Banach Contraction Mapping Principle

Oksana Bihun

March 2, 2010

Department of Mathematics and Computer ScienceConcordia College, Moorhead, MN

Stefan Banach

Metric Spaces

Banach Contraction Mapping Principle

Applications to Ordinary Differential Equations

Applications to Numerical Solutions of Linear Systems

Stefan Banach

Born: March30, 1892, in Ostrowsko, nearby Krakow(now Poland). Never met his mother.

Father Stefan Greczek arranged for hisson to be brought up by Franciszka Plowa.

Received early education from aFrench intellectual Juliusz Mien. Graduatedfrom Henryk Sienkiewicz GymnasiumNo. 4 in 1910 without distinction.

Felt that nothing new can be discovered inmathematics and chose to study engineeringat the Lviv Polytechnic (1910-1916).

Lviv Lvov Lwow Lemberg Leopolis

Lviv was founded in 1256 by King Danylo Halytskiy of theRuthenian principality of Halych-Volhynia, and named in honor ofhis son, Lev (Lion).

Throughout history, Lviv belongedto Halych-Volynia, Lithuania,Poland, Austrian-HungarianEmpire, and SovietUnion. Now it belongs to Ukraine.

Asof 1910 Lviv and Krakow belongedto Austro-Hungarian Empire.

Banach and SteinhausIn his memoirs, Hugo Steinhaus wrote: “During one such walk Ioverheard the words“Lebesgue measure”. I approached the parkbench and introduced myself to the two young apprentices ofmathematics... From then on we would meet on a regular basis,and ... we decided to establish a mathematical society.”

Banach solved a problemSteinhaus posed, since then theywrote many papers together. In1920 Banach became an assistantto Lomnicki at Lviv Polytechnic.

Banach’s thesis “On Operationson Abstract Sets and theirApplication to Integral Equations”is sometimes said to markthe birth of Functional Analysis.

In 1922 Banach was awardedhabilitation by Lviv University.

The Scottish Cafe in Lviv

The cafe was a meeting place for many mathematicians includingBanach, Steinhaus, Ulam, Mazur, Kac, Schauder, Kaczmarz, andothers. Problems were written in a book kept by the landlord andoften prizes were offered for their solution. A collection of theseproblems appeared later as the Scottish Book.

R. D. Mauldin,The Scottish Book, Mathematicsfrom the Scottish Cafe (1981).

In 1936 Mazurposed an approximation problemand offered live goose to theone who solves it. The problem wassolved only in 1972 by a Swedishmathematician Per Enflo. He wasawarded a live goose in Wroclaw.

Metric Spaces

A metric space (X , d) is a set X together with a functiond : X × X → R that satisfies the properties

1. d(x , y) ≥ 0 for all x , y ∈ X ;

2. d(x , y) = 0 iff x = y ;

3. d(x , y) ≤ d(x , z) + d(z , y) for all x , y , z ∈ X (triangleinequality);

4. d(x , y) = d(y , x) for all x , y ∈ X .

The function d is called a metric on X . The metric on X gives away to measure distance on X .

Examples of Metric Spaces

1. (R2, d) with d(X ,Y ) =√

(y1 − x1)2 + (y2 − x2)2 forX = (x1, x2)T and Y = (y1, y2)T .

2. (S2, d), where S2 is the 2-sphere in R3 and for every pair ofpoints x , y on S2, d(x , y) is defined as the length of theshortest arc of the great circle that passes through x and y .

3. (C [0, 1], d), where C [0, 1] is the set of all continuousfunctions on [0, 1] and for every pair of continuous functionsf , g : [0, 1]→ R the distance between them is defined to bed(f , g) = max

x∈[0,1]|f (x)− g(x)|.

Convergent sequences and Cauchy sequences

Let (X , d) be a metric space. An ε-neighborhood of x ∈ X is theset Bε(x) = {y ∈ X : d(x , y) < ε}.

Let {x1, x2, . . .} = {xn}∞n=1 be a sequence on X .

The sequence {xn}∞n=1 converges to x ∈ X , limn→∞ xn = x , ifevery ε-neighborhood of x contains all but finitely many terms of{xn}∞n=1.

The sequence {xn}∞n=1 is a Cauchy sequence if for every ε > 0there exists an index N such that the distance d(xn, xm) < ε aslong as n,m > N.

The metric space X is complete if every Cauchy sequence in Xconverges to an element of X .

