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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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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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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66 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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69 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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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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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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115 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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109 views

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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50 views

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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49 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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47 views

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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90 views

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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81 views

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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72 views

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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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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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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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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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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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 = ...
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How to find fixed point for Hamilton equations

I am learning the dynamical system. I start with Taylor-Greene-Chirikov map as follow \begin{eqnarray*} && I_{n+1} = I_n + K\sin(\theta_n), \\ && \theta_{n+1} = \theta_n + I_{n+1} ...
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Application of Brouwer fixed point theorem, why is compactness not required here??

Define a map $K:X_n \to X_n$ where $X=\text{span}(v_1, ..., v_n)$ where $v_i$ are basis functions some Hilbert space $H$. So $X_n$ is finite-dimensional. $B_r(0)$ denotes the ball of radius $r$ ...
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Trouble with fixed point iteration.

Find the fixed point(s) of $g(x) = (1/2)x^2 + (1/2)x$. Does the fixed point iteration(s) converge(s) to the the fixed point(s) if you start with a close enough approximation? Then choose $x_0 ...
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The number of periodic orbits

Let $\{a_n\}_{n=2}^\infty$ be a sequence of nonnegative integers. How to construct a continuous map $f$ on the 2-dimensional closed disk such that the number of $n$- periodic orbits is $a_n$? So far, ...
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236 views

Fixed Point Iteration, does it converge?

Find the fixed point(s) of $g(x) = x^2 + 3x - 3$. Does the fixed point iteration(s) converge(s) to the fixed points if you start with a close enough first approximation? I set $g(x) = x$ and got ...
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How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
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why do we take this interval?

I am looking at an exercise,where given that $a_{n}=\sqrt{2+\sqrt{2+...+\sqrt{2}}}$,I have to show that $\lim_{n \to \infty}a_{n}=2$. We find that $a_{0}=0,a_{1}=\sqrt{2},a_{2}=\sqrt{2+a_{1}} \text{ ...
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Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
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79 views

Successive approximation (Banach's fixed point theorem)

Let (X,d) be a complete metric space and $T\colon X\to X$. Moreover it exists a $n\in\mathbb{N}$, so that the n-th power of $T$ is $q$-contractive. Yesterday my task was to Show, that $T$ ...
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84 views

If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one ...
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137 views

Proving $\displaystyle\lim_{n\to\infty}a_n=\alpha$ $a_1=\frac\pi4, a_n = \cos(a_{n-1})$

Let $a_1=\frac\pi4, a_n = \cos(a_{n-1})$ Prove $\displaystyle\lim_{n\to\infty}a_n=\alpha$. Where $\alpha$ is the solution for $\cos x=x$. Hint: check that $(a_n)$ is a cauchy ...
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Fix-Point Theorem Proof.

Firstly, the assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow [a,b]$ be monotone increasing. Show that if $x:= \mathbf {sup}\{y \in [a,b] \| \ y ≤ f(y)\}$ ...
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Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to ...
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630 views

Fixed-Point Theorem Proof

Merry Christmas everybody. Let $a,b\in\mathbb{R}$ and $a<b$. Prove that: If $f: [a,b] \rightarrow [a,b]$ is continuous, then there is a fixed-point in $f$. So basically, if f is continous I ...
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Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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47 views

Using Banachs fixpoint-theorem to show the uniqueness of a solution of a non-linear integral equation

Show, that under the conditions (1) $-\infty<a<b<\infty, f\in C([a,b]\times\mathbb{R}), F\in C([a,b]\times [a,b]\times\mathbb{R})$ (2) $\exists~1>c\geq 0~\forall x\in [a,b]~\forall ...
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Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$. Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n ...