# 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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### Does this iterative sequence converge?

Let $f:[0,1] \rightarrow [0,1]$be a continuous function. Choose any point $x_0 \in [0,1]$ and define a sequence recursively by $x_{n+1}=f(x_n)$. Suppose $\lim_{n \rightarrow \infty}x_{n+1}-x_n =0$, ...
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### How to prove convergence of a sequence maximizing a sum of exponential distances?

I want to find the argument $x$ that maximizes $f(x)=\sum_i e^{-(x-d_i)^2/c}$ for some data values $d_i$ and an arbitrary positive constant $c$. I assume that $f(x)$ has only a single maximum (most ...
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### $f:A\rightarrow B$ and $a_n=f(a_{n-1})$ $L=[a_1,a_2,…a_n,…]$ For which $f$ and $a_0$ is $\bar L=B$?

This is a generalization of a question I posted a few days ago regarding a particular recursive sequence . I am trying to find general conditions (if they do exist) on which the problem has a ...
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### An inverse problem for a parabolic equation and fixed-point theorem

I'm reading the paper "The determination of a parabolic equation from initial and final data" http://www.ams.org/journals/proc/1987-099-04/S0002-9939-1987-0877031-4/S0002-9939-1987-0877031-4.pdf. A ...
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### Whether there exists some compact manifold whose continuous map must have a fixed point?

Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point. I mean that any continuous map from $M$ to $M$ must have a fixed point. I ...
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### Quotient Spaces Defined By Bijection

I was working with a question in topology and came to the following statement that I can't seem to figure out: Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism with no fixed points. ...
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### On the in-comparability of the Gauss-Seidel and Jacobi iteration schemes.

Given the system $x_1 + x_2 = 2$ $-x_1 + x_2=0$ $x_1 + 2x_2 - 3x_3=0$ the Jacobi iteration converges and Gauss-Seidel iteration diverges. Is there a way we can derive these two facts using the ...
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### Every nonempty, compact convex set $M$ in a locally convex space has fixed point property

I need to prove that "Every nonempty, compact convex set $M$ in a locally convex space has fixed point property". In the book the reference has been given to "Eisenack & Frenske, 1944, page 44". ...
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### Finding an interval of convergence for a given $g(x)$

I am trying to do a fixed point iteration on the function: $f(x) = x^2 -3x+2$, analyzing different forms of $g(x)$. I solved for the actual roots and they equate to $x=1$ and $x=2$. I am currently ...
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### Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
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### Establish the sufficient condition $|g'(x)| < 1$ for convergence of an iteration using the Banach fixed point theorem?

If $x_n = g(x_{n-1})$ is an iteration, it converges if $g$ is continuously differentiable and $|g'(x)| < 1$. The Banach FPT says that if $T$ is a contraction on a complete metric space $X$ then it ...
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### Does any analytic function from the unit disk to a compact subset of itself have a fixed point?

I was reading a proof by Beardon of the Wolff-Denjoy Theorem in Complex Dynamics. In the proof, a family of maps $$f_\epsilon=(1-\epsilon)f(z)$$ is used, where $f(z)$ is an analytic map from the unit ...
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### Range of parameter values for a stability of a fixed point for this 2d map

So I am trying to do a linear stability analysis for a very simple 2d discrete system: \begin{aligned} x_{n+1} &= y_{n}\\ y_{n+1} &= -\frac{x_{n}}{2} + ay_{n} + y_{n}^{3} \...
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### Degree of infinite dimensinal antipodal map

Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ . $L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$? In fact ,I just know that the algebra define ...
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### Prove there is no contraction mapping from compact metric space onto itself

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a ...
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### Large-time limit of the general solution of an ODE is a fixed-point. Is the fixed-point stable?

This question might well have an obvious affirmative answer (or an obvious counterexample!), which at present I cannot see. Suppose I have a first-order ODE $$u'(t)=f(u)$$ whose general solution ...
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### 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 ...
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### while $d(x_n, x_{n+1})$ is converges to $0$, proving there exists an $\varepsilon >0$ such that $d(x_{{m_k}-1},x_{n_k}) < \varepsilon$

I study fixed point theory from Kirk and Khamsi's An Introduction to Metric Spaces and Fixed Point Theory and I couldn't understand a proof. STEP 1: Let $\{d(x_n,x_{n+1})\}$ be a monotone ...
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### Nonlinear contraction on Hilbert space

Let $C\subset H$ be a nonempty closed convex subset of a Hilbert space $H$ and let $T:C\rightarrow C$ be a nonlinear contraction; i.e. $$|Tu-Tv|\leq|u-v|\quad\forall u,v\in C.$$ Let $(u_n)$ be a ...
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### Show that $|T(x) - T(y)| \lt |x-y|$ when $x \neq y$ but the mapping has no fixed points.

Consider $X = \{ x \in \mathbb{R} \mid 1 \le x \lt \infty \}$, taken with the usual metric of the real line, and $T \colon X \to X$ defined by $x \mapsto x +x^{-1}$. Show that $|T(x) - T(y)| \lt |x-y|$...
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### Existence of a solution of a limit of Fixed point equations

I am considering a setting where I am given an iid sample of symmetric positive matrices $\{S_i\}$, $i=1,\dots, n$, of a matrix valued random variable $S$ with distribution $F$. The support of $F$ is ...
Let $P$ be a chain-complete poset with a least element, and let $f_1,f_2,\ldots,f_n$ be order-preserving maps $P\to P$ such that $\forall i,j: f_i \circ f_j = f_j\circ f_i$. Claim. The functions $f_1,... 1answer 62 views ### Poset where every monotonic function has a least fixed point Let$P$be a poset such that every order-preserving map$f:P\to P$has a least fixed point. Must$P$be chain-complete? 1answer 72 views ### Chain-complete and least element iff every order-preserving map has least fixed point Let$P$be a poset. I want to show the following are equivalent.$P$is chain-complete and it has a least element. For every order-preserving map$f:P\to P$, the set$P_f$of fixed points of$f$has ... 1answer 52 views ### Least fixed point of restricted function Let$P$be a poset with the property that every order-preserving map$f:P\to P$has a least fixed point$\mu(f)$. Now for any$p\in P$, the poset$\downarrow(p)=\{x\in P|x\leq p\}$must also have ... 1answer 91 views ### Fixed point property of spaces having same homotopy type Suppose X and Y have same homotopy type.X as a topological space has fixed point property.Can we conclude anything about fixed point property of Y? 1answer 53 views ### Fixed point without the constant If$d(Fx,Fy)<d(x,y)$for all$x,y$in a closed bounded subset$X$of Euclidean space and$F\colon X\rightarrow X$then there is a unique fixed point$x_0$and$\lim \limits _{n\to\infty} F^n(x)=...
I have a question on the derivative test for showing that a function is a contraction. The proposition that I know is in this version: Let $I$ be a closed bounded set of $\mathbb{R}^l$ and \$f : I →...