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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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Proving a particular function is surjective on a Banach space.

Let $(E,\|x\|)$ and let $f: E \to E$ such that $f+Id$ is a contraction ($Id$ is the identity map). Prove that $f$ is surjective and prove that, if $f$ is linear, then $f$ is a homeomorphism. The ...
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For what $x$ does $g(x)$ converge to a fixed point?

Show that the iteration: $x_{k+1}=2x_k-αx_{k}^2$ where $α > 0$ converges quadratically to $\frac{1}{α}$ for any $x_0$ such that $0 < x_0 < \frac{2}{α}$. I have been able to prove that it ...
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Finding a Möbius Transformation given constraints

I am trying to solve this problem, but am running into very complicated solving, and think that there is a simpler approach that I am missing. Find a Möbius transformation $M(z)$ that satisfies ...
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Is this function necessarily a contraction?

If $(X, d)$ is a compact metric space satisfying $d(f(x), f(y)) < d(x, y)$ for all $x, y \in X$ such that $x \ne y$, is $f$ necessarily a contraction? I know an analogue of the Banach Fixed Point ...
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Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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How to prove that this iterative formula converges to the Gram points for the constant $c=0$?

We intend to find values of the $n$-the point $t$ such that: $\Re\left(\zeta \left(i t+\frac{1}{2}\right)\right) \neq 0$ and $\Im\left(\zeta \left(i t+\frac{1}{2}\right)\right) = 0$ which should ...
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Fixed point property for the total space of the canonical line bundle over $\mathbb{C}P^{2n}$

It is well known that the even dimensional complex projective pace $\mathbb{C}P^{2n}$ has the fixed point property. What about the total space of the canonical line bundle over $\mathbb{C}P^{2n}$? ...
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Show that the only intervals having the fixed point property are the closed intervals.

By Fixed Point Theorem, I know that it deals with closed interval, for eg, [0,1]. And this theorem will be false if [0,1] is replaced by (0,1). The counter example will be $f:(0,1)\rightarrow (0,1)$ ...
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If a map on a complete metric space has a contraction property, it has a unique fixed point

I am stuck on the following problem: Prove that if $(X, d)$ is a complete metric space and $f : X\rightarrow X$ is a function with the property that there is a number $A < 1$ such that $d(f(x),f(y)...
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Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & \frac{dx}{dt}=...
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Proof: Contraction Mapping and Cauchy Sequence

Question: Let $(X, d)$ be a metric space, let $f : X → X$ be a contraction, and let $a_o \in X$. Let $a_1 = f(a_o)$ and $a_{n+1} = f(a_n)$ for $n \geq 1$. Prove that $(a_n)$ is a Cauchy ...
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Studying Muller's example of IVP with unique solution whose Picard iterates do not converge.

I have read about the following example from Muller: $(M) \begin{cases} x' = f(t,x) \\[1mm] x(0) = 0 \end{cases}$ where $f: \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ is the function: $f(t,...
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Fixed point in plane transformation.

Some one give me a idea to solve this one. It's a problem from Vladimir Zorich mathematical analysis I. Problem 9.c from 1.3.5: A point $p \in X$ is a fixed point of a mapping $f:X \to X$ if $f(p)=p$...
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Sufficient and necessary condition for the global uniqueness of fix-points

https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...
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Brouwer Fixed Point Theorem implies Invariance of Domain Theorem

Is there a simple and elementary proof that Brouwer's Fixed-Point Theorem implies Invariance of Domain Theorem? By 'simple and elementary' I mean proof that: Do not use tools such as degree theory or ...
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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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Show that $f(x) :=\cos(\cos(x))$ is a contraction mapping. Find the fixed point.

Show that $f(x) := \cos(\cos(x))$ is a contraction mapping. Find the fixed point. I know that $|\cos(x)|\leq 1$ which means that $|\cos(\cos(x))| \in [\cos 0,\cos 1]$ I need to find $\alpha <1$ s....
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Contraction Map (Fredholm integral type)

I have been stuck on this problem for a little while. I think the proof might be similar to that of proving the Fredhom integral of the second type, however I am not sure. I am a little confused on ...
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539 views

An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, y(0)=...
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Fixed point location for functions

How are fixed points calculated? Are intersections of $ y = f(x) ,y = f^{-1} (x) $ graphs give real fixed points for all $f$ ?
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IVP- Has at most one solution

Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem $$ \left\{ \begin{array}{l} x'(t)=f(t,x) \\ x(t_0)=x_0 \end{array} \right. ...
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Fixed point of matrix

Suppose that $a$ is a fixed point of matrix $A$, what that means? What is a fixed point of matrix? Thank you!
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Prob. 7 (a), Sec. 28, in Munkres' TOPOLOGY, 2nd ed: A contraction of a compact metric space has a unique fixed point

Here is Prob. 7, Sec. 28, in the book Topology by James R. Munkres, 2nd edition: Let $(X, d)$ be a metric space. If $f$ satisfies the condition $$ d\big( f(x), f(y) \big) < d(x, y) $$ for ...
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176 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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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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How to predict which fixed point a mapping converges to for initial conditions?

Say there is a population of mass 1 in which individuals can choose one of three traits (1,2,3). The population shares of traits 1, 2, and 3 at time $t+1$ is mapped from the shares at time $t$ via: $$...
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Newton method iteration

I am trying to solve non-linear systems and since I can't download matlab on this device, I was wondering if there is a way I can set it up in excel. I know the formula for x$^{k+1}$=x$^k$-[Df(x)]$^-...
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Apply the picard iteration to the first-order equation

Apply the picard iteration to the first-order equation $x'=2t-2 \sqrt{max(0,x)} \,\, , x(0)=0 $. Does it converge? In this case should we separate in cases"$x>0$ and $x<0$ ", and resolve the ...
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Fixed Point Iterations of F(X) = (S + KX) / (K + X), S > 0

Assume S and K are rational numbers. So F is a function from the rational numbers to the rational numbers. Assume K > 0 (corresponding results when K < 0) By using google spreadsheets, I came up ...
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Fix points of integral transfroms

After I have worked a little bit with different integral transform, especially with the Laplace transform, I was confronted with the fact that there are some functions which remain of the same type ...
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Prove $f: \mathbb{R}\rightarrow\mathbb{R}, x_{n+1}=f(x_n) $ with $|f'(x)|< \theta < 1$ converges for $n \rightarrow \infty$ and it has a fixed point.

Show that $f: \mathbb{R} \rightarrow \mathbb{R}$ $x_{n+1} :=f(x_n)$ with arbitrary $x_0 \in \mathbb{R}$ with $|f'(x)|< \theta < 1$ converges such that $\lim_{n\to\infty} x_n=x^*$ and $f(x^*...
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Let $x_1 = 2$, $x_{n+1}=2+\frac{1}{x_{n}}$, $n \in \mathbb{N}$. Does the sequence converge or diverge?

Let $x_1 = 2$, $x_{n+1}=2+\dfrac{1}{x_{n}}$, $n \in \mathbb{N}$. Does the sequence converge or diverge? How can I solve this question using Monotone Convergence Theorem or any other better method? I ...
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A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...