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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
40 views

Using Banach's Fixed Point Theorem on an Integral Equation

I have been studying the solutions $f(x)$, $x \in [0,1]$ to integral equations of the form \begin{equation} f(x) = \int_{0}^{1}K(x,y)\frac{f(y)}{\sqrt{h^{2}+(f(y))^{2}}}dy \end{equation} where the ...
user avatar
2 votes
1 answer
24 views

Prove that in a given coloring of a square, there exists a sub-square with a certain coloring.

I was given the following question: There is a square which is divided to sub-squares by edges which are parallel to the edges of the big square (vertices are connected by the edges). The vertices of ...
user avatar
  • 81
0 votes
0 answers
25 views

Bivariate fixed points

Let $f(x):\mathbb{R}\rightarrow\mathbb{R}$, be a strictly decreasing, convex and continuous function in $x$ with $f(x)>0, \forall x>0$, with $\underset{x\rightarrow\infty}{\textrm{Lim}} f(x)=0$. ...
user avatar
  • 53
6 votes
3 answers
162 views

Showing that $\mathbf{X}^{2} + \mathbf{X} = \mathbf{A}$ has a solution

Show that there exists some $\epsilon >0$ s.t. for all $\mathbf{A}\in \mathbb{R}^{2\times 2} $ with $|( \mathbf{A})_{i, j}| < \epsilon $ for all $i, j$ (let the space of all such matrices be $E$)...
user avatar
  • 471
1 vote
0 answers
46 views

Does there exist a continuous function which satisfies Kannan contraction but not Banach contraction?

If possible, give an example of a continuous function defined on a convex subset of a Banach space $X$ satisfies Kannan contraction but does not satisfy Banach contraction. Definitions. Let $C$ be ...
user avatar
  • 21
0 votes
0 answers
32 views

How can I prove that the next sequence converges? [closed]

Let $f:[a,b]\to[a,b]$ such that is continuously differentiable, $f^{-1}$ exists and $$\underbrace{\min\text{ }x}_{x\in[a,b]} |f'(x)|>1.$$ How can I prove that the sequence $x_{k+1}=f^{-1}(x_k)$ ...
user avatar
0 votes
1 answer
96 views

Fixed-Point iteration method fails on converging on equation.

I'm looking to solve the following equation in $x$. $$\frac{Wa}{b} = \left((\frac{a}{b}+x)\Phi(\frac{a}{b}+x)+\phi(\frac{a}{b}+x)\right)-\left(x\Phi(x)+\phi(x)\right)$$ , where $W, a$ and $b$ are ...
user avatar
1 vote
1 answer
128 views

Is there a constructive proof of Brouwer's fixed-point theorem that does not rely on triangulation?

I'm aware of the constructive proof of Brouwer's fixed-point theorem via Sperner's Lemma, and I love it for its simplicity, directness, and constructiveness. However, I still have a lingering ...
user avatar
  • 1,055
0 votes
0 answers
22 views

Regularity of Random Fixed Point

Suppose we have a measurable random dynamical system $(X,\mathfrak{B})$ covering a metric dynamical system $(\Omega, \mathfrak{F}, \mathbb{P}, \sigma)$ where $\mathfrak{B},\mathfrak{F}$ denote the $\...
user avatar
  • 655
3 votes
2 answers
203 views

Existence of a fixed point to a vector-valued mapping

I have the following system: $$\begin{cases} F_1(x_1,x_2) \colon= (c_1 - a_{11}x_1 - a_{21}x_2)^{a_{11}}(c_2 - a_{12}x_1 - a_{22}x_2)^{a_{12}} = x_1 \\ F_2(x_1,x_2) \colon= (c_1 - a_{11}x_1 - a_{21}...
user avatar
  • 11.7k
2 votes
1 answer
75 views

If $f\in \mathcal{C}(\mathbb{S}^n,\mathbb{S}^n)$ and $\mathrm{deg}_2(f)=0$, then $f(x_0)=x_0$ and $f(y_0)=-y_0$ for some $x_0, y_0$.

If $f\in \mathcal{C}(\mathbb{S}^n,\mathbb{S}^n)$ and $\mathrm{deg}_2(f)=0$, where $\mathbb{S}^n=\left \{x\in\mathbb{R}^{n+1}:\|x\|=1\right \}$. Prove that there exist two points $x_0, y_0\in \mathbb{S}...
user avatar
  • 8,589
0 votes
1 answer
31 views

Leray-Schauder Degree for Periodic Functions

Let $X$ be a Banach space, $Z \subset X$ a closed, linear subspace and $$ S : U \cap Z \to X $$ where $U \subseteq X$ is open. Question: Can I define the Leray-Schauder degree of $S$? If I had $S : U \...
user avatar
4 votes
1 answer
60 views

What are some examples of fixed point problems where transfinite constructions elucidated the problem?

