Let f’Po P be a compact map

(i)

Iff ^(x) v

^2x

for

^{every x}

S

^{and every2}

>=

^{1, theni(f}

^Pp)

^1.

(ii)

If

^{there existsan elementp >} 0 such that x

f ^(x) v

2p

for

^{every x}

S

and every2 ^>_ 0, theni(f,

Po)

^O.

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Proof

^{(i) Define acompactmap h’[0,}

^1]

^x

Po

^{P by h(z,}

x)..= zf(x).

Then, by thehomotopyinvarianceand thenormalizationproperties,

i(f

Po)

^i(O,

Po)

^1.

(ii) Let #..=sup

{llf(x)

[x eP

o}

^{and let}

21

^{> (p}

+ #)/IPll.

Then, by the homotopy invariance,

i(f

Po)

i(f

+

^21p,

Po)"

If i(f,

Po)4:

^{O, then there exists an element x}

Po

^{such that x}

Consequently,

Ilxlt ^>_-21

P

IIf(x)

__>21[p -/ >P, which isimpossible.Hence i(f.

Po)

^0.

THEOREM₊ 12.2. Let

f’Po

^{--* P be a compact map such that}

^{f(x) va}

^2x

^for

xeS_o and 2 ^>_ 1. Then

f

^hasa

fixed

^{point in}

Proof

^Thisfollows from Lemma 12.1 and thesolutionproperty of the fixed point index.

Observe that the condition of the above theorem is satisfied if

f(x) =

^{x for}

allx e

Sf.

More generally,it

sumces

toknow that forsome

majorantf

off,

f ^(x) ;

^x

for all xe

Sf. ^A

^similarremark appliestothe following theorem andits corollaries.

THEOREM 12.3. Let

f" Po -

^Pbeacompactmap, and leta, z e(0, p]^witha4: z.

Supposethat

(i)

f ^(x)

^{=/: 2x}

for

^{every x}

S

^{+ and2}

^>__

^l,

(ii) ^{there exists an} element p > 0 such that x

f ^(x)

^{:/: )p}

for

^{every x}

S

⁺

and2 ^>_O.

Then

f

^{has atleastone}

fixed

^pointxwith

min

(a, z)

<

xll

<^max

(a, z).

Proof

^Let Po ^min(a,

z)

_{and P} max(a,

z).

Then by the additivity pro-perty of thefixedpoint index,

i(

f Po,

^\

Poo)

ⁱ⁽

f Po

ⁱ⁽

^{f Poo)}

Hence,

by Lemma 12.1, i(f,

Po,\Poo)

^{isequal to} if Pl a,and equal^{to if} P ^z.In eachcasetheassertionfollowsfrom thesolution property. ^V]

COROLLARY 12.4

(on

the compression ofa

cone).

Let

f’Po

^Pbeacompact

map,and let 0 < a < p.Suppose that

f ^(x)

^x

for

^x

S

^and

^{f (x)}

^x

^for

^x

^S+

Then

f

^hasa

fixed

^{pointxwitha <}

^]]xl]

^{< p.}

COROLLARY 12.5

(on

the expansion ofa

cone).

^Let

f’Po

^{P be a compact}

map,^Thenand let 0

f

^hasa

fixed

< ^apoint x with< p.Supposeathat< [x[I <

f ^(x)

^$_p.^x

for

^x

S2

^and

^{f (x)} =

^x

^for

^x

^S+

Remark 12.6. It should be observed that the above theorems remain true if Eis anarbitrary Banach space andPis aclosed wedgeinE, that is,P hasall the properties ofa conebut P

fl (-P) {0}. (In

^thiscase, the relation < is onlya preorderingin

E.)

Moreover,

itshould be observed that theproofof Theorem 12.3 shows that the fixedpoint indexof

f

^overthe "conicalshell"

Po, \Poo

^{is either}

⁺

^{or 1. This}

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additional information can be very usefulⁱⁿapplications. For example,suppose itisknown that there existsan isolated fixed point of

f

ⁱⁿ

Pp,\Ppo

whose local index isdifferentfrom

+

^{or 1,}respectively. Then,by the additivity of the fixed point index, there ^mustbeasecond fixed point in this conical shell.

