Banach fixed point theorem

Statement

Definition. Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that

d ( T ( x ) , T ( y ) ) ≤ q d ( x , y )

for all x, y in X.

Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x₀ in X and define a sequence {x_n} by x_n = T(x_n−1), then x_n → x*.

Remark 1. The following inequalities are equivalent and describe the speed of convergence:

d ( x ∗ , x n ) ≤ q n 1 − q d ( x 1 , x 0 ) , d ( x ∗ , x n + 1 ) ≤ q 1 − q d ( x n + 1 , x n ) , d ( x ∗ , x n + 1 ) ≤ q d ( x ∗ , x n ) .

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(x), T(y)) < d(x, y) for all x ≠ y is in general not enough to ensure the existence of a fixed point, as is shown by the map T : [1, ∞) → [1, ∞), T(x) = x + 1/x, which lacks a 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(x, T(x)), 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) ⊆ X.

Banach's original proof

Let x₀ ∈ (X, d) be arbitrary and define a sequence {x_n} by setting: x_n = T(x_n−1). Banach's original proof can be broken down into several Lemmas:

Lemma 1. For all n ∈ N, d(x_n+1, x_n) ≤ qⁿd(x₁, x₀).

Proof. We will proceed using induction, the base of the induction (n = 1) holds:

d ( x 1 + 1 , x 1 ) = d ( x 2 , x 1 ) = d ( T ( x 1 ) , T ( x 0 ) ) ≤ q d ( x 1 , x 0 ) .

Suppose the statement holds for some k ∈ N. Then we have

d ( x ( k + 1 ) + 1 , x k + 1 ) = d ( x k + 2 , x k + 1 ) = d ( T ( x k + 1 ) , T ( x k ) ) ≤ q d ( x k + 1 , x k ) ≤ q q k d ( x 1 , x 0 ) induction hypothesis = q k + 1 d ( x 1 , x 0 ) .

By the principle of mathematical induction, for all n ∈ N, the Lemma is proven.

Lemma 2. {x_n} is a Cauchy sequence in (X, d) and hence converges to a limit x* in X.

Proof. Let m, n ∈ N such that m > n.

d ( x m , x n ) ≤ d ( x m , x m − 1 ) + d ( x m − 1 , x m − 2 ) + ⋯ + d ( x n + 1 , x n ) triangle inequality ≤ q m − 1 d ( x 1 , x 0 ) + q m − 2 d ( x 1 , x 0 ) + ⋯ + q n d ( x 1 , x 0 ) Lemma 1 = q n d ( x 1 , x 0 ) ∑ k = 0 m − n − 1 q k ≤ q n d ( x 1 , x 0 ) ∑ k = 0 ∞ q k = q n d ( x 1 , x 0 ) ( 1 1 − q ) geometric series

Let ε > 0 be arbitrary, since q ∈ [0, 1), we can find a large N ∈ N so that

q N < ε ( 1 − q ) d ( x 1 , x 0 ) .

Therefore, by choosing m, n large enough we may write:

d ( x m , x n ) ≤ q n d ( x 1 , x 0 ) ( 1 1 − q ) < ( ε ( 1 − q ) d ( x 1 , x 0 ) ) d ( x 1 , x 0 ) ( 1 1 − q ) = ε .

Since ε > 0 was arbitrary this proves that sequence is Cauchy. By completeness, then, there must be a point x* ∈ X that the sequence converges to.

Lemma 3. x* is a fixed point of T.

Proof. Take the limit of both sides of the recurrence x_n = T(x_n−1),

lim n → ∞ x n = lim n → ∞ T ( x n − 1 )

Since T is a contraction mapping, it is continuous, so we may take the limit inside:

lim n → ∞ x n = T ( lim n → ∞ x n − 1 ) .

Thus, x* = T(x*).

Lemma 4. x* is the only fixed point of T in (X, d).

Proof. Suppose y also satisfies T(y) = y. Then

0 ≤ d ( x ∗ , y ) = d ( T ( x ∗ ) , T ( y ) ) ≤ q d ( x ∗ , y ) .

Remembering that q ∈ [0, 1), the above implies that 0 ≤ (1−q)d(x*, y) ≤ 0, which shows that d(x*, y) = 0, whence by positive definiteness, x* = y.

Shorter proof

Now we present a simpler proof that appeared recently in the Journal of Fixed Point Theory and its Application (see reference).

By the triangle inequality, for all x, y in X,

d ( x , y ) ≤ d ( x , T ( x ) ) + d ( T ( x ) , T ( y ) ) + d ( T ( y ) , y ) ≤ d ( x , T ( x ) ) + q d ( x , y ) + d ( T ( y ) , y )

solving for d(x, y) we get the ``Fundamental Contraction Inequality":

d ( x , y ) ≤ d ( T ( x ) , x ) + d ( T ( y ) , y ) 1 − q ,

and we note that if x and y are both fixed points then this implies that d(x, y) = 0, so x = y, proving that T has at most one fixed point. Now define the mapping Tⁿ by composing T with itself n times and note by induction that it satisfies a Lipschitz condition with constant qⁿ. It remains to show that for any x₀ in X, the sequence {Tⁿ(x₀)} is Cauchy and so converges to a point x* of X, which as noted above is clearly a fixed point of T. If in the Fundamental Inequality we replace x and y by Tⁿ(x₀) and T^m(x₀), we find that

d ( T n ( x 0 ) , T m ( x 0 ) ) ≤ d ( T ( T n ( x 0 ) ) , T n ( x 0 ) ) + d ( T ( T m ( x 0 ) ) , T m ( x 0 ) ) 1 − q , = d ( T n ( T ( x 0 ) ) , T n ( x 0 ) ) + d ( T m ( T ( x 0 ) ) , T m ( x 0 ) ) 1 − q ≤ q n d ( T ( x 0 ) , x 0 ) + q m d ( T ( x 0 ) , x 0 ) 1 − q = q n + q m 1 − q d ( T ( x 0 ) , x 0 )

since q < 1, the last expression converges to zero as n, m → ∞, proving that {Tⁿ(x₀)} is Cauchy. Note also that as m → ∞ gives us

d ( T n ( x 0 ) , x ∗ ) ≤ q n 1 − q d ( T ( x 0 ) , x 0 )

derived in the first proof that gives the rate at which {Tⁿ(x₀)} converges to x*.

Applications

A standard application is the proof of the Picard–Lindelöf theorem 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 space E; let I : Ω → E denote the identity (inclusion) map and let g : Ω → E be a Lipschitz map of constant k < 1. Then

1. Ω′ := (I+g)(Ω) is an open subset of E: precisely, for any x in Ω such that B(x, r) ⊂ Ω one has B((I+g)(x), r(1−k)) ⊂ Ω′;
2. I+g : Ω → Ω′ is a bi-lipschitz homeomorphism;

precisely, (I+g)⁻¹ is still of the form I + h : Ω → Ω′ with h a Lipschitz map of constant k/(1−k). A direct consequence of this result yields the proof of the inverse function theorem.

Converses

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 iterate fⁿ has a unique fixed point. Let q ∈ (0, 1), 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 T₁ topological space with a unique fixed point a, such that for each x in X we have fⁿ(x) → a, then there already exists a metric on X with 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 as immediate corollaries, which are of some interest for the sake of applications. Let T : X → X be a map on a complete non-empty metric space.

Assume that some iterate Tⁿ of T is a contraction. Then T has a unique fixed point.

Assume that T is a continuous function, and for all x and y in X,

Then T has a unique fixed point.

However, in most applications the existence and unicity of a fixed point 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.