# 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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### 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 ...
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### 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 ...
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### 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 $\underset{x\rightarrow\infty}{\textrm{Lim}} f(x)=0$. ...
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### 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 $E$)...
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### 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 ...
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### 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 $x_{k+1}=f^{-1}(x_k)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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&... 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 ... 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}\|... 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 ... 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 ... 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\... 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 ... 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 ... 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 ... 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 ... 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\... 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}}... 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 ... 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 ... 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 ... 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$X$... 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 ... 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 ... 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 ... 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 ... 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 ... 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$(X,\|\cdot\|)$, ... 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 ... 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\...
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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 $\#F$ ...
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)...