R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
E SPEDITO D E P ASCALE P JOTR P. Z ABREJKO
New convergence criteria for the Newton- Kantorovich method and some applications to nonlinear integral equations
Rendiconti del Seminario Matematico della Università di Padova, tome 100 (1998), p. 211-230
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
http://www.numdam.org/
for the Newton-Kantorovich Method and Some Applications
to Nonlinear Integral Equations.
ESPEDITO DE
PASCALE (*) -
PJOTR P.ZABREJKO (**)
ABSTRACT - This paper presents some new conditions for the convergence of New- ton-Kantorovich
approximations
to solutions of nonlinear operatorequations
in Banach spaces. The derivatives of the nonlinear operators involved are re-
quired
tosatisfy
a rather mildcontinuity
condition in a ball centered at the in-itial
approximation.
The abstract results are illustratedby applications
tononlinear
integral
equations ofUryson
type in theChebyshev
space C, theLebesgue
space and the Orlicz spaceLM .
1. Introduction.
The purpose of this article is two-fold.
First,
we aregoing
togeneral-
ize some results about the convergence of certain Newton-Kantorovich
approximations
discussedin [1-2]. Second,
wegive
newapplications
ofboth the old and new results to
Uryson integral equations
inLebesgue
and Orlicz spaces.
As it was shown
in [2],
whenapplying
the classical Newton-Kan- torovich method toUryson integral equations,
one meets astrange
situ-(*) Indirizzo dell’A.: Universita della Calabria,
Dipartimento
di Matematica,1-87036 Arcavacata di Rende (CS),
Italy.
(**) Indirizzo dell’A.:
Belgosuniversitet, Matematicheskij
Fakultet, BR-220050 Minsk, Belarus.
AMS (MOS) Classification: 65J15, 47H17, 47H19, 47H30, 45G10, 65R20.
ation.
Namely,
the standard theorem on the convergence of the Newton- Kantorovichapproximations
does notapply
in some spaces, forinstance,
in
Lp-spaces
for1 ~ ~ ~
2. Thisunpleasant phenomenon
can be over-come
by modyfying
the smoothnesshypotheses
in a suitable way. Some modifications of thistype
have beensuggested
in[1];
here we proposesome new variants which works «well» if the character of
continuity
ofthe derivative at the initial
approximation
isessentially
better(in
a sense that will become evident in thesequel)
than thegeneral continuity
of thederivative.
The new conditions
proposed
below may be checked rather effective-ly
forUryson integral equations
in the spaces C andLoo.
On the otherhand,
the situation is worse in the spacesLp
for finite p and in Orlicz spaces. Here we describe some class of kernel functions for which theapplication
of our convergence results to thecorresponding Uryson equations
is natural andsufficiently
effective.Generalizations of the convergence conditions introduced in this pa- per are in a different direction from that
exposed
in[1].
It ispossible
togive
a unified treatment of the results in[1]
and in this paper.Actually,
for sake of
simplicity,
weprefer
to confine ourselves to the variant pre- sented here.2. New convergence criteria for Newton-Ka,ntorovich
approximations
Let X and Y be two Banach spaces,
B(xo , R ) : _ ~ x : x eX, I ix - ro )) S
the closed ball centered at
Xo e X
with radiusR > 0,
andF : B(xo , R ) -~ Y
some(nonlinear) operator.
The Newton-Kantorovich method is one of the basic tools forfinding approximate
solutions of theoperator equation
In the
corresponding
iterative schemeone has to
require,
inparticular,
that the Frechet derivative of F at allpoints
rn exists and is invertible in the Banach space2(X, Y)
of all bounded linearoperators
from X into Y. Thenonnegative
numbersand
will be of
particular
interest to us in what follows.We suppose that the Fr6chet derivative
F’ (x)
of F satisfies at eachpoint
ofB( xo , R )
a condition of the formwhere
co : [ o , 00) ~ [0, oo)
ismonotonically increasing
withMoreover,
we assume that there is anothermonotonically increasing
function
8 : [ o, ~ ) ~ [ o, ~ )
such that 0 -0(r) % w( r), (0:::;
r ~R),
and
We define three scalar functions on
[0, R] by
and
As a
special
case of the main theorem of[6],
about the convergence of successiveapproximations,
weget
then thefollowing:
THEOREM 1.
