Positive eigenvectors of k-set contractions

Non-linear Analysis, Theory, Methods &Applications. Vol. 3. No. I. pp. 35-44 0 Pergamon Press Ltd. 1979 Prmted m Great Britain

POSITIVE

EIGENVECTORS

0362-546X/79/0101-0035

102.0010

OF R-SET CONTRACTIONS

IVAR MASSABO* Dipartimento

di Matematica,

Universita

CHARLES Department

de Mdthematiques,

della Calabria,

cone,

Ecole Polytechnique Fed&ale, CH-1007 Lausanne, Switzerland

positive

Italy

and A. STUART

(Received 17 January Key words: k-set contraction, elliptic equation.

Cosenza,

eigenvector,

Lausanne,

61 avenue de Cour,

1978) component

of solutions,

essential

spectrum,

INTRODUCTION THIS paper begins with a version of a theorem due to Krasnoselskii (Theorem 1.1, p. 243 of [l]) which is itself closely related to an early result of Birkhoff and Kellog [2]. Krasnoselskii’s theorem is given in the context of a compact mapping T: P -+ P where P is a cone in a real Banach space. It asserts the existence of an eigenvector of Ton the boundary of an open set Q containing 0, provided that I/Txll > 0 for all x E 80. Our result (Theorem 1.1) allows T to be a k-set contraction, but we suppose that P is normal. Other results related to the Krasnoselskii and BirkhoffKellog theorem are contained in [3-61. Let Y = {(x, 2) E P x [0, co]: x = ATx} and let %?be the component of Y containing (0,O). We use Theorem 1.1 to formulate an alternative concerning the global behaviour of %‘. This result (Theorem 1.2) is in the spirit of a theorem of Rabinowitz [7] concerning non-linear elliptic eigenvalue problems on bounded domains. In the context of abstract Banach space and T: P + P a compact mapping, the unboundedness of G??has been established by Dancer [S] and Amann [9]. The case where T: X + X is a k-set contraction is treated in [lo]. Our proof of Theorem 1.2 is based on Theorem 1.1 via a trick used by Kuiper [ll]. As is shown by the example in Section 6 of [lo], our alternative cannot, in general, be sharpened. In Section 2, we reformulate the alternative in the context of unbounded operators in an ordered Hilbert space. Cast in this form, the result is most easily applied to non-linear elliptic eigenvalue problems on unbounded domains. A typical application of this kind is given in Section 3. The results given in this article are primarily concerned with non-linear operators and some (Theorem 1.2, for example) are trivial in the case of linear operators since 0 E P is accepted as a positive eigenvector. For linear operators a positive eigenvector should be an element of P\(O). For compact linear operators the theorems of Krein and Rutman [18] deal with this situation, and some extensions to linear k-set contractions T: X -+ X with T(P) c P are given in [19]. In Section 4, we show how Theorem 3 of [ 193 can be obtained very easily from Theorem 1.1. Actually we need the additional assumption that the norm in X is monotone, but our result is more general in so far as T: X -+ X is only required to be homogeneous and monotone. * Partially

supported

by a C.N.R.

(Italy) Fellowship. 35

I. MASSABOAND C. A. STUART

36 1. EXISTENCE

OF

POSITIVE

EIGENVECTORS

The following notation and de~nitions are used in what follows. Let X denote a real Banach space, let 1).I/ d enote the norm in X and let R, denote the nonnegative real numbers. A subset 9 of X is said to be a cone if 9’ is closed, convex, invariant under multiplication by elements of R,, and if 9 n (- 9) = (0). Moreover, B is called no~~ff~ if there exists a j3 > 0 such that, for all x, ~~69, //x + y/l 2 j? max(//x//, jly(/). A mapping &:B + 9 is called a k-set contraction if it is continuous and for every bounded subset B of 9 c&d’(B)) < kc@), where CIis the measure of non-compactness of B in 9 (i.e. a(B): = inf(t: > O/B can be covered by finitely many subsets of 9 of diameter less or equal to F)). Now we can state the main theorem of this section. 1 .I. Let B be a normal cone in a Banach space X, let $3 be a bounded open subset of 9 containing 0 and let &: 9 + 9 be a k-set contraction. Suppose that

