Banach fixed-point theorem

The fixed point theorem of Banach, also Banach fixed point theorem (after Stefan Banach ) is a set of mathematics. It contains an existence and uniqueness statement for fixed point problems and convergence results and error estimates a generated by the so-called fixed point iteration sequence.

An illustration of the theorem provides a map on which the environment in which one finds oneself, is shown. If you see this card as a contraction of the environment, so do I find a point on the map, which coincides with the point directly below it in the real world.

Statements of the fixed-point theorem

Existence and Uniqueness: A contraction of a ( non-empty ) complete metric space has exactly one fixed point, ie a point.

Where:

• For example, each Banach space, and among them every normalized finite real or complex vector space, a complete metric space,
• A contraction of an image, which is Lipschitz continuous with a constant.

Construction: converges for each starting value with the result against.

Error estimate: There are the following estimates for the distance of the fixed point for recursive sequence:

• A- priori estimate

• A posteriori estimation

Is the contraction constant or Lipschitz constant.

Note: Often, this sentence does not generally on (complete) metric spaces, but on a Banach space B, ie a complete normed vector space, or a subset M ⊆ B is formulated it. The only difference is that the distance is then declared by the norm of the difference.

For the proof

Idea of ​​proof for normed spaces

To understand the proof of transition, it is helpful to first operate in a Banach space. The actual proof steps can then be carried out also in the metric space. We construct the recursive sequence with any starting point. This we take as partial sums of a ( telescopic ) on series whose terms the differences are. From the series, we show that it has a geometric series as majorant. Because of the completeness of the space it follows the convergence of the series, the point is the desired fixed point.

Proof in the metric space

We consider the problem in complete metric space. Again we construct about the recursive sequence of points. Analogous to the majorant of the series, you can now try out a majorant of the sequence found in the sense that the growth of the x -series are limited by the growth of the b -series,

It follows then via repeated application of the triangle, that the distances between any two x - followers are constrained by the distance corresponding to the b sequence,

From the convergence of the b -series to a limit now follows directly that the x - sequence is a Cauchy sequence. This then has a limit because of the completeness of the metric space. The convergence to this limit can be estimated again, it is

It is therefore to be determined from the condition of the contractility of the fixed point mapping is a monotonically increasing convergent sequence whose growth outvote those of the x -series. It turns out that these gains may be selected as a geometric sequence, because of

Follows by induction

Thus in particular

Therefore, one can choose as a b- sequence the sequence of partial sums of a geometric series, more

This has the limit

From this it follows that, as indicated above, the convergence of the x - row against a threshold value. For the estimates we obtain

If one forms in this inequality the limit with respect to n, it follows

So is actually a fixed point.

Η is valid for another possible fixed point, which is possible only when, ie there is exactly one fixed point.

Applications

This phrase is used in many constructive sets of analysis, the most important are:

• The inverse and implicit functions theorem
• The existence and uniqueness theorem of Picard - Lindelöf for ordinary differential equations

In the numerical analysis, the fixed point plays an important role. In addition, the rate plays an important role in convergence theory of Newton's method, and other numerical methods, such as the splitting method.