Suppose
that thefunction (9)
has aunique
zero r * EE
[ 0, R]
and that~p(R) ~
0. Thenequation (1)
has a solution x * EE B( xo , r * );
this solution isunique
in the ballB( xo , R).
As a matter of fact
(see
e.g.[1, 4]),
under thehypotheses
of Theorem1 the Newton-Kantorovich
approximations
need not converge.However,
thefollowing
is true.LEMMA 1.
Suppose
that thefunction (10)
has aunique
zero EE
[ 0 , R]
and that~(R ) ~
0 . Then the scalar sequence(rn)neN defined by
converges
monotonically
to p*.PROOF. For
simplicity
we consideronly
the case when all the scalar functions under consideration aredifferentiable;
thegeneral
case can bereached with usual considerations
involving monotonicity.
Since Q. is theonly
zero of the function(10),
this function isstrictly positive
on[ 0, ~o * ).
The same is true for the function r H - = 1 -bFo(r),
as may be seen from the
following reasoning.
First of all we have- bO’ (0) = 1. Suppose that lil ’ (j) = 0
for SinceO’(r) = w(r) - 1/b
isincreasing
in[ 0 , R ],
wehave q5’(0)
=0 O’ (r)
forr E [-Q, R 1. So O
isincreasing
and convex on[o, R]:
thehypotheses
0
together
witho(Q
= 0implies
= Q= R,
i.e.- 0’(r)
> 0 on(0, Q Now,
since 0- bQ’ (r)
= 1 - 1 -bO(r),
thefunction Y
defined
by
is alsostrictly positive
on(0, o*).
We claim that the map +
y(r)
isincreasing
on[ 0, Q .).
Infact,
its derivative satisfies for r E
[ 0 , Q . )
Indeed condition -
bQ’ (r)
1 -implies
that 1 +b(p’ (r) /(1 -
-
bO(r)) >- 0,
and themonotoniticity
of 0together
with theposivity of Q
imply
thevalidity
ofinequality Q(r) 8 ’ ( r) ;
0 .Now the assertion of the lemma follows
easily.
Infact,
the sequence(11)
ismonotonically increasing
and bounded aboveby
~:for rn ~ O * .
Consequently,
the sequence(11)
converges to some r * EE [0, Q,I.
But r* = r*+ 1jJ( r *) implies 1jJ( r *) = 0,
that is r* = o*.THEOREM 2. Under the
hypotheses of
Lemma 1 theapproxima-
tions
(2)
aredefined for
all n,belong
to the ballB(xo, Q . ),
are convery-ing
to a solution x *of (1)
andsatisfy
the estimatesand
PROOF. If for every n > 0 1- rn , then for m > n we have
I lxm -
rm - rn , andconsequently
isCauchy
sequenceconvergent
to alimit x *
inX,
which solvesequation (1) by (2)
andby
ourassumption
on F .Clearly Ilx * -
(2 * - rn and it is sufficient to prove(12).
For n = 0 the
equality (12) simply
readswhich is obvious.
Suppose
that(12)
holds for all n I~ . Thisimplies,
inparticular,
thatNow, by
definition(2),
we have:We conclude that
But the definition
(8)
of the function 65implies
thatcv(r - s) w(r) -
-
0(s),
henceThis shows that
(12)
holds for n = k aswell,
and so theproof
iscomplete.
Theorem 2 can be extended in a standard way
(see [1]) by replacing
the
assumption
on the function(10) by
theassumption
that the sequence(11)
converges. Thisextension, however,
is not veryeffective;
we there-fore
just
formulate thecorresponding
result withoutproof:
THEOREM 3.
Suppose
that the sequence(rn)n given by (11)
con-verges to some limit roo
(a).
