EIGENVECTORS OF NONLINEAR POSITIVE OPERATORS

AB~ THE LINEAR KREIN-RUTMAN THEOP~

BY

@

ROGER D, NUSSBAUM

Mathematics Department

Rutgers University

New Brunswick, New Jersey 08903

O, INTRODUCTION

In a recent pa~er [13] Massabo and Stuart prove an existence theorem for non-

zero eigenvectors of a nonlinear operator which maps a normal cone into itself. They

conjecture that normality of the cone is unnecessary; in Section I below we prove their

conjecture. Our proof is quite different from that of Massabo and Stuart and involves

some results from asymptotic fixed point theory. We hope that even the relatively

simple case considered here will illustrate the usefulness of these ideas.

In the second section of this paper, which is essentially independent of the

first, we prove a new fixed point theorem for nonlinear cone mappings. We then prove

that our nonlinear theorem implies as a corollary the most general versions of the

linear Krein-Rutman theorem. Finally, we discuss briefly an example of a linear ope-

rator which is best studied in non-normal cones.

Although the linear theorem we obtain is new, our central point is methodolo-

gical. The linear Krein-Rutman theorem has played an important role in the study of

nonlinear cone mappings, particularly in computing the so-called fixed point index of

such mappings. Our results cone full circle and show that the linear Krein-Rutman

theorem follows from a simple fixed point theorem. Partial results in this spirit

have been obtained before [3, 8, L3, 18, 20, 21], but here we avoid unnecessary hypo-

theses like normality of cones (see Section 5 of [18]).

The approach to the linear Krein-Rutman theorem given here is suita~ole for a

*Partially supported by a National Science Foundation Grant.

310

course on nonlinear functional analysis, and in fact that was our original motivation

for obtaining the results in Section 2. After development of the Leray-Schauder degree

theory, the most general versions of the Krein-Rutman theorem for linear compact maps

can be obtained in one lecture by our method.

1, EIGENVECTORS OF NONLINEAR CONE MAPPINGS

By a cone K in a Banach space X we mean a closed subset of X such that

(i) if x,y E K and k and ~ are nonnegative reals, then Xx+~y E K and (2) if

x E K-{O }, then -x ~ K. If K only satisfies (1), K is a "wedge". Notice that

induces a partial ordering on X by x < y if and only if y-x E K. A cone K is

"normal" if there exists a positive constant T such that for all elements x and

of K, Ilx+ylI ~ zlIxIl. The cone of nonnegative functions in C[O,I] or LP[0,1],

1 ~ p ~ ~, is normal; the same cone in ck[o,1], k ~ 1, or in a Sobolev space (other

than L p) is not.

We also need to recall Kuratowski's notion of measure of noneompactness [10].

If S is a bounded subset of a Banach space X (or, more generally, of a metric

space) define a(S), the measure of noncompactness of S, by

I n 1 ~(S) = inf d>O: S = U Si, n < ~ and diameter (Si) _~ d for l~i_~n . i=1

In general suppose that ~ is a map which assigns to each bounded subset S of X

a nonnegative real number ~(S). We will call ~ a generalized measure of noncompact-

ness if ~ satisfies the following properties:

(1 ~(S) = 0 if and only if the closure of S is compact.

(2 ~(c~(S)) : ~(S) for every bounded set S in X (c~(S) denotes the t

convex closure of S, i.e., the smallest closed, convex set which eontains

S).

(3 ~(S+T) ~ ~(S)+~(T) for all bounded sets S and T, where

S+T = {s+t: s E S, t E T}.

(4) ~(S U T) = max(~B(S), ~3(T)).

The measure of noncompactness ~(S) is well-known to satisfy properties 1-4. Only

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property 2, first proved by Darbo [5], presents any difficulties.

If D is a subset of a Banach space X, ~ is a generalized measure of non-

compactness, and f:D ~ X a continuous map, f is called a k-set-contraction with

respect to ~ if

$( f (S ) ) .e k#(S) (1.1)

for every bounded set S in D. If ~ = a we shall simply say that f is a k-set-

contraction. Now suppose that C is a closed, convex subset of X and that W is

a bounded, relatively open subset of C (so W = 0 n C for some open subset 0 of X).

Assume that f:W ~ C is a k-set-contraction with respect to ~ and that k < 1.

If {x ( W: f(x) = x} is compact or empty or (less generally) if f is a k-set-

contraction on W and f(x) $ x for x ( W-W, it is proved in [16] that there is

defined an integer ic(f, W), the fixed point index of f:W ~ C, which is roughly an

algebraic count of the fixed points of f in W. We shall only need a few facts about

the fixed point index. If ic(f , W) r 0, then f has a fixed point in W. If

fs(X) = sf(x) for 0 ~ s ~ 1 and fs(X) $ x for x ( W-W, then iC(fs,W ) is constant

for 0 ~ s ~ 1 and ic(f,W ) = iC(fo,W) = i if 0 ( W. If C is a cone (or a wedge),

x ~ ( C, ft(x) = f(x)+tx ~ for t > 0 and ft(x) # x for x ( W-W, then ic(ft,W ) is

constant for 0 ~ t.

