Second Two

J. oflnequal.&Appl., 2001, Vol. 6, pp.191-198 Reprints availabledirectlyfrom the publisher Photocopying permitted bylicenseonly

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Printed inSingapore.

Comparison of Two Definitions of

Lower and Upper Functions

Associated to Nonlinear Second Order Differential Equations

IVO

VRKO0*

Mathematical Institute,Academy ofSciencesof the Czech Republic, 115 67PRAHA1,’itn&25, Czech Repubfic

(Received25July1999; Revised 10October1999)

Thenotionsof lower and upper functions of the second orderdifferentialequations take theirbeginning from theclassical workbyC.Scorza-Dragoni and have been investigatedtill nowbecause theyplay an important roleinthetheory ofnonlinearboundary value prob- lems.Mostof them define lower and upperfunctionsassolutions ofthe corresponding second order differential inequalities. Theaimofthispaperis tocomparetwomoregeneral approaches. Oneisdue to Rachflnkov/tandTvrd ^{(Nonlinearsystemsof}differential inequalities andsolvability ofcertainboundary value problems(J.oflnequal.&Appl.(to appear)))whodefinedthe lower and upperfunctionsof the given equationas solutionsof associatedsystemsof two differentialinequalitieswith solutionspossiblynotabsolutely continuous.ThesecondbelongstoFabry and Habets(NonlinearAnalysis,TMA10(1986), 985-1007)and requires the monotonicity ofcertainintegro-differential expressions.

Keywords: Differentialinequalities; Second ordernonlinearordinary differential equation; Lowerand upper functions; Right derivative; Boundedvariation AMS SubjectClassification:^{34A40, 34B15}

The method oflower andupperfunctionsisaneffective tool in thetheory of nonlinearboundaryvalueproblemsfor the second order differential equation

u"-f(t,u,u’). (1)

*E-mail:[email protected].

191

Let

us note that the terminology ^{is not} uniform and some authors usethetermloweranduppersolutions. Until now,alotof definitions of these notions, less or more general, have been introduced. In

[3],

the authors made use of the following definition of lower andupperfunc- tions of

(1)

^where

f

fulfils theCarathodoryconditions on [a,

b] 2,

i.e.

fhas

^thefollowing properties: (i)for eachx and y thefunc- tion

f(.,

x,y) is measurableon[a, b]; (ii)for almost every t

[a, b]

the function

f(t, .,.

is continuouson

2; (iii)

for each compactsetKC

2

the function

mr (t)

sup,y)r

If(t,

x,

Y)I

^isLebesgue integrableon

[a, b].

By Car([a, b] 2)

^{we denote the} set of functions which satisfy the Carathodoryconditions on

[a, b] 2.

DEFINITION Functions

(or,

p)arecalledlower(upper)

functions of ⁽¹⁾

(on [a, b])/for

^{isabsolutelycontinuouson}

[a, b]

andphasaboundedvariation on

[a,

b], the singularpart

psing of

^pisnondecreasing(nonincreasing)^on

[a, b]

andtheyverify thefollowingsystem

of differential

inequalities:

a’ _t) p( t)

^{a.e. on}

[a, b],

p’(t) >_ f(t,a(t),p(t)) (p’(t) <_f(t, cr(t),p(t)))

a.e. on

[a,b]. (2)

Onthe otherhand, the authorsof

[1]

introduce somewhat different definition of lowerandupperfunctionsof

(1) withfcontinuous.

DEFINITION 2 A lower(upper)

function of ⁽¹⁾

^{is acontinuous}

function

a on

[a, b]

whichpossesses

for

^{all E[a,}

^b)

the right derivative

a+ ^(t)

^and

P_ (t)

^isright-continuous

for

^all

(a, b]

^the

left

^derivativea

^(t)

^{such that}

a+

on[a,

b),

aP_(t) ^< a+(t) ^{(a_(t) >} a+(t)) for

^all

^(a,b)

and

a(s), a+ ^(s))

^ds

a+(t) ^f(s, -a+ ^{(t) + f (s, a(s),} a+ ^(s)

^ds

isnondecreasingon

[a, b).

At

oneglancetherelationshipof Definitions and 2is notclear.Their comparisonwillbe given by the followingtwotheorems.

LOWERAND UPPER FUNCTIONS 193

THEOREM1 Let

f

^E

^{Car([a, b]}

^x

]I2).

Then,

if

^{a isalower}(upper)function

of(l)

inthesense

of Definition

^2,the

functions (a,

p),where

o-(t)

^on

[a, b]

^and

p(t) (t) /f [a, b),

lims--,b-

a+ ^{(s) /f}

^b,

(3)

arelower(upper)functions

of ⁽¹⁾

with respectto

Definition

^1.

