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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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If the initial points for secant iteration method are sufficiently close to the root, the iteration converges to the root

Well I wish to prove that in case I may choose $x_0,x_1$ close enough to the root $a$ of $f(x)$, then the secant method $x_{n+1} = x_n -\frac{x_n -x_{n-1}}{f(x_n)-f(x_{n-1})}f(x_n)$ converges to the ...
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Banach fixed point theorem application

I'm trying to use the Banach fixed point theorem to show that an intergral equation has a unique solution, but can't seem to make my answer work any help would be appreciated. Let $f:[a,b] \...
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Exactly one solution to integral equations

Show $\exists$ exactly one solution $U\in C([-1,1])$ to the intergral equation $U(x)=x\int_{0}^{x}t^{2}cos(U(t)) dt $ for $x \in [-1,1]$ Attempt I think I can use the contraction mapping theorem ...
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Showing uniqueness of a fixed point on $[0,1]$

Given $g(x)=-x\sin^2(\frac{1}{x})$ for $0<x\leq1$ My attempt: let fixed point given by $g(x)-x=-x\sin^2(\frac{1}{x})-x=0$ $$0=-x\left(\sin^2\left(\frac{1}{x}\right)+1\right) $$ Therefore only for ...
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Different definition of Veblen functions

Consider the Veblen hierarchy, where $\psi_0(x) = \omega^x$ and $\psi_1(x)$ is the x'th fixed point of $\psi_0$, $\psi_2(x)$ is the x'th fixed point of $\psi_1(x)$, and so on. We eventually get to $\...
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Showing a map has a unique fixed point

Show that the function $f:\mathbb{R}^{3} \rightarrow \mathbb{R}^3$ given by $(x,y,z) \mapsto \bigg(\frac{\sin y}{4},\frac{\sin z}{3}+1,\frac{\sin y}{4}+2 \bigg)$ has a unique fixed point. Attempt By ...
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Fixed point iteration.

I have a general question about fixed point iteration. I have used this method several times in my Numerical Analysis course and sometimes it won't converge to certain root even if the start guess is ...
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Fixed points of ordinal exponentiation for bases besides $\omega$

The first fixed point of the map $x \to \omega^x$ is the first epsilon number $\epsilon_0$, which is the supremum of $\omega, \omega^\omega, \omega^{\omega^\omega}, ... = \omega^{\omega^{\omega^{.^{.^{...
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Show root using Banach Fixpoint

I'm required to show that: $f(x) = e^x - 4x$ has a root in $(0,1)$ using the Banach Fixpoint theorem. The fact that $f((0,1)) \neq (0,1)$ confuses me. How do I proceed without knowing that $f$ isn't ...
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A Question on Fixed point theorem

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a map such that for $x\in X$ there exists a sequence $(a_n(x))\in [0, \infty)$ such that (A) $\lim _{n\to \infty} a_n(x)=a_{\infty}(x)<1$ ...
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How many fixed points can a function have?

For dimension one, it is easy to think in samples of continuous functions $f:[a,b]\rightarrow [a,b]$ with one, two, three,... fixed points. Or even, infinitely fixed points (take the idendity map). ...
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Let $u'(t)=Au^2-Bu$. Find conditions on A, B to guarantee global solution.

I'm taking a real variable course and we have just covered the Banach Contraction Principle. Our professor sometimes makes problems up on the spot for us to try and figure out together. This is one ...
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Does there exist a triple point map?

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinite double points.also by borsuk-ulam theorem this is true for each map $N:S^n\to \mathbb{R}^n, n\in \mathbb{N}$. A ...
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A question about fixed point theorem [closed]

how to prove this theorem how to prove first if the sequence is there that is Cauchy...thanks
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Does the sequence $(x_n)$ given by $x_{n+1} = -16+6x_n+\frac{12}{x_n}$ converge?

Question. If $x_0$ is sufficiently close to $2$, then will the sequence obtained as $$x_{n+1} = -16+6x_n+\frac{12}{x_n}$$ converge to 2 ? My attempt : I have shown that if $x_0$ is close to ...
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Banach fixed-point theorem : Existence of solution

We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\...
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Looking for an application of DeMarr's point-fixed theorem

I recently discovered the DeMarr theorem: In a vectorial space, if you got two non expensive maps from a convex compact to itself that commutes, they got a common fixed point. I have no example of any ...
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Prove that there is a unique continuous solution to the following integral equation.