In the situation described above, the fixed point index can be described in

= -- = co f(W R and let terms of Leray-Schauder degree. Define C 1 co f(W), C n Cn_ 1)

D denote a compact, convex set such that n C c D c C and f(W n D) c D (such n~l n --

a D exists). Let 0 be any bounded open set in X such that 0 rl D = W n D and

let g:0 ~ D be a continuous map such that glW N D = flW n D (g exists by virtue

of a theorem of Dugundji [6]). One can define ic(f, W) = deg(I-g, 0, 0) (observe

that the fixed point set of g in 0 is compact, so the Leray-Schauder degree can

be defined) and prove that the definition is independent of the particular D, g and 0

as above. Properties of the fixed point index now follow from properties of the Leray-

Schauder degree, and this is especially easy to see if f is compact.

We want to generalize now following theorem of Massabo and Stuart [18].

THEOREM (Massabo-Stuart [ L3 ] ) .

Let C

open subset of

that

be a normal cone in a Banach space X, let ~ be a bounded, relatively

C containing 0, and let f:C -~ C be a k-set-contraction. Suppose

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8 > kd/y ( ] . 2 )

where 6 = inf{l[f(z)[I: z E ~-~}, d = max{l[z[J: z E ~-~} and y is the constant appearing

in the definition of a normal cone. Then there exist t > 0 and z E 8K(~) (the

boundary of ~ as an open subset of K) such that f(z) = tz.

The chief tool we shall use is an "asymptotic fixed point theorem". Theorem l.l

below is a special case of Theorem 3 in [15] or Propositions 2.4 and 3.1 in [17].

(i.i) THEOREM

Let C be a closed, convex subset of a Banach space X, U a bounded, relati-

vely open subset of C and f:U~ U a k-set-contraction with k < i. Assume that

fn(u) is contained in U for some integer n. Then ic(f, U) is defined, Lgen(f),

the generalized Lefschetz number of f:U ~ U is defined and Lgen(f ) = ic(f , U). In

particular, if Lgen(f ) # 0, f has a fixed point in U.

We have not defined here Leray's generalized Lefschetz number [ii], but it

suffices to know a few facts. The generalized Lefschetz number agrees with the ordi-

nary Lefschetz number for f:U ~ U if the ordinary Lefschetz number is defined. If

fm(u) c y c U, f(y) c y and g denotes f:Y ~ Y, then Lgen(f) = Lgen(g ). In par-

ticular, if Y is homotopic in itself to a point, Lgen(f ) = i.

We can now show that the assumption of normality in Theorem i is unnecessary,

at least if the set ~ is "radial".

( 1 . 2 ) THEOREM

Let C be a cone in a Banach space X and let s be a bounded, relatively

open neighborhood of 0 in C. Assume that for each x E S d~f {x E C: Jlxll = I}

there is a unique real number t = t x > 0 such that tx E 8C(~), where ~c(~) denotes

the boundary of ~ as a subset of C. Let f:Sc(~) ~ C be a k-set-contraction and

suppose that

6 > kd ( 1 . 3 )

where 6 = inf{ Iif(z) ll: z E 8C(2)} and d = sup{ [Izll: z E ~C(~)}. Then there exists

t >_ 6d -I and z E 8C(~ ) such that f(z) = tz.

313

Proof

D e f i n e s = {x E C: Ilxlt = d}. For each x E 2, t h e r e e x i s t s a u n i q u e s = s x

~C(a ) , < 1 The assump- such t h a t sx ~ and by t h e d e f i n i t i o n o f d one has 0 < s x _ .

t i o n t h a t s x i s u n i q u e i m p l i e s e a s i l y t h a t R(x) d~f Sx x i s a c o n t i n u o u s map.

S ince t h e image o f any s e t A in ~ l i e s i n c-~(A U {0}) , we f i n d t h a t R i s a 1- d e f

s e t - c o n t r a c t i o n . I t f o l l o w s t h a t f l ( x ) = f (Rx) i s a k - s e t - c o n t r a c t i o n and

i n f IIf~ (x) ll = 5 x~3

I f we can p r o v e t h a t Theorem 1 .2 i s v a l i d f o r t h e s p e c i a l c a s e ~ = Ix ~ C: Ilxl] < d},

t h e d i s c u s s i o n above shows t h a t t h e r e e x i s t s x ~ 3 and s ~ 5d -1 such t h a t f l ( x ) = s x .

I f we w r i t e z = R(x) ( @ C ( a ) , i t i s c l e a r t h a t z i s an e i g e n v e c t o r o f f w i t h

e i g e n v a l u e t ~ 6d -1 .

