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)...

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Does this iterative sequence converge?

Let $f:[0,1] \rightarrow [0,1]$be a continuous function. Choose any point $x_0 \in [0,1]$ and define a sequence recursively by $x_{n+1}=f(x_n)$. Suppose $\lim_{n \rightarrow \infty}x_{n+1}-x_n =0$, ...
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1answer
28 views

How to prove convergence of a sequence maximizing a sum of exponential distances?

I want to find the argument $x$ that maximizes $f(x)=\sum_i e^{-(x-d_i)^2/c}$ for some data values $d_i$ and an arbitrary positive constant $c$. I assume that $f(x)$ has only a single maximum (most ...
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0answers
49 views

$f:A\rightarrow B$ and $a_n=f(a_{n-1})$ $L=[a_1,a_2,…a_n,…]$ For which $f$ and $a_0$ is $\bar L=B$?

This is a generalization of a question I posted a few days ago regarding a particular recursive sequence . I am trying to find general conditions (if they do exist) on which the problem has a ...
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25 views

Convergence criterion of vector fixed point iteration

As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$. For the vector variable version, it has been proved in the case when $...
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36 views

Proof that fixed points of a null field are zero

In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $ Suppose we have a ...
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2answers
25 views

Show that there is an equilibrium point

Show that if $x_{k+1}=f(x_{k})$, where $f$ is continuous, has two stable equilibrium points, then there's a third equilibrium point between then. I'm trying to approach this problem using the ...
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1answer
26 views

connection between Newton’s method and fixed point iteration

This is from my lecture slide I can understand Newton’s method, but I don't understand the context in red which requires rewriting th equation $x=g(x)$ as the Newton’s method require the right ...
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32 views

Number of fixed points of a meromorphic function

I would like to know whether a meromorphic function on the whole complex plane with at most one pole can have infinitely many fixed points or not. Many thanks in advance.
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37 views

Is every Boyd-Wong mapping also a contraction?

I know every contraction is a BW map, so I suspect the BW class of mappings is strictly larger. Can someone help me construct an example of a mapping that is not a contraction but is a BW map? Edit: ...
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22 views

Prove that if $X=[0,1]$ and $T:X \to X$ is defined as below then $d(Tx, Ty) \le \alpha (d(x, Tx) + d(y,Ty))$.

$Tx = x/4$ if $x \in [0, 1/2)$ and $Tx =x/5$ if $x \in [1/2, 1]$ and $\alpha$ is a constant such that $0 \le \alpha < 1/2.$ What I have tried so far: if $x,y \in [0, 1/2)$ then $d(Tx, Ty) = 1/4 ...
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44 views

An inverse problem for a parabolic equation and fixed-point theorem

I'm reading the paper "The determination of a parabolic equation from initial and final data" http://www.ams.org/journals/proc/1987-099-04/S0002-9939-1987-0877031-4/S0002-9939-1987-0877031-4.pdf. A ...
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1answer
32 views

Whether there exists some compact manifold whose continuous map must have a fixed point?

Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point. I mean that any continuous map from $M$ to $M$ must have a fixed point. I ...
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1answer
61 views

Quotient Spaces Defined By Bijection

I was working with a question in topology and came to the following statement that I can't seem to figure out: Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism with no fixed points. ...
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38 views

On the in-comparability of the Gauss-Seidel and Jacobi iteration schemes.

Given the system $x_1 + x_2 = 2$ $-x_1 + x_2=0$ $x_1 + 2x_2 - 3x_3=0$ the Jacobi iteration converges and Gauss-Seidel iteration diverges. Is there a way we can derive these two facts using the ...
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1answer
24 views

Every nonempty, compact convex set $M$ in a locally convex space has fixed point property

I need to prove that "Every nonempty, compact convex set $M$ in a locally convex space has fixed point property". In the book the reference has been given to "Eisenack & Frenske, 1944, page 44". ...
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38 views

Finding an interval of convergence for a given $g(x)$

I am trying to do a fixed point iteration on the function: $f(x) = x^2 -3x+2 $, analyzing different forms of $g(x)$. I solved for the actual roots and they equate to $x=1$ and $x=2$. I am currently ...
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82 views

Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
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30 views

Establish the sufficient condition $|g'(x)| < 1$ for convergence of an iteration using the Banach fixed point theorem?