13. Fixedpointsof ditferentiablemaps.Inordertoapplythegeneraltheorems of the preceding section,onehasto findmaps

f

^which,roughly speaking, pushthe elements of

S

^{+ either away from zero orcloser to zero.}

^In

^{order to} verify these conditions,it isreasonabletolinearize themap

f

^{at zero}and infinity andto study the behavior

off’

^{at zero and atinfinity.}

Throughout this section,we denoteby

(E, P)

an arbitrary

OBS,

andbyp a positive number.

LEMMA

13.1. Let

f’Po

^{P be a compact} map such that

f ^(O)=

^O.

^Suppose

that

f

^{hasa rightderivative}

^f’+(O)

^{atzero suchthat isnotaneigenvalue}

of ^f’+(O)

toa positive eigenvector. Then thereexistsa constant

ao ^(0,

^{p] such that}

for

^every

o

(0, ao],

(i) i(f,

P)= /f f’+(0)

has no positive eigenvector ^to an eigenvalue greater than one;

(ii) i(f,P,)= ⁰

/f f’+(0)

possesses a positive eigenvecor ^{to an} eigenvlue greater than one.

Proof.

^Since

^f’+(O)x

^lim_o+

,-f(x)

for every

xP,

it follows that

f’+(0)

L⁺

(P P). Moreover,

byTheorem 7.1,

f’+(0)lP

is acompletelycontinuous map.Henceid f’+(O)is^dosedondosed subsets ofP. Therefore, (id

f’+(0))(S +)

is a dosed set and

0

(id-

f’+(0))(S +)

^by the hypothesis of the 1emma. Hence thereexists apositiveconstant such that

x

f’+ ^(0)x

^{x for all xeP.}

Choose

ao

^{(0, p] such that}

^{f(x) f’+(0)xl[}

^x

/2

^{for all x}

P,o.

^Then

for every o

(0, ao],

every y P satisfying Ilyll

< o/2,

and every 2 [0,

1],

the map

(1 2)(f’+(0) +

^y)

+ 2f

^{possesses no} fixed point on ^$

+.

^{Indeed, for every}

IIx- ^{(1 2)(f’+(O)x +}

^y)-

^2f(x)ll

_>_

x

f’+(0)x f(x) f’+(0)x

Y

a( /2

Ilyll/) > 0.

Hence,

bythehomotopyinvariance property, i(

f ^P,)

ⁱ⁽

f ’+ ^{(0) +}

^y,

^P,)

(i) Inthiscase,wesety 0and observe that theequationx

2f’+(0)x

⁰ hasnopositivesolutionfor2

[0, ].

Hence bythehomotopyinvarianceand by the solutionproperty,

i(

f ^P)

ⁱ⁽

f ’+ ^{(O) P)}

^i(O

^P)

^1.

(ii) In ^this case, denoteby h S⁺ an eigenvector of

f’+(0)

^{to an} eigenvalue 2 > 1. Then we claim that, for every/3 >0, the equation x

f’+(O)x =h

^has

no positivesolution. Indeed,suppose that there existsa solutionx > 0 forsome

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fl

^{> 0.}Then there exists anonnegative number Zo such that x

>= ^zoh

^{and x}

;

^rh

for

>

_to.Henceweobtainthe inequalities

x

f’+(O)x + h ^{>__ f’+(O)oh} +/h >__ (to +/)h,

whichcontradictsthe maximality of

o.

Now,

by setting y

flh

^{with0 <}

fl

^<

^ae/2,

^the solution property implies

i(f

P)

i(f’+

(0) + flh, P)

^{O. I--1}

The following theorem ^is closely related to Theorem 12.3. It should be observed thatits proofalso gives some informationabout the value of thefixed pointindex inanappropriate conical shell.

TI-mOREM 13.2. Let

f "Pp

^{--. P be} a compact mapsuch that

f ^(O)

^{O. Suppose}

that

f’+(0)

hasaright derivative

f’+(0)

^{atzerosuch that is notaneigenvalue}

of ^f’+(O)

to apositive eigenvector.Letone

of

the following conditions be

satisfied"

(i)

f(x):/:

2x

for

^{every x}

S

^{and 2}

>=

^{1, and}

^f’+(O)

^{possesses a positive}

eigenvectortoaneigenvalue greater than one;

(ii) there exists an element p > 0 such that x

f ^(x)

^{:/: )p}

for

^{every x}

S

and2

>= ^O,

^and

^f’+(O)

^hasno positive eigenvectorto’aneigenvaluegreaterthanone.

Then

f

^{hasa positive}

fixed

^point.