The above discussions shows that it suffices to prove Theorem 1.2 in the case

that ~ = {x ~ C: llxll < d}. In this case define g:E ~ E by g(x) = dr(x) . If A

IIf(x)l l

i s any s u b s e t o f 3, i t i s c l e a r t h a t

g(A) c d S - i { t f ( x ) : 0 ~ t ~ 1, x ~ A}

c d6-ic-6-(f(A) U {o})

(1.4)

where for a set T, XT d~f {kx: x E T}. Using (1.4) and the basic properties of the

measure of noncompactness, one finds that

~ ( g ( A ) ) ~ kd6-1~(A) ( 1 . 5 )

so g is a c-set-contraction for c = kd5 -1 < i.

For e > 0 let U= {x ~ C: d-s < llxll < d+E} and define a retraction dx

r: U~ 3 by r(x) - . By reasoning like that above one can see that r is a

-1 Ilxrl -1 d(d-e) -set-contraction, so h(x) = g(r(x)) is a cd(d-e) -set-contraction.

lect ~ so small that c I = cd(d-s) -I < i and observe that fixed points of h:U ~ U

are the same as the fixed points of g:Z ~ E. Since fixed points of g correspond

to the desired eigenvectors, it suffices to find a fixed point of h. By Theorem i.i

and the remark immediately following it, h will have a fixed point in U and Lgen(h)=l

if Z is homotopic in itself to a point. To show 3 is homotopic to a point, let

p:K-{O} ~ E be radial retraction, fix x E s and define a deformation ~: 3x[0,1]~3 o

by

314

~(x,t) : p[(l-t)x+tx ]. (1.6) o

The fact that K is a cone insures that (1-t)x+tx ~ # 0

is well-defined and continuous. �9

for 0 _< t _< 1 and x ( Z, so

(1 .1 ) REMARK

If X is an infinite dimensional Banach space, S = {x E x:llxll = d > 0} and

f: S + S is a k-set-contraction with k < 1, then it follows from a remark on p. 373

of [15] that f has a fixed point in S. (Note that S is a continuous retract

of B = {x ( K: llxll ! d} when X is infinite demensional, so S can be deformed in

itself to a point). The proof of Theorem 1.2 follows by essentially the same trick.

(1.2) Remark

We have not proved Theorem 1.2 for general neighborhoods ~ of the origin in

K, although we conjecture it is true for such neighborhoods. In fact, Massabo and

Stuart use their theoren for general ~ in proving Theorem 1.2 oF [13]. However,

as is observed in [13], Theorem 1.2 of [13] is a global bifurcation result which can

be proved (for general cones) by using the fixed point index in the same way degree

theory is used to prove the classical Banach space version of Theorem 1.2 in [13].

(1.3) Remark

The problem of proving Theorem 1.2 for general ~ may be related to an exten-

sion problem for certain functions. I am indebted to Heinrich Steinlein for a conver-

sation which led to the following observation.

(1.3) THEOREM

and

Let ~ be a bounded, relatively open neighborhood of the origin in a cone

f: 8K(~) ~ K-{0} a continuous map. Define

s 1= {x~ K: llxlt = 1}, d 1: inf{HxJl: x~ ~K(~)},

315

d 2 = sup{Hxll: x ~ ~K(~)} an d A = { ( x , X ) E S l X [ d l , d 2 ] :

D e f i n e a map cp:A -+ S 1 b y

f(Xx) ~p(x,X) = ~ .

Xx ff a K ( ~ ) } .

Assume that there exists a continuous map ~: SlX[dl,d2J ~ S 1 such that (i) ~IA =

and (ii) ~ is a c-set-contraction for some c < 1. Then there exists x ~ aK(~)

and t > 0 such that f(x) = tx.

Outline of proof

Define ~x(X) = ~(x,X). One can associate a fixed point index to ~x:S1 + S 1

and prove that this fixed point index equals one. Using this fact one can prove that

there exists a connected set C c SlX[dl,d2] , C c {(x,X): ~(x,~) = x}, such that C

has nonempty intersection with SlX{dl} and SlX{d2}. It follows that C must

intersect aK(~) , which is the desired result. W

Under the hypotheses of Theorem 1.2, one can prove that there exists an exten-

sion ~ as in Theorem 1.3, so Theorem 1.2 is a consequence of Theorem 1.3. Further-

more one can derive the original Massabo-Stuart theorem from Theorem 1.3. Thus the

known results reduce to extending ~ to �9 in such a way that �9 is a c-set-con-

traction, c < 1.

2i LINEAR AND NONLINEAR KREIN-RUTMAN THEOREMS

The linear Krein-Rutman theorem has been used to calculate the fixed point in-

dex of certain nonlinear cone mappings and to obtain fixed point theorems for such

mappings. We shall show here that the most general linear Krein-Rutman actually follows

by elementary arguments using the fixed point index for cone mappings. We wish to

emphasize the elementary nature of our proof. We do not need the apparatus of asymp-

totic fixed point theory, and (at least for compact linear operators) our proof is

suitable for a course in nonlinear functional analysis as an application of the Leray-

Schauder degree.

316

We begin by recalling a trivial but useful observation of Bonsall [1].