If $x_n = g(x_{n-1})$ is an iteration, it converges if $g$ is continuously differentiable and $|g'(x)| < 1$. The Banach FPT says that if $T$ is a contraction on a complete metric space $X$ then it ...
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Generalized Fixed Point Theorem

Suppose that $T: M \to M$ is a self map of a nonempty closed set $M$ in a complete metric Space $(X,d)$. Suppose further that $$d(Tx,Ty) \le k(a,b)d(x,y)$$ for all $x,y \in M$ with $0 \lt a \le d(x,y) ...
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1answer
11 views

Need help finding the smallest contraction constant.

I have to show $ T : X \to X$ given by $x \mapsto x/2 + 1/x$ is a contraction map, where $X = \{x \in R : x \ge 1 \}$ and find the smallest contraction constant. I have worked out that $|T(x) - T(y)|...
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1answer
40 views

Does any analytic function from the unit disk to a compact subset of itself have a fixed point?

I was reading a proof by Beardon of the Wolff-Denjoy Theorem in Complex Dynamics. In the proof, a family of maps $$f_\epsilon=(1-\epsilon)f(z)$$ is used, where $f(z)$ is an analytic map from the unit ...
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30 views

Range of parameter values for a stability of a fixed point for this 2d map

So I am trying to do a linear stability analysis for a very simple 2d discrete system: \begin{equation} \begin{aligned} x_{n+1} &= y_{n}\\ y_{n+1} &= -\frac{x_{n}}{2} + ay_{n} + y_{n}^{3} \...
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1answer
37 views

Degree of infinite dimensinal antipodal map

Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ . $L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$? In fact ,I just know that the algebra define ...
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3answers
125 views

Prove there is no contraction mapping from compact metric space onto itself

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a ...
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2answers
63 views

Large-time limit of the general solution of an ODE is a fixed-point. Is the fixed-point stable?

This question might well have an obvious affirmative answer (or an obvious counterexample!), which at present I cannot see. Suppose I have a first-order ODE $$ u'(t)=f(u) $$ whose general solution ...
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4answers
100 views

Fix point of $L:S^2\rightarrow S^2$

Let $L:S^2\rightarrow S^2$ be a bijective continue map. Is there exists $L$ such that $\forall x\in S^2 \Rightarrow Lx\neq x$ and $Lx\ne -x$? I mean that whether $L$ must have fix point? In fact ...
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1answer
38 views

How to find the number of iteration in Fixed point iteration method?

I want to know how to find the number of iterations in fixed point method. The book that i have, gives me 2 ways to find the number of iterations. The first one: $|p_n - p| \leq k^n max ${ $p_0 - ...
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3answers
48 views

Using fixed point iteration find the positive root of $f(x)=e^{-x}-x^2$

Consider $f(x)=e^{-x}-x^2$. I'm suppose to find the positive root using fixed point iteration. after drawing the graph, it's safe to set the interval from [0.25,1]. (I actually want to set it from [...
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1answer
28 views

Does a fixed point depend smoothly on the parameters?

Let $(X,d)$ be a complete metric space. A well-known theorem states that, for any map $G: X \to X$ satifying $d(G(x), G(y)) < Ld(x,y)$ for some fixed constant $L < 1$ and arbitrary $x, y \in X$, ...
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0answers
26 views

Computing the set of fixed points of a map

Compute the set of fixed points of the following map: $f(X) := (y,-x)$ when $X=(x,y)$ So for this, do I just have to solve the system of equation such as: $x=y$ $-x=y$? Plugging the first into ...
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1answer
50 views

Is this $x \mapsto k^{k^{\frac{-1}{x}}}$ a contraction?

Given that $k,x \in \mathbb{R}^+$ and $k > 1$, the function $f$ defined by $$f(x) = k^{k^{\frac{-1}{x}}}$$, generates a sequence $x_0,f(x_0), f(f(x_0)), \cdots$ is observed to converge to a fixed ...
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1answer
61 views

Fixed point of a dynamical system

What does a fixed point mean in a autonomous dynamical system, I mean I know the definition of it, but I keep hearing that if a dynamical system starts at a fixed point then it will remain there, why ...
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1answer
33 views

Fixed point of limit equal to limit of fixed points?

