# Questions tagged [fixed-point-theorems]

A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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### Constructive proof of the approximate Brouwer's Fixed Point Theorem for $\Delta^n$

The approximate Brouwer Fixed Point Theorem (aBFPT) for the standard $n$-simplex is: Let $f$ be a uniformly continuous function from $\Delta^n$ into itself. Then for each $\varepsilon>0$ there ...
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### Convergence of a nonlinear iterative sequence

I have the following iterative sequence: \begin{eqnarray*} a_{t+1} &=& (1+\alpha-\beta)^2a_{t} - 2\alpha(1+\alpha-\beta)b_{t} +\alpha^2a_{t-1}+\frac{L}{a_{t}}, \\ b_{t+1} &=& (1+\alpha-...
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### Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?

Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...
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### Convergence of estimator given by a fixed point

Let $X$ be a non-negative random variable with cdf $F$ and define $$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function. Let $s_0$ be the unique fixed point of $G$. Now let $X_1,\dots,X_t$ ...
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### On some characteristics of continuous maps $S^n \to \mathbb{R}^n$

I've asked this question about two month ago in math exchange but there were no answer to it. Any information or paper relating to this question is appreciated. By the Borsuk-Ulam theorem we know ...
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### Browder's Fixed Point Theorem in uniformly convex Banach spaces with non-identical image

This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my ...
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### Is there a fixed-point index theorem that treats the fixed points on the boundary?

Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points ...
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Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function $T:[ 0,1 ]\... 2answers 144 views ### On fixed point probability in discrete logarithm? Given integer$n>2$what is the probability that for a given$h\in\Bbb Z_n$there is no$x\in[0,\varphi(n)-1]\cap\Bbb Z$such that$h^{x\bmod\varphi(n)}\equiv x\bmod n$? 1answer 519 views ### Fixed point of a group action Let$\mathbb{R}^\infty$be the product of countably many real lines. Assume that a finitely generated group$\Gamma$acts on$\mathbb{R}^\infty$(linearly and continuously) and there is a nonempty ... 2answers 408 views ### Continuity of mapping sending a function to its (brouwer) fixed point Let$f:[0,1]^n \rightarrow [0,1]^n$be a continuous mapping. Brouwer's fixed point theorem says that$f$has a fixed point, i.e., some$x$such that$f(x) = x$. Suppose we have a continuous family, i.... 0answers 82 views ### Formal way to prove existence and continuity in an integral equation In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE$$\frac{\mathrm{d}^2 y}{\... 1answer 462 views ### Equivariant Riemann-Hurwitz The Riemann-Hurwitz formula starts with a genus$g$algebraic curve$Y$and a ramified cover$\pi\colon X\to Y$of degree$N$, with ramification indices$e_P$and computes invariants of$X$, such as ... 1answer 280 views ### Is it possible to find a$\mathbb{C}$-algebra endomorphism of$\mathbb{C}[x,y]$that fixes a given$w \in \mathbb{C}[x,y]-\mathbb{C}$? Let$w \in \mathbb{C}[x,y]-\mathbb{C}$and let$u \in \mathbb{C}[x,y]-\mathbb{C}[w]$. Is it possible to find a$\mathbb{C}$-algebra endomorphism$f$of$\mathbb{C}[x,y]$such that$f(w)=w$and$...
Let $P$ be a poset with a least element $\bot$ ($\forall x \in P.\ \bot \le x$). Let $M$ be a set of monotone maps $P \to P$. Call $x \in P$ reachable if $x = f_1(f_2(...f_n(\bot)...))$ for some ...