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, ...

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Kakutani's FP theorem for $F:X\to 2^Y$

The Kakutani Fixed Point Theorem is postulated for mappings of the form $F:X\to 2^X$, where $2^X$ stands for the powerset of $X$, $X$ is nonempty and $F(x)\neq \varnothing$ for all $x\in X$, $F$ is ...
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On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
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Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...
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Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
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45 views

Constructive fixed-point theorems where finite iteration yields the fixed point

I would like to show that $p$, a fixed point of some effective map $f : S\rightarrow S$, can be constructed effectively. Ideally, I would like there to exist a finite $n$ such that $p = f^n(p_0)$, ...
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How to show this is a contraction

Let's say I want to find a fun way to write a number $l$. I can procede by doing so: (as I saw here Can we get just $3$ from $\pi$?) Let $f(x) = \sqrt{2lx - l^2}$. The only fixed point is $f(x) = x ...
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Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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The constant of fixed-point iteration

I saw some sentences and proofs related to fixed-point iteration in my numerical method textbook: $$e_{k+1} = g'(\theta) e_{k}$$ if $|g'(x^\ast)|<1$, there is a constant such that $|g'(\theta)|\le ...
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Fixed Point of a Function

I am trying to answer the following question: Prove the function $f(x)=1-x^2$ has a fixed point on $[0,1]$. Find the value of this fixed point explicitly. I know how to prove that it has a fixed ...
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Aitkens Extrapolation

The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$ Verify it? And $X^n$ can be viewed being defined recursively by ...
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55 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
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Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
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65 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
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Heuristics for sequence convergence

Having a finite sequence of double precision floating point numbers (obtained using the fixed point iteration of a function), is there any algorithm which can be used to determine that this sequence ...
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fixed point iteration algebra problem

I am looking at an example which finds the root of: $$ f(x) = \cos(3x) \tag 1$$ using the fixed point iteration method. It uses $$ g(x) = \frac{2x+\cos(3x)}{2} \tag 2$$ However, it was my ...
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Strictly increasing on R

Choose correct options , more than one may be correct Let f be the function defined by $$h(x)=e^x (x-1)+x^2$$ we've : $h$ is positive on $(0,\infty)$ $h$ is negative on $(0,1)$ $h$ is ...
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Fixed Point (Differential Equation)

I want to study about the Fixed point before my class next week. The problem is that I could not find a good site online. If you guys know some sites which talk about this thing in the fundamental ...
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41 views

Application of Schauder fixed point theorem

Let $\Omega$ an open bounded in $\mathbb{R}^n$, and let $A(x,u)$ an Caratheodory function, bounded and coercive, i.e., $$|A(x,u)|\leq \beta$$ $$A(x,u) \geq \alpha I,\quad \alpha > 0$$ and let ...
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Proving Banach's fixed point theorem

The hint tells me how to proceed but I am stuck. I define the sequence ${z_n}$ as is stated in the hint, First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$ I add $z_{n+k-1}$ and ...
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Show that $x_{k+1}:=g^{-1}(g(x_k)-f(x_k))$ converges to a root of $f$

If $f:[-1,1]\to\mathbb R$ continously differentiable and $g:[-1,1]\to[-2,2]$ continuously differentiable and bijective such that, $|f'(x)-g'(x)|\le 1/2 \inf\limits_{y\in[-1,1]}g'(y)$ ...
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Calculate the Derivative of a univariable integral at a point $4$

Considering the function below: the objective is to calculate $F'(4)$ (the derivative of $F(x)$ in the point $4$)? we know that: and that: so if I try to replace $x$ by $4$ in $F'(x)$ I get ...
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65 views

Root of a function? (Proof with Banach theorem)

Given is a function $f$: $\left [ -1,1 \right ]\rightarrow \mathbb{R}$, which is continuous and differentiable. The function $g$: $\left [ -1,1 \right ]\rightarrow \left [ -2,2 \right ]$ is a ...
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On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
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68 views

Bug in Brouwer Fixed Point Theorem using Sperner?

so I am just trying to illustrate to an informal audiance how to prove the Brower Fixed Point Theorem using Sperner's lemma. I seem to have trouble with the iterative application of Sperner's lemma. ...
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generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
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81 views

Fixed-point iteration, Convergence of a sequence?

Given is the function $f(x)=x^{3}+x-1$ on $\mathbb{R}$. Use the Fixed-point iteration for $x\in \left [ 0.5 , 1 \right ]$ to show that the sequence $\left \{ x \right \}_{n}$ converges to the ...
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59 views

Limit of a monotone function

Let $f\colon [a,b] \to[a,b]$ be a non-decreasing function in a sense that $f(x)\leq f(y)$ whenever $x\leq y$. Although there may be several fixpoints of $f$, at least one does always exist and there ...
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Has this theorem (on existence of inverse) an analogous for unbounded operators?

