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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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}}...
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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 ...
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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 ...
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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 ...
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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 $X$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $(X,\|\cdot\|)$, ...
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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 ...
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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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Can you disprove this counterexample to the diagonal lemma?

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 $\#F$ ...
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Fail to get convergence on point iteration method

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)...
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Proving that Iterative Process of a Partition Converges

An Econ student doing their graduate thesis here, with no formal maths background so I'd be grateful if any kind soul can nudge me in the right direction. I am studying a partition of the space [0,1] ...
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Does Fixed Point theorem work on an integer to integer mapping?

I have an equation which I need to get its (least) fixed point for. Generally, we have: ...
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How to prove that Hilbert Cube has the Fixed Point Property without using Brouwer Fixed point theorem?

So these two statements might be equivalent, but still there is supposed an easier way to prove the former without knowledge in algebraic topology It's an exercise on my textbook after the chapter ...
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How to combine fix-point methods with linear equation systems for solving non-linear matrix equations?

A few days ago I revisited an old favorite problem of mine : How to find "fractional" discrete integration operators? My approach was that of a damped fixed point method $$M_{i+1} = (1-\...
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Convergence of iterative map

I have the following iterative mapping: $$x_{n+1} = (N-n)^{-1} \frac{x_n}{f(x_n)} \left(C - \sum_{i=1}^n f(x_i)\right)$$ defined for $n \leq N$ and where $C > 0$ is some constant. I am trying to ...
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Maximun of a Linear Functional on a Tetrahedron

I am looking for some help framing the solution to this question formally. Let $v_0, v_1, v_2, v_3$ be vertices of a 3-simplex T in $\mathbb{R}^3$ Let $f:\mathbb{R}^3\rightarrow\mathbb{R}$ be a ...
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When does a fixed point iteration converge and diverge?

According to Numerical Methods for Engineers, if $x_{i+1} = g(x_i)$ is the function used for iteration, the magnitude of the derivative, $|g'(x)|$, must be less ...
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Linear Multistep Method With No Spurious Fixed Points

For a general linear multistep method, i.e. a method of the form $\sum_{j=0}^s \alpha_j x_{n+j} = \triangle t \sum_{j=0}^s \beta_j f(x_{n+j})$ Is there any way to manipulate the coefficients $\alpha_j$...
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Using definition of Cauchy sequence in proof of Banach Fixed point Theorem

I'm going through the proof of Banach Fixed Point for a metric space $(X,d)$ in this pdf, the step I having trouble with in is the one after the geometric series is summed: $$ d(x_n , x_m ) < \...
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Proving that $\sin(\lambda x)$ only has one fixed point, if $\lambda$ is smaller than one.

I've been trying to prove that $\sin(\lambda x)$ only has a single fixed point (over all the real number) for $0< \lambda < 1$. I've thought of using the fixed point theorem, since it's obvious ...
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Continuous function on the unit interval with commuting compositions

Let $I$ be the unit interval and $f, g:I\to I$ be continuous functions. Assume that $f\circ g = g\circ f$. This post shows that if $f$ and $g$ in addition are assumed to be increasing then $f$ and $g$ ...
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Solve $\frac{dy}{dx} = x^2y-x$ using method of successive approximations where $\phi(0) =0$

I want to find the general expression for '''phi n(x) but the only way I can do that is by using an iterative product. Is there a better way to do it? Also i want to show that the sequence {\phi n(x)} ...
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Fixed Points of Weighted Sum of Two Mappings

I do not have a background in functional analysis but I am basing something on tuition here and would really appreciate some help with this problem. I have two mappings $F_1: X\rightarrow X$ and $F_2: ...
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Intuitive definition of "rate of attraction" of a fixed point

If possible, I would like to know if there is an easy and intuitive definition of the "rate of attraction" of a fixed point. I am especially interested in the difference between super-...
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When do continuous maps from $X$ to $X$ have a fixed point?

Let $f$ be a continuous map from $X$ to $X$ (compact metric space). I know that we always have a set $A$ such that $f(A)=A$. I want to know when it has a fixed point. Similarly, if $f$ is a continuous ...
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Uniform convergence of Q functions with respect to Bellman residual (fixed points with non-expansive operator)

I'm considering reinforcement learning (RL) problems. In RL, the Bellman equation for Q function is given as $$ Q(x, u) = \mathcal{T}Q(x, u) $$ where the Bellman operator $\mathcal{T}$ is defined as $$...
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Use of Zorn lemma in a proof of Kakutani theorem

I'm trying to understand the first sentence in the proof of the Kakutani fixed point theorem here : https://mathweb.ucsd.edu/~nwallach/haarmeasure.pdf (page 2) So let $K$ be compact in a locally ...
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Applications of Kakutani's fixed point theorem for two different functions

