Singular boundary value problems for quasi differential equations

[nternat. J. Math. & Mth. Sci. VOL. 18 NO. 3 (1995) 571-578

571

SINGULAR

BOUNDARY

VALUE

PROBLEMS

FOR

QUASI-DIFFERENTIAL

EQUATIONS

PAUL W. ELOE

JOHNNY HENDERSON

(’enter h)r I)v,amical

Systcns

and Nonlinear Studies

(’,(’ogia

Instit,tteof

qchnology,

Atlanta,

Georgia

30332

and

l)cpart,,,t

of1)iscretcand Statistical Scieccs

A,,burn _iniversity,

Auburn,

Alabama 36S.19

USA

(Received April 13, 1993)

ABSTRACT. Sol.tionsar,’obtainedofboundary valueproblemsfor

L,,+

f(x,

Loy

L,_.y),satisfyingL,y(O)= L,,_y(l) O,0

<

<

n-2, where

L,

denotes the

’

(lasiderivative,and wheref(.r,y y,_l) has singularitiesat y, O,

<

<

n- 1. KEYWORI)SAND PIIRASES.Q.asi-derivative, singularboundaryvalueproblem,cone. 1992

AMS CLASSIFICATION

C, OI)E.34B15.

1.

INTRODUCTION.

We

first definethequasiderivativeoperators,

L,,

0

<

<.,

inductively by,

l<i<n,

where

p,(x)

C’"-’)([0,1],

(0, o)),

0

<

<

n,andwe assume

p._,(x)=

on

[0, 1].

We

will beconcerned

with solutionsofboundaryvalueproblemsfor thequasi-differentialequation,

L,y

+

f(x, Loy, Ly

L,_y)

0,

(1.1)

satisfyingtheboundaryconditions,

L,y(O) 0,0

_<

_<

n-2, and

L,_y(1)

0,

(1.2)

where

f(x,

yi,...,y,_l) 0 hassingularitiesaty, 0, g

<

n- 1.

For

notation,welet

and

and weassumethroughou.t that,

(5,= inf p,(x),O<i<n,

O<x<l

A,

sup

p,(x),

0

<

<

n,

(A) f(z,yt !/,,-1) ((). (0.

)"-I

(0,vc) iscontiliuous; (!) f(.r.!/l !/,,_) s(l(,cl(,,si y,. forea(l

y,,),

_

n- 1"

((’)

IS

f(x,Y

y,,_)dx

<

.

for each fixed (y y,,_);

(1)) lira f(x,y y,,_)=

+

uniformlvonconpacl

subsctsof(0,1)x(0,)

’’-.

<<n-

1"

(E)

li

f(z.g 9,,_)-0 uiformlyoncopacl subsels of(0.1) x (0.)’’-e

-

1.

Weobservethat,fy isasolulion of(1.1),lhe by (A),

L,,y

<

0,sothat L,,_y is a concavefunction. Thesingularboundaryvalueproblem 1.1),(1.2) generalizesin some sensethe secondordernonlinear singularproblemsconsideredbyBobisud

[1],

Bobisud andLee

[2],

DoandLee

[3],

Garnerand Shivji

[4],

O’Regan

[51,

Tineo

[61,

and

Wang

[7]-[8].

Among

thoseworks,

[1],

[4],

and

[8]

usedsingular boundaryvalue

problemstonodel difl’usio prollemsarisingin physiology and physics, whilein

[5].

singular problems

included asspecialcases theTho,nas-Fermi and l,nden-Fowlerequations.

Moreover.

i,,

[1],

[2],

[3], [5],

[6],

and

[7]- [8],

a pr,om bounds are established on solutions, and the,, Schauder degr or Granas topological transversality applicationsyield solutions of the singularboundary valueproblems. Others

have also used these metlod (in theca.sep,(z) 1,0 n,so tha,

L,

),

tbringular boundary

valueproblems,withthe paperbyEloe and ltenderson

[9]

containingmanyreferences tothoseworks.