Maths is filled with various different kind of "fixed point theorems", sometimes even when they are not phrased as such, where we have an operation $f$ on objects of a certain kind and wish ...
user avatar
  • 2,609
1 vote
3 answers
77 views

Cardinality of set $\{f\in C^1 (\mathbb R)\mid f(0)=0,f(2)=2, |f’(x)|\leq 3/2\}$

The Cardinality of set $\{f\in C^1(\mathbb R)\mid f(0)=0,f(2)=2, |f’(x)|\leq 3/2\}$ is $1.$ empty set . $2.$ non empty finite set. $3.$ infinite set. $4.$ uncountable set . Function like $f(x)=x$ is ...
user avatar
  • 5,370
5 votes
1 answer
151 views

Conditions for $I-x A$ to be a convergent matrix for some $x\in \mathbb{R}$

I'm looking for interpretable necessary/sufficient conditions on $A$ which guarantee that $(I-x A)$ is a convergent matrix for some $x\in \mathbb{R}$ For instance, $A$ normal with non-zero eigenvalues ...
user avatar
3 votes
1 answer
56 views

If $f\in \mathcal{C}(\mathbb{S}^n,\mathbb{S}^n)$ and $f$ is not surjective, then $f$ has a fixed point

If $f\in \mathcal{C}(\mathbb{S}^n,\mathbb{S}^n)$ and $f(\mathbb{S}^n)\neq \mathbb{S}^n$, where $\mathbb{S}^n=\left \{x\in\mathbb{R}^{n+1}:\|x\|=1\right \}$. Prove that $f$ has a fixed point. This is ...
user avatar
  • 8,589
1 vote
0 answers
44 views

A proof of the sub-super solution method using Schauder's fixed point theorem

Firstly, I would like to define the notion of a sub solution and a supersolution and enunciate the Schauder's fixed point theorem. Consider the non-linear elliptic problem $$(P) \begin{cases} \begin{...
user avatar
  • 3,477
0 votes
0 answers
17 views

Is the closure of a connected component of an isotropy type submanifold a connected component of a fixed point submanifold

I am trying to prove a plausible statement from a book. Let for simplicity $G$ be a finite group acting on a smooth manifold $X$ by automorphisms. Let $H$ be a subgroup of $G$. Let $X_H$ be the $H$-...
user avatar
3 votes
1 answer
92 views

Convergence of Newton's method for Banach spaces

Let $F:U\subset V\rightarrow V$ be a $C^1$ function in a Banach space such that $F(x_*)=0$ and $DF(x_*)$ is invertible. I want to prove that there exists $r>0$ such that if $x_0\in B(x_*,r)$ then ...
user avatar
0 votes
0 answers
19 views

Proving the convergence of fixed point iteration algorithm

I've been trying to prove the convergence of the following fixpoint iteration: $$r = 0.5, \quad 0<c<1$$ Repeat: $$r = (1-r)^c$$ It looks simple, but since I am new to this field I can't seem to ...
user avatar
1 vote
0 answers
16 views

Practice Problems for Revising Basic Results from Analysis

Does anybody have a good resource with worked solutions for quick revision of some basic results from analysis? I am self-studying a maths course on Fixed-Point Theorems (Course notes here) over the ...
user avatar
2 votes
0 answers
49 views

Find the metric so that this map is a contraction

Let $y_1, y_2 >0$ and $y_{n}=\frac{2}{y_{n-1}+y_{n-2}}$ for $n\ge 3$. I know that this sequence is converging and that $y_n \to 1$ as $n\to \infty$. Let us write $Y_n = (y_n,y_{n+1})$ and $f(a,b)=(...
user avatar
  • 719
0 votes
0 answers
23 views

Quadratically convergent procedure so that the associated procedure function is without division

Hey I'm having trouble solving the following exercise. Do you have a solution? Let $a>0$ and $1/a > δ > 0$. Give a locally quadratic convergent procedure for the determination of $1/a$ on the ...
user avatar
0 votes
1 answer
66 views

Goedel's Fixed Point Theorem

Let's consider an arithmetic theory such as Peano Arithmetic. Suppose the Gödel codes of all the true sentences are even and all the false sentences are odd. Then, how can there be a fixed point for ...
user avatar
0 votes
0 answers
39 views

How to prove that the following is partial recursive?

