# 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: \|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: $$x_{i} = x_{i}^{-\alpha}\left(\sum_{n}x_{n}^{-\beta}\right) + x_{i}^{-\gamma}\left(\sum_{n}x_{n}^{-\delta}\right)$$ 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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### 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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### 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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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 $F_2: ... • 145 0 votes 0 answers 32 views ### Intuitive definition of "rate of attraction" of a fixed point If possible, I would like to know if there is an easy and intuitive definition of the "rate of attraction" of a fixed point. I am especially interested in the difference between super-... 1 vote 0 answers 41 views ### When do continuous maps from$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 ... • 1,390 0 votes 0 answers 14 views ### 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 $$... • 424 2 votes 1 answer 53 views ### Use of Zorn lemma in a proof of Kakutani theorem I'm trying to understand the first sentence in the proof of the Kakutani fixed point theorem here : https://mathweb.ucsd.edu/~nwallach/haarmeasure.pdf (page 2) So let K be compact in a locally ... 0 votes 0 answers 20 views ### Applications of Kakutani's fixed point theorem for two different functions Kakutani's fixed point theorem described in Theorem 2.6 in the paper (as shown below) and its application to prove the minimax theorem in Theorem 3.2 in the same paper seem like I can make the ... 2 votes 1 answer 24 views ### fixed point iterations I have a following question on the fixed point argument. Assume that we have some space X with a norm \|\cdot\|_X. Suppose also that there is an equation of type u = Au, were A is an operator. ... • 1,846 0 votes 1 answer 31 views ### Fixed Point Iteration and Order of Convergence of a function For the function f(x) = cos(cosx), does the fixed point iteration for finding the fixed point in [0,1] converge for all first points, p in [0,1]? If it does, what is the order of convergence? I am ... • 57 1 vote 0 answers 41 views ### Minimax theorem for two different objective functions I have a question regarding the minimax theorem (https://en.wikipedia.org/wiki/Minimax_theorem). The minimax theorem tells us that for x\in X, y\in Y where X, Y are compact and closed sets, if f(\... 0 votes 0 answers 34 views ### Stable solutions of quadratic matrix equations In this paper, the authors are interested in solutions X of the following matrix equation: $$F(X) = XBX + XA - DX - C = 0. \tag{1}\label{quadratic_equation}$$ According ... 2 votes 1 answer 194 views ### 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 ... 1 vote 0 answers 24 views ### 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 ... • 5,329 2 votes 3 answers 72 views ### 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 ... 1 vote 0 answers 23 views ### 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 ... • 191 0 votes 0 answers 25 views ### 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 ... • 325 3 votes 1 answer 79 views ### 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 ... 1 vote 1 answer 34 views ### 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 ... • 328 2 votes 1 answer 49 views ### 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 ... • 10.9k 1 vote 0 answers 68 views ### 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−\... 0 votes 1 answer 36 views ### 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 ... 2 votes 1 answer 70 views ### 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 \...
From Ciric fixed-point theorem, we know that if $T:X\rightarrow X$ is a quasicontraction mapping i.e., $d(T(x),T(y)) \le \alpha \max (d(x,y),d(T(x),y),d(x,T(y)),d(x,T(x)),d(y,T(y)),~\alpha \in [0,1)$ ...
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 ...