Banach Contraction Mapping Principle

Let (X , d) be a complete metric space.A map T : X → X is a contraction if there exists a nonnegativenumber ρ ≤ 1 such that

d(T (x),T (y)) ≤ ρd(x , y).

T is a strict contraction if ρ < 1.

A point x ∈ X is called a fixed point of T if T (x) = x .

TheoremEvery strict contraction T on a complete metric space (X , d) has aunique fixed point x. Moreover, for every x0 ∈ X , x is the limit ofthe sequence {x0,T (x0),T (T (x0)), . . .}, which can be definedrecursively:

xn = T (xn−1) for all n = 1, 2, 3, . . . .

Application to Ordinary Differential Equations

Consider the initial value problem{x = f (x),x(0) = x0,

(1)

where f : R→ R is a Lipschitz continuous function, i.e. thereexists L > 0 such that |f (x)− f (y)| ≤ L|x − y | for all x , y ∈ R.

We would like to prove the existence of a continuouslydifferentiable function x : [0, δ]→ R, where δ > 0, that solves theinitial value problem (1).

The latter problem is equivalent to the problem of finding acontinuous function x : [0, δ]→ R that solves the integral equation

x(t) = x0 +

∫ t

0f (x(s)) ds.

Existence of Solutions of ODEs

x(t) = x0 +

∫ t

0f (x(s)) ds.

Let X = C [0, δ] with the same distance d as before.Define T : X → X by T (x)(t) = x0 +

∫ t0 f (x(s)) ds for all

t ∈ [0, δ].Claim. T is a strict contraction provided δ > 0 is small enough.

|T (x)(t)− T (y)(t)| ≤∫ t

0|f (x(s))− f (y(s))| ds

≤ L

∫ t

0|x(s)− y(s)| ds

≤ Lδ maxs∈[0,1]

|x(s)− y(s)|.

Therefore, d(T (x),T (y)) ≤ Lδd(x , y) for all x , y ∈ X ; T is astrict contraction provided δ < 1

L .

Existence of Solutions of ODEs

TheoremIf f : R→ R is Lipschitz continuous, then there exists δ > 0 and aunique continuously differentiable function x : [0, δ]→ R thatsolves the initial value problem{

x = f (x),x(0) = x0,

where x0 ∈ R.

Applications to Numerical Solutions of Linear Systems

Consider the problem of solving a linear system

Ax = b,

where A is an n × n nonsingular matrix and b is an n-vector.Assume that the diagonal elements of A are equal to 1.

Gauss elimination method produces large errors when used on acomputer.Rewrite

(I − A)x + b = x .

Consider the (complete) metric space (Rn, d), whered(X ,Y ) = max

1≤i≤n|xi − yi | for X = (x1, x2, . . . , xn)T and

Y = (y1, y2, . . . , yn)T .

Applications to Numerical Solutions of Linear Systems

Define T (x) = (I − A)x + b for every x ∈ Rn.Because T (x)− T (y) = (I − A)(x − y).

max1≤i≤n

|T (x)i − T (y)i | ≤ max1≤i≤n

n∑j=1

|(I − A)ij ||xj − yj |

= max1≤i≤n

n∑j 6=i ,j=1

|Aij ||xj − yj |.

Thus, d(T (x),T (y)) ≤ max1≤i≤n

n∑j 6=i ,j=1

|Aij | d(x , y).

Applications to Numerical Solutions of Linear Systems

TheoremLet A be a nonsingular n × n matrix and let b be an n-vector. If anonsingular matrix A has diagonal dominance, that is,

n∑j 6=i ,j=1

|Aij | ≤ |Aii | for all 1 ≤ i ≤ n, then for every x0 ∈ Rn the

approximation scheme

xn = (A− I )xn−1 + b,

where n = 1, 2, . . ., converges to the solution of the linear systemAx = b.

References

I Michael Reed and Barry Simon, Methods of ModernMathematical Physics, I: Functional Analysis, Academic Press,1972.

I David Luenberger, Optimization by Vector Space Methods,John Wiley and Sons, Inc., 1969.

I Banach Biography from http://www.gap-system.org/ history/Biographies/Banach.html.

I Images: http://www.mathematik.de/ger/information/landkarte/gebiete/linearealgebra/bilder/banach.jpg (photo of Stefan Banach);

http://www.gap-system.org/˜ history/Miscellaneous/Scottish

Cafe.html (Scottish Cafe in Lviv).

I The Scottish Book,http://banach.univ.gda.pl/e-scottish-book.html.