Our

tnotivationfor the techniques used inobtainingsolutionsof

(1.1), (1.2)

arethe worksby

[10]

and

[],

followed

,y

t, pp,’ by

:oe

,,d H,,d,o,,

[e]

,,d

,,a.,-so,, ,,a w,,

It3].

’rhearguments

involve concavity properties,aniterative technique, andafixed point theorem dueto

[ll]

for mappings thataredecreasing withrespect to a cone in aBanach space.

In

Section2,westate someproperties ofa cone inaBanach space,followed bythe fixed point theorem.

In

Section 3, weconstructasuitablecone and define asequence of modifications of

f,

sothat noneofthese modifications havethe singularities

of

f.

For this sequence, weconstructasequence ofoperators,each of which satisfies thehypothesesof the fixed point theorem, hence obtMningasequenceofiterates in thecone. The sequenceisshownto

convergetoasolutionof

(1.1), (1.2)

inthecone.

2.

SOME PRELIMINARIES.

The following definitions and properties ofconesin a Banach spacecan befound in

Amann’s

[14]

treatise.

Let

B

beaBanach space, and h"aclosed, nonempty subset of

B.

h" isaconeprovided

(i)

au+v

K,

for all u,t’

K

and all a,/3

_>

0. and

(ii)

u.-u It" imply u 0. Given acone

It’,

apartial order,

SINGULAR BOUNDARY VALUE PROBLEMS 573

Were.hark lhal. ifh"isa or,l conein

B.

lhen cl)s,d orderi.lervalsareorbounded. We nowslate a fixed poittt theoret d,e to(;atica. Oliker, a.d W,lt.a

[I

1]

for operators lhat are lecreasing ilhrespecl loa co.e.

THEOREM 1. Lit

I

a H.nach .act.

K

a uormalcont ,n

L.

E

C_

A

..uch that. ,fx,y C

E

wth x

<_

!t, then

(x.

y)

C_ E, aud hi7"" !’.’-- h"

&

acouttnuous mapping that isdctrasiugw,th

reswct

to h’,

,ridwhich iscompac!on.ny clo.,iorthritt r’alcont.tncd in

E.

Suppos( thcrcexists

xo

E

such that

7

’-’.re,

T(I’xo)

ts

dJintd...d

both

7"Xo

ami

7’2o

art ordt.rcomparable toXo.

If

eithcr.

()

Tzo

<_

xo

..d

7’’.to

<_

x, or

()

x,

<_

"l’x, a.d.r,

<_ T’.ro,

thtn

T

has

fiJ’td lfint

i

E.

3.

SOLUTIONS

OF

(1.1),

(1.2).

In

tillssection, wewillapplyTheorem to asequence of operators thataredecreasing with respect

o

an approl)riatecone. We tlwn obtainasequence ofiterates from these fixed points whichconverges

1o asolution of

(1.1),

(1.2). Concavity ofthe (n 2)’’aquasiderivative ofa solutionplays arole in this construction.

Let the Banach space

B

C[’-’)[O, 1]

withnorm

IIII

mx{ILoyl

IL.-.yloo

},

where

I"

1oo

denotesthesupremum norm, and let

It’= {y

13

L,y(x)

>

0on

[0,1],0

_<

<

n-

2}.

K

isanormalconein/3.

We

alsonotethat,ifu,vfi/3andu

<

v

(wrt

K), then

L,u(x) <

L,v(x)

on

[0, 1],

0 g

<

_n-2.

In

addition,wewillhaveneed ofthesignof the

Green’s

function,

G(x,s),

for theproblem,

L,,y

O,

L,y(O)

0, 0g

<

_{n 2, andL,,_ty(1) 0.}

(3.1)

Eloe

[15]

has proved

(L,),:G(x,s) >

0on

(0,1)

x

(0, 1),

0

<

<

n 2.

(3.2)

By

asolution, y, of

(1.1), (1.2),

we mean y

C(")(0,

1)f3

C("-l)[0,

1],

y satisfies

(1.1)

on

(0,1),

y satisfies

(1.2),

and

L,y(x)

>

0on

(0, 1),

0

<

<

n-2.

