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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1answer
225 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
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2answers
463 views

Confused about a version of Schauder's fixed point theorem

I have read this: We have a map $S:W_0 \to W_0$. Moreover $W_0$ is not empty, convex, and weakly compact in $W$. Thus we can apply Schauder's fixed point theorem: Schauder's fixed point ...
3
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1answer
124 views

Fixed point in a metric space with distance at most 1

The question is: Suppose that $X$ is a complete metric space such that the distance function is at most 1, and $f:X\rightarrow X$ is such that $d(f(x),f(y))\le d(x,y)−1/2(d(f(x),f(y)))^2$. Prove that ...
3
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2answers
972 views

How to find fixed point

I have recurrence equations of \begin{align*} x_{n+1} &:= 0.5\cdot (x_{n}^2+2\cdot y_{n}-2\cdot x_{n}\cdot y_{n}-2\cdot y_{n}^2) \\ y_{n+1} &:= -0.8\cdot (x_{n}^2+2\cdot y_{n}-2\cdot x_{n}\...
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1answer
313 views

Getting a root of a continuously differentiable function by Banach's Fixed Point Theorem.

Banach Fixed Point Theorem: Consider a metric space $X = (X, d)$, where $X\neq \varnothing$. Suppose that $X$ is complete and let $T: X \to X$ be a contraction on $X$. Then $T$ has precisely one fixed ...
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1answer
72 views

Brouwer transformation plane theorem

Can somebody show that BPTT, version 2 is deduced from BPTT, version 1 [BPTT, version 1] Let $h$ be a fixed point free orientation preserving homeomorphism of $\mathbb{R}^2$ Every point $m\in\mathbb{...
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1answer
1k views

Convergence of fixed-point iteration for convex function

Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying $$ f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty. $$ I am confident to be able to argue that $...
3
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1answer
284 views

Fixed point of a non linear contraction in a convex set

Hi I'm stuck on the following problem of Haim's functional analysis book. Let $C\subseteq H$ ($H$ a Hilbert space) be a non-empty closed convex subset and let $T:C\rightarrow C$ be a non linear ...
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2answers
540 views

Some homework questions about a Lipschitz function (cauchy sequence)

Do you want to help me with my homework? The exercise is as follows: Consider a Lipschitz function, $h:\mathbb{R}\rightarrow\mathbb{R}$, satisfying for every $x, y$: $$\left| h(x)-h(y) \right| \...
3
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1answer
182 views

Fixed point: general case

This is the second part of the question Fixed point: linear operators. Here I would like to ask you about the general case. A lot of concepts can be described or even defined as a solution of a ...
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31 views

Existence of fixed points for this Markov operator.

Perhaps math overflow is a better place to put this but I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point. In particular ...
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61 views

Point-set topological proof of Brouwer's fixed point theorem

I have tried to understand the point-set topological proof of Brouwer's fixed point theorem presented in Cou11. But I couldn't clarify some parts. Here are the theorem and its proof. Theorem: There ...
3
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1answer
40 views

Prove this version of Banach differential fixed point

Let $I$ a closed interval and $f:I\to \mathbb{R}$ a differentiable function such that: $f(I)\subseteq I$ Exists $p\in\mathbb{N}$ and $c\in\mathbb{R}$ such that $g=f\circ f \cdots \circ f=f^p$ satisfy ...
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System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }...
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83 views

Fixed point method for non-homogeneous ode and pde

During my course in mathematical methods for physics, my professor introduced us to the "method of fixed point" for studying non-homogeneous systems of odes, odes and pdes. Although I think I ...
3
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1answer
122 views

Fixed point theorems involving isometries

A fixed point theorem is a theorem which establishes some conditions that guarantee a certain function has a fixed point. I would like to know if there exists a fixed point theorem involving metric ...
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104 views

Existence and Uniqueness of Equilibrium Points in Non-Linear Dynamical Systems

Let $\dot{X} = F(X)$ be a non-linear dynamical system. I'm interested in knowing if there are any existence theorems for an equilibrium point, that is, an $X^*$ that satisfies: $$ F(X^*) = 0 $$ I ...
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335 views

Equivalence between Brouwer fixed-point theorem and Borsuk-Ulam theorem. Is there a simple proof of equivalence between them?

