Math 2.0

Operator theory Nonlinear operators and fixed points Theorem

Banach fixed point theorem

Builds on Contraction mappingComplete metric spaceEpsilon characterisation of Cauchy sequences

Statement

Let (X,d)(X, d) be a nonempty complete metric space and let f ⁣:X→Xf \colon X \to X be a contraction with constant k∈[0,1)k \in [0, 1). Then ff has exactly one fixed point x∗x^*, that is, a point with f(x∗)=x∗f(x^*) = x^*. Moreover, for each x0∈Xx_0 \in X, the sequence xn+1=f(xn)x_{n+1} = f(x_n) converges to x∗x^*, and d(xn,x∗)≤kn1−k d(x1,x0)for all n≥0.d(x_n, x^*) \le \frac{k^n}{1 - k} \, d(x_1, x_0) \quad \text{for all } n \ge 0.

Remarks

Proof idea: d(xn+1,xn)≤knd(x1,x0)d(x_{n+1}, x_n) \le k^n d(x_1, x_0), so the sequence (xn)(x_n) is a Cauchy sequence by a geometric series bound. Its limit is a fixed point because ff is continuous. Two fixed points x,yx, y satisfy d(x,y)≤k d(x,y)d(x, y) \le k \, d(x, y), so x=yx = y. Completeness is needed: x↦x/2x \mapsto x/2 on (0,1](0, 1] is a contraction without a fixed point. The theorem gives the existence of solutions of ordinary differential equations in the Picard–Lindelöf theorem.

History

Stefan Banach stated the theorem in 1922.

Reading path

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered 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 spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionContraction mapping, definition, no evidence levelContraction mappingdefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinitionBanach fixed point theorem, statement, no evidence levelBanach fixed point th…statement
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  • no evidence level
  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Supremum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Convergence of a sequence in a metric space definition
  20. Lipschitz map definition
  21. Diameter of a set definition
  22. Contraction mapping definition
  23. Epsilon characterisation of Cauchy sequences statement
  24. Complete metric space definition
  25. Banach fixed point theorem statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Banach_fixed-point_theorem, Statement
About this page
Tag0VK6VEH345
URIm2://0VK6VEH345
Aliasescontraction-mapping-theorem, banach-fixed-point-theorem
Kindstatement, theorem
WritingW1 stated
MSC47H10
StatusThis copy of the site has no build data. Run m2 status 0VK6VEH345.
UsesContraction mapping, Complete metric space, Epsilon characterisation of Cauchy sequences
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