The contraction mapping principle and some applications

90 Pages·2009·0.498 MB·English

by  Brooks R.M., Schmitt K

Electronic Journal of Differential Equations, Monograph 09, 2009, (90 pages).
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
THE CONTRACTION MAPPING PRINCIPLE AND SOME
APPLICATIONS
ROBERTM.BROOKS,KLAUSSCHMITT
Abstract. These notes contain various versions of the contraction mapping
principle. Several applications to existence theorems in the theories of dif-
ferential and integral equations and variational inequalities are given. Also
discussedareHilbert’sprojectivemetricanditeratedfunctionsystems
Contents
Part 1. Abstract results 2
1. Introduction 2
2. Complete metric spaces 2
3. Contraction mappings 15
Part 2. Applications 25
4. Iterated function systems 25
5. Newton’s method 29
6. Hilbert’s metric 30
7. Integral equations 44
8. The implicit function theorem 57
9. Variational inequalities 61
10. Semilinear elliptic equations 69
11. A mapping theorem in Hilbert space 73
12. The theorem of Cauchy-Kowalevsky 76
References 85
Index 88
2000 Mathematics Subject Classification. 34-02,34A34,34B15,34C25,34C27,35A10,
35J25,35J35,47H09,47H10,49J40,58C15.
Key words and phrases. Contractionmappingprinciple;variationalinequalities;
Hilbert’sprojectivemetric;Cauchy-Kowalweskitheorem;boundaryvalueproblems;
differentialandintegralequations.
(cid:13)c2009byRobertBrooksandKlausSchmitt.
SubmittedMay2,2009. PublishedMay13,2009.
1
2 R.M.BROOKS,K.SCHMITT EJDE-2009/MON.09
Part 1. Abstract results
1. Introduction
1.1. Theme and overview. Thecontractionmappingprincipleisoneofthemost
usefultoolsinthestudyofnonlinearequations,betheyalgebraicequations,integral
or differential equations. The principle is a fixed point theorem which guarantees
that a contraction mapping of a complete metric space to itself has a unique fixed
pointwhichmaybeobtainedasthelimitofaniterationschemedefinedbyrepeated
imagesunderthemappingofanarbitrarystartingpointinthespace. Assuch,itisa
constructivefixedpointtheoremand,hence,maybeimplementedforthenumerical
computation of the fixed point.
Iteration schemes have been used since the antiquity of mathematics (viz., the
ancient schemes for computing square roots of numbers) and became particularly
usefulinNewton’smethodforsolvingpolynomialorsystemsofalgebraicequations
andalsointhePicarditerationprocessforsolvinginitialvalueandboundaryvalue
problems for nonlinear ordinary differential equations (see, e.g. [58], [59]).
TheprinciplewasfirststatedandprovedbyBanach[5]forcontractionmappings
in complete normed linear spaces (for the many consequences of Banach’s work see
[60]). At about the same time the concept of an abstract metric space was intro-
duced by Hausdorff, which then provided the general framework for the principle
for contraction mappings in a complete metric space, as was done by Caccioppoli
[17] (see also [75]). It appears in the various texts on real analysis (an early one
being, [56]).
In these notes we shall develop the contraction mapping principle in several
forms and present a host of useful applications which appear in various places in
the mathematical literature. Our purpose is to introduce the reader to several
different areas of analysis where the principle has been found useful. We shall dis-
cussamongothers: theconvergenceofNewton’smethod; iteratedfunctionsystems
and how certain fractals are fixed points of set-valued contractions; the Perron-
Frobenius theorem for positive matrices using Hilbert’s metric, and the extension
of this theorem to infinite dimensional spaces (the theorem of Krein-Rutman); the
basic existence and uniqueness theorem of the theory of ordinary differential equa-
tions (the Picard-Lindel¨of theorem) and various related results; applications to the
theory of integral equations of Abel-Liouville type; the implicit function theorem;
the basic existence and uniqueness theorem of variational inequalities and hence
a Lax-Milgram type result for not necessarily symmetric quadratic forms; the ba-
sic existence theorem of Cauchy-Kowalevsky for partial differential equations with
analytic terms.
