Rapid convergence of approximate solutions for first order nonlinear boundary value problems

(1)

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 3 (1998) 499-506

499

RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS FOR

FIRSTORDER NONLINEAR

BOUNDARY VALUE PROBLEMS

ALBERTOCABADAandJUANJ.NIETO Departamentode Anhlise Matemhtica

Facultadede Matemticas Universidade deSantiagodeCompostela

15706,Santiago deCompostela, SPAIN

SEPPO

HEIKKIL

Department

of Mathematical Sciences University of Oulu

90570Oulu57, FINLAND

(Received

August

20, 1996)

ABSTRACT. Inthispaperwestudytheconvergence of theapproximatesolutionsforthe following

firstorderproblem

u’(t)

f(t,u(t));t

[O,T],au(O)-bu(to)

=c,a,b

>

0,a+b

>

0,t0

(0,T].

Here

f-

I R ttissuchthat existsand is a continuous functionforsome k

>

1. Under some additional conditions on

-,

weprovethat it is possibleto construct two sequences of approximate

solutionsconvergingtoasolution with rateofconvergenceoforder k.

KEY WORDS AND PHRASES: Approximation of solutions, rapidconvergence.

1991AMSSUBJECTCLASSIICICATION CODES: 34A45,34B 15.

1. INTRODUCTION

The method of upper and lower solutions is a well-known theoretical procedure to prove the existenceofasolutionforagivennonlinearproblem. Underadditionalconditions,it ispossibletoapply themonotoneiterativetechnique that providesaconstructiveschemefor thesolutions. Moreoverone canuse the monotone iterates to giveerrorbounds. Forpractical

purpose

itwouldbe interestingto know the orderof convergence of thosemonotonesequencesof approximatesolutions.

Werecall thatforagiven Barmch space

(E,

II),

and aconvergentsequence

{x, }

zin

E,

it is said that theorderof convergenceis k 1,2, ifthere existg

>

0 and

no

Nsuch that

IIx./a

roll

-<

Xllz

xll

vn

_>

no,

Whenk 1

(k

2)

wesay that theconvergenceis linear(quadratic).

Itis not difficult to see[1],

[2]

that theconvergence

f

thesequenceof the approximate solutions given bythe monotone iterativetechniqueisveryslow. Indeed, that convergenceislinearbut ingeneral,

notquadratic. Under some convexity conditions, the method ofquasilinearization[3],[4],

[5]

providesa monotoneincreasingsequence converging uniformly and quadraticallytothesolution.

Itwouldbe importantto have somegeneral methods leadingto monotonesequences convergingto a

solutionwithorderofconvergencek

>

2.

Tobe specific, letusconsider thefollowingboundaryvalueproblem

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500 A.CABADA, J.J.NIETOANDS.HE1KKILA

where

f

Ix R ]Ris a continuousfunction,

to

E

(0,

T],

anda, b

>_

0, witha

+

b

>

0

As usual,wesaythataEC

(I)

is a lower solutionfortheproblem(1.1)if

Similarly,/ C

(1)

is anuppersolutionfortheproblem (I. I)if

If

t0

T,

itisprovedin

[6]

thata_</on

I

impliesthatthere existsatleastonesolutionuof(I I), with u

e

[c,/]

(v

E

C(1);

a(t)

_<

v()

_</(t),

t6

I)

Moreover,

ifthere exists

MI

>

0 suchthata be-M1t

>

_{0 and}

f(t, u)

4-

M1u

isnondecreasingin u

[a(t),/(t)],

I,

(1.2)

then it is possible to construct two monotone sequences

{a,)

and

{/),

which start in a and /

respectively,andconvergeuniformlytotheextremesolutions and of

(I. I)

on

[a,/].

Weinsistthat, ingeneral,theorderofconvergence ofthe monotonesequencesisatmostk Iin thespace

E

C(1)

with theusual uniform norm.

Nowwe obtainan extensionofthisresulttotheproblem(I.I)asfollows. First,weprovethatfor some m

> 0(a,

b

_>

0,a

+

b

>

0,a be

-’

>

0)

if

u’

+

mu

>_

0 in

I

and

au(O)

bu(to) >_

0 then

u

>__

0 inI. To provethis,weuseLemma2.3 in[6]andweobtainthatu

_>

0 in

[0, t0].

Thus,

u(to)

_>

0 whichimplies, using againLemma2.3 in[6],thatu

_>

0 in

It0, T]

and,inconsequence,u

_>

0 inI.