I am trying to prove that there is a unique continuous solution to the integral equation $$F(\alpha) = \int_{0}^{\alpha}F\left(\frac{t}{1-t}\right)\frac{dt}{t}; \qquad F(\alpha)=1 \text{ for } \alpha\...
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A question on Contraction and contractive map

how to prove this map is not contraction and have no fixed point and I am proved contractive by using mean value theorem
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Iterative method with squared function (edited)

For solving a certain equation I've come up with this iterative method $$x^{n+1}=g^2(x^{n}),$$ where $g$ is given by $$g(z)=\frac{1}{2}\left[\sqrt{\left(A\cdot\text{erf}(\sqrt{z}/2)\right)^2+B\cdot\...
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Fixed point iteration convergence

Was exploring the following fixed point iteration $$ x \leftarrow 1 + \frac{1}{c*log(1-x) - 1}$$ for $x\in(0,1)$ and $c>1$ and initial guess at $$x_0 = \sqrt{1 - \frac{1}{c}}$$ It seems to ...
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Kleene's fixed point theorem for cpo's

Let $D$ be a cpo with the Scott topology, then Kleene's fixed point theorem states that every continuous function $f:D\rightarrow D$ has a fixed point: $$ \operatorname{Fix}(f) = \bigsqcup_{n\in\...
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Is this simple demand-based prices game a submodular game?

I have this simple market game: $I=\{1,2...,n\}$ players $S_i$ strategy space of each player $i\in I$ $u_i(s_i,s_{-i})=R_i(s_i)-C(s_i,s_{-i})$ There's only one type of resource. The resource is ...
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Showing that $f:M\mapsto M$ has a unique fixed point if $M$ is compact and $d(f(a),f(b))<d(a,b)~\forall a,b\in M$

I'm not sure if this is a good approach (I know there is a different proof that involves a function $a\mapsto d(a,f(a))$ whose minimum is the unique fixed point): For any $a\ne b\in M$ we have that $...
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Fixed point iteration: how to finish proof of its convergence

For this assignment we want to determine the location $r = (x, y)$ with measure distances $d_i$ from two reference points $(x_i, y_i)$. The system of equations would then be $f(r) = \sqrt{(x-x_i)^2 + ...
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Write a C program to find a root of $x^3 - 3*x + 1 = 0$ by fixed point iteration method (including convergence check)

The code works fine. But I want to include the convergence criterion which is as follows: if the equation is written in the form $x=g(x)$, then condition of convergence is: $g'(x)<1$. Note that: ...
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Solving $f(x) = \frac{x^2}2 +x - \int_0^x f(t)dt, x\in[0,1] $ with Iteration Method

I have problem solving following integral equation $$f(x) = \frac{x^2}2 +x - \int_0^x f(t)dt, x\in[0,1] $$ using iteration method with initial approach $f_0(x)= \frac{x^2}2 +x$ I applied Picard ...
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Lefschetz number of constant map

Consider a map from a connected $n$-manifold $M$ to itself defined by $f(p) = c$ for some fixed $c \in M$. This is a Lefschetz map since the subset $M \times \{c \}$ intersects the diagonal $\Delta$ ...
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Guesses on fixed point existence

Let $\mathcal{X} \subset \mathbb{R}^n$ be a finite set and the mapping $\Phi : \mathcal{X} \rightarrow \mathcal{X}$ be defined as follows: $\Phi(x) := \{y \in \mathcal{X} \mid J(y,x) \leq J(z,x), \, \...
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Obtain an aproximation to $\sqrt{5}$ using other numeric methods

From the original problem: Find an approximation to $\sqrt{5}$ correct to an exactitude of $10^{-10}$ using the bisection algorithm. In which I have a function in Mathematica to do the ...
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Banach's Contraction Mapping Principle theorem

can you please some one explain me how the highlighted statement comes im understanding all but not the last one please
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Fixed point of unusual integral equation

I am a little rusty in this area so please forgive the slowness. I am trying to prove or disprove the existence of fixed points for the following integral equation. Throughout I am interested in the ...
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Continuity of integral of indicator function

I am trying to prove the existence of an equilibrium by applying Brouwer's fixed point theorem. In order to invoke this, I of course need my function to be continuous. The only missing step is finding ...
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When are quadratic integer programs easy to solve? [duplicate]

Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to maximize $f$ on $N$. $f$ has the following form $$ f(n) = \sum_i A_i n_i -\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^...
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Rate of convergence $x_{n+1}=2\sin(x_n)$

$f(x)=\sin(x)-\frac{1}{2}x$ , for $x>0$ We are trying to evaluate the root of function using the following fixed-point iterative method: $x_{n+1}=2\sin(x_n)$ What is the rate of convergence of ...
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Prove this version of Banach differential fixed point

Let $I$ a closed interval and $f:I\to \mathbb{R}$ a differentiable function such that: $f(I)\subseteq I$ Exists $p\in\mathbb{N}$ and $c\in\mathbb{R}$ such that $g=f\circ f \cdots \circ f=f^p$ satisfy ...
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1answer
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Fixed-point method for $x=x+wf(x)$

The equation $f(x)=x^3+x-1=0$ admites a unique solution $\alpha\in[0;1]$. I want to approximate the solution $\alpha$ using fixed-point method, for that, sitting $x=g(x)=x+wf(x)$ an equivalent of $f(...
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How can $\ln(x+2)$ have a fixed point in $(-2,-1]$?

