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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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) - ...
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37 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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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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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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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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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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99 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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32 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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45 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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24 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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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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49 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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60 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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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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62 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 $A$ ...
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44 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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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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49 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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56 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 ...
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73 views

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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46 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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40 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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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; ...
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36 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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42 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 ...
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58 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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70 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 ...
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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 ...
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89 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?
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51 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} ...
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35 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 ...
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35 views

Rate of Convergence of $x_{k+1} = \frac 14 (x_k^2 +sin(x_k) + 1)$

I have shown that the fix point Iteration $$x_{k+1} = \frac 14 (x_k^2 +\sin(x_k) + 1)$$ has exactly one fix point $x^*\in[0,1]$ for every $x_0 \in [0,1]$ with the Banach theorem. Now I want to find ...
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38 views

Proving a function is not a contraction

Let $\psi: C([0,1]; \mathbb{R}) \mapsto C([0,1]; \mathbb{R}) $, be such that if $x \in [0,1]$ then $$\psi(f)(x) = \int\limits_{0}^{x} f(t) dt $$ I am asked to show that $\psi$ is not a contraction ...
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57 views

Show that mapping is a contraction?

Show that the mapping $f:\Bbb R \to \Bbb R $, $f(x)=1-xe^x$ is a contraction. I tried everything i could think of but i cant get it to work. Witch is not much since i couldn't really find any ...
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Equilibrium points of $\dot x(t)=-2\cdot x^3(t)$

The following differential equation is given: $$ \dot x(t)=-2\cdot x^3(t)\qquad x(t)\in\mathbb R $$ I am asked to find the equilibrium points of the system. By definition, the equilibrium points are: ...
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53 views

Fixed point Iteration: Convergence & Divergence from geometrical figure

I want to understand the geometrical interpretation of Convergence & Divergence of Fixed point iteration method. The figure is here IMAGE: CLICK HERE In Figure 2.4(a) Left panel: if we draw a ...
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26 views

Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
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How can I prove $x_{n+1} = e^{-x_n}$ is convergent?

I'm doing a practice problem which asks to prove that the sequence defined by $x_{n+1} = e^{-x_n}$ is convergent (or rather "study the convergence of $(x_n)$"). So I'd like to try and find some ...
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60 views

Banach fixed point theorem - Find $x$ such that $f(x) = 0$.

Let $f: \mathbb R \to \mathbb R$ be a continuous function with a continuous derivative. In short $f \in C^1$. We know that $0<c\leq f'(x) \leq d < \infty$. We want to prove that $\exists! x_0 ...
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Algebraic fixed point theorem

I was wondering if there are some "algebraic" fixed point theorems, in group theory. More precisely, given a group $G$ and a group morphism $f : G \to G$, what conditions on $G$ and $f$ should we ...
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Picard's existence theorem, successive approximations and the global solution

Picard's existence theorem states that if $U$ is an open subset of $\mathbb{R}^2$ and $f$ is a continuous function on $U$ that is Lipschitz continuous with respect to the second variable then there is ...
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39 views

Proving a function to be a contraction

I'm studying contractions and I am trying to understand under which conditions a function is actually a contraction. I have understood that a continuous function $f: [a,b]\rightarrow\mathbb{R}$ is a ...
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52 views

Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
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The conjugate of a complex function

This may be a stupid question. I am trying to explain/prove the following: Let $f$ be a complex rational function of degree $d ≥ 2$ with fix point $z_0 \neq 0$. It is always possible to conjugate ...
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unique fixed point problem

Let $f: \mathbb{R}_{\ge0} \to \mathbb{R} $ where $f$ is continuous and derivable in $\mathbb{R}_{\ge0}$ such that $f(0)=1$ and $|f'(x)| \le \frac{1}{2}$. Prove that there exist only one $ x_{0}$ such ...
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65 views

Does any continuous function from the open unit interval $(0,1)$ to itself has a fixed point?

I know that the result is true for closed interval $[0,1]$ by using intermediate value property. But in the case where we consider open interval $(0,1)$ does the solution change?
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22 views

Stating if a function being a contraction

I have to state if a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a contraction on the interval $I\subset \mathbb{R}$ and say if it admits fixed points. The function is $f(x)=\left\{ ...
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47 views

$r\rightarrow1/r$ invariant

(Not sure the tags are appropriate, but can't think of better ones. Please suggest better.) Suppose you have a function $f(x,y,z,...;g(r))$ with the requirement that $r\rightarrow1/r$ leaves $f$ ...
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22 views

Problem in Topology related to continuous functions

Let $\mathbb{R-Q}$ be the subspace of $\mathbb{R}$ with the usual metric. Is there a function, $f:\mathbb{R-Q} \rightarrow \mathbb{R-Q}$ such that f is continuous and $f$ does not have a fixed ...
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1answer
46 views

Lefschetz number of a transformation of the sphere.

In differential topology the Lefschetz number of an automorphism of a compact manifold is the oriented intersection number of the graph of that automorphism with the diagonal. I would like a proof or ...