(2 .1 ) LEMMA (Bonsall [ 1 ] )

Let {am: m ~ i} be an unbounded sequence of nonnegative reals.

exists a subsequence {am. : i > i} such that i

Then there

(1) a ~ i m. i

(2) a >_ a. for 1 _< j _< m.. m. j i 1

If C is a cone in a Banach space and f:C ~ C is a map, we shall say that

f is "order-preserving" if 0 } u } v implies f(u) ~ f(v) for all u and v in

C. We shall say that f is "positively homogeneous of degree 1" if f(tu) = tf(u)

for all real numbers t ~ 0 and all u ~ C.

(2 .1 ) THEOREM

Let C be a cone in a Banach space X and f:C ~ C an order-preserving map

which is positively homogeneous of degree 1. Assume that there exists a generalized

measure of noncompactness ~ such that f is a k-set-contraction with respect to

and k < 1. Assume that there exists u E C such that {llfm(u) ll: m ~ 1} is unboun-

ded. Then there exists x E C with I[xll = 1 and t ~ 1 such that f(x) = tx, and

i f f ( y ) r y f o r Ilyll = 1 and U = {y ~ c: Ilylt < 1 } , i c ( f , U ) = O.

(2.1) REMARK

If C is a normal cone, Theorem 2.1 is a very special case of Proposition 6

on p. 252 of [18], so the whole point of the following argument is that it applies

to nonnormal cones.

(2 ,2 ) RE~IRK

Suppose that g:V = {x E C: Nxll < ~ ~ c is a k-set-contraction w.r.t. ~,

k < 1, and that f is as in Theorem 2.1. If tf(x)+(1-t)g(x) ~ x for Ilxll = R, and

0 ~ t ~ 1, then there exists k ~ 1 and x ~ C with Ilxll = R such that f(x) = kx.

If not, the homotopy gs(X) = sg(x), 0 ~ s ~ 1, shows that ic(g ,~ = 1, while the

homotopy tf(x)+(1-t)g(x) shows that ic(g,~ = ic(f,~ = 0, a contradiction.

317

Proof of theorem (2.1)

Let ~ = Ix ~ C= IIxll < i} and assume that there does not exist t ~ 1 and a

point x ~ C with llxl] = 1 such that f(x) = tx. If we define fs(X) = sf(x) for

0 ~ s ~ 1, the above assumption shows that fs(X) r x for llxl] = 1 and 0 5 s ~ 1.

The homotopy property of the fixed point index implies that

ic(fl, 6) = ic(f, ~) = ic(fo, ~) = 1. (2.1)

If we can prove that ic(f,~ ) = O, we will have a contradiction of our original

assumption and the theorem will be proved. It is well-known that I-fl~ is a proper

map (the inverse image of any compact set is compact), so there exists 6 > 0 such

that llx-f(x) ll ~ 6 for x ~ C, Hxll = 1 (since we are assuming x-f(x) r 0 for

IIx]l = 1). Let u be as in the statement of the theorem. Because f is homogeneous

of degree 1, we can, by multiplying u by a small positive constant, assume that

0 < ]lull < 6 and {[]fm(u) ll: m ~ 1} is unbounded. Define a function g(x) by

g(x) = f(x)+u . (2.2)

The homotopy f(x)+su for 0 5 s 5 1 has no fixed points x with ]]xl] = 1 so

i c ( f , ~ ) = i c (g ,~ ) . (2.3)

To complete the proof it suffices to show ic(g,~ ) = O; and to prove

ic(g,~ ) = O, it suffices to prove that g has no fixed points in ~. Thus we assume

that g(x) = x for x ~ ~ and try obtain a contradiction.

If g(x) = x = f(x)+u we have

x ~ u. (2.4)

In general:, for purposes of induction, assume that

x ~_ ~ ( u ) . (2.5)

It follows from the order-preserving property of f that

x = g ( x ) = f ( x ) + u > f ( x ) >_ f ( f m u )

= fm+l (u)

so e q u a t i o n ( 2 . 5 ) h o l d s f o r a l l m > 0 by m a t h e m a t i c a l i n d u c t i o n .

(2.6)

318

Now we apply Lemma 2.1. Define a = llfm(u) ll, an unbounded sequence of nonne- m

gative reals, and let {ami : i > 1} satisfy the conclusions of Lemma 2.1. I]efine

fm(u) and let s = _ {Vmi - : i ~ 1}. We claim that ~(Z) = 0, so that Z has V m

II fm (u) lJ compact closure. To see this observe that for j > ~ we can write (using homogeneity

of f)

m ,

S i n c e Hfmi-J(u) ll < Irf l ( u J I P

J Z = ~_~ {Vmi } U fJ(Tj)

i = 1

Z. J

def =

I l l m i ( u ) ] l > j}

for i t j, we have

2.7)

T . �9 B = {x e K: ilxl] ~ i} J

2.8)

Equations (2.7) and (2.8) imply that

J Z c U {Vm. } U fJ (B) (2 ,9

i=1 i

Since fJ is a kJ-set-contraction with respect to ~, equation (2.9) implies

~(Z) _~ kJ~(B) (2.10)

The right hand side of (2.10) approches zero as j + ~, so ~(Z) = 0, Z is compact,

def and for some s u b s e q u e n c e Vm. = wj we c a n a s s u m e t h a t wj c o n v e r g e s s t r o n g l y

J t o w. Of c o u r s e Nwll = 1 a n d w ~ K.

