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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Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
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Given $y_0\in(0,b)$ and $y_{k+1}=\frac{1}{2}y_k(3-y_k^2)$ converges. Find $b$.

Given $y_0\in(0,b)$ and $y_{k+1}=\frac{1}{2}y_k(3-y_k^2)$ converges to $1$. Find $b$. I know the sequence converges quadratically. But I have no idea how to find $b$.
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Finding roots by Fixed Point Iteration

How to know or how to find the root of the equation by Fixed Point Iteration? In FPI is there any definition/theorem of when root exists? Or is it correct that when ...
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Help me find a fixed point.

I want to find a fixed point of $1/(x+1)$. I set: $p = 1/(p+1)$ $p^2+p-1=0$ $p= (-1\pm\sqrt(5))/2$ But I plug in $f((-1 + \sqrt(5))/2)$ and $(-1 - \sqrt(5))/2$ but I don't get the same output to ...
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If $G<S_n$ is transitive, calculate $1/|G| \cdot \sum_{g \in G} f(G)$

$G<S_n$ is transitive calculate $1/|G| * \sum_{g \in G} f(g)$ where $G<S_n$ and $f(g) = |\{ 1 \le i \le n | g(i) = i \}|$ I tried to use the orbit stabiliser theorem but didn't get anywhere ...
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If $f:S^2\to S^2$ is homotopic to the identity does it have a fixed point? [duplicate]

Question: Let $f:S^2\to S^2$ be a continuous map that is homotopic to the identity. Does $f$ necessarily have a fixed point? I thought about this question after learning the proof of Brouwer Fixed ...
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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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Finding the condition number of an iterative method.

I'm trying to find the condition number on the function $A$ for the iterative method below however I'm struggling to begin. $$p_{n+1}=p_n-A(p_n)\frac{f(p_n)}{f'(p_n)}=g(p_n)$$ In particular, the ...
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Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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How does banach fixed point theorem related to matrix analysis?

As stated by Banach fixed point theorem, a contraction mapping has only one fixed point. In plain words it means that the contraction mapping T has only one solution that satisfy $Tx = x$. A ...
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Fixed point iteration, finding g(x)

I have struggle on finding this function g(x). Assume function $f(x) = 5x^3 -20x + 3$ and it is specified to find root in [0, 1]. So I guess, first thing is to find function g(x). $$g_1(x) = \sqrt[3]...
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Does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point?

As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance. To be clear, I'm using the statement of Brouwer's Fixed-Point ...
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Exam question on fixed point iteration

I am solving the following exam problem. Problem: An iterative scheme is given by $$ x_{n+1}= \frac{1}{5}\left(16-\frac{12}{x_n} \right).$$ Such a scheme with suitable initial approximation $x_0$ ...
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A question on the Banach Contraction Mapping Principle

The BCMP states that in a complete metric space $X$, a contraction mapping $T$ on $X$ has a unique fixed point, i.e. if $T$ satisfies $d(Tx, Ty) \le k d(x,y)$ such that $0 \le k < 1$, then $T$ has ...
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If $d(f(x), f(y)) \leq kd(x, y)$ for all $x, y \in X$. Then $f$ is continuous and has exactly one fixed point

Let $d$ be a complete metric for $X$. Let $f: X \to X$ be a function. Suppose there is a number $k$, with $0 < k < 1$, such that $d(f(x), f(y)) \leq kd(x, y)$ for all $x, y \in X$. Then $f$ is ...
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If two continuous maps of an interval commute, then they agree at some point

Let $f,g:[0,1] \rightarrow [0,1]$ be continuous functions such that $f\circ g =g\circ f$. Prove that there exists $x \in [0,1]$ such that $f(x)=g(x)$
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Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space ...
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Can some help me understand Zeidler's intuitive proof of Brouwer Fixed Point theorem

On pg53, Zeidler gives the Brouwer's Fixed Point Theorem The continuous operator $A: M \to M$ has a fixed point provided $M$ is a compact, convex and nonempty set in a finite dimensional normed ...
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Problem in Banach Fixed Point Theorem for a functional equation