Proof ^By

^{Lemma 13.1 thereexists a numbera}

^(0,

^{p)such that i(f,}

^P,)=

⁰

in case(i)andi(f,

Pp)

^incase(ii).

Hence,

by

Lemma

12.1 and the additivity of the fixed point index,

i(

f Po

^\

^P,,)

ⁱ⁽

^{f Po)}

ⁱ⁽

^{f P)}

isequalto incase(i)andequalto in case(ii).Therefore thesolutionproperty impliesthe assertion.

It

has been noted in the proofof

Lemma

13.1 that

f’+(O)eL+(P P)

and that

f’+(0)lP

is a completely continuous map. Let Te

L/(P- P)

be arbitrary.

Then

TIIP

^sup

^{ll TxII Ix

^e

P}

<

andit canbe shown that

rp(T) lim

Tk P/k

is awell-defined nonnegativenumber,theconespectralradiusofT.Itiseasytosee that

re(T r(T)

ifP gives an open decomposition ofE.

Suppose

that

TIP

^is

completely continuous

(that

is, T is P-compact). Then it has been shown by Bonsall

[79] (cf.

also

10]),

thatthe conespectralradiusofTis aneigenvalue ofT withapositive eigenvector, provided

re(T

> 0. Using thisresult,it is obviousthat condition(ii) of

Lemma

13.1 is satisfied provided

re(f’+(O))

> 1.

Moreover,

con-dition (i) is certainly satisfied ifP is total, that is, P P Eand

r(f’+(O))

< ^1.

Finally,ifweassumethatPisgenerating, then

(cf.

Theorem

1.6)

theconespectral radiusequalsthe spectralradius.

For

simplicityweonlyformulate one corollarytoTheorem 13.2 forarather special,thoughimportantcase.

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COROLLARY 13.3. Suppose that Phas nonempty interior and let

f "Po -

^Pbe

a compactmap such that

f(O)

^O.

Moreover,

suppose that

f

^{hasastronglypositive}

right derivativeatzero.Then

f

^{hasa positive}

fixed

^{point,providedone}

of

thefollowing conditionsis

satisfied"

(i)

f(x) g

x

for

^x

S

^and

r(f’+(0)) >

^1, (ii)

f ^(x)

^x

for

^x

S

^and

^r(f’+(0))

^{< 1.}

Proof ^It

^{sufficestoobserve}that, by^Theorem3.2,thespectralradius

off’+(0)

isan eigenvaluetoapositiveeigenvectorand,infact,theonlyeigenvalue with this property.

Hence

itis easytoverify thehypothesesofTheorem 13.2.

In

thefollowingwelinearizeatinfinityinordertoverifytheconditionsatS_o Thefollowinglemmaisthe counterpartatinfinitytoLemma 13.1.Sincetheproof is almost verbal, the same as the proofof

Lemma

13.1,we leave it to thereader

(cf.

also

[24,

Lemma

2]).

LEMMA

13.4. Let

f’P -

^{P be a completely} continuous, asymptotically linear map such that is not aneigenvalue

of ^f’()

^toapositive eigenvector. Then

there

exists a positivenumber tro suchthat,

for

^everytr

^>__

^tro,i(f,

^P)

¹

if ^f’()

^{has no}

positive eigenvectorto aneigenvaluegreaterthan one, andi(f,

P)

^0otherwise.

As

afirstapplication ofthis lemma,we obtainthe following theorem ^which alreadyhas been used intheproofof Theorem 7.4.

THEOREM 13.5. Let

f"

^{P-. Pbeacompletelycontinuous}asymptotically linear map. Then

f

^possessesa

fixed

^point

if ^{f’( )}

^doesnothave a positive eigenvectorto aneigenvaluegreaterthanorequaltoone.

Proof ^{By Lemma}

^{13.4,thereexists a}positive numberrsuchthati(f,

P)

1.

Hence

the assertionfollows from thesolutionproperty. ^V]

It is now obvious how Lemma ^{12.1 and}

Lemma

13.4 can be combined to provethe counterparttoTheorem 13.2, namely,toprove the existence ofapositive fixed point of

f

by imposing conditions for

f

^at

S

and conditions for

f’().

We

leaveit tothe readertoformulateandprovethe corresponding^statement.

THEOREM 13.6. Let

f"

^{P--. Pbeacompletelycontinuous}map such that

f ^(O)

^O.