Then theapproximations (2)
aredefined for all n, belong
to the ballB(xo , ( a ) ) ,
andsatisfy
the estimates(12)
and(13).
Our Theorems 1-3 are
generalizations
of the Theorems 1-3 contained in[ 1 ]
which may be obtainedby
thespecial
choice0(r)
=When
0(r) cv(r)
theapproximations (2)
coincide with the classicalones studied
by Kantorovich, Vertgeim
and others. When0(r)
is acostant in
(0, 1 ),
our conditions coincide with the so-calledMysovskikh
conditions
([11]).
We remark that the usefulness of Theorem 2 consists inreducing
the(hard) problem
offinding
zeros of a nonlinearoperator
ina Banach space to the
(possibly simpler) problem
offinding
zeros of ascalar function.
3.
Integral equations
ofUryson type.
The purpose of this section is to discuss and illustrate various
aspects
of our
preceding
Theorems 1-3by
means of the nonlinearintegral equation
We suppose that the
Uryson integral operator
defined
by
theright-hand
side of(14)
acts in some Banach spaceX;
thuswe may
put
Y = X and writeequation (14)
in the form(1)
withWe shall assume
throughout
that ,S~ is acompact
set in the Euclidean space, and k:Carath6odory
function(i.e. 1~( ~ , ~ , u)
is measurable on Q xQ,
andk(t, s , ~ )
is continuous onR ). Moreover,
wesuppose that the derivative
exists and is also a
Carath6odory
function. Wepoint out, however,
thatthe results of this section carry
over,without
essentialchanges,
also toarbitrary
setsQ, equipped
with someo-algebra
a of measurable subsets and somecountably
additive measure; inparticular,
our results are truefor infinite
systems
of nonlinearequations.
For
simplicity,
we choose xo = 0 in this and in thefollowing sections;
the case of
arbitrary xo E X
may be formulatedsimply by «shifting»
arguments.
°In what
follows,
as Banach space X we take either the space C == C( S~ ),
theLebesgue
spaceLp (Q)
or the Orlicz space Weremark, however,
that our results carry over the ideal spaces of measurable functions(see [6,13]):
A Banach space of measurable functions is called ideal if
implies x ( e X
and and the relations x e X and~
I
a.e. in S2imply yeX
andllxll.
For fixed
x e X,
we define a linearintegral operator L(x) by
and
put
It is natural to
expect
that the derivative of theoperator (15)
is related to theoperator (17),
and hence the derivative of(16)
is related to(18).
Infact,
thefollowing
is true:LEMMA 2. Let X be an ideal space or the space C.
Suppose
that
and the
operator
Lgiven by ( 17)
isdefined
on the ballB( 0 , R),
takes itsvalues in the space
2(X, X),
andsatisfies
a conditionwhere
Then the
operator (16)
isdifferentiable,
as anoperator
onX,
at everypoint
x EB( 0 , R),
andWe omit the
(rather elementary) proof
of the lemma(see
e.g.[2]).
The
following
lemma was beenproved
in[2]
as well:LEMMA 3.
Suppose
that theUryson integral operator (15)
acts inan ideal space X and admits a derivative
at zero. Assume that 1 and denote
by r(t, s)
the resolventkernel
for L(t ,
s ,0 ).
Thenand
the
inequality (25)
turns into anequality if
X = C or X =L ~ .
Recall that the resolvent kernel
r( t , s)
for agiven kernel L( t , s )
is de- fined as a solution to theequation
the condition is therefore necessary
(and, usually,
alsosufficient)
for the existence ofr( t , s). Furthermore,
the norm11r( t, s) 11.’X(x,
x> ,by definition,
coincides with the norm of thecorrispond- ing integral operator.
At least in the case X = C or X=Loo theequality
holds true.
In the case of
spaces X
different from C andL 00 ,
the calculation of thenorm of
r( t , s)
inx(X, X)
leads to seriousproblems.
We will return tothis
problem
in Sections 5 and 6 for the casesX = Lp
andX = LM .
