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.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related concepts (10)
Related courses (27)
Related lectures (239)
Related MOOCs (5)
Ce cours contient les 7 premiers chapitres d'un cours d'analyse numérique donné aux étudiants bachelor de l'EPFL. Des outils de base sont décrits dans les chapitres 1 à 5. La résolution numérique d'éq
Ce cours contient les 7 premiers chapitres d'un cours d'analyse numérique donné aux étudiants bachelor de l'EPFL. Des outils de base sont décrits dans les chapitres 1 à 5. La résolution numérique d'éq
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number such that for all x and y in M, The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces.
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions.
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.
This course covers the statistical physics approach to computer science problems ranging from graph theory and constraint satisfaction to inference and machine learning. In particular the replica and
Ce cours introduit les systèmes dynamiques pour modéliser des réseaux biologiques simples. L'analyse qualitative de modèles dynamiques non-linéaires est développée de pair avec des simulations numériq