I was recently presented this within the context of topological spaces: I am asked to show that there exists a unique continuous function $ f\colon \left[0,\frac{1}{2}\right] \rightarrow \Bbb R $ ...
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$A^2$ has a fixed point implies $A$ has also a fixed point

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. Assume that $A^2$ admits a fixed point in $X$ i.e. there exists $x_0\in X$ such that $A^2x_0=x_0$. Does this mean that $A$ has also a fixed ...
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A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of $g$...
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If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
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Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ n\...
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A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
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Reference request: equivalence between formulas in fixed point and first-order logic

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in some sort of fixed-point logic to have an equivalent first-...
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Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
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Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in \...
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Convergence of sequence as $O(1/n)$ using damped iterations

I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps. Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta ...
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Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
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Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known fixed-...
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Application of fixed point theory in Physics

Is there any application of fixed point theory in Physics?
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Find all real solutions to the following system of equations (involving fixed point iteration)

From the 1996 Canada National Olympiad. I have emphasised the real point of the question. Find all real solutions to the following system of equations. Carefully justify your answer. $\...
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Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
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Banach contraction principle: closed sets mapped to itself

I'm shoring up my understanding of basic real analysis and encountered this problem. Consider the operator $$K(x)(t) = \int_0^2 B(t,s)x(s) ds + g(t)$$ where $B$ and $g$ are continuous and $|B(t,s)| &...
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What kinds of functions have fixed points?

Among continuous functions, can we characterize those which have fixed points and those which do not? Geometrically, these are the functions that intersect the line $f(x) = x$. Is that the most ...
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Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} \exp\left(-\frac{1}{2}({\...
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1answer
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Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ \begin{array}{l@{\quad:\quad}l}...
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Contraction on a metric.

Let $M=\{x\in \mathbb{R}|x\geq 1\}$ with the absolute metric, being a metric space. Show that: a) The mapping $f:M\to M$ with $f:M\to M,~f(x)=\frac{1}{x}+\frac{x}{2}$ is a contraction ...
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Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
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Counterexamples of Brouwer fixed point theorem applied on the close unit ball

Brouwer fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least one fixed point. Brouwer fixed point theorem applies in particular on the ...
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multidemensional fixpoint iteration

We want to solve the following system of equation, which shows an intersection between a circle and an ellipse . $x^2+y^2=5$ $\frac{x^2}{16}+y^2=\frac{5}{4}$ We can expres this system as a fixpoint ...
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Leray-Schauder fixed point theorem and remark

Leray-Schauder fixed point theorem from Gilbarg and Trudinger book is quoted below. I do not understand remark below this theorem. Could you explain? Theorem 11.2 from this text is Schauder fixed ...
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Convergence of a particular fixed point iteration scheme

Setup I have the following non-linear system of equations: $$ \mathbf{x} P(\mathbf{x}) = 0 $$ where $\mathbf{x} \in \mathbb{R}_{>0}^n$ is a probability distribution, i.e., $\sum_i x_i = 1$, and $...
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1answer
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Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
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Show that a zero of $f$ is a fixed point of $g$

I want to show that a solution of the equation $x^2+cos(x)-10x=0$ is a fixed point of $g(x)=(x^2+cos(x))/10$. I tried using the quadratic equation but my solution doesn't simplify nicely in $g$. I'm ...
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1answer
79 views

Prove that the function in $[0,\pi]$ defined by $f(x)=\sin(x)/x$ and $f(0)=1$ is a contraction

Let $f$ be a function $f:[0,\pi]\to\mathbb{R}$ such that: $$f(x)=\left\{ \begin{array}{ll} \frac{\sin(x)}{x} & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x = 0 \end{array} \right.$$ I want ...
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51 views

A fixed-point theorem by Zamfirescu

I am having a trouble with understanding the proof of a fixed-point theorem by Zamfirescu. Could somebody please explain how the inequality in the inner, pink rectangle is obtained from the previous ...
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40 views

Fixed points and infinite series

Consider the formula $1 + \frac{y}2$. This has a fixed point at $y = 2$. And if we use the equation $y = 1 + \frac{y}2$ to substitute for $y$ in our formula, we get $1 + \frac{1 + \frac{y}2}2$, or $1 +...
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2answers
45 views

Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?