Let $S,T:X\to X$ be bounded linear operators, where $X$ is a Banach space. It's a consequence of the Banach Fixed Point Theorem that if $T$ is invertible and $\|T-S\|\|T^{-1}\|<1$ than $S$ is ...
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102 views

Showing that $f$ has exactly one fixed point

Let $\gamma$ be the circle $\{z \in \mathbb{C}: \lvert z\rvert=1 \}$. Suppose $f$ is a function analytic on an open set containing $\gamma$ and its interior and that $\lvert\, f(z)\rvert<1$ for ...
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What is the method for solving this class of problem?

$\frac{dx}{dt} = y$ $\frac{dy}{dt} = -\partial y - \mu x - x^2$ Find the fixed points and discuss stability. I'd at least like to know what I should be googling, any help is appreciated.
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Confused about a version of Schauder's fixed point theorem

I have read this: We have a map $S:W_0 \to W_0$. Moreover $W_0$ is not empty, convex, and weakly compact in $W$. Thus we can apply Schauder's fixed point theorem: Schauder's fixed point ...
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The most important theorems in fixed point theory

What are the most important theorems in fixed point theory and why are they so important? I know some: Banach's contraction principle, Brouwers fixed point theorem, caristi fixed point theore... I ...
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Weighted Average Fixed Point Theorem

I was wondering if someone can help with the following question. I am pretty sure I have to apply the Intermediate Value Theorem for the solution, just I am not quite sure exactly how to set the ...
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42 views

Show an iterative fixed point method does not converge

I was asked the following question: let $g(x)=\frac{30}{1+x}$. Notice that $g(5)=5$. Is there an $\epsilon >0$ such that the series $\{x_k\}_{k=0}^{\infty}$ defined by $x_{k+1}=g(x_k)$ and ...
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How to use the COUNTING THEOREM to determine patterns? [duplicate]

This question tests your understanding of the Counting Theorem. A flower has 6 identical petals, equally spaced. Each petal is to be coloured either red or yellow. Use the Counting Theorem to ...
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68 views

Perron-Frobenius Theorem. A particular case?

Let $\{a_{i,j}\} =A \in \mathbb{R}^{N \times N}$ be a non-negative matrix, such that: $a_{i,i} = 0 ~~ \forall i \in \{1, \ldots, N\}$ $a_{i,j} \geq 0 ~~ \forall i \neq j$ Given the previous ...
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Fixed point in a metric space with distance at most 1

The question is: Suppose that $X$ is a complete metric space such that the distance function is at most 1, and $f:X\rightarrow X$ is such that $d(f(x),f(y))\le d(x,y)−1/2(d(f(x),f(y)))^2$. Prove that ...
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Is every convergent limit of an iteration a fixed point as well?

Let $f(x)$ be a function and suppose $\lim_{n \to \infty}f^n(a)=L$ for some $a$ in the domain of $f$. What are the sufficient conditions for $L$ being a fixed point of $f$? Is the continuity of $f$ ...
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Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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Cone metric spaces and fixed point theory

These days cone metric spaces, as a generalization of metric spaces, started to be very interesting... A special interest for this space is its application to the fixed point theory. On the last ...
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Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
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Is there a method for finding the fixed point of logarithmic functions?

I am faced with this function (warning, I am not good at math) $x(t+1)=0.5 \ln x(t)+1$ initial condition = 1 . I know the fixed point is 1 because $0.5 \ln (1)+1=1$ but I wanted to know the ...
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Fixed points of multivariable calculus

I have discrete case. $z=1-x-y$; $x=a_1(x^{2}+2yz)+a_2(z^{2}+2xy)+a_3(y^{2}+2xz)$; $y=b_1(x^{2}+2yz)+b_2(z^{2}+2xy)+b_3(y^{2}+2xz)$; $z=c_1(x^{2}+2yz)+c_2(z^{2}+2xy)+c_3(y^{2}+2xz)$; where ...
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139 views

How to find fixed point

I have recurrence equations of \begin{align*} x_{n+1} &:= 0.5\cdot (x_{n}^2+2\cdot y_{n}-2\cdot x_{n}\cdot y_{n}-2\cdot y_{n}^2) \\ y_{n+1} &:= -0.8\cdot (x_{n}^2+2\cdot y_{n}-2\cdot ...
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Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some ...
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27 views

Problems finding fixed point.

$x_n=f(x_{n-1})={ax_{n-1} +e^{-x_{n-1}}\over 1+a}$ for $a>0$ Setting $f(x)=x$. The problem is while trying to find the fixed point of $x={ax +e^{-x}\over 1+a}$ I only get $x=e^{-x}$. What's the ...
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36 views

Applying Schauder fixed point theorem to a map (explanation needed)

Let $F:L^2(\Omega) \to L^2(\Omega)$ be continuous map. Let $D$ be a function space. Since $F(L^2(\Omega)) \subset D$, and $D \subset L^2(\Omega)$ is a compact embedding, $F$ is a compact operator ...
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42 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...
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50 views

Is there a name for this theorem about the convergence of a function?

Let $f(x)$ be a continuous function over $\mathbb{R}$ such that for all $a < b$, we have $a < f(a) < b$. Then, for any $x < b$, the sequence $\{t_n\}$ defined by $t_0 = x, t_n = ...