THEOREM

6 > kd/fl,

(1.1)

and /? is the constant appearing in the where 6: = inff //&‘(z)j/: z E Xl>, d = maxi //z //:z~Xlf definition of normal. Then there exist ,u > 0 and z E 3Q such that z = p&‘(z). Proof. Suppose that there are no p > 0 and z E 13stsuch that z = p&(z). Hence ind(p&, Q) is defined for all 0 < .Dc k -I, where ind(p&‘, Q) denotes the fixed-point index for k-set contractions (see for example [12]; our ind(@, Q) = i&&, 0) in notation of [12]). Since the homotopy h(z, ,u): = p&(z) is admissible, in the sense of [12], for all 0 < p < k- ‘, ind(p&, R) is independent of p for all 0 < ,u < k-‘. In particuIar, since OEQ ind(O, Q) = 1, where 0 stands for the mapping which is zero everywhere. Therefore, ind(p&‘, Q) = 1 for all 0 Q p < k- ‘. On the pther hand, choose E 1 0 such that 6 > (k + ~)d/fl (1.2) and set ,u,,: = (k + 4-l. If u is any vector in g\(O), consider the homotopy h(z, t): = pO&‘(z) + EU,t E [0, I], and suppose that there exist z E aQ and t E CO,1J such that z = ,u,,&‘(z) + tu. Therefore d 2 I/z/I = ([,L+,&‘(z)+ tv /I 3 p~~pcl,-aul(z)~~ (the last inequality follows from the normality of 9). But, by our choice of p0 and (1.2), bl/p,,&(z)II > &l/k + E) 6 > (P/k + E) . (k + c/B) d = d, which is impossible. By our choice of p0 and by the above argument, it follows that h is an admissible homotopy (in the sense of [12]). Therefore, ind&&, Q) = ind(p& + U, Q). Let now p E 9\{0>. Then there exists o! > 0 such that w - ~$9 for all w E X with j/w (/ < c(. Let IE= d/a and set v = np. Hence, for all z EQ, I(z/ n /I < CYand so z/n - p # 9. This implies that z # ,+Q?(z) + np for all z E Sz. Therefore, ind(,+& + np, 0) = 0, and so ind&,&, C?)= 0 with 0 < cl0 < k- l contradicting the fact that ind(p&, Q) = 1 for all 0 < p < k- ‘. Hence, there exist ,U> 0 and z E 80 such that z = p&(z). Remark. If the constant j3 can be taken to be one. the norm in X is said to be monotone

with

respect to the partial ordering induced by the cone 9. We do not know whether the assumption that the cone be normal is necessary for the truth

Positive

of Theorem

1.1. Indeed,

PI*

Let us note, however,

eigenvectors

in the case of a compact that the condition

Example. Let I2 denote

the Banach

31

of k-set contractions

A(i.e. k = 0) it is certainly

operator

(1.1) is optimal

in the following

space of all real sequences

unnecessary

sense.

x = (x,, x2, . . .) with norm

lixll = (“Z1 xY. Let P = {x E Z2: xn 3 0 for all n E N}. Then P is a cone in l2 and the norm with respect to the corresponding partial ordering. Let T: P + P be the mapping defined by ifllxll

T(X): = (+Jm1,x2,xg,...)

(0, xi,x2,x3,.

in I2 is monotone

d 1

if lixll > 1.

. .I

It is easily seen that T is a l-set contraction. Let S(r) = {xE 12: l[xll = r} n P, 6(r) = inf{ IIT(x)11: x E S(r)} d(r) = max{ llxll :x E S(r)}. Clearly d(r) = r and 6(r) =

1 r

Ol

.

Hence we see that (1.1) holds on S(r) if and only if 0 < r < 1. But the eigenvalue Ancan easily be solved explicitly. For llxli = r < 1 we must solve (x1. x2,. . .I = l(Jqp, This implies

and

r2 = f

A2”(1 - r2) =

i=l

x =

X1’ X2’. ..I.

that xn = A”- 1 x1,x1 = &/m xf = f

problem

(I - r2)A2 1 _ A2

i=l

HenceA = irandx = +r,/g(l, &r,r2, fr3 ,... ).ThusA = randx = rJ~(v2(1,r,r2 ,...) is the only solution with x E P. On the other hand, for llxll = r > 1 we must solve (x,, x2, . . .) = A(0, xi, x2, . . .). But this implies that x1 = 0 and x,, = ;In-lxr which contradicts the facts that llxli = r 2 1. Hence we see that the problem x = ATx, has a solution

x E S(r)

and I E R

if and only if 0 < r < 1. i.e. if and only if (1.1) holds.