THEOREM 2 Letthe

functions (a,

p)be lower(upper)

functions of ⁽¹⁾

ⁱⁿ

thesense

of Definition

1. Then the

function

^a

defined

^by

a( t) or( t) for

^E

[a, b]

isalower(upper)function

of(l)

inthesense

of Definition

^2.

Let

usstartwith theproofof Theorem 1.

To

this aimthe following three lemmasarehelpful.

LEMMA

andassume

Leta

function

^{g be}

defined

^andcontinuous ontheinterval

[a, b) (i) g (t)

^exists

for

^all

^[a, b),

(ii)

g t)

^isright-continuouson

[a, b)

and

(iii) thereexistsa

function

^{hcontinuouson}

^{[a, b)}

andsuch that

g (t) h(t)

isnondecreasing^on

[a, b).

Then the

function

^{g is absolutely continuous on every interval}

^{[a, c],} c(a,b).

Proof

^Choosec

^{(a, b).}

^Condition

⁽ⁱⁱⁱ⁾

^implies

g+(t)- h(t) < g+(c)- h(c)

^{for t}

[a, c]

sothat

g (t) < g (c) + h(t) h(c) ^<_ g_ (c) + max[h(t) h(c)].

tE[a,c]

Similarly,for E

[a, c]

^wehave

g+(t) ^>_ g_(a) + h(t) h(a) > g+(a) +

^min

[h(t) h(a)],

tE[a,c]

194 I.

VRKO

i.e.

g’ +(t)

^{isbounded.Define}

(t) g+(s)

^{ds for E}

[a,b).

Due to (ii) (continuity from the right of

g_(t)),

for any

E[a,b)

the derivative

_ ^(t)

^isdefined and

g+

-!

(t) g+ (t).

Now,

the proof ofthe lemma will becompleted bytheproofof the followingrelation:

g(t) ,(t)

^=_

g(a) (a)

^on

[a, b).

Denote

A(t)--g(t)-(t).

^Then

A(t)

^{is continuous on}

[a,b)

^and

A_ (t)

0 for any

[a, b). Assume

^{that thereis apoints}

(a, b)

^such that

A(s) > A(a)

anddefine

p(t) [A(s) A(a) + ^{A(s)- A(a)} (t- a)]- (A(t)- A(a)).

s--a

(4)

Certainly, p(a)>0 andp(s)<0. Let

t*

be the greatest point in

(a,s)

fulfilling

p(t*)

0. Equality

(4)

yields

1A(s) A(a)

>

0,

p+ (t*)

s a

Sincep(t)

<

^{0 for}

> t*,

^wehave

p(t) -p(t*)

t* <0

and hence

p+ (t*) ^_<

0,acontradiction with

(5).

^Thecase

A(s) < A(a)

is symmetric.

LEMMA2 thelimit

Letabealower

function ^of(l)

^accordingto

Definition

^2.Then

’(b-)

^lim

ex&tsandis

finite.

LOWERANDUPPERFUNCTIONS 195

Proof ^Let

^usdefine the functionr

[a, b)

bythe relation

r(t)

^for

[a,b)

a+(t) f(s,a(s),a+(s))ds +

^E

(6)

DuetoDefinition2,ris nondecreasing andright-continuouson

[a, b).

Consequently,the limits

lim

a_(s) a+ ^(t-),

^E

^{(a, b) (7)}

existand

a_(t-) ^_< a_ (t)

^on

(a, b). (8) Now,

let

liminf

t-.b-

a+(t) <

^limsup

Letusfixthe numbers c,₂⁽ insuchawaythat

liminfa_(t) <

1

<

C2

<

limsup Lettl,t2^G

(a, b)

be such that

a.’+/-(t) ^_>

^{c2 and}

a+(t2) ^_<

^cl.

Considerafamily2 ofclosedintervalsI=

[r, 7"2]

^fulfilling

IC

[t,t2], a+(n) ^>

^c2,

a_(7"2)<

cl.

(9)

Let

In

^{be a decreasing sequence} of intervals from 2". Its intersection I

NnIn

^{is a nonempty interval}

^I=[c,2

and the existence of the limits

(7)

togetherwith the relations

(8)

^and the continuity from the right of

a+

ensure that conditions

(9)

^are fulfilled for/, either, i.e.

I 2".

It

means that a minimal interval Imi,=[z,z2] ^{exists in the} family 2-. Minimality yields

a+(t) (Cl, c2)

^{for E}Imi,. We conclude

(put K=_supt[a,b]

]a(t)])

^that

This is in contradiction with the Carath6odory property of

f

^since

infinitelymanyofsuch disjoint intervalscanbe constructed.Hence liminf

tb-

a+(t)=

^limsup

tb-

a+ ^t) t-b-lim ^{a+ (t).}

If

lim

t--b-

0+ ^c

werevalid, thenby

Lemma

wewould have

a’_ (b)= limtb_ a(b)b ^{lim- a(t)-}

tb-

(ln- ^a()7- )

lim

(s)

ds

t---b- 7"-

acontradiction.