Kakutani's fixed point theorem described in Theorem 2.6 in the paper (as shown below) and its application to prove the minimax theorem in Theorem 3.2 in the same paper seem like I can make the ...
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fixed point iterations

I have a following question on the fixed point argument. Assume that we have some space $X$ with a norm $\|\cdot\|_X$. Suppose also that there is an equation of type $u = Au$, were $A$ is an operator. ...
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Fixed Point Iteration and Order of Convergence of a function

For the function f(x) = cos(cosx), does the fixed point iteration for finding the fixed point in [0,1] converge for all first points, p in [0,1]? If it does, what is the order of convergence? I am ...
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Minimax theorem for two different objective functions

I have a question regarding the minimax theorem (https://en.wikipedia.org/wiki/Minimax_theorem). The minimax theorem tells us that for $x\in X, y\in Y$ where $X, Y$ are compact and closed sets, if $f(\...
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Stable solutions of quadratic matrix equations

In this paper, the authors are interested in solutions $X$ of the following matrix equation: \begin{equation} F(X) = XBX + XA - DX - C = 0. \tag{1}\label{quadratic_equation} \end{equation} According ...
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Defining a Non-Linear System of Equations involving max and min with as a contraction

Suppose I have a vector of strictly positive variables $X \in \mathbb{R}_{++}^{J}$ and a vector of strictly positive variables $Y \in \mathbb{R}_{++}^{I}$ and I want to find the values for which a ...
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Contraction principle type theorems for uniform spaces

I was recently wondering whether there are conraction principle like theorems for uniform spaces but which are not metric spaces. The theorem relies heavily on the notion of completeness, which exists ...
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A question on contraction mapping theorem and fixed point iteration

First of all, thank you for taking the time to read my post. Secondly, this is a question I got as a part of homework. However, the professor allows us to work in groups so I'm hoping that this is ...
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Stability of Banach Contraction Principle w.r.t parameters

I am curious about the following question, originated in little discussion I had with a colleague this afternoon. Let $X,Y$ be complete metric spaces and consider a map $F: X \times Y \to Y$ such that ...
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Show that $x_{k+1} = x_k - \alpha f(x_k)$ converges

How do I show that the fixed point iteration method $x_{k+1} = x_k - \alpha f(x_k)$ converges for $\alpha \in (0, \frac{2}{M}]$ when $0 < f'(x) < M$ for $x \in [a,b]$. I was thinking about using ...
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Proving convergence of Newton-Raphson using contraction mapping theorem

I'm trying to prove there exists an $\epsilon \in \mathbb{R}$ such that for a root, $p$, of a function $f \in C^2$, we have that $\forall q \in (p - \epsilon, p + \epsilon)$, $q$ converges to $p$ due ...
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understanding relation between correct decimal digits and logarithm

Assume that $k<1$ and an iteration satisfies $|a_{n+1}− t| < k|a_n − t| \Rightarrow |a_{n}− t| < k^n|a_0 − t|$ And here the book declares that iteration step contributes at worst roughly the ...
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Relation between a fixed point and being a well-order

I've been trying to prove the following, but with no particular success: Given a linear order $\leq$ on $A$, define $\pi:2^A\to 2^A$ by $X\mapsto\{y\in A: (\forall x < y)(x\in X) \}$. Let $A_0$ be ...
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Solving for range of c values in Fixed Point Iteration

For $x^3−2x^2−13x+30 = 0$, with root r = 3. I am supposed to add $cx$ to both sides of the equation before dividing by $c$ to obtain the fixed point equation $$g(x)=x$$ where $$g(x) = \frac{1}{c}x^3−\...
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Basic confusion about Banach fixed point theorem

Banach fixed point theorem says that if $T$ is a mapping and there is a $q \in [0,1)$ where $|Tx - Ty| \leq q |x-y|$ for all $x,y$ then $T$ has a fixed point. But if $T$ is a linear operator, then ...
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2 votes
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Proof of Bourbaki's Fixed Point Theorem

I am studying GTM 139 and troubling about the proof of Bourbaki's fixed point theorem. To quote from that book: Let $X$ be a poset such that every well ordered subset has an lub in $X$. If $f: X \...
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Composition of quasicontraction mapping

From Ciric fixed-point theorem, we know that if $T:X\rightarrow X$ is a quasicontraction mapping i.e., $d(T(x),T(y)) \le \alpha \max (d(x,y),d(T(x),y),d(x,T(y)),d(x,T(x)),d(y,T(y)),~\alpha \in [0,1)$ ...
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Why is the solution of the periodic KdV equation unique?

Bourgain proved that the periodic KdV equation $$\begin{align} \partial_t u+\partial_x^3 u+u\partial_x u&=0\\u(0,x)&=u_0(x)\end{align}$$ is locally well-posed in $H^s(\mathbb T)$ in [1]. Here ...
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