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)...

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Rate of Convergence of $x_{k+1} = \frac 14 (x_k^2 +sin(x_k) + 1)$

I have shown that the fix point Iteration $$x_{k+1} = \frac 14 (x_k^2 +\sin(x_k) + 1)$$ has exactly one fix point $x^*\in[0,1]$ for every $x_0 \in [0,1]$ with the Banach theorem. Now I want to find ...
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Proving a function is not a contraction

Let $\psi: C([0,1]; \mathbb{R}) \mapsto C([0,1]; \mathbb{R}) $, be such that if $x \in [0,1]$ then $$\psi(f)(x) = \int\limits_{0}^{x} f(t) dt $$ I am asked to show that $\psi$ is not a contraction ...
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Show that mapping is a contraction?

Show that the mapping $f:\Bbb R \to \Bbb R $, $f(x)=1-xe^x$ is a contraction. I tried everything i could think of but i cant get it to work. Witch is not much since i couldn't really find any ...
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Equilibrium points of $\dot x(t)=-2\cdot x^3(t)$

The following differential equation is given: $$ \dot x(t)=-2\cdot x^3(t)\qquad x(t)\in\mathbb R $$ I am asked to find the equilibrium points of the system. By definition, the equilibrium points are: $...
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53 views

Fixed point Iteration: Convergence & Divergence from geometrical figure

I want to understand the geometrical interpretation of Convergence & Divergence of Fixed point iteration method. The figure is here IMAGE: CLICK HERE In Figure 2.4(a) Left panel: if we draw a ...
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Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
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How can I prove $x_{n+1} = e^{-x_n}$ is convergent?

I'm doing a practice problem which asks to prove that the sequence defined by $x_{n+1} = e^{-x_n}$ is convergent (or rather "study the convergence of $(x_n)$"). So I'd like to try and find some ...
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64 views

Banach fixed point theorem - Find $x$ such that $f(x) = 0$.

Let $f: \mathbb R \to \mathbb R$ be a continuous function with a continuous derivative. In short $f \in C^1$. We know that $0<c\leq f'(x) \leq d < \infty$. We want to prove that $\exists! x_0 \...
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130 views

Algebraic fixed point theorem

I was wondering if there are some "algebraic" fixed point theorems, in group theory. More precisely, given a group $G$ and a group morphism $f : G \to G$, what conditions on $G$ and $f$ should we ...
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Picard's existence theorem, successive approximations and the global solution

Picard's existence theorem states that if $U$ is an open subset of $\mathbb{R}^2$ and $f$ is a continuous function on $U$ that is Lipschitz continuous with respect to the second variable then there is ...
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40 views

Proving a function to be a contraction

I'm studying contractions and I am trying to understand under which conditions a function is actually a contraction. I have understood that a continuous function $f: [a,b]\rightarrow\mathbb{R}$ is a ...
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2answers
64 views

Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
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The conjugate of a complex function

This may be a stupid question. I am trying to explain/prove the following: Let $f$ be a complex rational function of degree $d ≥ 2$ with fix point $z_0 \neq 0$. It is always possible to conjugate $f$...
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343 views

unique fixed point problem

Let $f: \mathbb{R}_{\ge0} \to \mathbb{R} $ where $f$ is continuous and derivable in $\mathbb{R}_{\ge0}$ such that $f(0)=1$ and $|f'(x)| \le \frac{1}{2}$. Prove that there exist only one $ x_{0}$ such ...
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73 views

Does any continuous function from the open unit interval $(0,1)$ to itself has a fixed point?

I know that the result is true for closed interval $[0,1]$ by using intermediate value property. But in the case where we consider open interval $(0,1)$ does the solution change?
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22 views

Stating if a function being a contraction

I have to state if a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a contraction on the interval $I\subset \mathbb{R}$ and say if it admits fixed points. The function is $f(x)=\left\{ ...
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47 views

$r\rightarrow1/r$ invariant

(Not sure the tags are appropriate, but can't think of better ones. Please suggest better.) Suppose you have a function $f(x,y,z,...;g(r))$ with the requirement that $r\rightarrow1/r$ leaves $f$ ...
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22 views

Problem in Topology related to continuous functions

Let $\mathbb{R-Q}$ be the subspace of $\mathbb{R}$ with the usual metric. Is there a function, $f:\mathbb{R-Q} \rightarrow \mathbb{R-Q}$ such that f is continuous and $f$ does not have a fixed ...
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49 views

Lefschetz number of a transformation of the sphere.

