Math 2.0

Operator theory Definition

Contraction mapping

Builds on Metric spaceLipschitz map

Definition

Let (X,d)(X, d) be a metric space. A map f ⁣:X→Xf \colon X \to X is a contraction when there is a real number kk with 0≤k<10 \le k < 1 such that d(f(x),f(y))≤k d(x,y)for all x,y∈X.d\bigl(f(x), f(y)\bigr) \le k \, d(x, y) \quad \text{for all } x, y \in X.

Remarks

A contraction is a Lipschitz map with a constant less than 11. Thus it is uniformly continuous. The map x↦x/2x \mapsto x/2 on ℝ\mathbb{R} is a contraction with k=1/2k = 1/2. The condition d(f(x),f(y))<d(x,y)d(f(x), f(y)) < d(x, y) for all x≠yx \neq y is weaker. The map x↦x+1/xx \mapsto x + 1/x on [1,∞)[1, \infty) satisfies it, but it is not a contraction and has no fixed point.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionContraction mapping, definition, no evidence levelContraction mappingdefinition
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  10. Cartesian product definition
  11. Subset definition
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  13. Binary relation definition
  14. Function definition
  15. Metric space definition
  16. Lipschitz map definition
  17. Contraction mapping definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Contraction_mapping, lead section
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DefinesContraction mapping
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