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Questions tagged [banach-fixed-point]

For questions involving the Banach Fixed Point Theorem and its applications

Let $(X,d)$ be a metric space and $T$ a map from $X$ to itself. The map $T$ is called a contraction if there is some $k\in[0,1)$ such that$$(\forall x,y\in X):d\bigl(T(x),T(y)\bigr)\leqslant kd(x,y).$$The Banach fixed point theorem states that if $(X,d)$ is complete and $T$ is a contraction, then $T$ has one and only one fixed point (that is, a point $x\in X$ such that $T(x)=x$).

The fact that the fixed point is unique is very easy to establish, even without the assumption that $X$ is complete. In fact, if $x$ and $y$ are fixed points, then$$d(x,y)=d\bigl(T(x),T(y)\bigr)\leqslant kd(x,y)$$and therefore, if $d(x,y)\neq0$, we would have $1\leqslant k$, which is false. So, $d(x,y)=0$, which means that $x=y$. On the other hand, the completeness of $X$ is essential for the existence of a fixed point.

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Prove existence of unique fixed point

Let $f(x)$ be a strictly decreasing function on $\mathbb{R}$ with $|f(x)-f(y)|<|x-y|$ whenever $x\neq y$. Set $x_{n+1}=f(x_n)$. Show that the sequence $\{x_n\}$ converges to the root of $x=f(x)$. Note that the condition is weaker than what is…
Zhang Edison
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Completeness can be omitted from Banach Fixed Point Theorem?

In Kreyszig's Functional Analysis, page no. 303, exercise no. 3 says that completeness cannot be omitted from Banach's Fixed Point Theorem. But if we take $f(x)=x^2$ from an incomplete metric space $(-1/3,1/3)$ to $(-1/3,1/3)$, here $f$ is a…
Infinite
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does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?

I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (t) = t - \tanh \left( \frac{f (t)}{\dot{f} (t)}…
crow
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Showing that a function $f$ has a unique fixed point in a metric space.

Let $(X, d)$ be a compact metric space, and suppose $f : X → X$ satisfies $$d(f(x), f(y)) < d(x, y)$$ for all $x \neq y \in X$. Show that f has a unique fixed point. All I've gotten it so far is that we need to somehow use another function…
Gauss
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Every 1-Lipschitz function in the closed unit ball has a fixed point

I'm currently trying to solve the following exercise: Let B be the closed unit ball in $\mathbb R^n$ together with the euclidean metric. Show that every 1-Lipschitz function $f:B\to B$ has a fixed point. I think I am supposed to use the Banach…
lasik43
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Every $f : [a, b] → [a, b]$ has a fixed point where $f$ is continuous. Deduce the intermediate value theorem.

Every $f : [a, b] → [a, b]$ has a fixed point and $f$ is continuous (on $[a,b]$). Deduce the intermediate value theorem. I managed to show the other way, now I'm here. I know that $f(c)=c$ for some $c\in [a,b]$, and I need to show that for all $x\in…
GRS
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Contraction mapping with no fixed point using a incomplete metric space

I know that if $f:X\rightarrow X$ is a contraction, then $d(f(x),f(y))\leq \alpha d(x,y)$ for $0<\alpha<1$. I'm looking for a counter example, that is a metric space that's incomplete, and where there are contractions with no fixed point. Can…
GRS
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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 defined by $x_{n+1}=Tx_n$ converges to the fixed…
Aarna
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Proof about continuity of a function involving the Banach fixed point theorem

Be $X$ and $\Lambda$ metric spaces, with $X$ complete, and $f\in C(X\times\Lambda,X)$. Suppose that exists some $\alpha\in[0,1)$ and, for each $\lambda\in\Lambda$, some $q(\lambda)\in[0,\alpha]$ such that $$d(f(x,\lambda),f(y,\lambda))\le…
user173262
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Where is my mistake using the Banach theorem for $x^2 - 2 = 0$?

Consider example $x^2 - 2 = 0$. I can rewrite so I get $x^2 + x - 2 = x$. If I define $\phi(x) = x^2 + x - 2$, I need to solve $\phi(x) = x$. $\phi$ is Lipschitz-continuous, since it's differentiable. On $[-\frac34,-\frac14]$ we have, using the mean…
Stefan Hante
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Breaking Banach's Fixed Point Theorem

In trying to see how Banach's fixed point theorem would break down in an incomplete space, I tried to come up with an example of a function: $f: \mathbb{Q} \longrightarrow \mathbb{Q} \ \ $such that $ \forall x,y \in \mathbb{Q}$ our function is…
Gridley Quayle
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Prove that $E=\{x=(x_n)\in \ell^\infty(\Bbb N): (x_n)_n~~\text{is periodic}\}$ is not complete.

Let $E=\{x=(x_n)\in \ell^\infty(\Bbb N): (x_n)~~\text{is periodic}\}$ Defintion: $x=(x_n)$ is periodic means there exists $p\in \Bbb N$ such that, $x_{n+p} =x_n ~~~\forall ~~n\in\Bbb N.$ we define on $E$ the distance $$d(x,y)…
Guy Fsone
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Finding the fixed point of $T(\phi)=\int_{0}^{x} \phi (t) dt$

Define $T: C[0, 1]\rightarrow C[0, 1]$ as follows: for $\phi \in C[0, 1]$ $$T(\phi)=\int_{0}^{x} \phi (t) dt$$ How to show that $T$ is not a contraction but have a fixed point. Thought: $$\begin{align} \bigl\lvert(Tf)(x)-(Tg)(x)\bigr\rvert &=…
1256
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Banach fixed point problem and dependence of the fixed point by a parameter.

Let $(X,d)$ be a complete metric space and let $(\Lambda,d_\Lambda)$ be a metric space. Suppose we have a continuous map $F:X \times \Lambda \to X$ such that there exist $0
Hugo
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Existence of a fixed point of a nonlinear integral operator

This is an elaboration on my post on MO. In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE $$\frac{\mathrm{d}^2 y}{\mathrm{d}…
user1337
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