Thesenoteshavebeencollectedoverseveralyearsandhave,mostrecently,been
used as a basis for an REU seminar which has been part of the VIGRE program
of our department. We want to thank here those undergraduate students who
participated in the seminar and gave us their valuable feedback.
2. Complete metric spaces
In this section we review briefly some very basic concepts which are part of
most undergraduate mathematics curricula. We shall assume these as requisite
knowledge and refer to any basic text, e.g., [15], [32], [62].
EJDE-2009/MON. 09 THE CONTRACTION MAPPING PRINCIPLE 3
2.1. Metric spaces. Given a set M, a metric on M is a function (also called a
distance)
d:M×M→R =[0,∞),
+
that satisfies
d(x,y)=d(y,x), ∀x,y ∈M
d(x,y)=0, if, and only if, x=y (2.1)
d(x,y)≤d(x,z)+d(y,z), ∀x,y,z ∈M,
(the last requirement is called the triangle inequality). We call the pair (M,d) a
metric space (frequently we use M to represent the pair).
A sequence {x }∞ in M is said to converge to x∈M provided that
n n=1
lim d(x ,x)=0.
n
n→∞
This we also write as
limx =x, or x →x as n→∞.
n n
n
We call a sequence {x }∞ in M a Cauchy sequence provided that for all (cid:15)>0,
n n=1
there exists n =n ((cid:15)), such that
0 0
d(x ,x )≤(cid:15), ∀n,m≥n .
n m 0
A metric space M is said to be complete if, and only if, every Cauchy sequence in
M converges to a point in M.
Metric spaces form a useful-in-analysis subfamily of the family of topological
spaces. We need to discuss some of the concepts met in studying these spaces. We
do so, however, in the context of metric spaces rather than in the more general
setting. The following concepts are normally met in an advanced calculus or foun-
dations of analysis course. We shall simply list these concepts here and refer the
readertoappropriatetexts(e.g. [32]or[72])fortheformaldefinitions. Weconsider
a fixed metric space (M,d).
• OpenballsB(x,(cid:15)):={y ∈M:d(x,y)<(cid:15)}andclosedballsB[x,(cid:15)]:={y ∈
M:d(x,y)≤(cid:15)};
• open and closed subsets of M;
• bounded and totally bounded sets in M;
• limit point (accumulation point) of a subset of M;
• the closure of a subset of M (note that the closure of an open ball is not
necessarily the closed ball);
• the diameter of a set;
• the notion of one set’s being dense in another;
• the distance between a point and a set (and between two sets).
Suppose(M,d)isametricspaceandM ⊂M. IfwerestrictdtoM ×M ,then
1 1 1
M will be a metric space having the “same” metric as M. We note the important
1
factthatifMiscompleteandM isaclosedsubsetofM,thenM isalsoacomplete
1 1
metric space (any Cauchy sequence in M will be a Cauchy sequence in M; hence
1
it will converge to some point in M; since M is closed in M that limit must be in
1
M ).
1
The notion of compactness is a crucial one. A metric space M is said to be
compact provided that given any family {G : α ∈ A} of open sets whose union
α
is M, there is a finite subset A ⊂ A such that the union of {G : α ∈ A } is M.
0 α 0
(To describe this situation one usually says that every open cover of M has a finite
4 R.M.BROOKS,K.SCHMITT EJDE-2009/MON.09
subcover.) We may describe compactness more “analytically” as follows. Given
any sequence {x } in M and a point y ∈M; we say that y is a cluster point of the
n
sequence {x } provided that for any (cid:15)>0 and any positive integer k, there exists
n
n≥k suchthatx ∈B(y,(cid:15)). Thusinanyopenball, centeredaty, infinitelymany
n
terms of the sequence {x } are to be found. We then have that M is compact,
n
provided that every sequence in M has a cluster point (in M).
In the remaining sections of this chapter we briefly list and describe some useful
examples of metric spaces.
2.2. Normed vector spaces. Let M be a vector space over the real or complex
numbers (the scalars). A mapping k·k : M → R is called a norm provided that
+
the following conditions hold:
kxk=0, if, and only if, x=0(∈M)
kαxk=|α|kxk, ∀ scalar α, ∀x∈M (2.2)
kx+yk≤kxk+kyk, ∀x,y ∈M.
