Fixed point theorems concern maps f of a set X into itself that, under certain conditions, admit a fixed point, that is, a point x ∈ X such that f(x) = x. The knowledge of the existence of fixed points has relevant applications in many branches of analysis and topology. contraction must map a point to itself, that is it must have a fixed point, and even more, it is unique. Theorem 2 (Banach's Fixed Point Theorem) Let X be a. There is a nice general book by the MAA that is cheap called Fixed Point Theorems (I think). In my field they are used to prove existence and uniqueness of.
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Proof of Brouwer's fixed point theorem for a disk using the No Retraction Theorem.
Fixed Point Theorem -- from Wolfram MathWorld
Assume there is no fixed point theorem point and use the intersection of the line from x to f x with the boundary circle to map x into the boundary as shown below. Let point A be an arbitrary point in the disk.
Point B represents the point A is mapped to. A line is drawn from A to B and continued fixed point theorem it intersects the boundary of the disk at point C. Now the mapping under consideration is the mapping of A to C.
This would be a continuous mapping of the interior onto the boundary. The existence of such a mapping is a contradiction of the No Retraction Theorem so the process must break down somewhere.
Therefore, since the assumption of no fixed point theorem point leads to a contradiction of the No Retraction Theorem there must be at least one fixed point. A topological space that is such that any continuous mapping of it into itself must have a fixed point is said to have the fixed point property.
Not all topological spaces have the fixed point property. The annulus shown below does not have the fixed point property.
Fixed Point Theorem
Fixed point theorem can easily be established by noting that the continuous mapping which rotates the annulus has no fixed point. If there were no hole in the disk then the center of the disk would be the fixed point of any rotation. But the center of the disk is not part of the annulus.
A physical example of a fixed point of a mapping is the center of a whirlpool in a cup of tea when it is stirred. The fact that the center of the whirlpool moves over time is just due to the fact that the mapping fixed point theorem changing over time.
There is a related problem involving the mapping of the points of a sphere into itself. This is a closed surface but not a convex surface. The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.
Fixed Point Theorems
A common theme in lambda calculus is to find fixed points of given lambda expressions. Every lambda expression has a fixed point, and a fixed-point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression.
In denotational semantics of programming languages, a special case of the Knaster—Tarski theorem is used to establish the semantics of recursive fixed point theorem. While the fixed-point theorem is applied to the "same" function from a logical point of viewthe development of the theory is quite fixed point theorem.
The same definition of recursive function can be given, in computability theoryby applying Kleene's recursion theorem.