Lipschitz continuity of vector-valued function

In summary, Lipschitz continuity of vector-valued functions refers to a property where a function maps vectors to vectors and satisfies a specific boundedness condition on the rate of change between input and output values. Specifically, a function \( f: \mathbb{R}^n \to \mathbb{R}^m \) is Lipschitz continuous if there exists a constant \( L \) such that for all \( x, y \in \mathbb{R}^n \), the inequality \( \| f(x) - f(y) \| \leq L \| x - y \| \) holds. This condition ensures that the function does not oscillate too rapidly, providing a uniform control over its
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I have a question about a proof on Lipschitz continuity of a vector-valued function.
I'm reading Ordinary Differential Equations by Andersson and Böiers, although this is more related to multivariable calculus. There is a Lemma regarding Lipschitz continuity which I have a question about. Below ##\pmb{f}:\mathbf{R}^{n+1}\to \mathbf{R}^n ## is a vector-valued function defined by ##\pmb{f}=\pmb{f}(t,\pmb{x})##, where ##\pmb x## is a vector in ##\mathbf{R}^n## that depends on ##t##.

Lemma. Assume that ##\Omega\subseteq \mathbf{R}\times\mathbf{R}^n## is a convex and bounded set, and that the function ##\pmb{f}## is continuously differentiable in a neighborhood of ##\overline{\Omega}##. Then ##\pmb{f}## is Lipschitz continuous in ##\Omega##.

Proof. The line segment between two points ##(t,\pmb{x})## and ##(t,\pmb{y})## in ##\Omega## is contained in ##\Omega## by the assumption of convexity. Hence \begin{align} \pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})&=\int_0^1\frac{\mathrm{d}}{\mathrm{d}s}\pmb{f}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))\mathrm{d}s \tag1 \\ &=\int_0^1\sum_{i=1}^n\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))(x_i-y_i)\mathrm{d}s,\tag2\end{align} by the chain rule. Set ##K=\max_i\sup_{\Omega}\left|\frac{\partial f}{\partial x_i}\right|## (which exists by assumption). Then $$|\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})|\leq\int_0^1 nK|\pmb{x}-\pmb{y}|\mathrm{d}s=nK|\pmb{x}-\pmb{y}|.\tag3$$

Here's my understanding of the notation. In ##(2)##, ##\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))## is a vector evaluated at ##(t,\pmb{y}+s(\pmb{x}-\pmb{y}))##. ##\frac{\partial}{\partial x_i}## denotes the partial derivative of ##\pmb{f}(t,x_1,\ldots,x_n)## with respect ##x_i## (a better notation in ##(2)## would probably be ##\frac{\partial}{\partial z_i}##, as ##\pmb{x}## is already being used to denote a vector). More precisely, if ##f_1,\ldots,f_n## are the components of ##\pmb{f}##, then $$\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))=\begin{pmatrix}
\frac{\partial f_1}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y})) \\ \vdots \\ \frac{\partial f_n}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))
\end{pmatrix}.\tag4$$

My question is that I do not really understand how ##(3)## is obtained. For the sake of brevity, I will write ##\frac{\partial \pmb{f}}{\partial x_i}## when I mean ##\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))##. Suppose ##n=2##. Taking the norm (##\ell^2## norm to be exact) of ##(1)## and using ##\left\lVert\int_a^b \pmb{f}\right\rVert\le\int_a^b\left\lVert\pmb{f}\right\rVert##, we get
\begin{align}
\lVert\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})\rVert_2&=\left\lVert\int_0^1\sum_{i=1}^2\frac{\partial \pmb{f}}{\partial x_i}(x_i-y_i)\mathrm{d}s\right\rVert_2 \tag5\\
&\leq \int_0^1\left\lVert\sum_{i=1}^2\frac{\partial \pmb{f}}{\partial x_i}(x_i-y_i) \right\rVert_2 \mathrm{d}s \tag6
\end{align}
Next, using the triangle inequality, the norm inside the integral gets bounded by
\begin{align}
\left\lVert\sum_{i=1}^2\frac{\partial \pmb{f}}{\partial x_i}(x_i-y_i) \right\rVert_2 &\leq\left\lVert \frac{\partial \pmb{f}}{\partial x_1}(x_1-y_1)\right\rVert_2+\left\lVert \frac{\partial \pmb{f}}{\partial x_2}(x_2-y_2)\right\rVert_2. \tag7
\end{align}
The above terms on the right-hand side equal
\begin{align}
\sqrt{\left(\frac{\partial f_1}{\partial x_1}\right)^2+\left(\frac{\partial f_2}{\partial x_1}\right)^2}|x_1-y_1|+\sqrt{\left(\frac{\partial f_1}{\partial x_2}\right)^2+\left(\frac{\partial f_2}{\partial x_2}\right)^2}|x_2-y_2|.\tag8
\end{align}
Letting ##K=\max\limits_{i,j\in\{1,2\}}\sup\limits_{\Omega}\left|\frac{\partial f_i}{\partial x_j}\right|##, ##(8)## gets bounded by
\begin{align}
\sqrt{2}K|x_1-y_1|+\sqrt{2}K|x_2-y_2|=\sqrt{2}K(|x_1-y_1|+|x_2-y_2|) \tag9
\end{align}
which is clearly not equal to ##nK|\pmb{x}-\pmb{y}|##. I suspect I am doing something wrong, but I'm unable to spot it.
 