Now,

we definea0 a, =/, andforn

>_

1,a,and/,aregivenasthe uniquesolutionof thefollowing linearproblem:

u’

+

MlU

f(t, rl)

+

Mrl,

au(O)

bu(to)

c (1.3)

withr] a,_ andr]=/5,,_ respectively. Condition(1.2)implies that a

<_

cq s c,,

_<

/

s/5

andthetwosequencesconvergeuniformlytothe extremal solutionsof(1.1)

Onthe otherhand,notethat if exists and it is continuous in

n

{(t, u)

e

x

.;

,(t) <

u

</(t)},

then condition on

f

in(1.2)isequivalenttothe following requirement:

O..f

(t, u) >

M1

(t, u)

6 f. (1.4)

Recently

[7]

the methodof quasilinearizationwasgeneralized forthe initial

:alue

problem

(b

0)

by

notdemanding

f(t,

u)

tobeconvex inufor tE

I

but imposing the following lessrestrictive condition

thereexists

M2

>

0suchthat

f(t,

u)

+

M2u

2is convex inufor any t6I.

(1.5)

If existsand it is continuousfor every

(t,

u)

6f,then

(1.5)

holds andit isequivalentto

O2f(t,u)>

-2M2,

(t,u)

6f2 (16) Ou2

Ifthis lastcondition is satisfied, then there exists anondecreasingsequence startingatthe lower

solutionand converging uniformly and quadraticallytothe unique solution of the initial valueproblem [7].

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RAPIDCONVERGENCE OFAPPROIXIVlATESOLUTIONS 501 orderof convergenceis k providedthat there exists

---,

and it is continuous inf Notethat in this case there exists

Mk

>

0suchthat

i)k

f

(t,

u) >

(k!)Mk

(t, u)

f.

(1

7)

t:3uk

The periodic boundary value problem

(u(0)=

u(T))

is considered in [2]. There the authors

construct a sequence

{c,}

which converges quadratically to a solution u

[a,/3]

ofthe periodic problem. Inthiscase, they supposethat

f

satisfies(1.6)and

a(T) <

6

<

0, where

(,())d,

(1 8)

Of

o()

and

[a,/].

Inthispaperwestudyproblem (1.1)andwe construct two monotonesequenceswhichconvergeto theextremalsolutions in

[a,/]

of

(1.1)

provided thatthere exists and it is acontinuousfunction in

and that for each

[a, ]

a-bea(t)

>

6

>

0. (1.9)

The following result from

[8]

isthebasictooltoproveourmainresult.

THEOREM1.1. If there exista_</3loweranduppersolutionsrespectively for theproblem

(1.1),

then thereexists a solution u

[a,/]

of

(1.1).

We finallynotethat wegeneralize previous knownresults.

2. MAINRESULT

Now, we obtain, in the following result, that if there exists acontinuous function in f and condition

(1.9)

isverified,then it ispossibleto construct twosequenceswhichconvergeto the extremal solutions and of(1.1) rapidly,that is, the order of convergenceisk.

THEOREM 2.1.

Suppose

that there exist a_</3lower and uppersolutionsrespectively for the problem(1.1).

Ifthere exists k

_>

1 such that is continuous inf2, and if condition(1.9)isverified,thenthere

exist two monotonesequences

{a.}

and

{/,, }

witha0 aand

&

=/3,whichconverge uniformlyto

theextremalsolutions

b

and of(1.1)in

[a,/3].

Thisconvergenceisof orderk.

PROOF. Wefirstnotethatproblem(1.1)has, byTheoremI.1, a solution in

[a,

].

Letusdenote 7 such asolution.

Toconstructthesequence

{a,},

lett

I

and

a(t) <_

v

<_

u

<_/(t).

Wefirstnotethat foragiven

tEI:

f(g,

u)

Oi

f

(t, v)

(u

v)’

O

k

f

(u

v)

k

(2.1)

+

(t,x(t))

k!

where

X(t)

E

Iv,

u].

Now,

since exists and is continuous inf, (1.7)isverified.

Thus,we define

g(t,

u,

v)

c3’f

(t,

v)

t-o

,U)

M(-

)k.

Inconsequence, using(1.7),we obtainthat

g(t,u,

v) <

f(t,

u),

for all t

I

and

a(t)

<

v

<

u

< (t).

Now,let us considerthe following boundary value problem

(2.2)

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502 A. CABADA, J. J NIETO AND$.HEIKKILA

’()

g(,(),(t)),

z,

(o)

.