If we now return to equation (2.5) and divide both sides by

Taking the limit as j +

theorem. R

am. we obtain i. J

(am.)-i x-wj ~ K (2..ii lj

yields that -w E K, and this contradiction proves the

Our first corollary generalizes Theorem 4.2 in [13] by removing the assumption

of normality of the cone. If the cone C is normal, however, Corollary 2.1 is an

easy corollary of an earlier result, Proposition 6 on page 252 of [18]. If the ope-

rator g below is compact and p = 1, Corollary 2.1 below is a result of Krein and

Rutman [9, Theorem 9.1].

319

( 2 . 1 ) COROLLARY

(Compare [13] and [ 1 8 ] ) . Let C be a cone in a Banach space X, ~ a g e n e r a l i -

zed measu re o f n o n c o m p a c t n e s s such t h a t ~ ( t S ) = t ~ ( S ) f o r e v e r y bounded s e t S in

C and e v e r y r e a l t T 0, and g:C + C a k - s e t - c o n t r a c t i o n w . r . t . 9- Assume t h a t

g i s o r d e r - p r e s e r v i n g and p o s i t i v e l y homogeneous o f d e g r e e 1 and t h a t t h e r e e x i s t s

u E C-{0} and e > k p such t h a t gP(u) ~ cu. Then t h e r e e x i s t s x ~ E C-{0} and

and X ~ c ( p - l ) such t h a t g(Xo) = kx o.

Proof

Let b be any real such that

k P < b < c .

-1 If q = q , define f(x) = b-qg(x)~ Using the homogeneity of f, it is clear that

fP (u ) ~ r u , r = cb -1 > 1 ( 2 . 1 2 )

and that f is a kl-Set-contraction w.r.t. 9, where k I = kb -q < 1.

By Theorem 2 .1 , f w i l l have an e i g e n v e c t o r in C o f norm 1 i f {llfm(u) ll: m~l}

i s unbounded . But e q u a t i o n ( 2 . 1 2 ) i m p l i e s t h a t

f J P ( u ) u > 0.

r j

I f I l fJP(u) ll were bounded , we would o b t a i n by l e t t i n g

a contradiction.

j ~ ~ in ( 2 . 1 3 ) t h a t

(2.13)

-u E C,

Theorem 2.1 t h u s i m p l i e s t h a t t h e r e e x i s t s x ( C, Ilxll : 1, and t ~ 1 such

t h a t f ( x ) = t x . W r i t i n g i n t e r m s o f g , t h e r e e x i s t s k ~ h q such t h a t

g ( x ) = Xx. ( 2 . 1 4 )

S e l e c t an i n t e g e r N such t h a t c_N-1 > kr .n -1 c - n = b , t h e above r e m a r k s show t h a t t h e r e e x i s t

such t h a t

If n ~ N and if we define

K, IlXnll = 1 and X > ( c - n - l ) q Xn n -

g(Xn) = XnXn. (2.15)

320

we define Z = ~ IXn: n ~ N}, �9 B = c-N -1 If and

that f kl-set-contraction with respect to

shows that

f(x) = B-qg(x), previous remarks show

for some k I < i. Equation (2.15)

E c ~%-(f(Z) U { 0 } ) , ( 2 . 1 6 )

Equation (2.16) implies that ~(s ~ klP(Z), so ~ is compact.

X > c q and a subsequence we can assume Xn. + x and kni 1

Therefore, by taking

f(x) = Xx, X ~ c q (2.17)

which is the desired result. �9

We now wish to show how Theorem 2.1 can be used to obtain a general version of

the linear Krein-Rutman theorem. First we need to recall some definitions. If X is

L:X ~X is a bounded linear operator, define r(L), the spectral a Banach space and

radius of L, by

If X is real and

I 1

r ( L ) d=ef lira HLni] n = inf I[Lntl n �9 (2.18)

n~ n_ > i

denotes the complexifieation of X, X = (x+iy: x,y E X1 with

IJx+iyll = sup I I(cos e) x+(sin e)yil o~ee2~

then L has an obvious linear extension L and Ilerl = I I L I I .

spectrum of L, then of course

If ~(g) denotes the

r(L) = sup{lzl: z ~ ~(~), z complex}.

