# Tag Info

Accepted

### Showing that $\mathbf{X}^{2} + \mathbf{X} = \mathbf{A}$ has a solution

Your approach is not correct, for the simple reason that you haven't showed $\Phi$ is a Banach contraction. You have provided upper bounds for both $\|X - Y\|_\infty$ and $\|\Phi(X) - \Phi(Y)\|_\infty$...
• 44.6k
Accepted

### Using Banach's Fixed Point Theorem on an Integral Equation

Let $A\colon \Bbb R\to \Bbb R$, $A(x)=\frac x{\sqrt{h^2+x^2}}$ and $$F\colon C([0,1])\to C([0,1]),\quad F(f)(x)=\int_0^1 K(x,y)A(f(y))dy.$$ Observe that $A$ is Lipschitz function satisfying with a ...
• 4,430
Accepted

### Is there a constructive proof of Brouwer's fixed-point theorem that does not rely on triangulation?

There is no constructive proof of Brouwer’s fixed point theorem at all. In particular, the following is not provable constructively: Intermediate Value Theorem: Let $f : [-1, 1] \to \mathbb{R}$ be a ...
• 22.5k