# 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.

133 questions
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### Proving a particular function is surjective on a Banach space.

Let $(E,\|x\|)$ and let $f: E \to E$ such that $f+Id$ is a contraction ($Id$ is the identity map). Prove that $f$ is surjective and prove that, if $f$ is linear, then $f$ is a homeomorphism. The ...
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### For what $x$ does $g(x)$ converge to a fixed point?

Show that the iteration: $x_{k+1}=2x_k-αx_{k}^2$ where $α > 0$ converges quadratically to $\frac{1}{α}$ for any $x_0$ such that $0 < x_0 < \frac{2}{α}$. I have been able to prove that it ...
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### Finding a Möbius Transformation given constraints

I am trying to solve this problem, but am running into very complicated solving, and think that there is a simpler approach that I am missing. Find a Möbius transformation $M(z)$ that satisfies ...
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### Is this function necessarily a contraction?

If $(X, d)$ is a compact metric space satisfying $d(f(x), f(y)) < d(x, y)$ for all $x, y \in X$ such that $x \ne y$, is $f$ necessarily a contraction? I know an analogue of the Banach Fixed Point ...
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### Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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### How to prove that this iterative formula converges to the Gram points for the constant $c=0$?

We intend to find values of the $n$-the point $t$ such that: $\Re\left(\zeta \left(i t+\frac{1}{2}\right)\right) \neq 0$ and $\Im\left(\zeta \left(i t+\frac{1}{2}\right)\right) = 0$ which should ...
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### Fixed point property for the total space of the canonical line bundle over $\mathbb{C}P^{2n}$

It is well known that the even dimensional complex projective pace $\mathbb{C}P^{2n}$ has the fixed point property. What about the total space of the canonical line bundle over $\mathbb{C}P^{2n}$? ...
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### Show that the only intervals having the fixed point property are the closed intervals.

By Fixed Point Theorem, I know that it deals with closed interval, for eg, [0,1]. And this theorem will be false if [0,1] is replaced by (0,1). The counter example will be $f:(0,1)\rightarrow (0,1)$ ...
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### Fixed point in plane transformation.

Some one give me a idea to solve this one. It's a problem from Vladimir Zorich mathematical analysis I. Problem 9.c from 1.3.5: A point $p \in X$ is a fixed point of a mapping $f:X \to X$ if $f(p)=p$...
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### Sufficient and necessary condition for the global uniqueness of fix-points

https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...
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### Brouwer Fixed Point Theorem implies Invariance of Domain Theorem

Is there a simple and elementary proof that Brouwer's Fixed-Point Theorem implies Invariance of Domain Theorem? By 'simple and elementary' I mean proof that: Do not use tools such as degree theory or ...
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### Fixed point location for functions

How are fixed points calculated? Are intersections of $y = f(x) ,y = f^{-1} (x)$ graphs give real fixed points for all $f$ ?
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### Let $x_1 = 2$, $x_{n+1}=2+\frac{1}{x_{n}}$, $n \in \mathbb{N}$. Does the sequence converge or diverge?
Let $x_1 = 2$, $x_{n+1}=2+\dfrac{1}{x_{n}}$, $n \in \mathbb{N}$. Does the sequence converge or diverge? How can I solve this question using Monotone Convergence Theorem or any other better method? I ...
### A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...