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Concept
Banach fixed-point theorem
Summary
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.
Definition. Let be a complete metric space. Then a map is called a contraction mapping on X if there exists such that
for all
Banach Fixed Point Theorem. Let be a non-empty complete metric space with a contraction mapping Then T admits a unique fixed-point in X (i.e. ). Furthermore, can be found as follows: start with an arbitrary element and define a sequence by for Then .
Remark 1. The following inequalities are equivalent and describe the speed of convergence:
Any such value of q is called a Lipschitz constant for , and the smallest one is sometimes called "the best Lipschitz constant" of .
Remark 2. for all is in general not enough to ensure the existence of a fixed point, as is shown by the map
which lacks a fixed point. However, if is compact, then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of , indeed, a minimizer exists by compactness, and has to be a fixed point of . It then easily follows that the fixed point is the limit of any sequence of iterations of .
Remark 3. When using the theorem in practice, the most difficult part is typically to define properly so that
Let be arbitrary and define a sequence by setting xn = T(xn−1). We first note that for all we have the inequality
This follows by induction on n, using the fact that T is a contraction mapping. Then we can show that is a Cauchy sequence. In particular, let such that m > n:
Let ε > 0 be arbitrary.
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