# 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.

1,413 questions
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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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### 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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### 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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### 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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### 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 ...
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### A Question on Fixed point theorem

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a map such that for $x\in X$ there exists a sequence $(a_n(x))\in [0, \infty)$ such that (A) $\lim _{n\to \infty} a_n(x)=a_{\infty}(x)<1$ ...
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### How many fixed points can a function have?

For dimension one, it is easy to think in samples of continuous functions $f:[a,b]\rightarrow [a,b]$ with one, two, three,... fixed points. Or even, infinitely fixed points (take the idendity map). ...
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### Let $u'(t)=Au^2-Bu$. Find conditions on A, B to guarantee global solution.

I'm taking a real variable course and we have just covered the Banach Contraction Principle. Our professor sometimes makes problems up on the spot for us to try and figure out together. This is one ...
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### Does there exist a triple point map?

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinite double points.also by borsuk-ulam theorem this is true for each map $N:S^n\to \mathbb{R}^n, n\in \mathbb{N}$. A ...
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### A question about fixed point theorem [closed]

how to prove this theorem how to prove first if the sequence is there that is Cauchy...thanks
### Does the sequence $(x_n)$ given by $x_{n+1} = -16+6x_n+\frac{12}{x_n}$ converge?
Question. If $x_0$ is sufficiently close to $2$, then will the sequence obtained as $$x_{n+1} = -16+6x_n+\frac{12}{x_n}$$ converge to 2 ? My attempt : I have shown that if $x_0$ is close to ...