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Questions tagged [lipschitz-functions]

For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.

For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.

1807 questions
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Is q-Lipschitz continuity necessary condition for fixed point iteration.?

Is q-Lipschitz continuity necessary condition for fixed point iteration? Meaning does it have to be so that $d(f(x),f(y))\leq qd(x,y)$ where $0\leq{q}<1$.
user305180
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Norm on direct product space

Here, page $127$, theorem $2.12$, states the following: Let $X$ be a Banach space. Then $\mathcal{F}(X)$ is Lipschitz-isomorphic to the space $\ker (\beta) \oplus X$. Moreover , these two spaces are linearly isomorphic if and only if the space $X$…
Idonknow
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Prove a function to be Lipschitz

With the condition $$ 0\le f(y)-f(x)-\langle f'(x),y-x\rangle \le \frac{L}{2} \|x-y\|^2, $$ I want to prove $f'(x)$ is Lipschitz with constant $L$ and is convex. It is easy to see that $f $ is convex from the first inequality. From the second…
Xiangru Lian
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If f' is Lipschitz, what about f?

Let $\|F'(\boldsymbol{\beta_1})-F'(\boldsymbol{\beta_2})\|_2 \leq M \| \boldsymbol{\beta_1}-\boldsymbol{\beta_2} \| _2,$ $\forall \boldsymbol{\beta_1},\boldsymbol{\beta_2} \in \overline{S(\boldsymbol{\beta}^*,\delta)},$…
john
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Jacobian determinant of Lipschitz function

Given a L-Lipschitz function $f:X \subseteq \mathbb{R}^n \to Y \subseteq \mathbb{R}^n$ is it true that $\det(J_xf) \leq L^n$?
Loreno Heer
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$\text{sign}(x)$ and one-sided Lipschitz

We say that a function $f:R^2\longrightarrow R$ satisfies one-sided Lipschitz condition with respect to x with constant $K$ if $$\langle f(x_{1},y)-f(x_{2},y),x_{1}-x_{2}\rangle \leq K||x_{1}-x_{2}||^2.$$ And my question is how can we check if this…
J. Mccain
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pointwise limit and continuity

having problems with this question. not sure where to start. Dont understand how is the monotone convergence theorem s helpful in here
ann
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Lipshcitz conditions for $f(x, y) = y^{1/2}$ and existence/uniqueness of differential problem.

(1) show that $f(x, y) = y^{1/2}$ does not satisfy a Lipschitz condition on the rectangle $x ∈ [−1, 1], y ∈ [0, 1]$; (2) show that $f(x, y) = y^{1/2}$ does satisfy the Lipschitz condition on the rectangle $x ∈ [−1, 1], y ∈ [c, d]$, where $0 < c <…
Joseph Aleshaiker
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