MHB Banach fixed-point theorem : Existence of solution

AI Thread Summary
The discussion focuses on applying the Banach fixed-point theorem to demonstrate the existence of a solution for the system defined by the equations involving \(x_1\) and \(x_2\) within the set \(G\). Participants explore the properties of the mapping \(\Phi\) and its Jacobian matrix to establish an upper bound for the maximum norm of the Jacobian, ultimately aiming to show that \(\Phi(G) \subseteq G\). They calculate bounds for the components of the Jacobian based on the constraints of \(G\), confirming that the maximum is less than 1, indicating that \(\Phi\) is a contraction mapping. The conclusion drawn is that these conditions imply a solution exists within the set \(G\). The discussion effectively illustrates the application of the theorem in this context.
mathmari
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Hey! :o

We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\infty}\leq 0.2\}$ where $\vec{c}=(0.2,1)^T$.

I want to show with the Banach fixed-point theorem that the system has a solution in $G$.

I have done the following:

Let \begin{equation*}\Phi (x_1, x_2)=\begin{pmatrix}\left (5+x_1^2+x_2^2\right )^{-1} \\ \left (x_1+x_2\right )^{\frac{1}{4}}\end{pmatrix}\end{equation*} The jacobi matrix is \begin{equation*}\nabla \Phi =\begin{pmatrix}-\frac{2x_1}{(x_1^2+x_2^2+5)^2} & -\frac{2x_2}{(x_1^2+x_2^2+5)^2} \\ \frac{1}{4(x_1+x_2)^{\frac{3}{4}}} & \frac{1}{4(x_1+x_2)^{\frac{3}{4}}} \end{pmatrix}\end{equation*}

Then we get \begin{equation*}\|\nabla \Phi \|_{\infty}=\max \left \{\frac{2x_1+2x_2}{(x_1^2+x_2^2+5)^2}, \frac{1}{2(x_1+x_2)^{\frac{3}{4}}}\right \}\end{equation*}

We want to get an upper bound for that, don't we?

How can we do that? Could you give me a hint?

(Wondering)
 
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Hey mathmari!

Can't we find an upper bound by using that $(x_1,x_2)$ is in $G$?
That is, $0\le x_1 \le 0.4$ and $0.8\le x_2 \le 1.2$. (Thinking)
 
Klaas van Aarsen said:
Can't we find an upper bound by using that $(x_1,x_2)$ is in $G$?
That is, $0\le x_1 \le 0.4$ and $0.8\le x_2 \le 1.2$. (Thinking)

We have that $$\|\vec{x}-\vec{c}\|_{\infty}\leq 0.2 \Rightarrow \|(x_1-0.2, x_2-1)\|_{\infty}\leq 0.2$$ That means that both are less than $0.2$, or not?
So we get $|x_1-0.2|\leq 0.2$ and $|x_2-1|\leq 0.2$, right? (Wondering)
 
mathmari said:
We have that $$\|\vec{x}-\vec{c}\|_{\infty}\leq 0.2 \Rightarrow \|(x_1-0.2, x_2-1)\|_{\infty}\leq 0.2$$ That means that both are less than $0.2$, or not?
So we get $|x_1-0.2|\leq 0.2$ and $|x_2-1|\leq 0.2$, right?

Yep. (Nod)
 
We have $$\frac{2x_1+2x_2}{(x_1^2+x_2^2+5)^2}\leq 2x_1+2x_2\leq 3.2 $$ or not? What about the second term? (Wondering)
 
mathmari said:
We have $$\frac{2x_1+2x_2}{(x_1^2+x_2^2+5)^2}\leq 2x_1+2x_2\leq 3.2 $$ or not? What about the second term?

Can't we do a little better?
$$\frac{2x_1+2x_2}{(x_1^2+x_2^2+5)^2}
\le \frac{2\cdot 0.4 + 2\cdot 1.2}{(0^2+0.8^2+5)^2} < 0.1006
$$
(Thinking)
 
Klaas van Aarsen said:
Can't we do a little better?
$$\frac{2x_1+2x_2}{(x_1^2+x_2^2+5)^2}
\le \frac{2\cdot 0.4 + 2\cdot 1.2}{(0^2+0.8^2+5)^2} < 0.1006
$$
(Thinking)

Ah yes! And for the other one we have $$\frac{1}{2(x_1+x_2)^{\frac{3}{4}}}\leq \frac{1}{2(0+0.8)^{\frac{3}{4}}}=0.591089$$

So the maximum is less than $0.591089$ which is less than $1$. Does this mean that there is a solution is $G$ ? (Wondering)
 
mathmari said:
Ah yes! And for the other one we have $$\frac{1}{2(x_1+x_2)^{\frac{3}{4}}}\leq \frac{1}{2(0+0.8)^{\frac{3}{4}}}=0.591089$$

So the maximum is less than $0.591089$ which is less than $1$. Does this mean that there is a solution is $G$ ?

Well, wiki lists:

Banach Fixed Point Theorem. Let (X,d) be a non-empty complete metric space with a contraction mapping $T\colon X\to X$. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element $x_0$ in X and define a sequence $\{x_n\}$ by $x_n = T(x_{n-1})$, then xn → x*.


So if we consider $G$ to be the set of the metric space and $\Phi$ the mapping, then we would need that $\Phi(G)\subseteq G$, wouldn't we? (Wondering)
And we also need to prove that $\Phi$ is a contraction mapping, don't we? (Thinking)
 
Klaas van Aarsen said:
Well, wiki lists:

Banach Fixed Point Theorem. Let (X,d) be a non-empty complete metric space with a contraction mapping $T\colon X\to X$. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element $x_0$ in X and define a sequence $\{x_n\}$ by $x_n = T(x_{n-1})$, then xn → x*.


So if we consider $G$ to be the set of the metric space and $\Phi$ the mapping, then we would need that $\Phi(G)=G$, wouldn't we? (Wondering)
And we also need to prove that $\Phi$ is a contraction mapping, don't we? (Thinking)


So for these two do we have to show that $\Phi (G)\subseteq G$ ? (Wondering)
 
  • #10
mathmari said:
So for these two do we have to show that $\Phi (G)\subseteq G$ ?

I believe so yes. (Thinking)
 

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