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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Numerical Stability of Fixed-Point Interation
The fixed-point iteration $x_{n+1} = \phi(x_n)$ for some Lipschitz-continuous function $\phi$ with Lipschitz-constant $L<1$ is one of the methods in numerical analysis to obtain a solution $x^*$ of ...
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Are Sperner's Lemma and Brouwer's Fixed Point Theorem equivalent?
On this Wikipedia link I found the following statement:
In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.
At first, I found ...
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Contraction of Fredholm equation
I am supposed to find fitting conditions for two fuctions $g(x)\in C([0,1])$ and $K(x,t)\in C([0,1]^2)$ such that the function
$$y(x)=g(x)+\int_0^1K(x,t)y(t)^2dt$$
has a unique solution. For this I ...
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Applying Picard-Lindelöf theorem
Consider the ODE:
$y'= x + \text{cos}(x+3y),\,\, y(0) = 1.$
Let $g(x,y) = x + \text{cos}(x+3y).$ Suppose $g$ is continuous on a neighbourhood of $(0,1)$ of the form $G = (-1,1) \times (0,2).$ I want ...
user715112
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Fixed points and system of equations
Suppose that $F : \mathbb{R}^n \rightarrow \mathbb{R}^n$. A fixed-point of function $F$ is such that
$$
F(x) = x.
$$
We can represent that fixed point as the solution to the system
$$
F(x) - x = 0.
$$
...
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Prove a continuous function on the unit ball vanishes, using Brouwer's fixed point theorem
Let $B^n\in\mathbb R^n$ be the closed unit ball, and $f:B\to\mathbb R^n$ be continuous. Suppose further that $f$ takes the boundary to itself, i.e., $|x|=1\implies f(x)=x$.
I would like to use Brouwer'...
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Banach fixed point theorem, $T: D \to V, f \to Tf$
I am working on a problem, but i am not sure if my solutions are correct.
To the Problem:
Let $ (V,‖·‖) = (C([0,1]),‖·‖∞)$ be a normalized vector space.
and let $D:=\{f∈V; \| f \| \leq 1/3\}$
We ...
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Proof of general Knaster-Tarski fixpoint theorem
Prove that the set of fixed points $Fix(f)$ of an order-preserving operator $f$ on a complete lattice
$(L, \sqsubseteq)$ is a complete lattice itself. Moreover, show that $\mathrm{Fix}(f)$ is a ...
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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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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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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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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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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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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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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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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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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} ...
- 147
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
- 4,439
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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 ...
- 1
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votes
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answer
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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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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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