In differential topology the Lefschetz number of an automorphism of a compact manifold is the oriented intersection number of the graph of that automorphism with the diagonal. I would like a proof or ...
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82 views

Edelstein Fixed Point Theorem

Let $(M,d)$ be a metric space, $M$ compact. If $f:M \to M$ is continuous and weakly contractive (i.e. $d(f(x), f(y)) < d(x,y) , \forall x,y \in M$), then $\exists x_0 \in M $ unique s.t $f(x_0)=...
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22 views

Proof there is no fixpoint

I want to proof that a certain function (in this case tan(x) on (0, 1/4)) doesn't have a fixpoint. Is there a general approach to try that? For example an "iff" fixpoint theorem?
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66 views

Existance of a fixed point of a bijective, smooth function:

Let $[a,b],\,\, [c,d]$ be two bounded closed intervals of $\mathbb{R}$ such that $$[a,b]\cap[c,d]\not=\emptyset$$ and $f:[a,b]\to[c,d]$ be a bijective, smooth function. We know that, if $[a,b]=[c,d]...
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60 views

Fixed point iteration methods lead to division by zero

So I need to prove that for $f(x) = ((x - x^*)^2)h(x)$ assuming $f''(x) \neq 0$ and assuming $f(x)$ be twice continuously differentiable, that Newton's method converges linearly in this case. So $g(x) ...
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19 views

Weak contraction mapping preserves order?

Suppose I have a function $\varphi$ on the unit interval such that $|\varphi (x)-\varphi (y)|<|x-y|$. Is it true that if $0 \leqslant c \leqslant 1$ then $\varphi (0) \leqslant \varphi (c) \...
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4answers
230 views

Fixed point iteration convergence of $\sin(x)$ in Java [duplicate]

Is it mathematically correct to say that $\sin(x)$ converges to zero as $x$ approaches $0$? If the $\sin(x)$ iteration is done starting at $\dfrac{\pi}{2}$ in Java, for $10^9$ iterations, the result ...
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52 views

What is meant when saying a function has a solution?

I have to code the solutions to these questions in Python. But before even thinking about coding, I have to understand what I have to do here. Table 1: Observed frequency distribution of Y $$\begin{...
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39 views

Solution of an equation involving Banach fixed point.

I find this queistion in my textbook: QUESTION) Let $T \in \mathcal{L}(\mathcal{B})$, where $\mathcal{B}$ is a Banach Space. If exists $n$ with $||T^{n}|| < 1$, show that, given $\eta \in \mathcal{...
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69 views

Newton's Method and Banach Fixed Point Theorem

This might be a dumb question, but I want to see how Newton's method can be understood in the context of Banach fixed point theorem (BFPT): A closed subset of a complete metric space is complete. If $...
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27 views

showing $Ax=b$ has a unique solution by finding the fixed points of a function

Let $n ∈ \mathbb N$. Consider an $(n×n)$-matrix A with real components and a column vector $b ∈ \mathbb R^n$. They give rise to an affine transformation $T : \mathbb R^n → \mathbb R^n$ with $T(x) = Ax+...
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Criteria for being below the least fixed point

Given a complete partial order $(D,\, {\leq})$ and a Scott-continuous function $F\colon D \rightarrow D$ and some fixed point $X = F(X)$. Are there any criteria that ensure that $X \leq \mathsf{lfp} ...
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68 views

Banach Fixed Point Theorem and the function $f(x)=x+e^{-x}$

let $X = [0,\infty)$ equipped with the standard metric $d(x,y) = |x-y|$. Let $f: X \rightarrow X$ be defined by $f(x)=x+e^{-x}$ Explain why this function doesn't contradict Banach's Fixed Point ...
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41 views

Fixed-Point Applications

Let $1<a<e^{1/e}$, and define $f(x)=a^x$ (b) Show that if $x_1$ is any point in the interval $(1,e)$ and $p$ is a fixed point of $f(x)$ in the interval $(1,e)$, then $$|f(x_1) - p | <|x_1 - ...
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60 views

“Comparing” fixed point Theorems.