I have to determine the fixed points of $\ln(x+2)$. So as a first step I plotted $\ln(x+2)-x$ and found that it does have two fixed points. One between $[0, \infty]$, which is perfectly fine and one ...
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stability of fixed/equilibrium points in system of differential equations

Given is the following system $(a>0)$: $$\dot{x}=x(1-x)-xy$$ $$\dot{y}=y(ax-1)$$ In order to find the fixed points I have set $\dot{x}$ and $\dot{y}$ equal to zero and found $(0,0)$, $(1,0)$ and $(...
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Existence of solution to $f(x) = 1 - [\int_{0}^{x} tf(t) dt]^2$

Prove the existence and uniqueness of $\text{ } f(x) \in C^1([0,1])$ such that $\text{ }f: [0,1] \rightarrow [0,1]$ where \begin{equation} f(x) = 1 - (\int_{0}^{x} tf(t) dt)^2 \end{equation} I was ...
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Is there a fixed point theorem I could use to solve this problem?

let $E = C([0,1]),\,\,$ $K : E \to E, \,\, (Kf)(x) = \int_0^1K(x,y)f(y)dy$ also $\|K\| \leq a < 1$ I want to prove that there for $g \in E$ there exists a unique $f_g \in E$ that satisfies the ...
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Proving $\sqrt{n}(x_n)$ converges when $x_n = \sin(x_{n-1}), x_1=1$ [duplicate]

This is a problem that showed up on a qual exam that I have been stuck on for a while. Let \begin{equation} x_n = \sin(x_{n-1}), x_1 = 1 \end{equation} Prove $\lim_{n \rightarrow \infty} \sqrt{n} x_n$...
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Pardon my ignorance, but isn't TREE(3) a finite number?

Pardon my ignorance, but isn't TREE(3) a finite number? -Dylan Thurston It is my understanding as well that TREE(3) is finite (Proof that TREE(n) where n >= 3 is finite?). However, I have seen ...
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Lefschetz fixed point theorem induced map on homology of $RP^n$

Consider a map $f: RP^n \rightarrow RP^n$ for n even. I want to show that $f$ has a fixed point by using the Lefschetz fixed point theorem. So we have to show for the Lefschetz number $l(f) \neq 0$. ...
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42 views

is this set invariant under a operator?

Let $E$ be a Banach space, and $T:E\rightarrow E$ a continuous bounded mapping. Let $x_0\in E$ and $x_n=T(x_{n-1})$, $U=\overline{conv}(x_0,x_1,...,x_n,...)$. Is $U$ invariant under the operator $T$...
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Is it possible that Successive Over Relaxation (SOR) method converges while Gauss-Siedel method does not?

Is it possible that Successive Over Relaxation (SOR) method converges while Gauss-Siedel method does not? Is the following statement correct? In trying to solve a linear system of equations, the ...
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1answer
23 views

Show that $T:\,c_0\to c_0\;\;$, $x\mapsto T(x)=(1,x_1,x_2,\cdots),$ has no fixed points

As a follow-up to my previous question Show that $T:\,c_0\to c_0\;\;$, $x\mapsto T(x)=(1,x_1,x_2,\cdots),$ is non-expansive. Let $X$ be a normed linear space and $X=c_0$ (the space of sequences of ...
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1answer
39 views

How to find g(x) and aux function h(x) when doing fixed point interation?

I'm learning fixed point iteration (first and second form). My teacher said there are two forms: g(x) = x - f(x) g(x) = x - h(x)f(x) where ...
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Derivative of an infinite composition of functions

Let $f(x)=g(g(g(...g(x))))$, where the function $g$ is applied to $x$ and infinite amount of times. I am assuming that $x$ is real. What is special about points at which $df/dx=0$? Are they related to ...
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2answers
82 views

Picard's method does not solve first order differential equation?

I have the following first order differential equation $$x^\prime(t)=-(x(t))^2+2x(t),\quad t\geq 0,\quad x(0)=1$$ Now I want to obtain an approximation of $x(t)$ by using Picard's method. Then the ...