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 $M$, $d(f(x),f(y))\leq k\,d(x,y)$.
61 questions
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A Fixed Point Theorem in Complete Metric Spaces for Self-Maps Satisfying This Contractive Condition
Let $(X, d)$ be a complete metric space, and let $f \colon X \longrightarrow X$ be a self-map of $X$ for which there exists a real constant $h \geq 0$ such that
$$
d \big( f(x), f(y) \big) \leq h \max ...
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Any Example(s) to Show the Independence of These Two Contractive Conditions?
Let $(X, d)$ be a metric space, and let $f \colon X \longrightarrow X$ be a mapping.
Then $f$ is said to be of $A$-type if there exists a positive real number $\alpha < 1/2$ such that
$$
d \big( f(...
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Majda and Bertozzi: Global existence for regularized solutions of Navier-Stokes equations
Theorem 3.2 in the book "Vorticity and incompressible flow" by Majda and Bertozzi" states that, given an initial condition $v_0 \in V^m$, $m \in \mathbb{Z}^+ \cup \{0 \}$, for any $\...
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If a self-map of a metric space satisfies the first condition, does it also satisfy the second? [closed]
Let $(X, d)$ be any metric space, and let $f \colon X \longrightarrow X$ be any function.
Does the first of the following two conditions on $f$ also imply that $f$ satisfies the second condition?
...
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Why is $y = \sin \left(\frac{y^s}{t} - k \right)$ outstandingly approximated by $y = - \sin k$, even for large $k$?
Equations of the form $$y = \sin \left(\frac{y^s}{t} - k \right)$$ are surprisingly well approximated as $$y = - \sin k$$ for a large range of $s$ $t$ and $k$ values:
Why? I understand why this is so ...
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Prove that the iterate $f^{n}$ is a constant function.
Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...
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Lipschitz contraction and covariance
I am reading this article by Chen and Eldan and on page 61 they make this argument:
$$
\text{Cov}(\text{tanh}(v+\alpha Y))\preceq\alpha^2\text{Cov}(Y)
$$
because $\text{tanh}$ is a contraction, where $...
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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 ...
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Is an injective contraction open?
Let $(X, d)$ be a metric space and $f:(X, d) \to (X, d)$ be an injective contraction.
Does any of the following statements hold:
f is an open map
If there exists a fixpoint $p \in X$, is it true that ...
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A problem about fixed point of a continuously differentiable function [closed]
QUESTION
Let $f: (0,+\infty)\to(0,+\infty)$ be a continuously differentiable mapping
and there exist $k$ s.t.$|xf'(x)|≤kf(x)$
then $f$ has unique fixed point
My approach
I want to prove that $f$ is ...
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Local surjectivity theorem
Let $f:\Omega \to \mathrm{R^n}$ be a continuous map, differentiable at $x_0$ and such that $Df_{x_0}$ is invertible. I want to prove there exists $\epsilon>0$: $f$ is surjective in $B_\epsilon(f(...
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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 ...
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Similar contractions
Let $T_1:\mathcal{H}_1\to\mathcal{H}_1$ and $T_2:\mathcal{H}_2\to\mathcal{H}_2$ two contractions on Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, i.e. bounded linear operators with (operator) ...
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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 ...
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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 $L$, $d(f(x),f(y))<Ld(x,y)$, for all $x,y \in O(a,r)$ and $d(f(a),a) \...
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