In mathematics, the Banach–Caccioppoli fixed-point 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 and Renato Caccioppoli, and was first stated by Banach in 1922. Caccioppoli independently proved the theorem in 1931.
Banach Fixed Point Theorem. Let be a non-emptycomplete metric space with a contraction mapping. Then T admits a unique fixed-pointx* in X. Furthermore, x* can be found as follows: start with an arbitrary element x0 in X and define a sequence by xn = T for n ≥ 1. 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 T, and the smallest one is sometimes called "the best Lipschitz constant" of T. Remark 2.d, T) < d → fixed point. However, if X 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 d, indeed, a minimizer exists by compactness, and has to be a fixed point of T. It then easily follows that the fixed point is the limit of any sequence of iterations of T. Remark 3. When using the theorem in practice, the most difficult part is typically to define X properly so that T ⊆ X.
Proof
Let x0 ∈ X be arbitrary and define a sequence by setting xn = T. We first note that for all n ∈ N, we have the inequality This follows [by induction">fixed point (mathematics)">fixed point. However, if X 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 d, indeed, a minimizer exists by compactness, and has to be a fixed point of T. It then easily follows that the fixed point is the limit of any sequence of iterations of T. Remark 3. When using the theorem in practice, the most difficult part is typically to define X properly so that T ⊆ X.
about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.
One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are bi-lipschitz homeomorphisms. Let Ω be an open set of a Banach spaceE; let I : Ω → E denote the identity map and let g : Ω → E be a Lipschitz map of constant k < 1. Then
Ω′ := is an open subset of E: precisely, for any x in Ω such that B ⊂ Ω one has B, r) ⊂ Ω′;
I+g : Ω → Ω′ is a bi-lipschitz homeomorphism;
It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third order method.
It can be used to prove existence and uniqueness of solutions to integral equations.
Several converses of the Banach contraction principle exist. The following is due to Czesław Bessaga, from 1959: Let f : X → X be a map of an abstract set such that each iteratefn has a unique fixed point. Let q ∈, then there exists a complete metric on X such that f is contractive, and q is the contraction constant. Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if f : X → X is a map on a T1topological space with a unique fixed point a, such that for each x in X we have fn → a, then there already exists a metric on Xwith respect to which f satisfies the conditions of the Banach contraction principle with contraction constant 1/2. In this case the metric is in fact an ultrametric.
Generalizations
There are a number of generalizations. Let T : X → X be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are:
Assume that some iterate Tn of T is a contraction. Then T has a unique fixed point.
Assume that for each n, there existcn such that d, Tn) ≤ cnd for all x and y, and that
In applications, the existence and unicity of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map T a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on fixed point theorems in infinite-dimensional spaces for generalizations. A different class of generalizations arise from suitable generalizations of the notion of metric space, e.g. by weakening the defining axioms for the notion of metric. Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.
