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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Similar statement for Banach fixed point theorem.

Let$(X, d)$ be complete space $,$ a $T: X \rightarrow X$ mapping such as, $\dot{\text { ze }} T(X)=X .$ Prove that if exist $T$ with special conition (exist $r>1 $ such that $ d(T(x), T(y)) \geq ...
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Fixed Point theorem with Lipschitz continuous mapping.

How can we prove that the below function does not have fixed point? Define $S:=\{(x_m)\in l^1\mid \sum^\infty |x_i|\leq 1)\}$, and consider the self map $\Phi$ on $S$ defined by $\Phi((x_m)):=(1-\sum^...
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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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Using Banach fixed point theorem to prove that $\vec{x} = A\vec{x} + \vec{b}$ has a unique solution

The problem is to use Banach's fixed point theorem to prove that $\vec{x} = A\vec{x} + \vec{b}$ has a unique solution, where $b = [3, -1, 2]^T$ and $$ A = \begin{bmatrix} 1/4 & -1/4 & 2/15 \\ ...
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Functional equation question involving fixed points

Let $g(x)$ be a quadratic function such that the equation $g(g(x)) = x$ has at least three different real roots. Then there is no function $f : R → R$ such that $f ( f (x)) = g(x)$ for all $x ∈ R$. ...
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Criterion for a contractive mapping to map its domain into itself

While going through the proof of Banach's fixed point theorem, this question came to my mind. I have no clue if it is true or not. I was unable to prove or disprove myself and didn't get any reference ...
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Are there any path-connected sets (of $\Bbb R^2$) that guarantee two or more fixed points for any continuous bijections mapping them onto themselves?

We know by Brouwer‘s fixed-point theorem that any continuous bijection mapping the closed unit circle to itself must have a fixed point. My question: are there any path-connected sets (preferably ...
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Infinitely iterating the cosine function yields the Dottie number

For simplicity’s sake, Let’s define our function to be cos(x). For any value of x, iterating this function will yield some constant, take a calculator and try it. But quite surprisingly, I recognized ...
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Iterating a continuous or differentiable function

I would like to know whether the following statement or a weaker / similar version of it is true. Intuitively, it seems to me that it should be true but I'm unable to prove it. Does anyone know a ...
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How to draw a Graph of the proof: Succession generated from an iterative function, converges to the root of the function.

in the following question: Prove that succession generated from an iterative function, converges to the root of the function. the Theorem : If $g(x) : [a,b] → \mathbb R$ continuous and such that $a ...
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How can you show that a function has a fixed point when its domain is a half-space or a convex cone?

Suppose that $f: \mathbb{D} \subseteq \mathbb{R}^n \to \mathbb{R}^n$, where $\mathbb{D}$ is a half-space or a convex cone. We wish to show the existence of a fixed point $x = f(x)$. Suppose we know ...
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Show that explicit formula for fixed-point does not depend on the initial point

I am looking at an example application of the Banach fixed-point Theorem. The problem is to find solutions to $Id(n) - A=y$ where $A \in M(n)$ and $Id(n)$ is the $n \times n$ identity matrix. The ...
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Am I doing well this fixed point problem?

I would like to know if I'm doing well the following exercise. I think that I'm doing something wrong (specially in part (3)) but I don't know what... Having $F(x)=x^2+\frac{3}{16}$ I have to answer ...
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Prove that succession generated from an iterative function, converges to the root of the function.

The problem of finding the root of the function $f(x)$, i.e. the value of $x^{\ast}$ such that $f(x^{\ast}) = 0$, can be reformulated as $x = g(x)$, and therefore the root $x^{\ast}$ will be the value ...
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Proving that an equation has only a fixed point

I only need a hint to solve this problem: Show that the equation $x^4+8x^3+32x-32=0$ has only a fixed point at the interval $[0,1]$. I know that as $\mathbb{R}$ is complete with the usual metric and $[...
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How can I prove that $F_1(x)=\frac{x^2+2}{3}$ can be used to solve $f(x)=x^2-3x+2=0$ with the fixed point method?

Let we have the following equation: $f(x)=x^2-3x+2=0$ How can I prove that $F_1(x)=\frac{x^2+2}{3}$ can be used to solve that equation with the fixed point method? I have calculate the roots of f(x) ...
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Composite mapping with $p$ factors is a contraction

Let $(X,d)$ be a complete metric space and let $f:X \to X$ be a mapping such that, for some $p \geq 2$, the composite mapping $f \circ f \circ \cdots \circ f$ with $p$ factors is a contraction. Note ...
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Proving integral transform using banach fixed-point theorem

I'm currently working on the following problem: Let $K: [0,1]^2 \to \mathbb{R}$ be continuous with $|K(x,y)| < 1$ for all $(x,y) \in [0,1]^2$. Prove the existence of a function $f \in C([0,1])$ ...
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Jacobian with zero eigenvalues - Fixed points

I'd like to ask, if possible, for some good and not excessively long material about classification of fixed points in which the Jacobian has zero eigenvalues. Many thanks in advance!
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Fixed Point for functions of several variables

Can someone solve, or help me solve this problem in Numerical Analysis in chapter Fixed Point for functions of several variables. Chapter 10, exercise set 10.1 problem 4. The problem is explained in ...
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Common fixed point of continuous affine linear functions on a convex compact metric space

Im stuck with an exercise, which drives me crazy. Let X be a Banachspace and $D \subset X$ nonempty, convex and compact. a) Let $\Lambda$ be an index set and for $\lambda \in \Lambda$ let $F_\...
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What's $-2^{-2^{-2^{-2…}}}$?