(2.4)

NOW,

e

r,

,(o)

b,(o)

and

a’()

f($,())

g(,(),a()),

I,

aa(O)-ba(to)<_c,

thatis,aand7 are loweranduppersolutions for

(2.4)

respectively.

Theorem 1.1showsthat there existsatleastone solutionof(2.4)

ax

E

[a,

7].

Now,supposewehave constructeda0 a

<

al

<

< an <

7, with

an

asolution of

u’(t)=g(t,u(t),an_x(t)),

EI,

au(O)-bu(to)=c,

lyingin

[a, "7].

Inthiscase,wehave that

7’(t)

f(t,7(t)) >

g(t,

7(t),an(t));

(o)

3(to)

and

an(t)

g(t,

an(t),an-x(t)) < f(t,a(t))

g(t,

an(t),a(t)),

,,.(o)- bc,,,(to)=.

We conclude,using again Theorem 1.1, thatproblem

u’(t)

g(t,u(t),an(t)),

I,

au(O)

bu(to)

c (2.5)

hasasolution

an+x

Jan,

7].

The so obtainedsequence

{an }

isnondeereasingandboundedinC’1

(I),

whence itconvergesin

C(I)

tosome continuous function

[a,

7].

Since

(tl

(ol

+

(,(,-(1),

wehavethat

b(t)

(0)

+

g(s,

(), )(s))ds

(0)

+

f(8,

whichimplies that

’(t)

f(t,

(t)).

Furthermore, since

aan(0)-

ban(to)=

c for all n

>

1, we conclude that

a(0)-

b(to)=

c.

That is,

p

[a,

7]

is asolution of

(1.1).

Since 7 is anarbitrary solution of

(1.1)

it isclear that isthe

minimal solutionof(1.1)in

[a,/],

which existsby(1.9).

Now,

weprove that theconvergenceisorderk. Forit, using(2.1),wehave that

’(t)

f(t,

(t))

,=o

(t,,:,,,(t))

!

+

(t,p(t)) !

a(O)

bp(to) c,

p,, E

[,,,,

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RAPIDCONVERGENCE OFAPPROIXMATESOLUTIONS 503

Letw,, a, and

a

a,+l _t2n. Thus,wehave that

l=l

(t)

Ou

O/(t’p’(t))

k! +Ma(t),

aw.+(O)-.+(to)

O.

Now,

thereexists

Nk

_

0suchthat

okf

(t,X)

<

(k!)Nk

_for_all

EIand x E

[a(t) (t)].

Ou (2.6)

Furthermore.

a(t)

< wn(t),

for all n N and

e

I.

B’

(A

B)

’-

A’-

1-B

wecanwritethat

Finally, using that for all

A,

B

6

R,

2=0

w’+(t)

p(t)wn+,(t) <_

where

Ck

N

+ M

>

0and

()

:.,

--()()

=1 ₃₌₀

In consequence,it is verifiedtt

where

a.(t)

f

p.(a)da. Thus,

w.+,(t)

ea"(>

wn+(O)+Ok

e-"(’>w(a)da

(2 7)

Now,

using expression(2.7)for

to

dthe

ui aw.+(O)

bw+(to),

_since_a,b _0, _we conclude that

Now,

duetothe

Nct

that

,

w

0 n

,

dusingte

eresin

f

,

endition(1 9)

implies thatthere

ests

Nch that e(tl

>

/2 >

0for 1n

no.

Thus,

o

where

A

is apositiveconat. Notethat the

prous

inequities holdsincethere

ests

in fld they e eominuous nions for 1, k. Thus, since

a

[a,],

we have

m ere

ests

a constt

D

>

0 such

tt

I(t)l

5 D

for 1n

.

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504 A. CABADA, J. J. NIETO ANDS.I-IEIKKILA

Here

Mk

and

N

arenonnegativeconstantsgiven by(1.7)and

(2.6)

respectively Thus,it iseasytoseethat

h(t,

u,

v)

>

f(t, u),

forall EIand

a(t) <

u

<

v

< (t).

Now,

let

.

Forn

>_

1 wedefine/, by induction,as a solution of the followingboundary

value problem

=’(t)

h(t,=(t),#_(t)),

Z,

(o)-=(to)

.

(2 9)

Indeed, using

(2.8)

it iseasytoseethat/_isanuppersolutionand₇isalower solutionfor (2.9). In consequence,7

<

,,

<

_1

<

=/

forn

>

1, and

{/}

convergesuniformlyto

,

where is the maximal solution in

[a,

]

of(1.1).

