Tagged Questions

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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Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
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Contraction and Fixed Point [duplicate]

How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$. I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ ...
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Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
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On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
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Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
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If $f^N$ is contraction function, show that $f$ has precisely one fixed point. [duplicate]

If $f$ is a mapping of a complete metric space $(X, d)$ into itself and $f^N$(composite $f$ for $N$ times) is a contraction mapping for some positive integer $N$, then $f$ has precisely one fixed ...
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Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...
Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, \frac{1}{3+\... 3answers 264 views Iterates of f_b(x) = x - \log_b(x)  - for \log(b) \approx 0.399: convergence to accumulation points or chaos? In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm):$$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$with the ... 3answers 763 views Prove that f has a fixed point . [duplicate] For f:[a,b]\rightarrow [a,b] is a continuous . Prove that f has a fixed point . Is that true if we change [a,b] by [a,b) or (a,b). 0answers 453 views Equivalence of Brouwers fixed point theorem and Sperner's lemma I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ... 1answer 457 views In a complete lattice every monotone function has a fixpoint (Knaster–Tarski Theorem) L is a complete lattice, so every subset has a supremum and infimum. In addition, there exists a function f:L \rightarrow L such that a \leq b implies f(a) \leq f(b). Prove that there exists ... 4answers 217 views Fix-Point Theorem Proof. Firstly, the assignment: Let a,b \in\mathbb{R} and a < b. Furthermore let f: [a,b] \rightarrow [a,b] be monotone increasing. Show that if x:= \mathbf {sup}\{y \in [a,b] \| \ y ≤ f(y)\} ... 1answer 62 views Fixed points of contractions in metric spaces How do I prove that all contractions on a complete, non-empty metric space has exactly one fixed point? What I know: I know that all contractions are continuous and that completeness of A means ... 2answers 843 views Homeomorphic to the disk implies existence of fixed point common to all isometries? A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ... 2answers 961 views Contraction mapping in an incomplete metric space Let us consider a contraction mapping f acting on metric space (X,~\rho) (f:X\to X and for any x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1). If X is complete, then there ... 2answers 208 views eventually constant maps Let f:[0,1]\to [0,1] be a continuous function with a unique fixed point x_{0} Assume that \forall x\in [0,1], \exists n\in \mathbb{N} such that f^{n}(x)=x_{0}. Does this implies ... 2answers 737 views What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface? Let X be a compact Riemann surface of genus g>1, f\in Aut(X), a biholomorphism of X onto itself, x\in X a fixed point of f. Since tangent map of a holomorphic map (on the real tangent ... 0answers 73 views Generalized Fixed Point Theorem Suppose that T: M \to M is a self map of a nonempty closed set M in a complete metric Space (X,d). Suppose further that$$d(Tx,Ty) \le k(a,b)d(x,y)$$for all x,y \in M with 0 \lt a \le d(x,y) ... 2answers 436 views Fixed points in category theory Is there a usual way to express the concept of fixed points in category theory? What would I say to express that a morphism has a fixed point? Thank you! 2answers 908 views Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem http://en.wikipedia.org/wiki/Diagonal_lemma I am wondering about the proof of the "Fixed-Point Lemma" \text{Mod } \Sigma is the class of all models of  \Sigma. \text{Th Mod } \Sigma is the set ... 2answers 79 views When does Newton-Raphson Converge/Diverge? Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: ... 1answer 270 views Do there exist sets A\subseteq X and B\subseteq Y such that f(A)=B and g(Y-B)=X-A? This is a little exercise I've been fiddling with for a while now. Let f\colon X\to Y and g\colon Y\to X be functions. I want to show that there are subsets A\subseteq X and B\subseteq Y ... 2answers 338 views How to figure out the starting point for this Mandelbrot? My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, f(z)\mapsto z-\log_b(z). In particular, for ... 1answer 328 views Proof that the solution to cosx = x, is the limit of a recursive sequence. So I've got this question. Exists a sequence a_n such that:$$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$Prove that \lim_{n\rightarrow\infty} a_n = \alpha Where \alpha is the solution to ... 2answers 450 views Is there a simple proof of Borsuk-Ulam, given Brouwer? (Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point. Given this lemma, is there a simple proof of: (Borsuk-Ulam) Any continuous ... 2answers 853 views Convergence of fixed point iteration for polynomial equations I'm looking for the solution x of$$x^n+nx-n=0.$$Thoughts: From graphing it for several n it seems there is always a solution in the interval [\tfrac{1}{2},1). For n=1, the solution ... 1answer 134 views Fixed point combinator (Y) and fixed point equation In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 to the fixed-point theorem states: In \lambda and CL: for every Z and n \ge 0 the equation$$xy_1..y_n = Z$$can ... 1answer 730 views If f: \mathbb R^n \to \mathbb R^n is a contraction, then x-f(x) is a homeomorphism I am stuck in following homework question. Let f : \mathbb R^n \to \mathbb R^n be a uniform contraction and g(x) = x - f(x). Investigate whether g : \mathbb R^n \to \mathbb R^n is a ... 1answer 64 views Is this a valid way to show \chi(SL_n(\mathbb{R}))=0? Why does \chi(SL_n(R))=0? I'm going about it like this. Let X:=SL_n(R). Define a map f:X\to X such by A\mapsto BA, where B is the identity matrix, except with an extra 1 in the upper ... 1answer 143 views Fixed point property of Cayley plane I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed ... 1answer 368 views Generalization of Banach's fixed point theorem I wanted to show that if f:X\to X is a function from a complete metric space to itself and if f^k is a contraction, then f has a unique fixed point (say p) and for any x in X f^n(x)\... 1answer 61 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{... 1answer 258 views Every increasing function from a certain set to itself has at least one fixed point I need a hint for the following question: Let S be a nonempty ordered set such that every nonempty subset E\subseteq S has both a least upper bound and a greatest lower bound. Suppose f:S \... 4answers 223 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 ... 1answer 96 views Brouwer's fixed-point theorem and the intermediate value theorem? In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ... 1answer 48 views Question about fixpoints and zero's on the complex plane. Define property A for an entire function f(z) as 1) f(z)=0 has exactly one solution being z=0 2) f(z)=z has exactly one solution =>z=0 (follows from 1) ) 3) f(z) is not a ... 4answers 99 views Fix point of L:S^2\rightarrow S^2 Let L:S^2\rightarrow S^2 be a bijective continue map. Is there exists L such that \forall x\in S^2 \Rightarrow Lx\neq x and Lx\ne -x? I mean that whether L must have fix point? In fact ... 3answers 526 views Prove the sequence x_n = \frac{x_{n-1}}{2}+\frac{A}{2x_{n-1}} \rightarrow \sqrt{A} as n \to \infty Show that if A is any positive number, then the sequence defined by:$$x_n = \frac{x_{n-1}}{2}+\frac{A}{2x_{n-1}} for any $n \geq 1$ converges to $\sqrt{A}$ whenever $x_0 > 0$.
For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, ...