If ~(L) is defined by

a(L) = inf{c ~ O: L is a c-set-contraction}

it is clear that ~(L) ~ I I L I [ .

and we can define p(L) by

Because a(LiL2) 2 ~(Li) ~ (L2) , lim

1 1

~(L) = lira (a(Ln))n = inf (a(Ln)) n. n-~o n~l

If L is as above, one can prove that ~(L) = ~(L).

o(L), the essential spectrum of L, and it is proved in [14] that

There is a subset

(2.19")

1

(~(Ln)) n exists,

( 2 . 2 0 )

ess(L) of

p(L) = sup{ I~1: ~ ~ e s s ( L ) } . ( 2 . 2 ~ )

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Actually, there are several inequivalent definitions of the essential spectrum, but

(2.21) is valid for all of them. It is proved in C14] that if s > O, there is at

most a finite number of complex numbers z such that z E d(L) and Izl ~ p(L)+e.

If z is such a complex number and F a simple closed curve in ~ which contains

no other point of ~(L) in its interior or on F, then the spectral projection P

corresponding to z,

if F

is finite dimensional.

If K is a cone in X

L(K) c K, define numbers IILII K

and L:X ~X is a bounded linear operator such that

and aK(L) by

IILIIK def= sup{tlgull: u E K, rlull ~ 1}

aK(L ) = inf{c ~ O: LIK is a c-set-contraction}.

( 2 . 2 2 )

In analogy r(L) and p(L), define rK(L), the cone spectral radius of

PK(L), the cone e s s e n t i a l s p e c t r a l r ad ius of L, by

1 def

rK(L ) = lira (IILnN K )n

n ~ ( 2 . 2 3 )

1

PK(L ) def lim (~K(Ln)) n . (2.23) n~

As was remarked in [18] it is easy to see that

PK(L) ~ p(L); if the cone K

prove that PK(L) 2 rK(L ).

if LIK is compact.

L, and

p(L) A r(L), rK(L ) 2 r(L) and

is "reproducing" (so X = {u-v: u,v E K}) one can also

Note also that p(L) = 0 if L is compact and PK(L) = 0

We need to recall one more definition. If K is a cone in a Banach space X

with norm If'If, define Y = {u-v: u,v ~ K}. Define a norm I-l on Y by

lyl : i n f { I/ull+llvll: u, v ~ K, y = u-v} . (2 .24)

It is remarked in [2, 22] (and is not hard to prove) that Y is a Banach space in

this norm, IIYH ~ IYl for y ~ Y, and IJYll = IYl for y E K. If L:X~X is a

bounded, linear operator such that L(K) c K, then L(Y) c y. Furthermore, if ILIy

denotes the norm of L considered as a map from the Banach space Y to itself, it

is easy to see that

322

so that

[LIy = [ILil K ( 2 . 2 5 )

1 1 -- def

lim ILnJ n : ry(L) : rK(L) : lira I[LnIIK . n~oo Y n~o~

( 2 . 2 6 )

In (2.26), ry(L) denotes the spectral radius of L as a map from Y to Y.

With these preliminaries we can prove our linear Krein-Rutman theorem.

( 2 . 2 ) THEOREM

Let X be a Banach space, K a cone in X and L:X ~ X a bounded linear

operator such that L(K) c K. If PK(L) = v and rK(L ) = K are defined by equations

(2.22) and (2.23), assume that ~ < R. Then there exists x E K-{0} such that

Lx = bx.

Proof

d e f i n e

Let s n be a sequence of real numbers such that v < s n

gn(X) = snlL(x). Our assumptions imply that

< b and s -~ ~ and n

-i -1 1 < s ~ = rK(gn). (2.27) pK(gn) = s n v < n

For notational convenience, fix n, write g = gn' s = s n and select N so large

that ~K(g m) < i for m > N. Just as in the proof of Proposition 7 in [18], define

a generalized measure of noncompactness by

N-1 1 j~0 ~(gj (A)) (2.28) ~3 (A) = ~

One can easily check that ~ is a generalized measure of noncompactness, that

~(XA) = IX[~(A) and that there exists a constant c < 1 such that

~(g(A) ) ~ c~(A) (2 .29)

for every bounded set A a K.

323

If Y is the Banach space defined immediately before the statement of Theorem

2.2, we have seen that rK(g ) = ry(g) = the spectral radius of g as a map of Y to Y.

Since rK(g ) > i, it follows that Ignly is unbounded. The uniform boundedness prin-

ciple on the Banach space Y implies that there exists y E Y such ~hat Ign(y) l

is unbounded. It follows that there exists u E K such that Ignul = llgnull is

unbounded. Theorem 2.1 now implies that there exists t ~ i and x E K, Iixll = 1,

such that

g(x) = t x . ( 2 . 3 0 )

= Of course t = t n and x = x n depend on n, and if we write ~n Snt n we have

L(Xn) = ~nXn . ( 2 . 3 1 )

< < = We must have Sn -< ~n ~, so lim ~n ~. Exactly as in the last paragraph of the n->oo

p r o o f o f P r o p o s i t i o n 7 in [18] o r as in t h e p r o o f o f C o r o l l a r y 2 . 1 , {Xn: n _> 1} has

compact c l o s u r e , so one can assume by t a k i n g a s u b s e q u e n c e t h a t x -~ x E K, ]lxH = 1, n

and t a k i n g t h e l i m i t as n ~ ~ o f ( 2 . 3 0 ) g i v e s

Lx = ~x ( 2 . 3 2 )

which is the desired result. �9

(2 .3) REMARK

If LIK is compact, so ~K(L) = O, Theorem 2.2 generalizes a result of Bonsall

[1,2]. The original Krein-Rutman theorem deals with the case that L is compact as

a map of X to X, K is total and r(L) > O, and one might believe that, at least

for total cones, Bonsall's result is equivalent to the Krein-Rutman theorem. However

Bonsall gives a simple example which shows this hope is false.