Pretty simple question that I know someone has probably asked before: What's $-2^{-2^{-2^{-2...}}}$? Or, more specifically, what is the number that this repeated sequence approaches? I have ...
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Conditions for a minimization mapping to be a contraction

Let $\Omega$ be a set of elements $\omega\in\Omega$ and let $A(\omega)$ be a random variable. Consider the function $$ K(\omega,A(\omega),P) $$ where $P>0$. Assume that $K$ is strictly positive, ...
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Nonexpansive Mapping Proof $(|G(x)-G(y)|<|x-y|)$

Function This function is supposed to show that strict nonexpansivity is not sufficient to guarantee the existence of a fixed point. But how exactly would one work out, i.e. show the inequality as ...
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Banach fixed point theorem application for f(x)=3πx-cos(pi*x/2)

I feel extremly stupid right now. So we did Banach fixed point theorem and i thought i got it. Basically if an intervall maps onto itself and is monotonous and contracts then you have a unique fixed ...
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convergence of fixed point-iteration for positive definite symmetric matrix

Let $A\in\mathbb{R}^{n\times n}$ be a positive definite symmetric matrix and consider a decomposition $A=B-C$ such that C is positive definite and symmetric as well. Show that the iteration method $...
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Rate convergence to fixed point of a function

Suppose I have two functions $f(x) = 1 - (1 + x)^{-m}$ and $g(x) = 1 - (1 + x)^{-n}$. And I use fixed point iteration to obtain a fixed point ($f(x) = x$ and $g(x) = x$). How can I compare the rate ...
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Limit in Proving Brouwer's Fixed Point using Sperner's Lemma

I'm trying to understand the proof of Brouwer's Fixed Point Theorem using Sperner's Lemma. For example, pg.10-11 in A Combinatorial Approach to The Brouwer Fixed Point Theorem. I was able to follow ...
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How to get more arrangements out of an equation?

I am trying to converge to the root of this equation $$2 x^3 + 4 x^2 – 2 x – 5 = 0$$ through 3 different arrangements so far tried 4 and only one worked which is this one $$(x/2 + 5/4 - 0.5 x^3)^{1/...
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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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Exercise 5.22 in Brezis, “Functional Analysis Sobolev Spaces and Partial Differential Equations”.

Let $H$ be a Hilbert space, $C\subseteq H$ a nonempty closed convex set and $T:C\to C$ a nonlinear contraction, that is $$ (*)\qquad|Tu - Tv| \leq |u-v|. $$ Let $(u_n)$ be a sequence in $C$ such that $...
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Proof that there exists a unique $x^* \in X$ such that $T(x^* ) = x^*$ .

Suppose $(X, \rho)$ is a complete metric space, and suppose the function $T : (X, ρ) \rightarrow (X, ρ)$ is such that $T_n = T ◦ T ◦ · · · ◦ T$ (n times) is a contraction map for some $n \ge 2$. Prove ...
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Prove that the function $f :\Bbb R \to \Bbb R$ defined by $f(x) = e^{-\cos(x)^2}$, for all $x \in\Bbb R$, has a unique fixed point on $\Bbb R$.

Hint: some arguments might be simpler if you recall the trigonometric formula $2\sin(x)\cos(x) = \sin(2x)$. Remember also that $\cos$ and $\sin$ are $2\pi$-periodic functions. I am a bit lost with ...
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Brouwer's fixed point theorem in an infinite-dimensional space

I am wondering if the Brouwer's fixed point theorem can also be applied in an infinite-dimensional space. For example let $E = [0, 1] \times [0, 1] \times [0, 1] \times \dots$ be an infinite ...
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Orbital Continuity of Suzuki contractions in spaces without the triangular inequality

I am working on Suzuki contractions in non-triangular metric spaces and I want to know if there are certain conditions under which such maps become orbitally continuous. For reference, the formal ...
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Showing that all fixed points of a homotopy are isolated - possible using fixed point index?

I have a function $F(X,y)$, with $F: [0,1]^n \times [0,\infty)\rightarrow[0,1]^n$ i.e. essentially a self-map on a compact subset of $R^n$ with one parameter $y$. $F$ ist real-analytic. I know that ...
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Determining a Cauchy's problem

$$X=\Bigg\{ \varphi \in C^0\left( \left[ 0,\dfrac{\pi}{2}\right]\right): \varphi (t)\geq 0\,\, \forall t\in \left[ 0,\dfrac{\pi}{2}\right] \Bigg\} $$ For each $\varphi \in X$, $T:(X.||\cdot||_\infty)\...
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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 ...
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Brouwer's fixed point theorem using category theory and “a computation involving covering spaces, that $\pi(S^1) = \mathbb Z$”

My latest folly has been to open a book about Category Theory (CT in Context). My scribbles in the following picture reveal my consternation in the highlighted blue line. "a computation involving ...
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Is a continuous function of an upper hemicontinuous correspondence continuous?

Suppose $X$ is a compact subset of $\mathbb{R}^n$, and $Y$ is a compact subset of $\mathbb{R}$. $f:X \rightarrow Y$ is a nonempty compact-valued upper hemicontinuous correspondence. $g:X \rightarrow \...
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Invoking Banach fixed-point theorem in space with fixed points at boundary

The setup of my problem is as follows. I have a mapping $f:X \rightarrow X$ where $X$ is a general space that may be changed through adjusting a set of parameters that define it. $f$ is always a ...
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Lloyd's Algorithm and convergence conditions

I need help understanding the following theorem about Lloyd's algorithm (Du's Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations, Theorem 2.6). Lloyd's algorithm is used ...
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What is the geometrical meaning of the space?

Consider the Banach space of all lipschitz functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\...

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