Suppose $f\left(x,\alpha\right)$ is a parameterized function. $f:\,D\times\left[0,1\right]\rightarrow D$ where $D$ is a convex subset of $\mathbb{R}^{n}$. Suppose $x^{*}$ is a fixed point ...
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1answer
63 views

Fixed points of contractions in metric spaces

How do I prove that all contractions on a complete, non-empty metric space has exactly one fixed point? What I know: I know that all contractions are continuous and that completeness of $A$ means ...
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3answers
63 views

Banach fixed point theorem

Given $(X,d)$ complete with $A \subset X$ closed, and $f: A \to A$ satisfying $$ d(f^{n}(x),f^{n}(y)) \leqslant a_{n}d(x,y) \hspace{3mm} \forall x,y \in A \hspace{3mm} n \in \mathbb{N}$$ where $...
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1answer
36 views

while $d(x_n, x_{n+1})$ is converges to $0$, proving there exists an $\varepsilon >0$ such that $d(x_{{m_k}-1},x_{n_k}) < \varepsilon $

I study fixed point theory from Kirk and Khamsi's An Introduction to Metric Spaces and Fixed Point Theory and I couldn't understand a proof. STEP 1: Let $\{d(x_n,x_{n+1})\}$ be a monotone ...
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1answer
51 views

Nonlinear contraction on Hilbert space

Let $C\subset H$ be a nonempty closed convex subset of a Hilbert space $H$ and let $T:C\rightarrow C$ be a nonlinear contraction; i.e. $$|Tu-Tv|\leq|u-v|\quad\forall u,v\in C.$$ Let $(u_n)$ be a ...
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1answer
58 views

Show that $|T(x) - T(y)| \lt |x-y|$ when $x \neq y$ but the mapping has no fixed points.

Consider $X = \{ x \in \mathbb{R} \mid 1 \le x \lt \infty \}$, taken with the usual metric of the real line, and $T \colon X \to X$ defined by $x \mapsto x +x^{-1}$. Show that $|T(x) - T(y)| \lt |x-y|$...
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When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $...
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2answers
47 views

Setting Lineard Equation ODE into standard form, proving existence of unique solutions

I'm working on proving existence and uniqueness of a local solution to ODE's of the Lienard variety $$y'' + f(y)\,y' + g(y) = 0$$ I'm trying to put this into standard $y' = f(t,y)$ system So I see ...
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1answer
42 views

How to write new algorithm of root finding by combining 2 or 3 standard algorithms(bisection, fixed, etc)

I just learned about Bisection Method, Fixed-Point Iteration Method, Newton- Raphson Method, and Secant Method. My prof wants us to be able to write new Algorithm of root finding by coming 2 or 3 ...
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0answers
15 views

Solution Space for a Fixed Point Problem

Hi I need to find the criteria for which the following has a solution: $$X= K_1 (a_1-b_1X)^{c_1} (a_2-b_2 X)^{c_2} (a_3-b_3X)^{c_3} X^{c_4}$$ where $K_1>0; a_1>0; c_1>1; (b_2 b_3)\leq0; c_2&...
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0answers
37 views

Existence of a solution of a limit of Fixed point equations

I am considering a setting where I am given an iid sample of symmetric positive matrices $\{S_i\}$, $i=1,\dots, n$, of a matrix valued random variable $S$ with distribution $F$. The support of $F$ is ...
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1answer
47 views

Common fixed point of commuting monotonic functions

Let $P$ be a chain-complete poset with a least element, and let $f_1,f_2,\ldots,f_n$ be order-preserving maps $P\to P$ such that $\forall i,j: f_i \circ f_j = f_j\circ f_i$. Claim. The functions $f_1,...
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1answer
62 views

Poset where every monotonic function has a least fixed point

Let $P$ be a poset such that every order-preserving map $f:P\to P$ has a least fixed point. Must $P$ be chain-complete?
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1answer
72 views

Chain-complete and least element iff every order-preserving map has least fixed point

Let $P$ be a poset. I want to show the following are equivalent. $P$ is chain-complete and it has a least element. For every order-preserving map $f:P\to P$, the set $P_f$ of fixed points of $f$ has ...
3
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1answer
52 views

Least fixed point of restricted function

Let $P$ be a poset with the property that every order-preserving map $f:P\to P$ has a least fixed point $\mu(f)$. Now for any $p\in P$, the poset $\downarrow(p)=\{x\in P|x\leq p\}$ must also have ...
4
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1answer
91 views

Fixed point property of spaces having same homotopy type

Suppose X and Y have same homotopy type.X as a topological space has fixed point property.Can we conclude anything about fixed point property of Y?
2
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1answer
53 views

Fixed point without the constant

If $d(Fx,Fy)<d(x,y)$ for all $x,y$ in a closed bounded subset $X$ of Euclidean space and $F\colon X\rightarrow X$ then there is a unique fixed point $x_0$ and $\lim \limits _{n\to\infty} F^n(x)=...
0
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2answers
37 views

Derivative test for contraction mapping on open sets

I have a question on the derivative test for showing that a function is a contraction. The proposition that I know is in this version: Let $I$ be a closed bounded set of $\mathbb{R}^l$ and $f : I →...