For such,

weseek afixed pointof theintegral operator,

But

becauseofthesingularitiesin

f

given by

(D),

wecannotdefine

T

onall of thecone

K. We

nextlet

"[0,

11

[0,

c)bethe solution of

for eacl 0

>

O,

notelhal,for 0

>

O,a,d0

_<

<

,

2.

HII(]

L,u(Ol:O. 0(__<,-2, ,u(O) 1,

L,r.i .r

dr

dr.., dr,

p,+,( r, )--. p,,_..,(r._,)p,,_, (r)

(3

L._,9(.r) 0on

[0.1],

so lat9( h’,al(! in fact L,9(x)

>

0on(0,

1],

0

5

n 2.

Assume

for the remainderoft]wpaper.

(1") Foreach 0

>

O.

[

f(.,Log,(x),L,g,(x) L,,_g(x))dx

<

.

Finally,wedefine

D

h" by

D=

{B]

there

exits0()>0such

thatg

(wrt

K)},

anddefine

T" D

h"by

It followsfi’om

(A)- (F)

and properties of

G(x,s)

that,if

e

D,

then

T

satisfies

(1.2), L,(T)(x) >

0 and increasingon

(0,

1],

0 n 2,

L_(T)

iseoneeve on

[0,1],

and that

r

D.

On

the otherhand,if

C(0,

1)n

C("-[0,1]

is asolution of

(1.1), (1.2),

with

,(x) >

0on

(0,1],

0 n-2,itagainfollows from the concavityof

_

thet

D.

Consequently,

D

isasolutionof

(1.1),

(1.2)

iff

T

.

Our

first result of thissectiongivesapriori boundson

L_,

for all solutions of

(1.1), (1.2),

that

belongtoD.

THEOREM2.

Assume

that

(A)-(F)

are

satisfied.

Then,theexists an

R

>

0suchthat

JL,_]

R,

for

all solutions

of

(1.1), (1.2),

thatlongto

D.

PROOF. Assume

the conclusion tobe false.

Then,

there is asequence

{t}

D

ofsolutionsof

(1.1), (1.2).

suchthat Jim

[L,,_t[

.

We

mayassume

that,

for each 1,

IL,-=tl

IL,-t+l.

(3.5)

From

the equation

(1.1), L,_t(x) >

0 and decreasing on

[0,1),

and from

(1.2),

L,t(x) >

0 and increasingon

(0.1],

0 n-2. Itfollowsthat,for each

,

SINGULAR BOUNDARY VALUE PROBLEMS 575

In

atltlilion,theconcavilv and posilivityof ..,timplylhat

L,,_..,t( )) d.

L,,_

7t( )-x

<

L._.t(x),

0

<

x

<

1. p,,_(.)

So, from the monoto,it’ityin (3.5)and (3.6),ifweset/9

L,,_.,(1),

then

L,,_..,:/,(x)

<_

L,,__t(z)

on

[0,],:

_>

I.

From the conditions satisfio(l l)y g and t at x 0, upon multiplying successively by

(p,(x))

-t _and

integrating,weol)tain

.q<t(wrt K), for nile> 1.

Now,

set

0

<

M

sup

(L_)G(z,s).

[0,]xt0.H Then

(B)

and

(F)

yield that, for0

<

x

<

and/

>

1,

L,,_..,qat(z)

L,,_.(Tpt)(x)

(L,,_.)a(x,s)f(s, Lo:t(s)

<_

M

I(,o(s),...,_,.e()Id

forsome0

<

N

<

o.

But

0

<

x

_<

and

t

>

werearbitrary.

So

IL.-2loo

_<

N,

for all

t

>

1,

which contradicts lim

L,_t

Ioo=

.

Theproofiscomplete.

COROLLARY. Assume

(A)-(F)

are

satisfied.

Thenthereexistsan

>

0such that0

<_

L,(x)

_<

,,_

8,+,

(n

TZ

)

on

[0,1],

for

0

g

g

n 2, and

I111

0sis,-sup

,-

,+

’

for

all solutions

of

(1.1), (1.2)

that long to

D.

In rticular,

(n-

2)

x"-

(wrt K),

for

allsolutions

:

O

4(1.1),

(1.2).