Consider now a k-set contraction T: P -+ P where P is a cone in the real Banach space X. of 9’ containing (0,O). Let 9’: = {(u, I)EP x R,..u = A 7b) and let W be the component THEOREM 1.2. Let T: P + P be a k-set contraction and let the norm in X be monotone toP.If [Ijull:(u,~)~%} #R+,then sup{l:(u,A)~%} > k-’ ifk > 0 =cO

ifk = 0.

with respect

I. MASABO AND

38

C. A. STUART

Proof We need only consider the case k > 0. Suppose that { llull: (u, A) E U} # R, and that sup{,?: (u, A) E%?} < k- ‘. Then there exists a bounded open subset G of P x R, such that V c 9 and sup {A :(u, A) E 9Y; < k- ‘. Since T is a k-set contraction, it is easily seen that Y n qis a compact set. Since V and Y n &ii!!are disjoint closed subsets of 9’ n 3, it follows from a result from general topology (e.g. [ 13]), that there exists an open subset Zz of % such that V c n and 9’ n t3R = 0. Let m = sup{ ilull: (u, A) E Xl} and s = sup{A: (u, A) E 8Q). Then m < cc and 0 < s < k-‘. Setting E = (I/Zm),/K’, we now consider the following norm on X, IIuI/~: = eIIuI/. With respect to this new norm, T is still a k-set contraction. Furthermore I/ IIE is monotone with respect to P. Let 9: = P x R, and let &:B -+ Y be defined by &(u, A): = (T(u), 1). Then 9 is a cone in X x R endowed with the norm II(u, A)I/,’ = llu/lf + A2 and this norm is monotone with respect to 9. In addition, &:9 --, 9 is a k-set contraction. Setting 6 = inf(I/&(u,A)IIE:(u,

A) E KI}

and c? = sup{ /I(% n)ll&

2) E an>,

we see that 6 > 1 and that c( d (.z2m2 + s~)I!~. Our choice of Eensures that kcc -c 6 and so Theorem 1.1 may be applied to _& on 8sZ. Thus there exist ,u > 0 and (u, A) E afl such that (u, 2) = ~J&u, 2) = (T(u), 1). Hence 1 = p and ATu, which means fi and so the result is established.

that (u, A) E Y n all. This contradicts

our construction

of

Remark. In deducing the above result from Theorem 1.1 we are forced to assume that the cone P is normal. In fact, by using the method of Theorem 2.3 of [lo] and the fixed point index instead of the topological degree, this result can be proved without the assumption of normality.

2. EQUATIONS

In this section space. Let H be a real in H is monotone = Q > 0 where assume also that

we apply Theorem

IN

HILBERT

1.2 to equations

SPACE

of the form Su = 2(,4(u) + B(u)} in Hilbert

Hilbert space with norm II.II and let P be a cone in H. We suppose that the norm with respect to P. Let S: 9(S) -, H be a self-adjoint operator in H with inf up(S) aJS) denotes the essential spectrum of S. If oe(S) = @we set Q = + co. We 0 # o(S), the spectrum of S. The domain of S with the graph norm llullf = [lull2 + I/Sul12

foruEQ(S)

is a real Hilbert space which we denote by H,. Then P, : = P n H, is a cone in H, and the topoand by logy on P, is that induced by H,. Since 0 4 a(S), S: H -+ H is a linear homeomorphism [14, Corollary 1.21, S-’ considered as a mapping from H into itself, is a Q-‘-set contraction. If se(S) = is; then S- ’ : H -+ H is compact. In addition we suppose that S-‘(P) c P,. Let A: P + P be a l-set contraction and let B: P -+ P be a compact mapping. Let 9, : = {(u, ,i) E P, x R, : Su = A[A(u) + B(u)]} and let %Ylbe the component of 9, containing (0,O).

Positive

eigenvectors

39

of k-set contractions

THEOREM2.1. If (IIuII,: (u, A))%?i} # R,, we have sup{l: (u, J)E%~} > Q. Proof: Since the mapping $:P, x R, -+ P x R, defined by &u, 1) = (Su, A), is a linear homeomorphism, it is easy to check that %: = &G.?J is the component of the set ((u, 1) E P x R, : u = 1[AoS-‘(u) + BoS-l(u)]} containing (0,O). Let T:P -+ P be the mapping defined by T(c):= AoS-‘(u) + BoS-‘(u). Since AoS-’ is a Q-‘-set contraction and BoS- ’ is a compact mapping, we have that T is a Q-‘-set contraction. Let us assume that {IIu~(~:(u,I)E%~) # R,, then {Ilull:(u, A)E%?}# R,. Hence Theorem 1.2 applies to the equation u = AT(u) = A(AoS-‘(u)

+ BoS-l(u)},

and we have that sup(A:(u, J.) E %?}2 k-’

= Q.

