# 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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### Navier-Stokes, admit space of divergence free or no?

Consider the problem of Navier-Stokes $$\frac{du}{dt}-\mu \Delta u+u \cdot \nabla u+\nabla p=f$$ such that $u(0)(x)=u_0(x)$ for $x \in \mathbb{R}^N$ in a adequate space (for example $L^p(\mathbb{R}^N)$...
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### Least fixed point and monotone mappings in complete lattice

Let $(E_1,\leq_1)$ and $(E_2,\leq_2)$ be two complete lattices. also let $f_1 : E_1 \times E_2 \rightarrow E_1$ and $f_2 : E_1 \times E_2 \rightarrow E_2$ be mappings monotonic with respect to their ...
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### How to prove this problem related to fixed point theorem?

$M$ is a bounded,convex and closed subset of Banach space $X$, $A:M\rightarrow M$ satisfies: $$||Ax-Ay||\le ||x-y||$$ for all $x,y\in M$ Show that $\forall \varepsilon>0$,there exist $x\in M$ such ...
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### What can we know about the fixed points of the action of a strongly continuos one-parameter subgroup of unitary operators?

My motivation is physical, but my question in mathematical. In quantum mechanics, time evolution of states is the action of a one-parameter subgroup of projective-unitary transformations of a ...
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### Will this function always have a fixed point?

$f:[0,\infty) \to [0,\infty)$ where $|f(x)-f(y)| \le \frac{1}{2}|x-y|$ Does this function always have a fixed point? My attempt: The function is continuous. If it becomes differentiable then it will ...
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### Are the fixed points of a continuous map from a compact, convex subset of the Euclidean space to itself always isolated? If not, is there a condition?

Let $C \subset \mathbb{R}^n$ be a compact, convex subset of $\mathbb{R}^n$. Let $f: C\to C$ be a continuous map that's not the identity map (see Qiaochu Yuan's comment below). If necessary here, ...
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$$x'\ =\ r\cdot x\cdot \ln\left(\frac{ {K} }{x}\right)$$ Analyze fixed points and their stability depending on the parameters. I found one fixed point $x = K$ and the stability of this point. If $r &... 0 votes 1 answer 30 views ### If a contractive map has a fixed point, is it necessarily unique? Is it true that if$T:X\to X$is contractive and has a fixed point, then this fixed point is unique (even if$X$is not complete)? If$(X,d)$is a complete metric space and$T:X\to X$is a contractive ... 0 votes 0 answers 31 views ### Speed of convergence of an iterative method Problem: Let$f: \mathbb{R} \to \mathbb{R}$be an analytical function with a simple root$\alpha$. Given an iteration $$x_{r + 1} = x_r - \frac{f(x_r)^2}{f(x_r + f(x_r)) - f(x_r)},$$ prove that the ... • 321 1 vote 1 answer 39 views ### Find position knowing 2 points and distances to those points I have a problem that I am trying to solve and I don't know how to approach it. I am trying to position myself in 2d space using Bluetooth Beacons. Basically i have my 2 beacons: BEACON 1 at ... 0 votes 0 answers 44 views ### Is the Brouwer's fixed point theorem applicable for convave functions? In my book it says that Brouwer's fixed point theorem requires the set to be compact and convex. Yet, in one of the exercises (Let f: [0,1] -> [0,1] be continuous and concave. Show that if f(0)>... 4 votes 0 answers 39 views ### Application of Kannan Fixed Point theorem to the integral equations Let$(X,d)$be a complete metric space. The map$T:X\longrightarrow X$is called a Kannan type contraction if $$d(T x, T y) ≤ α[d(x, T x) + d(y, T y)], ∀x, y ∈ X,\alpha\in[0,\frac{1}{2}).$$ Kannan's ... • 1,250 1 vote 0 answers 37 views ### Characterization of normal structure of a Banach space I'm reading the Kirk-Goebel's book, "Topics in metric fixed point theory" and I don't get one implication of an equivalence proof. I'm talking about the Lemma 4.1. It sais as follow: A ... • 360 0 votes 1 answer 41 views ### Choosing initial approximation and the function in Fixed point iteration method In Numerical analysis, to solve an equation of the form$f(x)=0$in$[a,b]$, fixed point iteration method is useful. To this end, we can write$f(x)=0$in the form$g(x)=x$and try to find a fixed ... • 23 0 votes 3 answers 42 views ### Fixed points for increasing function Let$f(x):[a,b] \to \mathbb{R}$a real-valued function that is strictly increasing in$x$. Further,$f(b)>b$. I would like to show that the function has at most two fixed points; is that true? ... 2 votes 0 answers 46 views ### Can you solve$y'(x) = y^2(x)\;\land\;y(x_0) = y_0 $on$]-1, 1[$using the Banach contraction theorem? I would like to solve the following Cauchy problem: $$\begin{cases} y'(x) = y^2(x)\\ y(x_0) = y_0 \end{cases}\tag 1$$ In my opinion, using the Banach contraction theorem it can only be solved in$I:=... 23 views

