# 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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### 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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### 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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### 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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### 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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### 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, ...