Therefore, we must have sup{A:(u, A)E %?& B Q, too.

3. APPLICATION TO ELLIPTIC DiFFERENTIAL

EQUATIONS

We consider differential equations of the following type, -Au(x) + q(x)u(x) = n{u(x, u(x)) + b(x, u(x), Vu(x))} for x E R”,

(3.1)

where q:R” -+ R, a:R”+l -+ R and b:R2”+’ + R are given functions whose properties are to be described later. We suppose that n > 2, the case of an ordinary differential equation being treated in [lo]. Since both the function u: R” -+ R and the real number I are unknown quantities, we consider a solution of (3.1) to be an ordered pair (u, A). For p E [ 1, co), we denote by LP(R”)the real Banach space of all equivalence classes of measurable functions u : R” + R which are such that (u(Pis integrable over R”. The norm in Lp(R”)is denoted by IIuil, = &&x)(Pdx)l’p. A natural cone in C(R”) is P: = {u E E(R”):u(x) > 0 for almost all x E R”}. We note that the norm in J?(R”) is monotone with respe t to this cone. For p E [ 1, co) and m a positive integ 5r, [LP(R”)]m is a real Banach space with norm defined by

llullp=

{i~lljql~~}‘~p where u = to,, u2, . . . , u,,,). This

norm is equivalent to that defined by

{JR,[ ,F ui(x)2]P’2dx}lip. We=n! state our hypotheses concerning the functions 4, a and b. (Hl) The function 4 : R” + R is continuous and 0 <

inf q(x) < TGR”

,sy; q(x) < co.

(H2) The function a:R”+’ + R is continuous and a(x, t) > 0 for x E R” and c > 0. Also atO) E C(R”) and there exists a constant k 2 0 such that la(x, t) - a(x, s( d kit - s( for all x E R” and t, s E R.

I. MASSABO AND C. A. STUART

40

(H3) The function b : R2”+ ’ + R is continuous and b(x, t. t) >, 0 for x, < E R” and t >, 0. The growth of b is limited by the following two requirements. (i) There exist a constant ,a E L?(R”) such that

p E [l, n/(n - 2)), a constant

c > 0 and a continuous

function

Ib(x, r])l < g(x) + c/qIp for all x E R” andallyER”+r where

(ii) Given

E > 0, there exist a constant

p E [l, n/(n - 2) and an R 3 0 such that (b(x, 0) - b(x, q)( $ a(~(~

forallxER”suchthatIxl

> Randforallr]ER”+‘.

Let Wz(R”) denotes the usual Sobolev space of all real-valued functions in C(R) whose partial derivatives (in the sense of distributions) of order less than or equal to two can be regarded as elements of Ii?( The norm in Wi(R”) may be defined by

and Wz(R”) is a real Hilbert space with this norm. Furthermore, 9(S) = Wi(R”) and Su(x) = -Au(x) + q(x)u(x) for almost all x E R” defines a self-adjoint operator in I?(R”) and the graph norm of S on 9(S) is equivalent to the norm I/. )I,2j just defined (This follows from Theorem 5.4 of Chapter V of Kato [15] together with the a priori estimate (2.1) of Browder [16].) The positivity of 4 implies that a(S) c (0. 00) and the boundedness of q implies that Q: = inf a,(S) < co. (If se(S) = 4, then S-‘:L2(R”) --t E(R) is compact and so S’S: Wz(R”) + J?(R) compact. Since W@“) is not compactly embedded in L?(R”) we conclude that (T (S) # 4.) of the classical Finally we note that S-‘(P) c P, : = P n Wi(R”). This is an easy consequence maximum principle. Remark. In applying the results of the preceding sections to the differential equation (3.1), we shall use only the properties of the self-adjoint realisation S listed above. These properties hold true in much greater generality. A reader who is interested in replacing -Au + qu by a more general second order linear elliptic operator and in replacing R” by an open (ungunded) subset of R” should consult the paper of Browder [16]. Those who are particularly Interested in allowing the potential 4 to have singularities are referred to Chapter V.3 of Kato [15] and the references therein. A means of determining Q from the behaviour of q is also given in Kato [ 151. Let us now consider LEMMA

the non-linear

terms in the differential

equation

3.1. For u E L2(R”), let A(u)(x) = a(x, u(x))

for almost

all x E R”.