Supposethat

f

^isasymptotically linear and right

differentiable

^atzero, such that 1 is not aneigenvalue

of f’( )

^or

of ^f’+(O)

^toa positive eigenvector. Then

f

^{hasat least}

one positive

fixed

^{point,providedone}

of

^{thefollowing twoconditions is}

satisfied"

(i)

f’+ ⁽⁰⁾

^hasno positive eigenvectortoaneigenvaluegreaterthan one, whereas

f’( )

possesses such^apositive eigenvector

(ii)

f. ⁽⁰⁾

^possessesa positive eigenvector to aneigenvaluegreater thanonebut thisisnotthecase

for f’().

Proof ^Lemma

^{13.1 and}

^Lemma

13.4 imply the existence of real numbersp andawith0 < p < ^trsuch that

i(f, Po)

^andi(f,

^P)

^{0 in case(i),andi(f,}

Po)

0 andi(f,

P)

^incase(ii).

Hence,

ineach case,

i(f

P\Pp) ^{i(f P)}

^i(f

Pp) ^O,

and the solution property of the fixedpointindeximpliesthe existence ofafixed point x such that p < [xl[ ^{<tr. V]}

Suppose,inaddition, thatPhas nonempty interior and that

f’+(0)

and

f’()

are strongly positive. Then

(cf.

the proof ofCorollary

13.3),

conditions (i) and (ii)of the above theoremcanbereplaced by

(i’) r(f’+(O))

< and

r(f’(c))

> 1, (ii’)

r(f’+(O))

> and

r(f’())

< 1.

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Moreover,

inthiscase, 1 is never aneigenvalue

off’+(0)

^{or of}

f ^’(or)

^{to apositive}

eigenvector. Finally, if it is supposed that

r(f’+(O))

< 1

(or r(f’())

< 1), then

f’+(0) (or f’())

is not requiredto be stronglypositive.

It

should be noted that in the one-dimensional case, the theorems of this paragraph have straightforward easy geometrical interpretations. For example, the lasttheorem expressesthe obvious factthatthegraphofacontinuousfunction

f:+

^---,

+

^with

^f(0)

0 intersects the45-lineata positive distance from zero, providedoneofthe followingconditions issatisfied"

(i)

f’(0) >

^and

f

^hasanasymptote withslopeless than one;

(ii)

f’(0) <

and

f

^hasanasymptote withslopegreater thanone.

The one-dimensional case suggests also the useof minorants andmajorants for theproofof fixed point theorems. The following theorem is an example for suchasituation

(cf.

Theorem

7.4).

THEOREM 13.7. Let

f

^{:P P be a map which is compact on} order intervals.

Suppose

that

f

^{possessesacompletely} continuous, increasing,asymptotically linear majorant

f’P

P such that

f’()

^does not have a positive eigenvector to an

eigenvaluegreaterthanorequaltoone. Then

f

^hasa

fixed

^point.

Proof. ^By

^{Theorem 13.5,}there existsafixed point j)

off.

Consequently,since

f

^{is an} increasing majorant for

f,

the map

f

^maps the order interval

[0,

p] into itself.Hence theassertionfollows from Schauder’s fixedpointtheorem. ^[3

14.

A

multiplicity result.

Let f:[9, 93] -

^{be a}continuous functionon some nontrivial interval

[, ]

such that (f(y)-

)(.9 -f())

> 0. Then the intermediate value theorem implies the existence ofa fixed point

off

ⁱⁿ

^{(, ).}

This elementary fact has been generalized by the theorem on the compression of a cone and the theorem on the expansion of a cone, respectively.

In

fact, in the one-dimensional case, these two theorems reduce precisely to the above assertion.

However,in practicalcases,it turns outthat thehypothesesof Theorem 12.3 arerather difficulttoverify.

In

fact,in ordertoestablishthesehypotheses,onehas tofindapriori boundswhich,inthecaseof nonlinear

BYP’s,

say,is avery difficult task.

For

astraightforward generalization of the one-dimensional case, itseems to be naturaltoreplacethe interval

[,

j)]byan order interval andnotbyaconical shell. Indeed, if we suppose that f:[y, p]--,

E

is increasing and compact, then Corollary 6.2

(or

^Schauder’s

theorem)

guarantees theexistence ofa fixed point, provided

=<

^{f(y) andf(p)}

=< ^p.

^The advantage of this theorem is that one has to verify only two conditions, namely one has to show that the "endpoints" of the order interval [y, p] are mapped inside of [y,

:9]. In

general, this is a much easierproblem thantoverify the hypotheses ofthetheorem on thecompression ofa cone, where one has to verify infinitely many conditions.

