Brouwer fixed point in one dimension
Proof (Following ) We can assume that and , since otherwise there is nothing to prove. Then, consider the function defined by . It satisfies
so by the intermediate value theorem, there is a point such that , i.e., .
Theorem 2 Suppose is a function that satisfies the following condition:
for some constant , we have for each ,
Then has a unique fixed point in . In other words, there exists one and only one point such that .
Remarks The fixed point in Theorem 2 can be found by iteration from any as follows: first choose some . Then form , then , and generally . As , approaches the fixed point for . More details are given on the entry for the Banach fixed point theorem. A function that satisfies the condition in Theorem 2 is called a contraction mapping. Such mappings also satisfy the Lipschitz condition (http://planetmath.org/LipschitzCondition).
- 1 A. Mukherjea, K. Pothoven, Real and Functional analysis, Plenum press, 1978.
|Title||Brouwer fixed point in one dimension|
|Date of creation||2013-03-22 13:46:25|
|Last modified on||2013-03-22 13:46:25|
|Last modified by||mathcam (2727)|