Next,

for each

_>

1,let

bt

"[0,1]

[0,

cx)

be definedby

(

a(,,II(,,

,)e.

Withassumptions

(A)-

(E),

wehave

0

< L,bt+(x) < Lbt(x) on(0,1),

0

<

<

n 2.

Furthermore,

lin

LiPt(x)

0uniformlyon

[0, 1],0

<

_<

n 2.

For f

>

1,

ft

is continuousand satisfies

(B).

Also,from (B),we have,for each{

>

1,

1,I1(|

f,(x.y, y,_)

<_

f(x,

vt ,j,,_)on

(0,1)

x (0,.)

’’-.

ft(x,y, y,_)

<

f(x,

Lo$t(x)

L,,_.g:t(z))

on(0,1)

(0,

c)"-’

THEOREM 3.

Assume

that conditions

(A)-(F)

are

satisfied.

Then the Mundary value pwblem

(1.1),

(1.2)

hasasolut,on,y, such that

L,y(z)

>

0 on

(0,1),

0 n-2.

PROOF. We begin by defining asequence of compactmappings

Tt

K

K

by

Tt:(x)

G(x,s)ft(s, Lo:(S)

L,,_:(s))ds,O

x 1,:

e

K.

For g and

e

K,

L,_(Tt) >

0is concave on

(0,

1),

Tt

satisfies the boundaryconditions

(1.2),

and from

(L,)G(x,s)

>

0,0 n-2,wehave

L,(T)

>

0 and increasingon

(0,1),

0

g

g

n-2. Since each

ft

satisfies

(B),

it follows that

Tt

isdecreasingwith respect to thecone

K,

for each 1.

Also,0

Tt(0) (wrt

K)and 0

T(0)

(wrt K),

andsobyTheorem 1, for each

,

thereestsat

e

K

suchthat

Ttt

_t.

From

ourobservationsabove,

By

essentiallythe samearguments in Threm 2,it follows that thereis an

R

>

0 such that,for each 1,

[Ln-2tl

R

and

IItl[

,

where isgiven in the

Coronary.

Our

next claimisthat there efists k

>

0 such that k

]L,-et],

all 1.

Assume

the claim isfMse.

Thenpsingtoasubsequence and relabeling,wehave without loss ofgenerality that lim

IL,_t]

0,

which implies,alongwiththeboundaryconditions

(1.2),

lim

L,t(z)

0uniformlyon

[0,1],

0 n-2.

(3.7)

Let0

<

<

be fixed and let

By

(D),

thereestsy

>

0 suchthat,if$z 1-$and0<y,<,forlgin-l,then 2

(,,...,,_) >

m

From

(3.7),

thereexits {o

_>

such that,for

_>

Q,

0

<

IL,:t(x)l < u/2,

0

<

<

1, 0

<

<

n-2.

Also,forsome{1

>

{0itfollows that,if{

_>

{1,

SINGULAR BOUNDARY VALUE PROBLEMS 577

So,forg>t? an(lb

<.r

<

l-a.

l.,,_-_,Cx)

L

....

2(;Cx.s)ftCs.

LotCs)

2tCs))ds

1-J

G(z,.s)ft(s, Lot(s)

2

-

f(,

/2

/2)

>1.

Thisis acontradiction to(3.fi). Titus, thereisa k

>

0 such that k

Ia,,-,l,

forall

.

With and applying(1.2),weobtain

L,go(x)

<_

L,t(x)

on

[0, 1],

0

<

<

n-2,

_>

1.

Wehave

or,inparticular,

go

<

t

< z"-"(wrt

K),g

>

1,

(.

2)!

k

{,}

C_

<go,

(n

2)!

’x"-’>

C_ D.

Vhen restrictedtotheclosedorder interval,

<9,

(n-

2)i

z"-=>’

T

is acompact mapping.

So,

thereis a subscquenceof

{Tt},

whichwerelabel astheoriginal sequence, whichconvergestosolne

o"

K;

that

k

To

completethe proof,we showthat

lIThe

t]l

0, asg oo. With 0

,

let

>

0 begiven,

and let

P=

_0<,<,-2max sup

L,G(x,s)}.