If M is a vector space and k·k is a norm on M, then the pair (M,k·k) is called
a normed vector space. Should no ambiguity arise we simply abbreviate this by
saying that M is a normed vector space. If M is a vector space and k·k is a norm
on M, then M becomes a metric space if we define the metric d by
d(x,y):=kx−yk, ∀x,y ∈M.
Anormedvectorspacewhichisacompletemetricspace,withrespecttothemetric
d defined above, is called a Banach space. Thus, a closed subset of a Banach space
may always be regarded as a complete metric space; hence, a closed subspace of a
Banach space is also a Banach space.
We pause briefly in our general discussion to put together, for future reference,
asmallcatalogueofBanachspaces. WeshallconsideronlyrealBanachspaces, the
complex analogues being defined similarly.
Inallcasestheverificationthatthesespacesarenormedlinearspacesisstraight-
forward,theverificationofcompleteness,ontheotherhandusuallyismoredifficult.
Manyoftheexamplesthatwillbediscussedlaterwillhavetheirsettingincomplete
metric spaces which are subsets or subspaces of Banach spaces.
Examples of Banach spaces.
Example 2.1. (R,|·|) is a simple example of a Banach space.
Example 2.2. We fix N ∈N (the natural numbers) and denote by RN the set
RN :={x:x=(ξ ,...,ξ ), ξ ∈R, i=1,...,N}.
1 N i
There are many useful norms with which we can equip RN.
(1) For 1≤p<∞ define k·k :RN →R by
p +
N
(cid:16)X (cid:17)1/p
kxk := |ξ |p , x∈RN.
p i
i=1
These spaces are finite dimensional lp− spaces. Frequently used norms are
k·k , and k·k .
1 2
EJDE-2009/MON. 09 THE CONTRACTION MAPPING PRINCIPLE 5
(2) We define k·k :RN →R by
∞ +
kxk :=max{|ξ |:1≤i≤N}.
∞ i
This norm is called the sup norm on RN.
Thenextexampleextendstheexamplejustconsideredtotheinfinitedimensional
setting.
Example 2.3. We let
R∞ :={x:x={ξ }∞ , ξ ∈R, i=1,2,...}.
i i=1 i
ThenR∞,withcoordinate-wiseadditionandscalarmultiplication,isavectorspace,
certainsubspacesofwhichcanbeequippedwithnorms,withrespecttowhichthey
are complete.
(1) For 1≤p<∞ define
X
lp :={x={ξ }∈R∞ : |ξ |p <∞}.
i i
i
Then lp is a subspace of R∞ and
∞
(cid:16)X (cid:17)1/p
kxk := |ξ |p
p i
i=1
defines a norm with respect to which lp is complete.
(2) We define
l∞ :={x={ξ }∈R∞ :sup|ξ |<∞}.
i i
i
and
kxk :=sup{|ξ |, x∈l∞}.
∞ i
i
With respect to this (sup norm) l∞ is complete.
Example 2.4. Let H be a complex (orreal) vectorspace. Aninner product on H
is a mapping (x,y)7→hx,yi (H ×H →C) which satisfies:
(1) for each z ∈H h·,zi:H →C is a linear mapping,
(2) hx,yi = hy,xi for x,y ∈ H ( hx,yi = hy,xi if H is a real vector space and
the inner product is a real valued function),
(3) hx,xi≥0, x∈H, and equality holds if, and only if, x=0. If one defines
p
kxk:= hx,xi,
then (H,k·k) will be a normed vector space.
If it is complete, we refer to H as a Hilbert space. We note that (RN,k·k ) and l2
2
are (real) Hilbert spaces.
Spacesofcontinuousfunctionsarefurtherexamplesofimportantspacesinanal-
ysis. The following is a brief discussion of such spaces.
Example 2.5. WefixI =[a,b], a,b∈R, a<b,andk ∈N∪{0}. LetK=R, orC
(the reader’s choice). Define
Ck(I):={f :I →K:f,f0,...,f(k), exist and are continuous on I}.