Onthe other hand,^it iseasytogiveexampleswhichshowthatamapwhich mapsthe "endpoints" oftheorderinterval

[, 39]

outside of

[., ],

^{does notalways} have a fixed point. Consequently, it does not seem to be possible to generalize the one-dimensional result on the "expansion of an order interval" to higher dimensions. Nevertheless, ^it is the purpose of the following considerations to show that,given certain additionalproperties,it ispossibletogeneralize the one-dimensional case inastraightforwardmannertoorderintervals.

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The multiplicity results of this paragraph are based upon the following generallemma.

LEMMA 14.1. Let Xbearetract

of

^someBanach space and let

f

^{X X bea}

compactmap.SupposethatX and

X2

^{aredisjointretracts}

of ^X,

^{and let}Uk,^k 1,2, be open subsets

of

^{X such that U}k c Xk, ^k 1, 2.

Moreover,

suppose that

f ^(Xk)

c X_kand that

f

^has

nofixed

^pointson

Xk\

Uk,^k 1, 2. Then

f

^hasatleast three distinct

fixed

^pointsx,Xl,x2 with_Xk Xk,k 1,2, andx

X\(X1 X2).

Proof. ^By

the additivity property for the fixed point index,

2

i(f

X\(U1 U U2), X)

i(f X,

X)

i(f Uk,

X).

k=l

The permanence property implies i(f, Uk,

X)-

i(f, Uk,

Xk),

and the excision propertygivesi(f Uk,

Xk)

i(f Xk,

Xk).

Consequently,

2

i(f,

X\(UI U2), X)

i(f, X,

X)

i(f,Xk,

Xk).

k=l

Itiseasytosee

(cf.

^theproofofSchauder’sfixedpoint

theorem)

thati(f, C,

C)

for every compactself-map

f

^{ofanarbitraryretractC.}

^Hence

^i(f,

^{X, X)}

^i(f,

X

1)

i(f, X2,

X2)

and, consequently,

i(f,X(U1

^(.J

U2),X)

1.

The assertion followsnowfromthe solution property of the fixed point^index.

THEOREM 14.2. Let

(E, P)

beanOBSwhosepositive conehasnonemptyinterior.

Supposethat there exist

four

^pointsYk,

k ^E,

^{k 1,2,with}

andacompact,

_Y

strongly

_-< f(;),

increasing

f()

map<

1, f" ^{.Pl, ]}

<

f(Y), -

^{Esuch that}

f(J) --< .

Then

f

^{has at} least three distinct

fixed

^{points x, Xx,}

x

^{such that}

^{1 <-}

xl <<

f;2 ⁽⁽

_Proof

_X2

2, _Let

and_X

f2 _{[1, 2]} =

^X

^1"

and

Xk

^{[Yk,.gk], k 1,2. Then}

^X,

^{X1, and}

X2

areretractsofEwith

Xk

^{c Xand}

X1 ^f’l X2

" ^Hence

^{Xk,k 1, 2,}is a retract ofX. Moreover, since

f

^is increasing, the hypotheses imply ^that

f(X)

^{X and}

f(Xk) =

^{Xk, k 1, 2. Since}

f

^{is strongly} increasing and

f(l)

<

.91,

^it follows fromCorollary 6.2 that

f

^{has a}maximal fixed point

1

ⁱⁿ

X1

^and

1

^<<

.91.

Con-sequently, X has nonempty interior U in X and

f

^{has no} fixed point on the boundary X ^\U ofX inX. Similarly, theexistence ofaminimal fixed point of

f

ⁱⁿ

X2

and the fact that

f

^{is strongly} increasing imply that

X2

has nonempty interior

U2

^{in X and that}

f

^hasno fixed point on X

z\U2.

^Hence

^Lemma

^14.1

is applicableand the assertion follows. ^[-1

Forthesimplicityof thestatement we have chosen the somewhat restrictive hypothesesgiven above.Itiseasyto seethat thesameproof applies^{to more}general

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situations.Forexample, itsufficestoassumethat

f

^{isincreasing(insteadof being}

strongly increasing), ifit isknown that the maximal fixed point

1 off

ⁱⁿ

^{[1, 1]}

satisfies << and that theminimalfixed point

2

ⁱⁿ

^{[2, 2]}

^satisfies

Y2

^<<

2.