Let X = {x E C[0,1]: x(O) = O} and define L:X +X by (Lx) (t) = x(�89 Bonsall

constructs, for each y > O, a total cone Ky such that L(Ky) c KT, LIK T is compact, IT

and rKT(L) = (~) . Pe rhaps most s u r p r i s i n g i s t h e f a c t t h a t t h e cone s p e c t r a l r a d i u s

can v a r y f o r d i f f e r e n t t o t a l c o n e s . B o n s a l l p r o v e s , however , t h a t i f L i s compact

as a map o f • i n t o i t s e l f , t h e n rK(L ) = r ( L ) f o r e v e r y t o t a l cone K such t h a t

L(K) c K. Our n e x t p r o p o s i t i o n i s a g e n e r a l i z a t i o n o f t h i s f a c t , and t h e a rgument we

g i v e i s a g e n e r a l i z a t i o n o f B o n s a l l ' s a rgument f o r t h e c a s e L compact .

324

( 2 .2) COROLLARY

(Compare [1].) Let X be a Banach space and L:X ~ X a bounded linear map.

Assume that p(L) < r(L) (p(L) and r(L) defined by (2.18) and (2.20)). Then if

K is any total cone in X such that L(K) c K and rK(L ) is defined by (2.23),

one has

In particular, if

r z ( L ) : r ( L ) .

r : r ( L ) , t h e r e e x i s t s x ( K-{O}

If X is the dual space of X

exists f ( ~ such that

such that

( 2 . 3 3 )

Lx = rx. ( 2 . 3 4 )

K ~ - and - {f E X : f(x) ~ 0 for all x ~ K}, there

L*(f) = rf. (2.35)

( 2.4 ) REMARK

For reproducing cones, the latter half of Corollary 2.2 was proved in [7] by

a linear argument like that used by Krein and Rutman.

Proof

Suppose we can prove that rK(L) = r(L). Since we clearly have PK(L) ~ p(L), it

will then follow from Theorem 2.2 that there exists x ( K satisfying (2.34). Sin~e

K is total, it is easy to see that K* is a cone and L*:K* ~ K*. It is proved in

[14] that p(L*) = p(L), so that pK,(L*) ! p(L). If we can prove that rK,(L*)~r=r(L),

the existence of f satisfying (2.35) is a consequence of Theorem 2.2 again. However,

if x satisfies (2.34), a version of the Hahn-Banach theorem implies that there exists

g ( K* with <g,x> > 0 (where < , > denotes the bilinear pairing between X* and

X) and one obtains

< ( L * ) n g , x > = <g , Lnx> = r n < g , x > . ( 2 . 3 6 )

Equation (2.36) easily implies that rK,(L* ) ~ r.

325

exists

Thus to complete the proof it suffices to prove (2.33).

x ( K such that

llnxH i im sup > 0.

n-~ IILnH

We claim that there

( 2 . 3 7 )

If (2.37) holds for some x, it is easy to see (using the fact that K is total)

that there exists u ( K such that

II Lnull lim sup > O. (2.38)

n-+~ llLnll

Inequality (2.38) implies that rK(L ) > r(L), while the opposite inequality is imme-

diate.

Thus it suffices to prove (2.37).

H Lnx rl lim sup

n-~ IILn[I

We argue by contradiction and assume

= O for all x ( K.

Select numbers Pl and P2 such that

p : p(L) < Pt < P2 < r : r ( L ) .

If B = {x: ]Ix]] ~ i} and a denotes the measure of noncompactness, select

that

~(Ln(B)) < pic~(B) _< 2Pi, n >_ N O .

We can also assume that N O

is so large that

2 o P 2 < r -

By definition of the measure of noncompactness, there exist sets

such that

L N~ (B) m No : U Si, diameter (Si) -< 2p i .

i:i

N O N O Select r with 0 < r < 2(p 2 -Pl ) and for each i, i <- i -< m, select

and N. such that 1

IITnxil] < ~l]Tnll , n >_ N . . 1

N such O

Si, i _< i -< m,

x. (S. 1 1

326

Finally, select an integer K ~ such that

N d~f k N ~ max N.. o o l!i!m i

I f y E B, there exists an integer i, 1 _< i _< m, such that

N N liT ~ <- 2P10.