We note that
C0(I):=C(I)={f :I →K:f is continuous on I}.
6 R.M.BROOKS,K.SCHMITT EJDE-2009/MON.09
For f ∈C(I), we define
kfk := max |f(t)|
∞
x∈[a,b]
(the sup-on-I norm). And for f ∈Ck(I)
k
X
kfk:= kf(i)k .
∞
i=0
With the usual pointwise definitions of f +g and αf (α ∈ K) and with the norm
defined as above, it follows that Ck(I) is a normed vector space. That the space is
also complete follows from the completeness of C(I) with respect to the sup-on-I
norm (see, e.g., [15]).
Another useful norm is
k
X
kfk∗ :=sup |f(i)(x)|.
x∈I
i=0
which is equivalent to the norm defined above; this follows from the inequalities
kfk∗ ≤kfk≤(k+1)kfk∗, ∀f ∈Ck(I).
Equivalent norms give us the same idea of closeness and one may, in a given
application,use,ofequivalentnorms,thatwhichmakescalculationsorverifications
easier or gives us more transparent conclusions.
Example 2.6. Let Ω be an open subset of RN, and let K be as above; define
C(Ω):=C0(Ω):={f :Ω→K such that f is continuous on Ω}.
Let
kfk := sup|f(x)|.
∞
x∈Ω
Sincetheuniformlimitofasequenceofcontinuousfunctionsisagaincontinuous,
it follows that the space
E :={f ∈C(Ω):kfk <∞}
∞
is a Banach space.
If Ω is as above and Ω0 is an open set with Ω¯ ⊂Ω0, we let
C(Ω¯):={the restriction to Ω¯ of f ∈C(Ω0)}.
If Ω is bounded and f ∈C(Ω¯), then kfk <+∞. Hence C(Ω¯) is a Banach space.
∞
Example2.7. LetΩbeanopensubsetofRN. LetI =(i ,...,i )beamultiindex,
1 N
i.e. i ∈N∪{0} (the nonnegative integers), 1≤k ≤N. We let |I|=PN i . Let
k k=1 k
f :Ω→K. Then the partial derivative of f of order I, DIf(x), is given by
∂|I|f(x)
DIf(x):= ,
∂i1x1...∂iNxN
where x=(x ,...,x ). Define
1 N
Cj(Ω):={f :Ω→K such that DIf ∈C(Ω), |I|≤j}.
Let
j
X
kfk := maxkDIfk .
j ∞
|I|≤k
k=0
EJDE-2009/MON. 09 THE CONTRACTION MAPPING PRINCIPLE 7
Then, using further convergence results for families of differentiable functions it
follows that the space
E :={f ∈Cj(Ω):kfk <+∞}
j
is a Banach space.
The space Cj(Ω¯) is defined in a manner similar to the space C(Ω¯) and if Ω is
bounded Cj(Ω¯) is a Banach space.
2.3. Completions. In this section we shall briefly discuss the concept of the com-
pletion of a metric space and its application to completing normed vector spaces.
Theorem 2.8. If (M,d) is a metric space, then there exists a complete metric
space (M∗,d∗) and a mapping h:M→M∗ such that
(1) h is an isometry d∗(h(x),h(y))=d(x,y), x,y ∈M)
(2) h(M) is dense in M∗.
We give ashortsketchofthe proof. We let C be the set ofallCauchysequences
in M. We observe that if {x } and {y } are elements of M, then {d(x ,y )} is a
n n n n
Cauchy sequence (hence, convergent) sequence in R, as follows from the triangle
inequality. We define d :C×C →R, by
C
d ({x },{y }):= lim d(x ,y ).
C n n n n
n→∞
The mapping d is a pseudo-metric on C (lacking only the condition
C
d ({x },{y })=0 ⇒ {x }={y }
C n n n n
from the definition of a metric). The relation R defined on C by
{x }R{y }, if, and only if, lim d(x ,y )=0,
n n n n
n→∞
or equivalently,
d ({x },{y })=0,
C n n
is an equivalence relation on C. The space C/R, the set of all equivalence classes
in C, shall be denoted by M∗. If we denote by R{x }, the class of all {z } which
n n
are R− equivalent to {x }, we may define
n
d∗(R{x },R{y }):=d ({x },{y }).
n n C n n
This defines a metric on M∗.
