Is There A Generalization Of Brouwer's Fixed Point Theorem?

What are some applications of this theorem? I've already seen it in the proof of the Picard–Lindelöf theorem and the Inverse function theorem. So any applications besides those two would be good. Thanks!

(scalar version) Let Ф: Ą ---> Ą be a contraction with constant k <1 then there exist a unique fixed point X* = Ф(x*) must be Ą show that the contraction constant k is given by k = max |Ф' (ε) | (max &epsilon...

Given a right tetrahedron, where the right angles of 3 right triangle faces share a common vertex, let A, B, C be the areas of those 3 right triangle faces, and D be the area of the 4th face. Then the 3D generalization of the Pythagorean theorem is:...

Help show me that Gödel's theorem isn't that big of a deal. A while back, I was surprised by: Infinity--more precisely the axiom of infinity--stalks every page. This axiom says that the collection of all natural numbers exists as a set,...

I know that A right triangle has line symmetry, but does It have Point Symmetry? The Def. of Point Symmetry is Point of rotation (according to The Teacher). Point of Rotation: A figure in a plane has rotational symmetry if the figure can be mapped onto...

If 'P' be a moving point on the ellipse x^2/25 + y^2/16 =1 in such a way that the tangent at 'P' intersect x = 25/3 at Q then circle on PQ as diameter passes through a fixed point.Find the fixed point.

One of the strategies that we use to facilitate learning and understanding foreign (as in unknown) concepts is generalization and association. We create schemas or mental maps. I think this is a wonderful, effective, built-in tool. But I think that in...