On the fixed point theorem of Brouwer
I. The fundamental result in the direction fixed point theorems of compact type is the famous theorem of LEJ Brouwer (1912). This theorem is usual in applications of the analysis, as it has influenced the development of several branches mathematics, particularly topology algébrique.Ce theorem has undergone several generalizations in Banach spaces of infinite dimension alone, mostly by Schauder (1930) and Darbo (1955) (by the technical concept of the extent of non-compactness ).
Both versions of Brouwer's theorem is formulated as follows:
Theoreme1. Let E be a
Banach space of finite dimension and K is a subset of E, non-empty, convex and compact. Then any continuous mapping f of K in K has at least one fixed point.
Theoreme2.
Let ℝ (n) Euclidean space and B the closed unit ball. Then any continuous mapping f from B to B admits at least one fixed point.
It easily demonstrated that these two versions are equivalent.
The proof of the theorem Brouwer's fixed point in the real line is easy for the intermediate value theorem:
Let f be a continuous mapping of [a, b] in [a, b], where [a, b] is a bounded closed interval of ℝ. Then f has at least one fixed point.
But the demonstration in ℝ (n) is not immédiate.C is the important outcome of the theory of fixed point the oldest and most delicate démontrer.On remember that there are several methods to demonstrate this theorem, we note that its proof is an immediate consequence of the theory of algebraic topology and the theory of degree topologique.Aussi can be read by direct demonstration of simplex method and the lemma of non-retraction.
II.Remarques.Exemples.
i) In the théorème1 it Clearly
Fix (f) = {x in K: f (x) = x} is compact,
but it is not necessarily reduced to a point, as shown in the following examples:
a) E = ℝ , K = [0,1] and f (x) = x ^ (2). Fix (f) = {0,1}
b) E = ℝ, K = [-1,1] and f (x) = x in S defined by Tx =-x; possible fixed point is the vector 0 which does not belong to S.
b) K is only bounded and convex, as an example, we take E = ℝ (2), equipped with the standard Euclidean
and K = Int (B) (= the open unit disk), and we consider the application
T defined on Int (B)
T (x, y) = ((x + [(1-y ^ 2)] ^ (1 / 2)), y), the only fixed point possible is the vector (x, y) belongs to S.
c) K is only convex and closed.
As an example, we take E = ℝ, ℝ ^ = K (+) and we consider applications that stabilize continuous
ℝ ^ (+) and admitting no fixed point, such as applying exp (x).
iii) The following example, due to Kakutani (1943) shows that if E is infinite dimensional Brouwer's theorem is not valid.
Let the space of sequences (2) with the usual standard:
For x = {x (n)} , X (1), x (2), ...)
It is obvious that B is stable by T, T is continuous and Fix (T) is empty (checked).
Nota.Pour article references the authors cited in this text, see the message in this blog (References 02/02/2007).