In order to
apply
the convergence results and error estimates ob- tained in Theorems 1-3above,
we have to calculate or estimate the con-stants a and b defined
by (3)
and(4), respectively,
and to «catch» the functionscv(r)
and0(r)
for which theinequalities (5)
and(7)
hold. It is easy to see that the«optimal»
choice for these functions isand
respectively.
In factimplies
and
consequently
The last
inequality gives
theoptimality
of 0 defined in(27).
Theproblem
of
calculating a
and b was studied in detail in[2].
As a matter offact,
thearticle
[2]
contains also some information about how to calculate the functionsco(r)
and0(r).
We return to thisproblem
in thefollowing
sec-tions ;
here we restrict ourselvesonly
tomentioning
thefollowing impor-
tant lemma whose
proof
can be obtainedby reasoning
as in theproof
ofTheorem 5 in
[2].
Before to state the next
lemma,
we want to recall that an ideal space X is calledrearrangement-invariant
if any twoequimeasurable
functionsf and g
in X(i.e.
for every h > 0lf(s) I
e Q , I
>h} ))
have the same norm.LEMMA 4. Let X be a
rearrangement-invariant
ideal space withfundamental function x(t). (x(t)
isdefined by
theformula x(t) =
where
xD is
the characteristicfunction of
a measurable subset Dof S2,
with mes D =t) . Suppose
thatfor
everyfixed
Tor more
generally
Then the condition
(6)
can besatisfied
in case(28) only if
thefunction
1 = 8k/8u does not
depend
on u, and in case(29) only if
1satisfies
thecondition
Roughly speaking,
the statement of Lemma 4 means that the New- ton-Kantorovichmethod, applied
to theUryson operator (15),
worksonly
insufficiently
«small» spaces. As was noticedin [2],
any conver- gence result for the Newton-Kantorovich method in the classical Kan- torovichsetting (i.e. (t)(r)
=kr)
is useless in the spacesLp
for1 p 2;
asimilar situation occurs in
Vertgeim’s setting (i.e. cv(r) = kra, 0 a 1 )
in the spaces
Lp
for 1 ~p %
1 + a(see [1])
4. The case L 00 .
Now we return to the
problem
ofcalculating (or estimating)
the func-tions
cv(r)
and0(r).
Thesimplest
case is that of the space X = L ~ . Thereason lies in the very
pleasant
fact that one may calculate both func- tions inexplicit
form.THEOREM 4.
Suppose
that the kernelk(t,
s,u) satisfies
thefollow- ing
three conditions:Then
(34)
and
PROOF. The
inequality
is evident. The reverse
inequality
is astraightforward
consequence of theimportant equality
which is a modification of the
general equality
Here one should notice that the sup in the left-hand side of
(37)
and(38)
is meant as supremum in thespace ,S
of measurable functions. Theproof
of the formula
(38)
can be found in[14] (see
also[2]).
The formula(37)
may be
proved similarly.
As a matter of
fact,
the spaceL ~
has rather «bad»properties.
As aconsequence, whenever
possible,
one tries to avoid the space and toconsider, instead,
the space C of continuous functions.Surprisingly, stating
ananalogue
to Theorem 4 for the spaceC,
is es-sentially
morecomplicated
than for the spaceL 00.
The basic reason for this is thecomplexity
of conditions under which theUryson integral
op- erator(15) (and
even linearintegral operators)
acts in the space C.THEOREM 5.
Suppose
that the kernelk(t,
s,u) satisfies
thefollow-
ing
three conditions:Moreover,
assumethat, for
each continuousfunction x(s) satisfying
~ x(s) ~ ~ R,
thekernel l(t,
s,x(s) ) defines
a continuous linearintegral operator L(x)
in the space C. Thenand
Of course, Theorem 5 is
only
arepetition
of Theorem 4containing,
inaddition,
some cumbersomeassumption
on the kernelsL(t ,
s ,x( s ) )
for||x||C R.