In class, I saw Banach's (Picard) fixed point theorem: Given a complete metric space and a contractive mapping, it admits a unique fixed point. And Brouwer's: Given a continuous function in ...
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Free $\mathbb{Z}_{2}$ action on the plane

Motivated by the following question we ask: Is there a free action of $\mathbb{Z}_{2}$ by homeomorphism on $\mathbb{R}^{2}$? Lie groups with no free $\mathbb{Z}/2\mathbb{Z}$ action
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Application of Banach Fixed Point Theorem to sequences

I have the following question Show that there is unique real bounded sequence $(a_n: n \in \mathbb{N})$ such that $$a_n = \frac{n+1}{n}+\sum^\infty_{m=1}\frac{\sqrt{3a^2_{m+n}+1}}{4m^2}$$ for all $n \...
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161 views

What is the relationship between eigenvector and computing PageRank?

I read several papers about PageRank and didn't get stuck in understanding the idea of PageRank because it is simple, but I got stuck in computing PageRank, those papers talk about some mathematical ...
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Every finite group of congruences of the $n$-dimensional Euclidean space has a fixed point

Is there an "absolute" proof of the fact that if $G$ is a finite group of congruences of the $n$-dimensional Euclidean space, then $G$ has a fixed point?
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45 views

Second order DE with Banach fixed point theorem

Let's consider such a problem: $$\left\{\begin{matrix} &u''-u=f(u)+g(x) & \\ &u'(0)=u'(1)=0 & \end{matrix}\right.$$ for $x\in(0,1)$, where $u=u(x)$ isn't known and $f \in C(\...
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52 views

Fixed Point Iteration Method - Starting Point

If $g(x):=21^{1/2}x^{-1/2}$, then $21^{1/3}$ is a fixed point of $g$. Question: Using the Fixed Point Iteration Method, are there conditions on the starting point $x_0$ in order for the method to ...
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32 views

Prove that there is a unique $\phi \in C([a,b])$ such that $\phi (x) = A(x) + \int_a^xK(x,y)\phi(y)dy$

Contraction Map Question I was reviewing another question on here (linked above), and I wanted to expand on it with another question (I hope this is the correct way to do such a thing). Here is my ...
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31 views

Set of all contraction maps

Consider $(X, d)$ be a compact metric space, and let $Con(X)$ denote the set of all contraction maps on $X$. We shall define the “distance” between two maps $f, g ∈ Con(X)$ as follows, $$d_{Con(X)}(f, ...
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217 views

Contractive composite function

A question arises when we are dealing with a function that is not known to be contraction but the composites of that function to itself is a contraction (it is called eventually contractive). More ...
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181 views

Is fixed point property a topological property?

Is fixed point property a topological property? I already came up with some examples, and think the answer may be yes. But I don't know theoretical proof to get that.
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22 views

Fixed point in a special continuous mapping

Show that a continuous mapping $f:[0,1]\to [0,1]$ which satisfies $f(f(x))=x$ for each $x \in [0,1]$, and for which $f(x)$ does not equal x for at least one x in [0,1], must have exactly one fixed ...
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78 views

Contraction Map (Fredholm integral type)

I have been stuck on this problem for a little while. I think the proof might be similar to that of proving the Fredhom integral of the second type, however I am not sure. I am a little confused on ...
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How can I find the roots of x^2+px+q=0 by using contractive mapping theorem?

Basically, Contractive mapping theorem guarantees that if F is contractive on some interval, then there exists a unique fixed point. However, $x^2+px+q=0$ has two different solutions as long as $p^2&...
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77 views

Continuous mapping and fixed points

Does a continuous mapping $f\colon \mathbb R \to \mathbb R$ which satisfies $f(f(x))=x$ for each $x \in \mathbb R$ necessarily have a fixed point?
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48 views

Diffeomorphism on $S^2$ to $S^2$ with at least two fixed points

Let $f:S^2\to S^2$ diffeomorphism orientation-preserving such that for every fixed point of $f$, $1$ is not an eigenvalue of $df_p$. Then $f$ admits at least two fixed points. I'd to construct a ...
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Convergence of autonomous system / time-scale separation

Let $x \in \Delta_n$, where $\Delta_n = \{u \in \mathbf{R}^n_{\ge 0} \mid \sum u_i = 1\}$ is the probability simplex. Suppose that I have an (autonomous) dynamical system $$ \dot{x} = A(x) \cdot x $$ ...
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58 views

Convergence of fixed point iteration when $g'(p)=1$.

I am dealing with a function $f(x)=e^{-\frac{1}{x^2}}$, which has a root $p$ of infinite multiplicity at 0. I am struggling with the convergence rate of the resulting standard Newton fixed point ...