

Fixed point theorem for three-point recursion?

Let's say we have some three term recursion $$x_{n+1} = f(x_n, x_{n-1})$$ where $f : \mathbb R^2 \to \mathbb R$ is sufficiently differentiable. If for instance $f$ only depends on $x_n$ and $\Vert \...
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21 views

Fixed point/lipschitz constant

Let $M \subseteq \mathbb{R}$ be closed and the mapping $T : M \rightarrow M$ fulfills $$ |T(x)-T(y)| \leq |x-y| $$ $\forall x,y \in M, x \neq y$ Prove or disprove that T has exactly a fixed point. So ...
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Prove that mapping of $Tf(x) = \int_{0}^x{(x-t)f(t) dt}$ is well defined [closed]

This question has been given to me in a Real Analysis class. In complete metric space $C[0,1] = \{f: [0,1] \rightarrow \mathbb{R} \;|\; f\; \text{is continuous} \} $ with metric $d(f,g) = \max\{|f(x)-...
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How to find a quadratic convergence function on fixed point iteration method on root finding?

I've read several references, and it is true that: A point is called a fixed point if $f(x_0) = x_0$. It can further be reduced to find root of a non-linear function $f(x) = g(x) - x = 0$ The fixed ...
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Fixed point system convergence

I am writing a computer program that solves a fixed-point system and I need to determine the convergence criteria. I am implementing an algorithm from a journal article, and the author states the ...
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29 views

Proving existence of roots

I have the following arbitrary function which is the result of solving an iterative map for any period two fixed points (ie. for $g(x_n) = x_{n+1}$, I am trying to find $k$-values for which g(g(x)) = ...
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20 views

banach points in different metrics

is it possible to have a transformation $T: X \to X$such that there is contraction for T in $(X,d_1)$ but not in $(X,d_2)?$ I tried defining the function $T(x)=x/2$ and $d_1=|x-y|$ but I cannot seem ...
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'Contraction-like' inequality: how to deal with the boundary term?

I am interested in the following problem. Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
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65 views

Performing a fixed point Iteration upon $f(x)$

Let us say we have the function $f(x) = (e^x - 1)^2$. I want to perform a fixed-point iteration upon this function, such that $x_{n+1} = g(x_n)$. How can I transform this particular function into a ...
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65 views

Converse of Banach's fixed-point theorem

While I was reading about Bessaga's converse to Banach's fixed-point theorem, I found this lecture on the internet. But I had a doubt over here. Let $f : X \to X $ given by $f(x)=x^3$ where $X=(-1,1)$,...
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it is possible to not consider the condition q < 1 in the Banach Fixed Point Theorem (No contraction basically?)

it is possible to not consider the condition q < 1 in the Banach Fixed Point Theorem (No contraction basically?) and still find a fixed point? Any particular example of a function? f: R -> R
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93 views

Proof of a fixed-point lemma

I'm trying to prove the following fixed-point lemma. Let $\mathcal X$ be a Banach space and $A \neq \emptyset$ a closed, bounded and convex subset of $\mathcal X$. Further let $g: \mathbb R^+ \to \...
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A textbook with a proof of the Rank Theorem using only Banach's Fixed Point Theorem.

It is well known that the Rank Theorem for $C^1$ maps can be obtained as a consequence of the Implicit Mapping Theorem and the Inverse Mapping Theorem for $C^1$ maps. See for example Zorich vol 1. It ...
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55 views

Brouwer's fixed point theorem with deformation retraction instead of retraction

The standart way to prove the theorem is to assume that there are no fixed points for a function $f: D^{n} \rightarrow D^{n}$ and from that obtain a retraction $r: D^{n} \rightarrow \partial D^{n}$, ...
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49 views

Find a unique $f^*\in C([0,1])$ s.t. $g+R(f)=f$ where $f,g,R(f):[0,1]\to\mathbb{R}$ are continuous functions

Here $g:[0,1]\to \mathbb{R}$ is a continuous function and so is $f: [0,1]\to \mathbb{R}$. $R(f)$ is a little bit more complicated: \begin{equation} R(f)(x):=\int_0^x k(x,y)f(y)dy \end{equation} for a ...
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45 views

Infinite transitive permutation group where every element has a fixed point

In this question, it we see that a transitive permutation group acting on a finite set with two or more elements must have a fixed-point-free element. I was wondering whether or not this result could ...
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26 views

Unstable Nash equilibrium (at a boundary point)

Let $x=(x_i)_{1\leq i\leq n}$ be the "actions" of $n$ players, where $x_i\in[0,1]$ is determined by player $i$ seeking to maximize its objective function $\pi_i(x)=\pi_i(x_i,x_{-i})\geq 0$. ...
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36 views

Prove that the equation $\phi(x)=\int_{0}^{x}\frac{(x-t)^{n-1}}{(n-1)! }\phi(t) dt $ have trivial solution for each $n\in \mathbb{N}$

Prove that the equation $$\phi(x)=\int_{0}^{x}\frac{(x-t)^{n-1}}{(n-1)! }\phi(t) dt $$ have trivial solution for each $n\in \mathbb{N}$ in $C[0,\alpha]$ Proof I want prove that exists a ...
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101 views

Does the coproduct preserve limits?