We next connect this to M. There is a natural mapping of M to C given by
x7→{x}
(the sequence, all of whose entries are the same element x). We clearly have
d ({x},{y})=d(x,y)
C
andthusthemapping x7→R{x}(whichwenowcallh)is anisometryofM toM∗.
That the image h(M) is dense in M∗, follows easily from the above construction.
Remark 2.9. We observe that (M∗,d∗) is “essentially unique”. For, if (M ,d )
1 1
and (M ,d ) are completions of M with mappings h , respectively h , then there
2 2 1 2
exists a mapping g :M →M such that
1 2
(1) g is an isometry of M onto M ,
1 2
(2) h =g◦h .
2 1
8 R.M.BROOKS,K.SCHMITT EJDE-2009/MON.09
The above is summarizedinthe following theorem, whose proofis similarto the
proof of the previous result (Theorem 2.8) recalling that a norm defines a metric
and where we define (using the notation of that theorem)
k{x }k := lim kx k,
n C n
n→∞
for a given Cauchy sequence, and
kR{x }k∗ :=k{x }k .
n n C
It is also clear that the set X∗ becomes a vector space by defining addition and
scalar multiplication in a natural way.
Theorem 2.10. If (X,k·k) is a normed vector space, then there exists a (essen-
tially unique) complete normed vector space (a Banach space) (X∗,k·k∗) and an
isomorphism (which is an isometry) h:X →X∗ such that h(X) is dense in X∗.
2.4. Lebesgue spaces. In this section we shall discuss briefly Lebesgue spaces
generated by spaces of continuous functions whose domain is RN, N ∈N. If
f :RN →K, K=R or C
we define the support of f to be the closed set
supp(f):={x:f(x)6=0}.
Wesaythatf hascompactsupportwheneversupp(f)isacompact(i.e.,closedand
bounded) set and denote by C (RN) the set of all continuous K− valued functions
0
defined on RN having compact support. (More generally, if Ω is an open set in
RN, one denotes by Cj(Ω) the set of all Cj− functions having compact support
0
in Ω.) This is a vector subspace of C(RN), the space of all continuous K− valued
functions defined on RN.
We first need to define the Riemann integral on C (RN). To do this, without
0
gettingintotoomanydetailsofthisprocedure,weassumethatthereaderisfamiliar
with this concept for the integral defined on closed rectangular boxes
B :={x=(ξ ,...,ξ ):α ≤ξ ≤β , 1≤i≤N},
1 N i i i
(B = QN [α ,β ]), where the numbers α , β , 1 ≤ i ≤ N, are fixed real numbers
i=1 i i i i
(for each box). We observe that if f ∈ C (RN) and if B and B are such boxes,
0 1 2
each of which contains supp(f), then
Z Z Z
f = f = f,
B1 B1∩B2 B2
B ∩B alsobeingaboxcontainingsupp(f). ThisallowsustodefinetheRiemann
1 2
integral of f over RN by
Z (cid:16) Z (cid:17) Z
f = f := f,
RN B
where B is any closed box containing supp(f).
The mapping f 7→ R f is a linear mapping (linear functional) from C (RN) to
0
K, which, in addition, satisfies
• if f is non-negative on RN, then R f ≥0,
EJDE-2009/MON. 09 THE CONTRACTION MAPPING PRINCIPLE 9
• If {f } is a sequence of non-negative functions in C (RN) which is mono-
n 0
tonically decreasing (pointwise) to zero, i.e.,
f (x)≥f (x), n=1,..., lim f (x)=0, x∈RN,
n n+1 n
n→∞
R
then f →0.
n
Definition 2.11. For f ∈C (RN) we define
0
Z
kfk := |f|.
1
It is easily verified that k·k is a norm - called the L1-norm- on C (RN). We
1 0
now sketch the process for completing the normed vector space (cid:0)C (RN),k·k (cid:1) in
0 1
such a way that we may regard the vectors in the completion as functions on RN.