(3.1).

41

Positive eigenvectors of k-set contractions

The A : L?(Rn) -+ L?(R”) and IIA(u) - A(v)ll, = kllu - ul12. Furthermore

A(P)&

P.

Proof: For U. u E L?(R”) /a(~, u(x)) - a(x. tj(x))(< k(u(x) - t&x)( for almost

all x E R”.

Hence A(u) - A(v) EL?(R”) and I/A(U) - A(u)/I 2 d kllu - vJ12. But A(O)EL?(R”) and so A(u) = A(u) - A(0) + A(O)EL?(R”).

by hypothesis Q.E.D.

LEMMA 3.2. For u E V z(R”), let

B(u)(x) = b(x,u(x), Vu(x)) for almost Then B: Wi(R”) --t L2(R”) is continuous Proof We begin by showing

and compact.

all x E R”.

Furthermore

that B: Wf(R”) + L?(R”) is bounded

B(P,) c P. and continuous.

For u E Wi(R”), let J(u(x)) = (u(x), Vu(x)) for almost Then it follows from the Sobolev that

embedding

theorem

all x E R”.

(Lemma

5 of Browder

[ 161, for example)

J: W;(R”) --f [L’(R”)]“+’

is a bounded linear map provided that 2 d r < 2n/(n - 2). Now choose and fix p E [f , n/(n - 2)] in such a way that the inequalities (i) and (ii) of (H3) hold. For u E [L?‘(Rn)ln+r, let C(u)(x) = b(x, u(x)) for almost all x E R”. From the basic result of Vainberg on Nemytskii operators (see for example Proposition 2.5 of Chapter 5 of Martin [17]) it follows that C: [L?‘(R”)]“+’ ---f I.$R”) is a bounded and continuous mapping. Noting that B(u) = C(J(u)) for u E Wi(R”) we deduce that B: W:(R”) + L?(R”) is bounded and continuous. We now show that the hypothesis (H3) (ii) implies that B: Wz(R”) + J?(R”) is compact. Let V be a bounded subset of W$R”). It is sufficient to prove that given any F > 0 there exists a compact mapping CE: Wz(R’) + L’(R”) such that )IB(u) - C,(u)11 Q E for all u E k! Now given E > 0, there exists R > 0 such that g(x)2 dx < c2 s B’(R) and

\b(x,0) - b(x, JU(X))~ < EIJU(X)IJ’for almost

all x E B’(R)

where B’(R) = ix E R”: jxj 3 R}. Hence Ib(x, Ju(x))12 dx < 2~~ + 2z2 1Jb4-4)~2p dx s B’(R) s B’(R) < 2E2 + 2E211Jull;; < 2E2 + 2&++4(zzq where c1denotes

the norm

of J: Wz(R”) + [L?p(Rn)]n+l.

I. MASSABO AND C. A. STUART

42

Let CE: W$R”) + I?(R”) be defined by

It follows easily from the compactness of the embeddings of Sobolev spaces over bounded in R” that C,: W:(R”) + L?(R”) is compact. Furthermore,

I/B(n) - CE(U)I12=

Ib(x, Jn(x))(* dx s B’(R) Q.E.D.

d 2s2(1 + .l(ull$),. Definition. such that

A solution

of the differential

domains

equation

(3.1) is defined

to be a pair (u, A) E P, x R,

su = I{A(u) + B(u)}. Let 9’ be the metric space of all solutions endowed with the metric (0,O) E 9’ and we denote by V the component of Y containing (0,O).

of W;(R”) x R,.

Clearly

THEOREM 3.3. If { (IuII(~):(u, A) E %?} # R, then sup{1:(u,A)~G?}

This result follows directly

from Theorem

> Q/k

ifk > 0

=cO

ifk = 0.

2.1.

