# 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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### Banach fixed-point theorem: Prove that given non-linear system has exactly one solution

Question: For $1\leq i, k \leq n$ you are given some real numbers $b_i$ and $c_{ik}$ so that: $$\sum_{i,k=1}^{n} c^2_{ik} < 1$$ Show, using Banach fixed-point theorem, that the following non-linear ...
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### Banach fixed point theorem, prove singular solution

I'm really having trouble understanding how to apply Banach's fixed-point theorem in this exercise. Let $b_i$ and $c_{ik}$ be real numbers with $1 \leq i,k \leq n$ such that the following equation ...
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### Constructive proof of approximate Brouwer's Fixed Point Theorem for $\Delta^n$ via Sperner's lemma

Brouwer's Fixed Point Theorem (BFPT) is not provable in Bishop-style constructive mathematics (BISH). For quick orientation, BISH is obtained from classical mathematics by removing the Law of Excluded ...
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### An example such that the continuous function defined on the sequences space dose not have a fixed point.

I want to find an example such that in this space: $l^2 = [(x_n):x_n \in R, \sum^\infty_nx_n^2<\infty]$ with the $L^2$ norm, a continuous function $f(x_n)$ maps the closed unit ball $B$ to itself ...
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### How to prove the 1 Lipschitz function defined on the closed unit ball has a fixed point

I need to prove that the 1 Lipschitz function has a fixed point: $\|f(x)-f(y)\|≤ \|x-y\|$ for all $x,y\in B$, where $B$ is the closed unit ball in the $R^n$. I want to apply the contraction mapping ...
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### Fixed point theorem of Markov-Kakutani

Let $V$ be a Hausdorff topological vector space and $C \subseteq V$ a non-empty, compact, convex subset. Let $\mathcal{T}$ be a collection of continuous affine maps $C \to C$ such that every two maps ...
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### Brouwers Fixed Point Theorem Proof using Winding Numbers

Hi. I've been reading Visual Complex Analysis and have been trying to prove Brouwer's Fixed Point theorem on the unit disc as set out in one of the exercises, using winding numbers/Rouche's Theorem. ...
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### How to find examples of periodic points of the (complex) exponential-function $z \to \exp(z)$?

Background: By considering the question which asks whether a certain summation-method $\mathfrak M$ for the (extremely divergent!) sum $\mathfrak M: S(z)=z + e^z + e^{e^z}+e^{e^{e^z}} + ...$ might be ...
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### Projective plane, fixed point [duplicate]

How to show, that for every continuous $f: X\rightarrow X$ there exists $x \in X$, such that $f(x) = x$, where X is a real projective plane $\mathbb{R}P^2$. In other words: every continuous map of ...
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### Existence and uniqueness of a non-linear system

I need to find the conditions under which the following system of equations has a unique solution: x=max{0,A-Bx} Where A and x are positive vectors, and B is a matrix with 0's at the diagonal and ...