Definition 2.12. A subset S ⊂ RN is called a set of measure zero provided that
for any (cid:15)>0 there exists a sequence of boxes {B }∞ such that
n n=1
S ⊂∪∞ B ,
n=1 n
∞
X
vol(B )<(cid:15),
n
n=1
where vol(B)=QN (β −α ) for the box B =QN [α ,β ].
i=1 i i i=1 i i
We say that a property holds “almost everywhere” (“a.e.”) if the set of points
at which it fails to hold has measure zero.
Theproofsofthefollowingtheoremsmaybefoundinaverycompletediscussion
of C (RN) and its L1− completion in [37, Chapter 7].
0
Definition 2.13. Asequence{x }inanormedvectorspace(X,k·k)issaidtobe
n
a fast Cauchy sequence if
∞
X
kx −x k
n+1 n
n=1
converges.
Theorem 2.14. If {f } is a fast Cauchy sequence in (C (RN),k·k ), then {f }
n 0 1 n
converges pointwise a.e. in RN.
Definition 2.15. ALebesgueintegrablefunctiononRN isafunctionf suchthat:
• f is a K valued function defined a.e. on RN,
• thereisafastCauchysequencein(C (RN),k·k )whichconvergestof a.e.
0 1
in RN.
Theorem 2.16. If f is a Lebesgue integrable function and if {f } and {g } are
n n
fast Cauchy sequences in (C (RN),k·k ) converging a.e. to f, then
0 1
Z Z
lim f = lim g .
n n
n→∞ n→∞
R
In light of this result we may define f by
Z Z
f := lim f ,
n
n→∞
10 R.M.BROOKS,K.SCHMITT EJDE-2009/MON.09
where {f }isanyfastCauchysequence in(C (RN),k·k ), converginga.e. tof on
n 0 1
RN. The resulting map
Z
f 7→ f
is then defined on the space of all Lebesgue integrable functions L1(RN) and is a
linear functional on this space which also satisfies
Z Z
| f|≤ |f|, ∀f ∈L1(RN).
Theorem 2.17. The mapping
Z
f 7→kfk := |f|, f ∈L1(RN),
1
is a seminorm on L1(RN), i.e., it satisfies all the conditions of a norm, except that
kfk =0 need not imply that f is the zero of L1(RN). Further L1(RN) is complete
1
with respect to this seminorm and C (RN) is a dense subspace of L1(RN).
0
Usually we identify two elements of L1(RN) which agree a.e.; i.e., we define an
equivalence relation on L1(RN)
f ∼g
whenever the set A∪B has measure zero, where
A:={x:f(x) or g(x) fail to be defined},
B :={x:f(x),g(x) are defined, but f(x)6=g(x)}.
This equivalence relation respects the operations of addition and scalar multiplica-
tion and two equivalent functions have the same seminorm. The vector space of all
equivalence classes then becomes a complete normed linear space (Banach space).
This space, we again call L1(RN),
Remark2.18. 1. Weagainreferthereaderto[37]foracompletediscussionofthis
topic and others related to it, e.g., convergence theorems for Lebesgue integrals,
etc.
2. The ideas above may equally well be employed to define integrals on open
regions Ω⊂RN starting with
C (Ω):={f ∈C(Ω):supp(f) is a compact subset of Ω}.
0
The resulting space being L1(Ω).
3. One also may imitate this procedure to obtain the other Lebesgue spaces
Lp(RN), 1≤p<∞, by replacing the original norm in C (RN) by
0
(cid:16)Z (cid:17)1/p
kfk := |f|p , f ∈C (RN).
p 0
And, of course, in similar vein, one can define Lp(Ω), 1≤p<∞.
4. For given f ∈L1(R) define the functional T on C∞(R) as follows
f 0
Z
T (φ):= fφ.
f
The functional T is called the distribution defined by f. More generally, the set of
f
all linear functionals on C∞(R) is called the set of distributions on R and if T is
0
such, its distributional derivative ∂T is defined by
∂T(φ):=−T(φ0), ∀φ∈C∞(R),
0