Banach fixed-point theorem

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.

Statement

Definition. Let

(

X

,

d

)

{\displaystyle (X,d)}

be a metric space with metric

d

(

x

,

y

)

{\displaystyle d(x,y)}

. Then a map

T

:

X

→

X

{\displaystyle T:X\to X}

is called a contraction mapping on

X

{\displaystyle X}

if there exists a nonnegative constant

q

<

1

{\displaystyle q<1}

such that

d

(

T

(

x

)

,

T

(

y

)

)

≤

q

d

(

x

,

y

)

{\displaystyle d(T(x),T(y))\leq q\,d(x,y)}

for all

x

,

y

∈

X

.

{\displaystyle x,y\in X.}

Banach fixed-point theorem. Let

(

X

,

d

)

{\displaystyle (X,d)}

be a non-empty complete metric space with a contraction mapping

T

:

X

→

X

.

{\displaystyle T:X\to X.}

Then

T

{\displaystyle T}

admits a unique fixed point

x

∗

∈

X

{\displaystyle x^{*}\in X}

, meaning

T

(

x

∗

)

=

x

∗

{\displaystyle T(x^{*})=x^{*}}

. Furthermore,

x

∗

{\displaystyle x^{*}}

can be found as follows: start with an arbitrary element

x

0

∈

X

{\displaystyle x_{0}\in X}

and define

x

n

=

T

(

x

n

−

1

)

{\displaystyle x_{n}=T(x_{n-1})}

for

n

≥

1.