Moreover,

the hypothesis that

f

^be increasing can be completelydropped if itis known that

f

maps each of the order intervals

X,

X1,and

X2

^{intoitselfsuch that}

f

hasnofixedpointsontheboundaries of

Xk

^{inX, k 1, 2.}For example,this can be guaranteed if the existence of strongly increasing majorants and minorants ispresupposed

(cf. [19.

^Thm.

4]).

Ingeneral,it cannot be assertedthat the thirdfixed point xliesbetween

1

and

Ys.

^{In fact,}it cannot evenbe shown that

1

^{< x}

^<

^{2, where}xl isthemaximal fixed point

off

ⁱⁿ

^[/1,/91] andff2

^isthe minimal fixed point

off

ⁱⁿ

[Y2, J2]" However,

since byCorollary 6.2

f

^{also has}a minimal fixedpoint anda maximal fixedpoint in the large order interval

[Yl, P2],

we obtain the followingtrivial corollary to Theorem 14.2.

COROLLARY 14.3. Let the hypotheses

of

Theorem 14.2 be

satisfied.

^Then

f

hasatleast three distinct

fixed

points xl ^x2,x3such thatxl <<x2 << x3.

Suppose

it isonlyknown that the compactstrongly increasingmap

f

^maps

oneorder intervalintoitself.Then, by Corollary 6.2,

f

^hasaminimal fixed pointff and amaximal fixed point

. ^Suppose,

ⁱⁿaddition, that itis known thatff<

.

Is ittrue that

f

^{has a}third fixed point in this situation? The following theorem gives an answertothisquestion.

THEOREM 14.4. Let

(E, P)

be an OBS whose positive cone has nonempty in-terior.

Suppose

that

< 19

^{and let f’[y,}

19]

^{E be a strongly} increasing compact map such that <

f

^{(y) and}

f ^{() <} .

^{Suppose, in}addition, that the minimal

fixed

point and themaximal

fixed

^{point are} distinct, and that

f

^{has stronglypositive}

derivatives at and2, respectively. Then

f

^{possesses at} least three distinct

fixed

points provided

r(f’())

4 and

r(f’())

4: 1.

Proof

^It follows from Proposition 7.8 thatff and are weakly stable fixed points.

Hence

the above hypotheses imply that

r(f’(2))<

and

r(f’(Y))<

1.

Since

f

^isstrongly increasing,2<<

,

^andLemma7.5 implies the^existenceof points

Pl

^and

Y2

^{such that 2 <}

Pl

^<

Y2 ^< , f(Pl) </91,

^and

f(Y2)

>

Y2. ^Hence,

^an application of Theorem 14.2 with

1 Y ^< Pl ^< Y2 ^< 2 ;--

gives the asser-tion. 1

By

specializing the above theorem to the one-dimensional case, it is easily seenthat thehypothesesconcerning thespectralradiicannotbe omitted.

15. Applications to nonlinear elliptic boundary value problems. Throughout this paragraph, we assume that the regularity hypotheses

(R)

of 9 are satisfied.

Weconsidermildlynonlinearelliptic

BVP’s

of the form Lu

f ^{(x, u)}

^in,

(1)

Bu

0 on

c,

andwe usethe notationsandconventionsof 9.

Firstweprove the existence ofapositivesolutionfor the BVP

(1)

ⁱⁿthecase thatzero is a solutionand that

f(x,

^isasymptotically linear.

THEOREM 15.1. Suppose that the

function f ^{C( +)} satisfies f

^(.,

^O)

⁰

and

f ^O,

^{and let}

f

^havea continuouspartial derivative

Dzf

^{in a} neighborhood

of

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668 HERBERT AMANN

zero such that

Daf(’, 0) C;’(). Moreover,

suppose that there exists a

function fo C()

^{such that}

lime_, - ^if(

^x,

^{) f(x),}

^uniformlyinx

^.

Finally, suppose that

D2f

^(x,

^O)

^{> 0and}

^fo(x)

^{> O}

for

almost all x

.

Denote by

20

the principal eigenvalue

of

^{the linearEVP}

Lv 2D₂

f ^O)v

^in),

By 0 on

,

and let

2

^bethe principal eigenvalue

of

^thelinearEVP

Lv-fv

^in,

By 0 on).

ThentheBVP

(1)

^{has atleast}one positive solutionprovidedeither (i)

o

^>

^and,

^{<1 or}

(ii) ²0 <

and,

> ^1.