We
point
out that theassumption
on the kernelsl(t,
s ,x( s ) )
is com-plicated only
because of itshigh generality:
thisassumption
covers infact all
possible
situations in which one can considerequation (14)
in thespace C. There are
simple
sufficient conditions under which the kernels s ,x( s ) ) generate,
for continuous linearintegral operators
in C. One such condition iswhere
a(Q)
denotes thea-algebra
ofLebesgue-measurable
subsets ofIn
fact,
the condition(45) guarantees
that all linearintegral operators
and the
Uryson integral operator (15)
acts from Loo toC,
and theidentity
holds in the space
C)
of bounded linearoperators
from£ 00
to C.Another
(rather strong)
sufficient condition on the kernelsx( s ) )
togenerate
continuous(in fact, compact)
linearintegral
op- erators in C isr
We omit the
proofs
of these well-known facts which may befound,
forexample,
in[9].
5. The case
Lp.
The
simplicity
of Theorem 4 and Theorem 5 is astrong
motivation tostudy Uryson integral equations
asoperator equations
in the spaceL~>~
or in the space C.
Unfortunately,
it mayhappen
that thecorresponding Uryson operator
does not act in the spaceL 00
orC,
or that one isjust
in-terested in unbounded solutions. In these cases it is natural to
study
thegiven integral equation
in some other Banach spaceX;
a classical choice is here X =Lp
for 1 ~ p oo .However,
the use of these spaces leads toserious difficulties. As was shown in
[2],
the classical theorems on the convergence of the Newton-Kantorovichapproximations [6,8,
and also15-17]
can beapplied
to theUryson integral equation (4)
inLp only
incase p *
2.Similarly,
thegeneralizations
of the classicaltheorems,
where the
Lipschitz
condition for the derivative of the nonlinear opera- tor involved isreplaced by
a Holder condition withexponent
a(0
a1), apply only
inLp
forp ~
1 + a[1].
In Section 3 we have seen that it isimpossible
to use the spaceL1
even in the case when the derivative of theoperator
involved isuniformly
continuous on the ballR ) (see
Lem-ma 4 and
following paragraph).
The lastassumption
seems to be «maxi-mal» in this kind of theorems.
Unfortunately,
the utilization of the spaceLp (1
poo )
in theanalysis
of theUryson equation (14)
is connected withyet
another diffi-culty.
As a matter offact,
in the spacesLp
for 1 p o0 one cannotgive explicit
formulas for the functionscv(r)
and0(r),
for thesimple
reasonthat one does not
know, except
for trivialspecial
cases,explicit
formulasfor the norm of a linear
integral operator
in these spaces.All this
emphasizes
the need ofchanging
the statement of theproblem
ingeneral.
Below we will restrict ourselves to thedescription
of a class of nonlinearities
k(t,
s,u)
for which thecorresponding
Uryson integral equation (14)
can be studiedsuccessfully
in the spaceLp
for 1 p oo .To this
end,
we need some standard definitions and notation. Let 1 ~~ ~ , q:::; 00.
Asabove, by 2(Lp,
we denote the space of all bounded lin-ear
operators
with the usual norm, andby ~,(Lp, Lq)
the kernel space with norm induced from2(Lp, Moreover, by Lq )
we denotethe
(Zaanen)
space of kernels whichgenerate regular (see [9]
for thedefinition) integral operators
fromLp
intoLq ,
endowed with thenorm
Calculating
these norms forgiven
kernelsis,
ingeneral,
a difficultprob-
lem. In most
application, however,
one may use somesimple
classical in-equalities (see [6,9,12,13]), namely
the direct Hille-Tamarkininequali- ty
the dual Hille-Tamarkin
inequality
or the Schur-Kantorovich
inequality
here the numbers ro , rl ,
( o , oo )
and I E(0, 1 )
are connectedby
the conditions
There are more
complicated inequalitites
based onsophisticated
inter-polation
theorems like the classical Marcinkiewicz-Stein-Weisstheorem;
we refer the reader to
[9].