Similarly, if

lim

a+ ^(t)

^-o

t--b-

held,wewould obtain

a’_ (b)

^-o,againacontradiction.

LEMMA3 Letabealower

function of(l)

^accordingto

Definition

^{2 and}

let p begivenby

(3).

Then p hasaboundedvariation on[a,

b].

Proof ^Lemma

^{2yieldsthat the}nondecreasingfunctionrgivenby

(6)

hasafinite limit

r(b-)= limt ^b_r(t),

^{i.e. ithasa}bounded variationon [a,

b].

Denoting

h(t) f (s, a(s), p(s)

ds,

LOWERAND UPPER FUNCTIONS 197

wecanwrite

Ip(ti+l) p(ti) h(ti+l) -+- ^{h(ti)l p(b) p(a)} h(b) + h(a) r(b-) r(a).

foranarbitrary partition

{

ti}of the interval[a,

b].

Thus

varba

^p

^< var6a

^h

^{+ r(b-) r(a).}

Proof of

^{Theorem 1}

^Let

^abealower functionof

(1)

with respectto Definition 2 and let the function rbe again given by

(6).

^{In the}proof of

Lemma

3 ithasbeenalreadyshown that thelimit

r(b-)

^existsandis finite.Furthermore, integrating

(6)

^{andmakinguseofLemma we}get

a(t) a(a) + f(s,a(s),a+(s))dsdT"

+ r(s)ds

^on

[a, hi,

i.e.aisabsolutelycontinuouson

[a, b]. Now,

^letrand0be definedby

(3).

Thus, ^isabsolutelycontinuouson[a,

b]

and, accordingto

Lemma

3,p has a bounded variation on [a,b].

Moreover,

in virtue of

(3)

^and Definition2,the function

p(t) f ^{(s, or(s), p(s)}

^ds

is nondecreasing on

[a, b).

^{Itmeans that the} couple

(or,

p)verifies the inequalities

(2)

and, moreover,thesingularpart ofp isalsonondecreas- ingon

[a, b) (cf.

^e.g.

[2,

Theorem

125]

or

[4, II.25]),

i.e.

(or,

p)arelower functions for

(1)

according^toDefinition1,either.

Analogouslywewouldargueinthecaseofupperfunctions.

Proof of

^{Theorem2 Let}

^(tr,

^p)be lower functions of

(1)

^withrespectto Definition 1. Sincethefunctionphasafinite variationandsince

+

^on

bl,

the limitsp(t+), p(t-) exist.

It

followsthat the function rhas the one- sided derivatives

cr+(t) ^p(t+)

^and

_(t) p(t-)

in each

E[a,b)

^or

isright E

(a, b]

respectively.

As

p(t+)isrightcontinuouson

[a, b), a+

continuouson

[a, b)

either.

Denote

paC

and

psing

theabsolute and singularpartsofp,respectively.

Since

(psing)’(t)=

0almosteverywhere

(see [4, II.25])

^wehave

(paC)t(t) >_f(t,r(t),p(t))

^{a.e. on}

[a,b]

andsince

psing

isnondecreasingon

[a,

b],fora

_<

_tl

<

t2

_<

bwehave

p(t2) p(tl) pae(t2) paC(tl) + psing(t2) psing(tl)

>_

insteadofpand considering the fact Substitutingainstead ofcrand

a+

thatp(t+)

:

^p(t)and hence also

a_ (t) : ^p(t)

^canhappen onlyinatmost countably many points E[a, b],wegetthe monotonicityon

[a, b)

of the function

a+(t)- f(s,a(s),a+(s))ds= ^p(t)- f(s,a(s),p(s))ds

requiredbyDefinition 2.Finally,the relationsa

_ ^{(t) <_ a+ (t),}

^E

^{(a, b),}

follow from the fact that

psing

isbyDefinition nondecreasingon

[a, b].

Analogouslywewould argueinthecaseofupperfunctions.

Acknowledgments

Thiswork wassupported by the grantNo.

201/98/0227

^{of the Grant}

Agency

of the CzechRepublic^andbythe

MMT

Grant No.96086.

References

1] Ch.Fabry andP.Habets,Lowerand upper solutions for second-order boundary value problemswith nonlinearboundary conditions,NonlinearAnalysis, TMA 10(1986), 985-1007.

[2] V.^Jamik.Integral CalculusII(inCzech),NakladatelstviCeskoslovensk6Akademie v6d, Praha, 1955.

[3] I. Rachflnkovfi and M. Tvrd, Nonlinear systems of differentialinequalities and solvability ofcertainboundary valuep.r.oblems,J. oflnequal.&Appl.(toappear).

[4] F.RieszandB. Sz-Nagy,Vorlesungen Uber Funktionalanalysis,VEBDeutscher Verlag der Wissenschaften, Berlin, 1956.