Proof.

^It follows from Theorem 4.3 that the principal eigenvalues

2o(D2f(., 0))

^and

2o ^2(f)

^{exist andare positive.}

By Lemma

9.7,the

BVP (1)

^isequivalenttothe fixed point equationu

f(u)

in

E,

where

f’P E

^{is completely} continuous. Clearly,

f(0)=

^{0, and}

(cf.

Lemma 9.1) f

^is asymptotically linear and right differentiable at zero with the strongly positive compact derivatives

f’. ⁺⁽⁰⁾

^and

^f’(),

respectively. Finally,it

r(f())

Hence the assertion is easily seen that

21= ^r(fe.+(0))

^and

2

follows from Theorem 13.6 and Theorem 3.2.

It is easy to remove the assumptions that

Dzf(x, 0)

> 0 and

fo(x) >

^{0 for} almost all x

.

^{Weleavethedetails tothereader.}

Itis importanttoobserve that the results of 12 and 13 donotpresuppose that

f

be increasing. ^This has the important consequence that those results are applicabletosemilinearelliptic

BVP’s,

that is,toproblemsof the form

Lu

f ^(x,

^{u, grad}

^u)

^infl,

Bu g on

.

For

simplicitywedonotgive further details.Forsomeapplicationsinthis direc-tion wereferto 11, Chap.

7].

By

applying Theorem 14.2

(more

precisely, Corollary

14.3)

^to the map

fe

weobtain thefollowingmultiplicity result.

THEOREM 15.2.

Suppose

that thereexist asubsolutiondp astrictsupersolution

1,

^{a strict}subsolution

b2,

^andasupersolution

t2for

^theBVP

Lu

f ^{(x, u)}

ⁱⁿ

,

(2)

Bu g on

such thatdpl <

11

^<

⁾²

^<

112.

^{Then theBVP}

(2)

^hasatleast threedistinctsolutions

ui, 1,2, 3,such that

c1 <=

^{ul < u2 < u3.}

^{<-_ 2.}

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Thefollowingtheorem appliestothe case where thereisonlyonesubsolution andonesupersolution butit isknown thattheminimalsolutionand the maximal solution

(cf.

Theorem

9.4)

are distinct. The proof again follows by a straight-forward applicationof Theorem 14.4tothemap

re.

THEOREM 15.3.

Suppose

that there exist a strict subsolution

d

^{and a strict}

supersolution

b for

^theBVP

(2)

suchtiat

d . ^Moreover,

suppose that the minimal solution u f and the maximal solution u₂ fi in the order interval

[b, ]

^are

distinct. Let

f

be continuously

differentiable

with respect to the second variable and suppose that the

functions

^x-o

DEf(X ill(X))

i- 1, 2, belong to

Ct’().

Then thereexists atleast onethirdsolutionu₃ such thatu _u3 u2,provided each

of

thelinearB^lp,s

Lv D_E

f ^(x

^{ui(x))v 0 in}

By 0 on

,

l, 2,has thetrivial solutiononly.

16. Notesantiremarks.Thefixedpoint^indexis, of course,awell-known tool in fixedpointtheorywhich canbedefined inmuch greater generality

(cf. [80]-[82]).

The permanence property is a special case of the more general commutativity property(e.g.,

[80]-[82]).

The possibility of thederivationof the fixedpointindex from the Leray-Schauder degree ^is well known to the specialists ⁱⁿ this field.

Itisimplicitly containedinthe papers byBrowder, Nussbaum,and others.

The theorems"onthe expansion and^onthe compression ofa

cone"

aredue to M.

A.

Krasnosel’skii (cf. [11], where a lengthy proofis given, which avoids degree theory). In the case wherethe positiveconehas interior points, adegree theoretic proofhas been given in

[83].

The short proofsgiven in thispaper have been discovered independently by T. B. Benjamin

[84]

^{and R. Nussbaum}

[85],

where the latter author had been motivated by Hamilton’s generalizations

[86]

of Krasnosel’skii’s theorems. Clearly, these theorems can be proved (by proofs whichareidenticaltotheonesgiven

above)

wheneverafixed point index is known toexist,hence,inparticular,inthecaseofset-contractions

(cf. [85], [87]).

Foraresult which iscloselyrelatedtoTheorem 12.3 andwhichcould easily beprovedby the abovemethods,wereferto

[88]. An

extensionof Krasnosel’skii’s theoremstoFr6chetspacesisgiven in

[89].