Let $g$ and $h$ be to two partial recursive functions and let us define the following function : $f(x,y) =\left\{ \begin{array}{ll} g(y) & \mbox{if} \ x=0,\\ f(x-1,h(x-1,y)) & \mbox{if} \ x&...
user avatar
  • 3,130
0 votes
2 answers
52 views

Help me understand why fixed point iteration works for backwards Euler's method

Euler's method for integration can be written as, $$ f(x) = x + g(x) $$ Assuming that $g$ has a Lipschitz constant which is $<1$, it is a contraction mapping and therefore has a fixed point by the ...
user avatar
  • 659
0 votes
0 answers
13 views

Given a fixed point operator with multiple variables, how to determine if it converges?

I encountered the following fixed point operator, but I am not sure if it converges to a unique point. $$ \mathbf{x}=T(\mathbf{x})=\left(\mathbf{I}+\frac{\lambda\mathbf{U}^T\mathbf{U}}{\|\mathbf{Ux}\|...
user avatar
  • 343
0 votes
1 answer
36 views

How prove the greatest post-fixed point of monotonic function is a fixed point of function?

I have monotonic function $f : \mathcal{P}(M) → \mathcal{P}(M)$ on $(\mathcal{P}(M),\subseteq)$ Is it possible to prove that greatest post-fixed point of $f$ is a fixed point of $f$ not using ...
user avatar
  • 1
0 votes
0 answers
22 views

Existence of cycles for a piecewise Lipschitz continuous map

Suppose that I have a map $f : [0,1]\rightarrow [0,1]$ which is piecewise Lipschitz continuous with Lipschitz constant $K < 1$. (i.e. $f$ is piecewise continuous and each piece is 1-Lipschitz). My ...
user avatar
  • 2,402
0 votes
0 answers
25 views

Inquiry of the convergence analysis for a specific fixed-point iteration $x(t+1)=(1-\alpha(t))x(t)+(\frac{\alpha(t)}{\sum_{i=0}^t\alpha(i)})$

I am currently working on the convergence analysis for a fixed-point iteration and would like to show the iteration w.r.t. the variable $x(t)\in\mathbb{R}$ can converge to $0$, i.e., $\lim_{t\...
user avatar
0 votes
1 answer
19 views

Intrepreting the correction equations $x_{n+1}=x_n+\lambda f(x_n)$ and $x_{n+1}=x_n+\lambda g(x_n)f(x_n)$ in the Variational Iteration Method

I am studying the Variational Iteration Method. There are some concepts related to this method that I don't understand very well. One of them is the correction function. For example, if we want to ...
user avatar
3 votes
2 answers
115 views

Importance of Fixed-point theorems [duplicate]

I have a more general question on the importance of fixed-point theorems. In mathematics youre being introduced to so many fixed-point theorems but i still could not figure out why they are so ...
user avatar
0 votes
1 answer
38 views

Proving Brouwer's fixed point theorem using fundamental groups

I am writing my bachelor thesis on the fundamental group $\pi_1(X)$ and homotopy theory. Now I was wondering if it is possible to prove Brouwer's fixed point Theorem in arbitrary dimensions using only ...
user avatar
  • 415
0 votes
1 answer
49 views

Why do w-FPP and FPP coincide in Reflexive Space?

Let $X$ be a Banach Space. We say that A mapping $T: C\subset X \to C$ is nonexpansive if $\|Tx - Ty\| \leq \|x-y\|$, for all $x,y \in C$. $X$ has the Fixed Point Property (FPP) if every ...
user avatar
5 votes
3 answers
66 views

Am I computing Jacobi Iteration wrong?

To solve the system $$2x_1-\hphantom2x_2+\hphantom2x_3=-1\\2x_1+2x_2+2x_3=\hphantom-4\\-x_1-x_2+2x_3=-5$$ with Jacobi iteration, we let $$A=2I_3,\qquad L+U=\begin{bmatrix}0&-1&1\\2&0&2\...
user avatar
  • 193
0 votes
1 answer
18 views

How can I show that $T : L^2(0, \pi/2) \to L^2(0, \pi/2)$ defined as $T(f) = x + \frac{1}{4}\cos(x)\int_{0}^{\frac{\pi}{2}}f(y)dy$ is a contraction?

I've tried the following: \begin{equation} \|T(f)-T(g)\| = \|\frac{1}{4}cos(x)\int_0^{\frac{\pi}{2}}f(y) - g(y)dy\| = \frac{1}{4}|\int_0^{\frac{\pi}{2}}f(y) - g(y)dy| \|cos(x)\| = \frac{\sqrt{\pi}}...
user avatar
0 votes
0 answers
31 views

Understanding Tarski's fixed-point theorem.