If n>-N we obtain that

n+N N liT ~ <- rlTnxilr + IlTnlrH T Oy_xil I,

_< r + 2pNOlrTnll (2.39)

< 2p~ o IITnll

Inequality (2.39) implies that for n ~ N

N IITn+Noil ~ 2P2OllTn[I . (2.40)

Using (2.40) one sees that there exists a constant B independent of the integer

k ~ I such that

kN N k liT ~ -~ B(2P2 ~ (2.41)

and (2.41) implies that

1 1 kN (k~--)

l i m sup liT ~ o _< 2 ~ 2 < r . ( 2 . 4 2 ) k ~

Inequality (2.42) contradicts the definition of r. �9

Many cones used in analysis are normal. We would like to close this paper by

discussing a linear operator which is best studied in nonnormal cones. The operator

L which we shall define below plays an important role in recent work by K Bumby on

the problem of finding the Hausdorff dimension of certain sets of real numbers defined

by properties of their continued fraction expansions [4].

Let M denote a finite union of closed, bounded intervals of reals, let

Xn, n ~ 0, denote the Banaeh space of real-valued maps x:M ~ ~ which are n times

327

continuously differentiable and let

X n. If x (X n define IlXNn by

K denote the cone of nonnegative functions in n

n

IIxlJ n : X maxlx (j) ( t ) I . j:0 te~

(2.43)

There is a natural map

S is a bounded set in

Jn:Xn ~Xo by Jn(X) : x (n), the n th- derivative of x.

Xn, we leave to the reader the straightforward proof that

If

(S) = a(Jn(S)). (2.44)

We use the same letter

on the left and in X O

in (2.44) to denote the measure of noncompactness in

on the right.

X n

We now define a linear map L by

( L x ) ( t ) : N

bj(t)xCzj(t)). j=i

(2.45)

We shall always assume about the given functions b. and ~. in (2.45) ] ]

HI. The functions b.:M ~ ~ are C and nonnegative for i ~ j < N and -- ]

N

j~l= b4(t)~ > 0 f o r a l l t ( M . The f u n c t i o n s ~ . :M4N] a r e C , ~ j (N) c M and

max (max ~j ( t ) l) = c < 1. L~j~N t(M

Under assumption HI L defines a bounded linear map of X to X for n ~ 0 n n

and L(Kn) c Kn. In our next theorem we discuss the spectrum of L:• n ~Xn; for reasons

of length we shall not prove Theorem 2.3 here.

(2.3) THEOREM

(Compare [4]). Let L be defined by (2.45) and assume that H1 holds. Consider

L as a map from X n = cn(M) into itself and let Pn denote the essential spectral

radius of L in X n and r n = rn(L) the spectral radius of L in X n. Then if

N N

A = max ~ Ibj(t) I , B = min ~ b~(t) and c = max ~(t) l < 1, one has t~M j=l t(M j=l j,t

Pn < Acn ( 2.46 )

328

r _> B. (2.47) n

Inequality (2.46) holds even if the functions b. are not nonnegative. If Pn < rn 3

(as will be true for large n), r n = ro; and if Pn < rn and D is a total cone

in X n such that L(D) c D, then rD(L ) = r n. There exists a nonnegative, C = function

u, not identically zero, such that

Lu = r u. (2.48) O

( 2 .5 ) REMARK

If b. is strictly positive for 1 ~ j ~ N, Bumby [4] proves the existence 3

of a continuous function which is strictly positive on M and satisfies (2.48). How-

ever, his proof does not apply under the weaker assumption H1, and indeed we do not

prove here that u is positive on M under H1, although we conjecture that this is

so. The fact that r n = r ~ = rD(L ) when Pn < rn and the estimates on Pn appear

to be new; if the b. are strictly positive, we can prove that r = r for all n. 3 n o

( 2 .6 ) REMARK

If the hypotheses of Theorem 2.3 are weakened slightly, then r may fail to o

be a positive eigenvalue of L with corresponding eigenvector in K o, For example,

if (Lx)(t) = tx(~t) for x ~ C[0,1], one can verify directly that the spectral ra-

dius r ~ of L is 0; any C function with support in [~, ] is a C eigenfunc-

tion.

If (Lx)(t) = b(t)x(t) for x E C[0,1], where b(t) is not constant on any

subinterval of [0,1], one can prove directly that the spectral radius of L equals

max ]b(t) I , but that L has no point spectrum. t

2.7 ) REMARX

The spectrum of L varies depending on what space L acts in. For example

if (Lx)(t) = x(ct), 0 < c < i, one can prove that the spectrum of L as a map of

cn[0,1] into itself is {cJ: 0 ~ j ~ n-l} U {z: z complex, Izl ~ cn}; cJ(0 ~ j ~ n-l)

t j Further- has algebraic multiplicity one and corresponds to the eigenvector xj(t) = . n

more if Izl < c , z is in the point spectrum of L:C n § C n and has infinite multi-

plicity; if 0 ~ z < c n, there are infinitely many nonnegative, linearly independent

eigenvectors corresponding to z.

829

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