Let $\mathcal{C}$ be a category with coproducts (+) and all limits. For a diagram $D : \mathcal{I} \to \mathcal{C}$ is the following true? $A+ \varprojlim_{i \in \mathcal{I}} D(i) \cong \varprojlim_{...
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Regularity of the fixed point of a Kernel

Take $S\subset \mathbb R^n$ bounded and be $k(\cdot|s)$ a probailiplity density kernel, i.e. $$\int_S k(s'|s)\ ds'=1\qquad s-a.e.$$ What are the hypothesis I have to put on $k$ so that, for any $b(s)\...
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Understanding a corollary of a theorem about fixed point in bicomplete quasi-metric spaces

I am studying a paper on quasi-metric spaces for the complexity space, and I found that Banach's famous result about fixed points in complete metric spaces can be extended to bicomplete quasi-metric ...
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Ordering of fixed points

Suppose one has two real functions $y = f_a(y)$ and $y = f_b(y)$ for which the contraction mapping theorem holds, such that there exist unique fixed points $y^*_a$ and $y^*_b$, respectively. Are there ...
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27 views

Fixed point theorems including some linear differential equations

I am having a trouble with applying the fixed point theorem. For instance, suppose I have three value functions. Two of them are linear differential equations where $r \in (0,1)$ is a discount factor. ...
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7 views

Convergence of non-linear vector fixed point iteration

Let $A$ be an $d\times d$, non-negative, symmetric matrix. Let $\mathbf{a}\in\mathbb{R}^d$ be a positive vector and $\mathbf{x}_0 = [1,1,\cdots,1]^T$. Under what conditions on $A$ and $\mathbf{a}$, ...
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33 views

Showing that $F(x) = x + f(x)$ defines a homeomorphism when $f : E \to E$, and where $E$ is a Banach space.

Let $E$ be a Banach space and $f : E \to E$ a contraction. Show that the equation $F(x)=x+f(x)$ defines a homeomorphism $F:E \to E$ that is Bilipschitz. Since $f$ is a contraction the following to ...
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36 views

Average of all $a_n$ where $a_0$=1, $a_{n+1}=\omega^{a_n}$, and $\omega=\frac{\pi i}{\ln(2)}$

Essentially, I've noticed that tetrations of $\omega=\frac{\pi i}{\ln(2)}$ seem to converge on a cycle of three fixed points. Specifically, if $a_0$=1, and $a_{n+1}=\omega^{a_n}$, then we find $$n\...
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25 views

Energy estimates of coupled system of PDE's

Let that I have a coupled system of non-linear partial differential equations. My goal is to derive energy estimates via Galerkin Method. The idea is to decouple the system and find energy estimates ...
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47 views

Prove there exists $c$ on $(a,b)$ such that $cf(c) = ab$ for all continuous $f$.

Let $a,b \in \mathbb{R}$ such that $ab > 0$ and consider $f : [a,b] \to [a,b]$ a continuous function. Prove there exists $c \in (a,b)$ such that $cf(c) = ab$ (from Berkely Math 104 Final). As ...
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40 views

Banach fixed point theorem in $\mathbb R^n$

I have a question on Banach fixed-point theorem. It supposes we are in a closed set $C \subset \mathbb R^n$ with the image of $C$ by a function is included in $C : f(C)\subset C$ and we also suppose ...
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58 views

Does the sequence $x_{n+1}=x_n+x_n^2$ converge to $0$ whenever $-1\lt x_0\lt0$?

My question concerns using the fixed-point iteration to find the fixed point of the function $f(x)=x+x^2=x(1+x)$ (this function has a single fixed point at $0$). The problem Given some fixed $x_0$, ...
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49 views

What is a contractive mapping vs contraction mapping?

This is an example from a text to show that this mapping does not have a fixed point because it is contractive but not a contraction: I am not sure what the difference is between contractive and ...
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42 views

Trapezoidal method using fixed point iteration

I am not sure how to apply the trapezoidal method using fixed point iteration, at each step, to this equation $\frac{dy}{dt}=\cos{(\frac{2y}{4})}$. Any help will be appreciated as I've been stuck on ...
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1answer
41 views

Fixed points for operators in Hölder spaces: Why is merely being a self-map of a ball not enough?

Let $D \subseteq \mathbb{R}^n$ be a bounded domain with smooth boundary and $C^{\alpha}(\overline{D})$, $\alpha \in (0,1)$, the space of Hölder continuous functions with the usual norm $\|\cdot\|_{C^{\...
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25 views

Help understanding Browder fixed-point theorem

I'm having some trouble wrapping my head around the Browder fixed-point theorem. The statement of the theorem is: If $X$ is a uniformly convex Banach space, and If $K \subset X$ is nonempty, convex, ...

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