I changed my question slightly. (Tarski Fixed Point Theorem). Let $X=\prod^{N}_{i=1} X_{i}$ where each $X_{i}$ is a compact interval of $\mathbb{R}$. Suppose $\phi : X \rightarrow X$ is an increasing ...
user avatar
0 votes
1 answer
34 views

Proof for fixed point under specific circumstances

Prove that every $f \in C(I, \mathbb{R})$ with $I := [-1, 1] \subset \mathbb{R}$ and $f(I) \subseteq I$ has a fixed point. This would be true if $f$ is a contraction on $I$, since then Banach's fixed ...
user avatar
  • 75
0 votes
0 answers
9 views

How to prove a set-valued function generates a closed graph?

I'm trying to understand the structure of my problem for Kakutani's Fixed-point Theorem. I have a set-valued function with two variables $\Gamma(u_{a},u_{b})$, where both arguments are from a closed ...
user avatar
4 votes
2 answers
109 views

A version of Brower's fixed point theorem for contractible sets?

Brouwer's fixed point theorem states that a continuous map $f:B^n\to B^n$ ($B^n\subset\Bbb R^n$ being the $n$-dimensional ball) has a fixed point. It is clear that we can replace $B^n$ with a space $X$...
user avatar
  • 28.3k
0 votes
1 answer
56 views

Fixed point iteration converges

I found an old problem from notes, which I was not able to solve. Assume that we have a given (arbitrary) norm $\| \cdot\|$ on $K$ and function $g:K \times K \rightarrow K \times K$ for some compact ...
user avatar
  • 189
0 votes
0 answers
38 views

Unique fixed point of contraction defined on a ball

In the case where $f : X \rightarrow X$ is not a contraction on the whole space $X$, but rather a contraction on some neighborhood of a given point $y$, In this case we restrict our function to a ...
user avatar
1 vote
0 answers
45 views

Unique solution to a specific Volterra's integral equation of the third kind

Consider an integral equation (Volterra's integral equation of the third kind) $$(d-cx) u(x) = \int_x^b u(y) dy, \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function ...
user avatar
1 vote
1 answer
39 views

On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
user avatar
0 votes
0 answers
14 views

Do we have unique fixed point for $x=J(x)$ when $J$ show: $\partial J_{i}/\partial x_{i} < \partial J_{i}/\partial x_{n}<0$?

I try to find a unique fixed point for: \begin{equation} x_{i} = x_{i}^{-\alpha}\left(\sum_{n}x_{n}^{-\beta}\right) + x_{i}^{-\gamma}\left(\sum_{n}x_{n}^{-\delta}\right) \end{equation} My idea is to ...
user avatar
  • 11
2 votes
0 answers
67 views

Assumptions in Schauder Fixed Point Theorem

I have a - maybe slightly stupid - question about the Schauder-Fixed-Point Theorem. The formulation I have in mind is: Let $A$ be a closed, convex, nonempty subset in a Banach space $(X,\|\cdot\|)$, ...
user avatar
  • 168
0 votes
2 answers
39 views

Show $\phi$ has a unique fixed point

Let $\phi : \mathbb{R} \rightarrow \mathbb{R}$ a function of classe $\mathscr{C}^{1}$ such that $$ \underset{x \in \mathbb{R}}{\text{sup}}\left|\phi'\left(x\right)\right|<1 $$ I need to show it ...
user avatar
  • 6,955
0 votes
0 answers
21 views

Generalization of Eaves' Theorem

Let $K\subset \mathbb{R}^n$ be a nonempty convex compact and $f:K\to K$ be a function. Let $g:K\to \mathbb{R}^n$ be $g(x)=f(x)-x$. Is there always a point $x_0\in K$ such that for all neighbourhood $U\...
user avatar
  • 449
1 vote
1 answer
51 views

Can you disprove this counterexample to the diagonal lemma?

I was looking at the Diagonal Lemma or Fix point theorem which states in every Theory $T$ every formula with one variable $ B(n) $ has a fix point: $T \vdash G \leftrightarrow B(\# G)$. Where $\#F$ ...
user avatar
  • 19
0 votes
1 answer
36 views

Fail to get convergence on point iteration method

I have the following formula: $$\lambda = \lambda(1-F(S-2)) + \frac{r}{c}p$$, where $S, p, r$ and $c$ are constants and $F(.)$ is the CDF function of a random (Poisson distributed) variable, so $F(S-2)...
user avatar

1
2 3 4 5
40