The theorems on cone-compressions and cone-expansions are, as a rule, difficult to apply since their hypotheses are not easy to verify. Applications to two-point

BVP’s

andtointegral equationsare contained in

I11

^{andinthepapers}

[90], [91

byLaetsch.For applicationstoperiodicsolutions ofordinary differential equations, we refer to

I11], [85], [88], [92].

Applications to partial differential equations are given in

I11], [93]. In

^thisconnection, we again point out the im-portance of the fact that there are no monotonicity assumptions for the map.

Consequently, these theorems are applicableto semilinearor even moregeneral quasi-linear elliptic

BVP’s

(cf.

[11], 93]).

Part (ii)ofLemma13.1 isduetothe author

[24]. Its

importanceliesinthefact that it gives information about the fixed point indices ofcertain fixed points

(cf.

the remarks preceding Theorem

13.2).

Theorem 13.2 and Corollary 13.3 arenew,but,of course,theyaretrivialapplicationsof thebasicLemmas 12.1 and

Downloaded 08/01/13 to 128.210.126.199. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

13.1.

Lemma

13.4 is due to the writer

[24].

Theorem 13.6 is contained in [11, Thm.4.11 and Thm.

4.16].

The above prooffollows

[24] (cf.

also

[87]).

^Theorem

13.7iscloselyrelatedto [11,Thm.

(4.9’)].

The results of 14aredue tothe author

(cf. [19],

where it isassumedthatP is a normal

cone).

Originally, Theorem 15.2 and Theorem 15.3 were provedby means ofa direct application of the Leray-Schauder degree theory

[94].

These resultsweremotivatedbycertainconjectures andnumericalfindingsin chemical reactortheory (e.g.,

[95], [96]). An

application of the above multiplicity resultsto suchaproblemwasgivenby

Parter [97].

The multiplicity results of 14 could also be appliedtoaproblem studiedby Rabinowitzin

[58].

Further applications are giveninChapterV.

CHAPTER

IV.

Nonlinear eigenvalueproblemsand bifurcation.Inthischapter, westudy fixed point equations in OBS’s in the presence ofa nonnegative real parameter.

By

using topological methods, we derive some important properties concerning theglobalstructureofthe solutionset. Inaddition, bycombining the topological methods with the results of Chapter

II,

we study the question of multiple positivefixedpointsatafixed parameter value.

In 17we prove that thesolution set ofan equation of the form x f(2,

x)

containsanonempty closed connected subset which is unbounded in

+

^P.In

18 westudythe situation

wheref(, 0)

0 for all2s

/.

^Inthis case, bifurcation from thelineof trivial solutions

/ {0}

canoccur.

We

give sufficient conditions for thistobe the case, andweshowthat, infact, thereexists an unbounded sub-continuum of positive solutions.

In

19 we discuss briefly the situation where

"bifurcationfrom infinity"canoccur, that is, the case of asymptotically linear maps.

The central section ofthischapter, orevenof thewholepaper,is 20.

Here,

bycombining the topological and the monotonicity results ofChapter

II,

wegive amoredetailed studyof thestructureof the solution set. These results enable us thentogive lower estimates for the number offixedpoints forafixedvalue of the parameter 2.

Section 21 containssome applicationstoelliptic

BVP’s,

and 22is devoted tohistoricalandbibliographicalremarks.

17. Global continua of positive fixed points. Let

(E, P)

be an OBS whose positiveconeis nontrivial, that is,P

: {0}.

^{Inthischapterwestudy}equations of the form

(1)

x

f(2, x),

where

f: +

^{x P Pis acompletelycontinuous map.}

^In

^{otherwords, westudy}

fixedpoints of one-parameter families ofcompletelycontinuous conemaps.

Forsimplicity,wesuppose that the parameter2 belongsto

N+.

^{There is not}

much loss of generality in this assumption. In fact, every half-open ^interval

[e,/3)

^can be considered as the image of

N+

^{by a smooth (in fact, analytic)}

strictly increasing function

. ^Hence

^{amap g:[e,}

^fl)

^{x P-. Pcan}be substituted by the map

f: N+

^{x P--, P defined by}

f(2, x):

g(q(2),

x),

where, roughly speaking,

f

has the same properties as g does.

In

the case of an open interval

(e, fl)

R or of a closed interval

[0, fl] ,

^{it is easy to see} how the following assertionsandproofshavetobe modified in ordertohold in thesecases too.

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