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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Find all the fixed points of the system $f(x,y)=-x+x^3, g(x,y)=-2y$ and use linearization to classify them.

Find all the fixed points of the system $f(x,y)=-x+x^3, g(x,y)=-2y$ and use linearization to classify them. I have found the solutions to be : $x = 0$ or $x = ±1$ and $y = 0 \implies 3$ fixed points ...
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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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On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
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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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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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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) = ...
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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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How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
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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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Help with a problem of fixed point (real analysis)

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 ...
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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),\ ...
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On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
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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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Reference request: fixed point and first-order logics

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in a fixed-point logic to have an equivalent first-order formula. ...
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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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Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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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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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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2answers
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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 ...
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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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Is the Knaster-Tarski Fixed Point Theorem constructive?

According to Tarski's Fixed Point Theorem, for a complete lattice $L$, and monotone function $f:L \rightarrow L$, the set of fixed points of $f$ forms complete lattice. Definition of $lfp(f)$ and ...
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1answer
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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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Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to ...
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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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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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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|}} ...
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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\{ ...
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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 ...
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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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Find Lyapunov function for $\dot{x} = -\sin(x)$

$$\dot{x} = -\sin(x)$$ Find the fixed points and also find out if it is attractive or repelling Find Lyapunov function for each of the attractive fixed points. I thought: Fixed points are ...
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1answer
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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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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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Proof of equality of least fixed point of two continuous function

The question is simple: given a set U, a continuous function (Scott continuity) $f \colon \mathcal{P}(U) \to \mathcal{P}(U)$ and function $g(X) := f(f(X))$, prove that $g$ is continuous and its least ...
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Stability of fixed points for a differential equation

Consider the differential equation $x'=x^2-9$ a. find the stability type of each fixed point To find the fixed points, I set this equal to $0$, right? Would someone mind explaining why I do ...
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1answer
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Unique Fixed Point for an Analytic Function

If $\Omega \subset \mathbb{C} $ is open and connected and $ f:\Omega \to \mathbb{C}$ is an analytic function, such that $\,\,K=\overline{\text{Im}\, (\,f)} \subset \Omega$ is a compact set. Prove that ...
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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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How to do fixed point iteration with matrices?

I am trying to follow solution to solve $$\min[\mathbf{z},\mathbf{q+Mz}]=0$$ by fixed point iteration. If $\mathbf{M=C+B}$ then a recursive algorithm with $k$ showing the iteration can be written as ...
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1answer
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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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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 ...