### Comparative statics for a fixed point or functional equation

I was wondering if I could get some references regarding the following problem. I have a functional equation: $$f(x;\alpha) = (T \circ f)(x;\alpha)$$ where $x \in [0,1]$. This equation involves a ...
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### What is the fixed point (in sense of a 2D function) of Radon transformation?

Looks that if I repetitively apply radon transform on image (use the output again as the input), I always get something like enter image description here I'm curious what kind of function is that.
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### Proving surjectivity of the exponential using Lefschetz's fixed point theorem

I've read in a few places (e.g., here), that one can prove Cartan's result on the surjectivity of the exponential map on compact Lie (throughout, assumed to be connected) groups using Lefschetz's ...
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### Does a function whose derivatives never attains values $\pm 1$ have a fixed point?

I started wondering if the following is true. Consider a differentiable function $f:\mathbb R \to \mathbb R$. If $f'(\mathbb R) \cap \{-1,1\} = \emptyset$, then $f$ has a fixed point. From Darboux ...
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### Lower bound of integration such that integral equals that bound [closed]

While doing some math I came up with the following task and I don't know how to solve it and whether it's solvable at all. Maybe anybody knows how to solve it or at least shows me the right direction. ...
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### Contractive mappings and fixed points

Hy friends, In a metric space, we now that $d(f(x),f(y))<d(x,y)$ is not sufficient for the existence of a fixed point for $f$. However, the results of Rakotch (A Note on Contractive Mappings, 1962) ...
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### Does $f'(x_*)=0\Rightarrow |e_{n+1}|<C|e_n|^2$ in the fixed-point iteration method?

(Multiple options could be correct!) Q. Let $f:[0,1]\to[0,1]$ be a twice continuously differentiable function with a unique fixed-point $f(x_*)=x_*$. For a given $x_0\in [0,1],$ consider the ...
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### Proof of non-convergence of fixed point iteration

Suppose we have a function $f$ which has a fixed point $\alpha$. Moreover, $|f'(\alpha)|>1$. I aim to prove that the fixed point iteration does not converge locally to $\alpha$ (only sequences ...
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### Fixed point theorem

Define $f : \mathbb R\rightarrow \mathbb R$ by $f(x) = (3x^2+1)/(x^2+3).$ Let $f^{\circ1} = f,$ and let $f^{\circ n} = f^{\circ(n-1)} \circ f$ for all integers $n \geq 2.$ Which of the following ...
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### Two contractions with the same fixed point

We have $f,g: \mathbb{R} \to \mathbb{R}$ two contractions with the same fixed point and $(x_{n})_{n\geq1}$ a sequence with real numbers with the propriety that $x_{n+1}\in \{f(x_{n}),g(x_{n})\}$, for ...
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### Order of a numerical iteration method

Suppose that we have the definition of order $p$ of a numerical method as in the first snippet below. Now I want to prove that for a one-point iterative method this order $p$ is a positive integer. To ...
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### Prove the existence of a fixed point for the sum of two mappings

Let $X$ be a real valued Banach-space and $A \subseteq X$ which is bounded, closed and convex. Furthermore let $f,g,h : A \mapsto X$ be continuous with $f=g+h$ and $g(A)+h(A) \subseteq A$. Lastly $g$ ...
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### Iterative way to "exactly" compute smallest eigenvalue of AA^T such that it is positive

I have a matrix $A$ and I'm interested in its smallest singular value, i.e. the smallest eigenvalue of $B = A A^T$ (which is clearly positive). Using common algorithms on $B$ to solve the eigenvalue ...
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