I wonder if Brouwer's fixed-point theorem and Borsuk-Ulam's theorem are equivalent. Brouwer's fixed-point theorem (simple form). Let $B_{\mathbb{R}^{n}}[0,1]=\{x\in \mathbb{R}^n: \|x-0\|\leq 1\}$ ...
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183 views

Fredholm alternative for elliptic PDE: Is $L_\mu^{-1}$ contractive?

In Evans' PDE, §6.2, in Theorem 4 he proves a Fredholm alternative based dichotomy for second order uniformly elliptic pde. The general idea: For $\mu\geq\gamma\geq0$ with $\gamma\geq0$ from Thm. 2, ...
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130 views

differentiable contractive mappings

Some time ago, I managed to show that $sin(x)$ and $cos(x)$ were both contractive almost everywhere and then I noted that $cos$ was the derivative of $sin$ so there might be a deeper connection. Let'...
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80 views

What is this numerical method for solving Ax=b called?

Following is the pseudo-code of a simple iterative method of solving $Ax=b$ where $A$ is an $n\times n$ matrix. ...
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196 views

Fixed-point iteration and continuity of parameters

Let $X$ a compact set and $A\subseteq \mathbb{R}$. Consider a continuous function $f\colon X\times A\to X$ and construct a fixed-point iteration as follows $$ x_{k+1}=f(x_k,a),\quad x_0\in X, a \in A.\...
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1answer
556 views

Using Fixed point iterations for solving system of linear equations

Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^...
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80 views

Extension of Kakutani's fixed point theorem.

Can the Kakutani's fixed point theorem's be extended to say that there exists a fixed point inside the set (not on boundary)(I am not sure how to formally state this). For a $n$-dimensional compact, ...
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118 views

Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
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75 views

Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
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313 views

fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle xt_{ij}\left(1-\frac{g}{r}\...
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0answers
451 views

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 ...
3
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1answer
466 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper semi-...
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5answers
1k views

Can't there be more than one fixed points in a contraction? or none?

I was going through the contraction mapping theorem in my book where it says, that if $\phi: G\to G$ is a contraction, then $\phi$ has a unique fixed point $\alpha$ on $G$. Sequence {$x_n$}, $x_{n+1} ...
2
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3answers
10k views

Fixed-Point Theorem Proof

Merry Christmas everybody. Let $a,b\in\mathbb{R}$ and $a<b$. Prove that: If $f: [a,b] \rightarrow [a,b]$ is continuous, then there is a fixed-point in $f$. So basically, if f is continous I ...
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3answers
2k views

Assume $f :[0,1] \rightarrow [0,1]$ is a continuous function. Show there exists a fixed point $x$, i.e. $f(x) = x$ [closed]

Assume $f :[0,1] \rightarrow [0,1]$ is a continuous function. Show there is $x \in [0,1]$ with $f(x) = x.$ Is there an example of a continuous function $g:(0,1) \rightarrow (0,1)$ where this is ...
2
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4answers
1k views

Fixed point iteration convergence of $\sin(x)$ in Java [duplicate]

Is it mathematically correct to say that $\sin(x)$ converges to zero as $x$ approaches $0$? If the $\sin(x)$ iteration is done starting at $\dfrac{\pi}{2}$ in Java, for $10^9$ iterations, the result ...
2
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4answers
7k views

Fixed Point Iteration, does it converge?

Find the fixed point(s) of $g(x) = x^2 + 3x - 3$. Does the fixed point iteration(s) converge(s) to the fixed points if you start with a close enough first approximation? I set $g(x) = x$ and got $g(x)...
2
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4answers
235 views

Banach Contraction mapping of $\Phi(f)(x)=\int_0^x \frac{1}{1+f(t)^2}dt$ , find a fixed point.

