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Questions tagged [contraction-mapping]

For questions related to contraction mapping. On a metric space $(M, d)$ it is a function $f$ from $M$ to itself, with the property that there is some real number $0\leq k<1$ such that for all $x$ and $y$ in $M$, $d(f(x),f(y))\leq k,d(x,y)$.

A contraction mapping, or contraction or contractor, on a metric space $(M, d)$ is a function $f$ from $M$ to itself, with the property that there is some real number $0\leq k<1$ such that for all $x$ and $y$ in $M$, $d(f(x),f(y))\leq k\,d(x,y)$.

More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if $(M, d)$ and $(N, d')$ are two metric spaces, then $f:M\rightarrow N$ is a contractive mapping if there is a constant $0\leq k<1$ such that $\displaystyle d'(f(x),f(y))\leq k\,d(x,y)$ for all $x$ and $y$ in $M$.

Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant $k$ is no longer necessarily less than $1$).

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Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and $f:X\to X$ is a contraction, that is $f$ satisfies…
nullUser
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Contraction Mapping question

Let X be the set of continuous real valued functions defined on $[0,\frac{1}{2}]$ with the metric $d(f,g):=\sup_{x\in[0,\frac{1}{2}]} |f(x)-g(x)|$. Define the map $\theta:X\rightarrow X$ such that $$\theta (f)(x)=\int_{0}^{x} \frac{1}{1+f(t)^2}…
hmmmm
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CLT as a special case of the Banach fixed-point theorem?

I'm wondering if the Lindeberg–Lévy CLT can be seen as a special case of the Banach fixed-point theorem. I have in mind some map $T:P\rightarrow P$ for some complete metric space of CDFs $P$ with finite variances. $T$ might be something like:…
DM-97
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If $g$ is contraction and $f+f'=g(f)$ then f(t) has a limit as $t\to\infty$

I would like to prove the following statement: Let $g:\mathbb{R}^n\to\mathbb{R}^n$ be contraction mapping. If function $f:\mathbb{R}\to\mathbb{R}^n$ satisfies: $$f(t)+f'(t)=g(f(t))$$ then the limit $\lim_{t\to\infty}f(t)$ exists. Below I write a…
Jakub Pawlak
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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 any $n\geq1$. Show that $(x_{n})_{n\geq1}$ is…
Stefan Solomon
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Does every contraction on $\mathbb{R}(X)$ possess a unique fixed-point?

Consider the field $\mathbb{R}(X)$ together with the total ordering given by: $a > b$ $\Leftrightarrow$ $a(x) - b(x)$ is eventually positive. Say that a map $f: \mathbb{R}(X) \to \mathbb{R}(X)$ is a contraction if there exists a constant $C \in…
Smiley1000
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Fixed point theorem on an open ball in complete metric space

Let $(X,d)$ be a complete metric space, $a \in X$ and $r>0$. Let $f:O(a,r)→X$ be a contraction with a contraction constant $L$, $d(f(x),f(y))
shwsq
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Prove that $T$ satisfies the hypotheses of the contraction mapping principle.

Let $T: C([0,1]) \to C([0,1])$ be defined by $Tf(x) = x + \int_{0}^{x}tf(t)\,dt$. Prove that $T$ satisfies the hypotheses of the contraction mapping principle. Show that the fixed point is a solution to the differential equation $f'(x) = xf(x) +…
beginner
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Finding invariant Borel probability measures for a contraction map

Let $X$ be a compact metric space. Let $f:X\rightarrow X$ be a contraction map. I need to find all $f$-invariant Borel probability measures. Thank you.
adtx11
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Proving that a function is not a Rakotch contraction

I want to prove that the function $f:[-1,1]\rightarrow [-1,1]$ given by \begin{equation*} f(x)=\frac{1}{2}x^2 \end{equation*} is not a Rakotch contraction. To give the definition of a Rakotch contraction I need to give the definition of a $\phi$-…
Aiswarya
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proof that the operator of integral equation is a contraction

The part I am reading is similar to the following questions, so if you are not familiar with integral operator and contraction, you could refer to them. Show integral operator is contraction mapping Reiterate Volterra integral operator is a…
Halk
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Prove that the space of contraction mappings in $[0,1]$ is compact.

I'm going through James Dugundji's Topology but I'm having trouble with chapter XI, section 4 problem 2: Let $\mathscr{F}$ be the family of all continuous maps $I\to I$ such that $|f(s)-f(t)|\leq |s-t|$. Define $$d^+(f,g)=\max_{0\leq t \leq…
Sebastián P. Pincheira
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Contraction mapping problem in $\mathbb{R}$

Problem:In the mapping $T:\mathbb{R} \rightarrow \mathbb{R}$ defined by $Tx=\begin{cases} x-\frac{1}{2}e^x & \forall x \leq 0 \\ -1/2+1/2x & \forall x>0 \end{cases}$ a contraction? Definition: A mapping f from a subset A of a normed space E into E…
neveryield
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What is a contractive mapping vs contraction mapping?

This is an example from a text to show that this mapping does not have a fixed point because it is contractive but not a contraction: I am not sure what the difference is between contractive and contraction. Doesn't the function satisfy all the…
Bill
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Proof for the existence of a compact attractor

Let $(X,d)$ be a complete metric space and let $f_1,\ldots,f_n$ be contractions with Lipschitz constants $q_i$. Then a unique non-empty compact set exists such that $K=\bigcup_{i=1}^n f_i(K)$. Now the way the proof usually begins is by proving that…
Dave the Sid
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