[0,]x[0,]

The integrabilitycondition

(F)

and theabsolute continuity of theintegral imply thereexists0

<

/i

<

suchthat

2P[

f(s,

Loga(s)

L._2g,(s))ds

+

f(s, Logo(s)

,L,,_go(s))ds] <

.

-6

Also,

thereexists

go

suchthat, for g

_>

eo,

L,t(x) <_

L,ga(x)

<_ L,t(x)

on

[8,1

ti],0 _<

_<

n 2.

Observe also that

ft(s,

Locpt(s)

L,_t(s))

f(s,

Lot(s)

L,_t(s)),

for 6 g s g 1- and

g_>go.

Thus,for0gi_<n-2,g>_g0, and0_<x_< 1,

+

f(s,

Lot(s),... ,L,,_2t(s))ds]

2P[

f(s,

Log(s),...,L,,_2g(s))ds

+

f(s, Logo(s)

-6

Therefore,

follows, inturn,that lin

liar

’11

O,sothat

"

(,

(,,_

z)?a.

c_

D,

a[Id

and lieproofis(’Ollit)lete.

REFERENCES

1.

BOBISUI), L. E.

Existenceofsolitionsofsomenolllineardiffusionproblems,

J.

Math. Anal. Appl.

168

(1992),

413-.12-1.

2.

BOBISUD,

L. E.

and

LEE,

Y. S.Existenceforaclass of nonlinearsingular boundaryvalueproblems, Appl. Anal. 38 (1990),45-67.

3.

DO, T. S.

and

LEE, II.

11. Existenceof solutions ofsingular nonlinear two-point boundary value problems, Kyungpook Math.

J.

31

(1991),

113-118.

,1.

GARNER, J. B.

and

SIIIVAJI, R.

Existence and uniqueness of solutions ofa (’lass of singular

diffusionproblems, Appl. Anal. 30

(1988),

101-114.

5.

O’REGAN, D.

Existencetheoremsforcertain classesof singular boundaryvalueproblems,

J.

Math.

Anal. Appi. 168

(1992),

523-539.

6.

TINEO,

A.Existencetheorens forasingulartwo-point Dirichletproblem,Nonlin. Anal. 19

(1992),

323-334.

r.

WANG, H. Y.

Solvability

or

someboundaryvalueproblemsfor thesingular equation

(p(t)r(t, y)y’)’/q(t)

f(t,y,

pry’),Chinese

Ann.

Math.

Set. A

12

(1991),

87-91

(Chinese).

8.

WANG,

H. Y.

Solvabilityofsomesingular boundaryvalueprol)lemsfor theequation

(p(t)r(t,

u)u’)’lq(t)

f(t,

u,pru’),

Nonlin. Anal. 18

(1992),

649-655.

9.

ELOE, P. W.

an(lIIENDEI{SON

J.

Existence of solutions ofsomesingularhigher order boundary valueproblems, Zeit.

Angew.

Math. Mech. 72

(1992).

10.

GATICA,

J. A., HERNANDEZ, G. E.,

and

WALTMAN,

P.

Radiallysymmetric solutions ofaclass ofsingular elliptic problems,

Prov.

EdinburghMath. So(:. 33

(1990),

169-180.

11.

GATICA,

J.

A., OLIKER,

V.,

and

WALTMAN, P.

Singularnonlinearboundaryvalueproblems for second-order ordinarydifferential equations,

J.

Diff.

Eqs.

79

(1989),

62-78.

12.

ELOE,

P. W.

and

IIENDERSON,

J.

Singularnonlinearboundary valueproblems for higher order

ordinary differential equations, Nonlin. Anal 17

(1991),

1-10.

13.

IIENDERSON,

J.

and

YIN, K. C.

Singular boundary value problems, Bull.

Inst.

Math. Acad.

Sinica19

(1991),

229-242.

14.

AMANN, It.

Fixed point equations and nonlineareigenvalue l)roblenisin ordered Banach spaces,

SIAM

Rev.

18

(1976),

620-709.