Let $(X,d)$ be metric space with $d(f,g)=\sup |f(x)-g(x)|$ where $X$ is the set of continuous function on $[0,1/2]$. Show $\Phi:X\rightarrow X$ $$\Phi(f)(x)=\int_0^x \frac{1}{1+f(t)^2}dt$$ has a ...
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2answers
2k views

Contradict the Contraction Mapping Theorem

I am trying to show that the function $f(x) = 2\pi+x-\tan^{-1}x$ is contractive but has no fixed points. Finally I wish to conclude that it does not contradict the contraction mapping theorem. $f$ is ...
2
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3answers
477 views

Where should I begin the study of fixed point theory, especially of multi-valued maps?

How should one begin one's study of fixed point theory, especially of multi-valued maps? What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have? ...
2
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4answers
197 views

Proving $\displaystyle\lim_{n\to\infty}a_n=\alpha$ $a_1=\frac\pi4, a_n = \cos(a_{n-1})$

Let $a_1=\frac\pi4, a_n = \cos(a_{n-1})$ Prove $\displaystyle\lim_{n\to\infty}a_n=\alpha$. Where $\alpha$ is the solution for $\cos x=x$. Hint: check that $(a_n)$ is a cauchy sequence ...
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3answers
409 views

Limit of a sequence of fixed points also a fixed point?

Suppose I have a continuous function $f : [0,1]^n \rightarrow [0,1]^n$ (maybe $n$ is infinite). Suppose I have a sequence $\{a_n\}_{n=1}^\infty$ of points in $[0,1]^n$ where each $a_n$ is a fixed ...
2
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3answers
715 views

Continuous function with no fixed point

I am searching an example of a continuous function $f: [0,1) \to [0,1)$ but f has no fixed point, that is, there is no point $x_0 \in [0,1)$ such that $f(x_0)\not= x_0 \forall x_0$.
2
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2answers
106 views

a fixed point theorem

A hiker starts to climb up from base B to the summit S on sat 6am one day,spends the night at S and starts to climb down at 6am the next day.Prove that there is a point on the path B-S (there is only ...
2
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3answers
741 views

Fixed point theorem

Is $|g'(x)|<1\ \forall x\in(a,b)$ is one of the hypothesis of the Fixed-Point Theorem? The answer is NO. Can someone please enlightened me about this? My teacher reason is this... Note that ...
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3answers
127 views

Proving $\sqrt{n}(x_n)$ converges when $x_n = \sin(x_{n-1}), x_1=1$ [duplicate]

This is a problem that showed up on a qual exam that I have been stuck on for a while. Let \begin{equation} x_n = \sin(x_{n-1}), x_1 = 1 \end{equation} Prove $\lim_{n \rightarrow \infty} \sqrt{n} x_n$...
2
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3answers
330 views

Prove that $f:[0,1] \to [0,1]$ has a fixed point

Let $f:[0,1] \to [0,1]$ such that $f$ has lateral limits at any point, is continuous at $0$ and $1$ and $$\lim_{t\nearrow x}f(t) \leq f(x) \leq \lim_{t \searrow x}f(t), \forall x \in (0,1)$$ Prove ...
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4answers
999 views

Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution

Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution. My solution: I was starting with a function $F(x) = \cos x - 1/2$ and the interval $[0,\pi/4]$ and trying to show that the ...
2
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1answer
510 views

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 ...
2
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3answers
887 views

Validity of this proof that any continuous function with domain and range in [0,1] must have a fixed point.

The following proof was given in a solutions manual to a question asking to prove that a continuous function with domain and range in $[0,1]$ must have a fixed point: Consider the function $F(x) = f(...
2
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1answer
763 views

If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $...
2
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1answer
326 views

Fixed point theorem on graphs?

I have a graph $G=(V,E)$ where to each vertex $v$ I have associated a value, $\hat{v}$ (ie I have a "network" in the terminology here http://snap.stanford.edu/snap/index.html ). Let $\phi : \hat{V} \...