Now,

thedefinition ofh, expression(2.1)and inequalities(1 7) and(2.6)implythattheconvergenceof

{

,

}

to isoforder k. F’I

REMARK

2.1. Condition (1.9) may seemvery restrictive but, as we will see in thefollowing

exhrnple,

in somecasesit is afundamentalcondition. Letusconsidertheproblem

u’(t)

f(u(t)),

E

I,

u(0)

u(to),

with

f

definedby

f

(u)

u if u

<

0 and

f

(u)

0 otherwise.

Notethata

1/2

and/ 0arelower and uppersolutionsrespectively forthisproblem. Analogously to theexample given in [2] weshow that thesequencesobtained viathe monotone

method converge linearly butnotquadraticallyto theuniquesolutionu 0. Ifweusethe function g(for

k

2)

as inTheorem2.1 (seeformula

(2.2))

weobtainthatctn+l

(2

V/’)an.

Clearly, thereexists aconstant)

>

0such that

II+all

<_ ,1111

if andonlyif c,,+

_<

(x/

2)/).

Thislast inequality

doesnothold.

Notethatin this case condition(1.9)reads

1-e2C(s)as

>

6

>

O, j(5

[-

1/2,

0].

This isnottruesincefor 0thisexpressionequalszero.

BOUNDARY CONDITIONS

au(tx)

bu(T)

c Inthissection weshall consider thefollowing problem

u’(t)

f(t, u(t)),

t

_

I,

au(tl)

bu(T)

c, a,b

>

0,a+b

>

0

(3.1)

where 0

<

t

<

T.

Forit wesay thatcris alowersolutionfor(3.1)if

d(t)

<

f(t,a(t)),

t

I,

aa(t

)-ba(T) <

c.

Analogously,wedefineanuppersolutionbyreversing the previous inequalities.

Thiscase canbereduced, byasimplechangeof variable,tothat consideredinprecedingsections, as

wewillseeinthe followingresult.

THEOREM3.1. If thereexisttr

and/

lower anduppersolutionsrespectively of(3.1)on

I,

with

<

a, there exists acontinuous functionin

{

(t, x);

E

I, (t) <

x

< or(t)

}

and

f

satisfies

b-ae(tt)

>

6

>

0, (3.2)

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RAPID CONVERGENCE OF APPROIXMATE SOLUTIONS 505

r

of

(,())d,

and 6

[/,

a].

Then thereexist two monotonesequences

{an)

and

{g/n

witha0 a and

&

B,

whichconverge uniformlytothe extremal solutions/,and of(3.1) Thisconvergenceisoforderk PROOF. Toprovethisresult we consider thefollowingmodifiedproblem

u’(t)=(t,u(t)),

tel,

bu(O)-au(T-tl)=

-c. _(3.3)

Here

f(,x)

f(T- t,x).

Using the concept oflower and upper solution for

(3.1)

it is clear that

(t)=

a(T-t)

and

/(t)

=/(T- t)

are an upper and a lower solution respectively for the problem (3 3), with

<

Furthermore, using (3.2), wehave that

f

satisfiescondition (1 9). Thus, we are intheconditions of Theorem2.1. Inconsequence thereexisttwo monotonesequences

{n)

and

(n),

whichconvergeto

the extremal solutions of (3.3) with rate of convergence k. The proof is completed defining

an(t)

",(T- t)

and

,,.,(t)

,.,(T- t).

El

REMARK3.1. Notethat it isnotpossibletoextend the results obtained inTheorems2 and3 to the conditions

au(to)

-bu(t)

c with 0

<

to

<

$

<

T. Inthis case (see [8]), thepresence of

lower and uppersolutions is not a sufficientconditiontoassurethe existence ofasolution.

Asimilarcommentis validfor the problem(1. l)witha_>/and

(3.1)

witha

_<

ACKNOWLEDGEMENT. A. CabadaandJ. J Nieto werepartially supported byEECprojectgrant ERBCHKX-CT94-09555 and bytheDGICYTprojectPB94-0610.

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V. and NIETO,

J.J.,

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&

Digest2(1995), 1-10

[3]

BELLMAN,

R.,Methods

of

NonlinearAnalysts, Vol.

II,

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[5]

SHAHZAD,

N.and

MALEK, S.,

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CABADA,

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!inear

and nonlinearboundary

conditions,Appl.Math.

Comp.

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(1994),

163-186.

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MALEK,

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