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  • #2
A basic upper bound for the value of an integral of [itex]F: [a,b] \to \mathbb{R}[/itex] is [tex]
\int_a^b F(s)\,ds \leq (b- a)\sup_{s \in [a,b]} F(s).[/tex] By extension, given [tex]I = \int_0^1 \sum_{i=1}^n\|\mathbf{F}_i(t,s)\||x_i - y_i|\,ds[/tex] we can obtain [tex]\begin{split}
I &\leq \sum_{i=1}^n \sup_s \|\mathbf{F}_i\||x_i - y_i| \\
&\leq \left(\max_i \sup_s \|\mathbf{F}_i\|\right) \sum_{i=1}^n |x_i - y_i|.\end{split}[/tex] Now [tex]
\begin{split}
\sum_{i=1}^n |x_i - y_i| &= \left|(1, \dots, 1) \cdot (|x_1-y_1|, \dots, |x_n - y_n|)\right| \\
& \leq \| (1, \dots, 1) \| \|(|x_1-y_1|, \dots, |x_n - y_n|)\| \\
& = \sqrt{n} \|\mathbf{x} - \mathbf{y}\| \\
& \leq n \|\mathbf{x} - \mathbf{y}\|.\end{split}[/tex] so that [tex]
I \leq \left(\max_i \sup_s \|\mathbf{F}_i\|\right) \sum_{i=1}^n |x_i - y_i| \leq \left(\max_i \sup_s \|\mathbf{F}_i\|\right) n \|\mathbf{x} - \mathbf{y}\|.[/tex]
 
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  • #3
Thank you for replying.

In the statement of the lemma, the authors stipulate that ##\pmb f## be ##C^1## in a neighborhood of the closure of ##\Omega## (I interpret ##\overline{\Omega}## to be the closure of ##\Omega##). Do you by any chance know why this is required? I see why we need the function to be ##C^1##, since we are integrating its derivative, but I do not see why this condition needs to be satisfied in the closure of ##\Omega##.
 
  • #4
To guarantee boundedness of the derivatives we require that they be continuous (hence [itex]C^1[/itex]) on a compact domain. This is the generalisation of the theorem that a function continuous on a closed, bounded interval is bounded, but a function continuous on an unclosed bounded interval may not be bounded (eg. [itex]x^{-1}[/itex] on [itex](0,1][/itex]).

For the Euclidean metric on [itex]\mathbb{R}^m[/itex], a subset is compact if and only if it is closed and bounded. We are given that [itex]\Omega \subset \mathbb{R} \times \mathbb{R}^n \simeq \mathbb{R}^{n+1}[/itex] is bounded, but we are not told that it is closed; its closure, however, is certainly closed and bounded.
 
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FAQ: Lipschitz continuity of vector-valued function

What is Lipschitz continuity for vector-valued functions?

Lipschitz continuity for vector-valued functions extends the concept of Lipschitz continuity from real-valued functions. A vector-valued function \( f: \mathbb{R}^n \to \mathbb{R}^m \) is said to be Lipschitz continuous if there exists a constant \( L \geq 0 \) such that for all \( x, y \in \mathbb{R}^n \), the following inequality holds: \( \| f(x) - f(y) \| \leq L \| x - y \| \), where \( \| \cdot \| \) denotes a norm in \( \mathbb{R}^m \). This means that the function does not change too rapidly, and the rate of change is bounded by the Lipschitz constant \( L \).

How can we determine if a vector-valued function is Lipschitz continuous?

To determine if a vector-valued function is Lipschitz continuous, one approach is to compute the difference \( \| f(x) - f(y) \| \) for arbitrary points \( x \) and \( y \) in the domain. If you can find a constant \( L \) such that the inequality \( \| f(x) - f(y) \| \leq L \| x - y \| \) holds for all \( x, y \), then the function is Lipschitz continuous. Additionally, if the function is differentiable, you can check if the norm of its derivative (the Jacobian matrix) is bounded by \( L \) in the relevant domain.

What are some examples of Lipschitz continuous vector-valued functions?

Examples of Lipschitz continuous vector-valued functions include linear functions such as \( f(x) = Ax \) where \( A \) is a bounded linear operator, and functions like \( f(x) = (x, x^2) \) or \( f(x) = (e^x, \sin(x)) \) defined on compact intervals. In these cases, the Lipschitz constant can often be determined through the maximum rate of change of the function across its domain.

What is the significance of Lipschitz continuity in analysis?

Lipschitz continuity is significant in analysis because it provides a controlled way to measure the stability of a function. It ensures that small changes in the input lead to small changes in the output, which is crucial in various applications such as differential equations, optimization, and numerical methods. It also guarantees the existence of solutions to certain types of equations and helps in proving convergence properties of iterative methods.

Can Lipschitz continuity be

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