Suppose
that s,u)
andl(t,
s,u) satisfy
aCarath6odory
condi-tion,
and let 0 a 1. We define a functionha
of four variablesby
this may be considered as a H61der
analogue
to the Hadamard function.Assume that the
corresponding (generalized) superposition operator
is bounded from the space
Lp
xLp
into the kernel space~,(Lp~~ 1 + a> ,
We have then
and therefore
The boundedness of the
operator (47) implies
thatand
These
simple
calculations are useful and effective in ourproblem
dis-cussed
above; however, they require
thecomputation
of the norms of in-tegral operators
between + a) andLp
in terms of their kernelswhich,
as we
already remarked,
is notpossible.
Here is a
pleasant exception,
where this can bedone;
thefollowing
theorem is in fact a direct consequence of the definition of the kernel class
Lq):
THEOREM 6.
Suppose
that the kernelk(t ,
s,u) satisfies
thefollow-
ing
three conditions:where
for j = 0 , 1, ... , m .
T henand
6. The case
LM.
Whenever one has to deal with
operator equations involving strong
nonlinearities
(for example,
ofexponential growth),
it is a useful device to consider theseequations
not inLebesgue
spaces, but in Orlicz spaces.The constructions and results of the
preceding
section carry over to Or- licz spaces almost withoutchanges. However,
since Orlicz spaces have a morecomplicated
structure thanLebesgue
spaces, it is notsurprising
that the
corresponding
calculations become more tedious.Therefore,
weomit the technical details in the
following discussion,
andpresent only
some
important
facts. Detailed information on the definition and the ba- sicproperties
of Orlicz spaces may be found in[7,12].
Given two Orlicz spaces and
LN,
the spaces2(LM, LN), x(LM, LN )
andLN )
are definedanalogously
as before forLebesgue
spaces. Let us call apair
ofYoung
functionsMl (u)
andM2 (u)
admissible if theproduct
Xl X2 of any two functions x, ELMl
and x2 ELM2
isintegrable.
In this case theYoung
function M =M1
0)M2
definedby
theformula
generates
an Orlicz spaceLM
which is called theO -product
of the Orlicz spacesLMl
andLM2.
In the same way one can define admissibletriples, quadruples,
andn-tuples
ofYoung
functions andcorresponding 0-prod-
ucts of several factors.
Let cv be a concave function
staisfying
the conditionWe associate with cv the function of four variables
in case we
get
of course the function(46).
Assume that thecorresponding (generalized) superposition operator
is bounded
from .the
spaceLM
xLM
into the kernel spaceX(LMw’ LM ),
where
= M(cv ~ -1 ~ (~c) ) .
Then we have~’
where
Of course, here we encounter
again
theproblem
ofcomputing
the normof an
integral operator
in terms of itskernel,
but this time in an Orlicz space which is still more difficult than in aLebesgue
space(see
e.g.
[7,12,13]). However,
also in this case we may formulate sufficient conditions whichgive
aparallel
result to Theorem 6 for Orlicz spaces:THEOREM 7.
Suppose
that the kernelk(t,
s,u) satisfies
thefollow- ing
three conditions:and
’tjJ j (u)
are continuousfunctions
such that the corre-sponding superposition operators satisfy
theacting
conditionsthe
Young functions M(u), Mw(u), Pj(u)
and areadmissible,
and
for j
=0 , 1,
... , rn . Thenand
The
acting
conditions(56)
areequivalent
to the relationsfor suitable aj and
(j
=0, 1,
...,m).
Some formulas useful to calculate the functions(57)
can be found in[3].
Aknowledgments.
This work wasperformed
under theauspices
ofCNR-Italia and MURST-Italia. The second author was also
supported by
the Belorussian Foundation of Fundamental Scientific Research and the International Soros Science EducationProgram.
We wish to thankour friend
Jurgen Appell
who read andimproved
all the paper, and Julia V.Lysenko,
who discussed with us the results obtained in the firstpart.
We are
obliged
to the referee for manysuggestions
andremarks,
which have
improved
the paper.REFERENCES
[1] J. APPELL - E. DE PASCALE - JU. V. LYSENKO - P. P. ZABREJKO, New results
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