p-Laplace equation – Fixed point approach
The -Laplace problem is the generalization of the usual Laplace problem and is defined for in the following way
We are not going into details regarding the functional spaces corresponding to and since in the end we are interested in showing a way to solve this equation numerically.
Shrinkable polygons
Here’s a nice problem inspired from a post on MathOverflow: link
We call a polygon shrinkable if any scaling of itself with a factor can be translated into itself. Characterize all shrinkable polygons.
It is easy to see that any star-convex polygon is shrinkable. Pick the point in the definition of star-convex, and any contraction of the polygon by a homothety of center lies inside the polygon.
Agreg 2012 Analysis Part 4
A Fixed Point Theorem
This parts wishes to extend the next result (which can be used without proof) to infinite dimension.
Theorem. (Browuer) Consider a finite dimensional normed vector space. Consider a convex, closed, bounded non-void set. If is a continuous application, then has a fixed point in .
1. In endowed with we consider the following application:
Prove that is continuous with values in the unit sphere of , but does not admit any fixed points.
2. Consider a normed vector space, a closed, bounded non-void subset of and a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)
i) Let . We can cover (which is compact) by a finite number of open balls of radius : with for every . For we define
Prove that is continuous and that there exists such that for we have .
ii) We introduce the application defined by
Prove that for every we have .
Darboux Functions with no Iterate Fixed Points
It is well known that there exist functions which have Darboux property and they have no fixed points. An example can be found in an earlier post of mine. Here is a generalization of that result.
There exist functions which have Darboux property and for which none of its iterates has a fixed point, i.e. for every and for every .
Brouwer fixed point theorem
Let be an open subset of such that is homeomorphic to the closed unit ball ( in ). If (continuous function on ) and , then has a fixed point in .
The Existence of a Triangle with Prescribed Angle Bisector Lengths
Prove that for any there exists a unique triangle (up to an isometry) such that are the lengths of bisectors of this triangle
Solved by L. Panaitopol & P. Mironescu in 1994, AMM 101, 58-60
Read more…
Fixed point for an operator
Suppose is a Hilbert space and is a linear operator on with . Prove that if and only if .
Solution: Using the Cauchy-Buniakovski inequality, we get
. Since this implies equality in the C-B inequality, we must have . We easily find that . The converse is equivalent to the implication above.
Fixed point theorem
Suppose is a compact convex set and is a function satisfying . Prove that has a fixed point ( i.e. with ). Read more…