# Tagged Questions

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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### Inverse function theorem - good proof

I am looking for a reference which give a full demonstration of the inverse function theorem (let say in Banach spaces) where we can have estimates of the bounds of the neighbourhoods that we build to ...
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### Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
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### Given $y_0\in(0,b)$ and $y_{k+1}=\frac{1}{2}y_k(3-y_k^2)$ converges. Find $b$.

Given $y_0\in(0,b)$ and $y_{k+1}=\frac{1}{2}y_k(3-y_k^2)$ converges to $1$. Find $b$. I know the sequence converges quadratically. But I have no idea how to find $b$.
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### Finding roots by Fixed Point Iteration

How to know or how to find the root of the equation by Fixed Point Iteration? In FPI is there any definition/theorem of when root exists? Or is it correct that when ...
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### Help me find a fixed point.

I want to find a fixed point of $1/(x+1)$. I set: $p = 1/(p+1)$ $p^2+p-1=0$ $p= (-1\pm\sqrt(5))/2$ But I plug in $f((-1 + \sqrt(5))/2)$ and $(-1 - \sqrt(5))/2$ but I don't get the same output to ...
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### If $G<S_n$ is transitive, calculate $1/|G| \cdot \sum_{g \in G} f(G)$

$G<S_n$ is transitive calculate $1/|G| * \sum_{g \in G} f(g)$ where $G<S_n$ and $f(g) = |\{ 1 \le i \le n | g(i) = i \}|$ I tried to use the orbit stabiliser theorem but didn't get anywhere ...
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### If $f:S^2\to S^2$ is homotopic to the identity does it have a fixed point? [duplicate]

Question: Let $f:S^2\to S^2$ be a continuous map that is homotopic to the identity. Does $f$ necessarily have a fixed point? I thought about this question after learning the proof of Brouwer Fixed ...
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### Fixed point property for the total space of the canonical line bundle over $\mathbb{C}P^{2n}$

It is well known that the even dimensional complex projective pace $\mathbb{C}P^{2n}$ has the fixed point property. What about the total space of the canonical line bundle over $\mathbb{C}P^{2n}$? ...
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### Finding the condition number of an iterative method.

I'm trying to find the condition number on the function $A$ for the iterative method below however I'm struggling to begin. $$p_{n+1}=p_n-A(p_n)\frac{f(p_n)}{f'(p_n)}=g(p_n)$$ In particular, the ...
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### Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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### How does banach fixed point theorem related to matrix analysis?

As stated by Banach fixed point theorem, a contraction mapping has only one fixed point. In plain words it means that the contraction mapping T has only one solution that satisfy $Tx = x$. A ...
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### A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
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### Reference request: equivalence between formulas in fixed point and first-order logic

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in some sort of fixed-point logic to have an equivalent first-...
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### Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
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### Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
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### Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known fixed-...
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### Application of fixed point theory in Physics

Is there any application of fixed point theory in Physics?
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### What kinds of functions have fixed points?

Among continuous functions, can we characterize those which have fixed points and those which do not? Geometrically, these are the functions that intersect the line $f(x) = x$. Is that the most ...
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### Finding a metric such that $\Phi$ becomes a contraction

Let  f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ \begin{array}{l@{\quad:\quad}l}...
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### Contraction on a metric.

Let $M=\{x\in \mathbb{R}|x\geq 1\}$ with the absolute metric, being a metric space. Show that: a) The mapping $f:M\to M$ with $f:M\to M,~f(x)=\frac{1}{x}+\frac{x}{2}$ is a contraction ...
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### Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
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### Counterexamples of Brouwer fixed point theorem applied on the close unit ball

Brouwer fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least one fixed point. Brouwer fixed point theorem applies in particular on the ...
We want to solve the following system of equation, which shows an intersection between a circle and an ellipse . $x^2+y^2=5$ $\frac{x^2}{16}+y^2=\frac{5}{4}$ We can expres this system as a fixpoint ...