4. LINEAR

OPERATORS

Let X be a real Banach space and P be a cone in X. We suppose that the norm in X is monotone with respect to the ordering introduced by P, (u > u if and only if u - u E P). THEOREM 4.1. Let T :X + X be a k-set contraction which is homogeneous of degree 1 (i.e. T(tx) t T(x) for all x E X and t > 0) and monotone (i.e. T(u) 2 T(u) if u 2 u > 0). Suppose that there elist u E P - P, p E N and c > kP such that TPu B cu and -u $ P. Then there exist x0 E P\(O) and p E (0, c- ‘lp] such that x0 = pTx,. Remark. These

is homogeneous

hypotheses imply and monotone.

that

T(P) c P. If T :X -+ X is linear

Proof We consider the Banach space X x R equipped \A\}. Then 9 = P x [0, co). IS a cone in X x R and the norm to 9. Since u E P - P, there exist V, w E P such that u = u u/n), 1) for x E P, I > 0. It is easily seen that An: 9 + 9’ is a

and T(P) c P, then

T

with the norm I/x, AI/ = max{ 11x1/, in X x R is monotone with respect w. For n E N, let A,,(x) 1,): = (T(x + k-set contraction.

Positive

eigenvectors

43

of /c-set contractions

Let c1, = %k-’ =c-l/P

+ c-l/P)

ifk > 0

+I

ifk = 0

and let R = {(x, 2) E S: I/(X,,I)[1< dk}. A,(x) A)I/ :(x, A) E LJ!~} 2 1 and max{ [1(x,n)ll :(x, J.) E an} = d,. Then min{ 11 and so according to Theorem 1.1, there exist (x,, An) E 8Q and pn > 0 such that (X”’2”) = &4Xn’ A”) = @(X”

Hence

kd, < 1

+ r/n), 1).

Therefore, A,, = p,, and x,, = A,, = i,,T (x,, + v/n). But T(x, + v/n) B T(u/n) = (l/n)T(u)

> (l/n)T(u)

and T(x, + u/n) > T(x,). Hence x,, B (l/n)AnT(u) The second inequality

implies

and xn 3 il”T(x,).

that x,, > iyTm(x,,)

If Y 2 0 and x,, > V(a), 2 lp-’ cyu. By induction, n

for all m E N.

then TPel(xn) 2 yTP(u) 2 cyu and consequently it is now easy to see that x,, > (l/n) (AEc)“u

xn 2 l;-lTP-l(~n)

for all m E N.

Hence n(,l;c)-mxn - u E P for all m E N. If n;c > 1, this implies that -u E P, contradicting our hypothesis. Thus we conclude that I,, E [0, c- llp] c [0, dJ. Since (x,, A,,)E 33 this implies that llxJ/ = dk for all n E N and by passing to a subsequence if necessary we may assume that co. A” -+ p < c-lp asn+ Let u({x,}) denote the measure of non-compactness in X of the corresponding sub-sequence {x,1. Then a({Q(x,)1) G pk4{x,J) and a({(~ - &,)W,)>)) = 0. Since 4(x,>) < a(WW,,)}) + a({(~ - &,)T(x,,)}) and pk < c- ‘lp k < 1, this implies that a({~,}) = 0 and so {x,} contains a subsequence converging to an element x0. It is easily seen that x0 = pT(x,), x,, E P and l/xOII = dk >O. This completes the proof. Remark. 1. If T is defined only on P rather than on all of X, the result remains course r4E P. 2. As the following example shows, the inequality c > kPcannot be weakened. Example. Let X and P be as in the example

in Section

valid provided

1. Let T: X + X be defined

of

by

7(x): = (x,, X1’ x2, $3 X4’. . .I where x = (xl, x2, x3, x 4,. . .). Clearly

T is a linear

l-set contraction,

T(P) c P and X = P - P.

44

I. MASSABO AND C. A. STUART

Let u EP be u = (1, 1, 0, 0, . . .). Then TPu 2 u for all p E N. Hence for all p E N there exist u E P - P and c 2 kP such that TPu 2 cu and - u 4 P. However, it is easy to check that T has no eigenvalues.

REFERENCES 1. KRASNOSELSKIIM. A., Topological Methods in the Theory of Non-linear Integral Equations. McMillan, New York (1964). 2. BIRKHOFF G. D. & KELL~C 0. D., Invariant points in function spaces, Trans. Am. math. Sot. 23,96-115 (1922). values of non-linear operators, Atti. Accad. Naz. Lincei, Rend. A. Sci. Fis. Mat. Natur. 3. REICH S., Characteristic 50 (8), 682-685 (1971). de1 Dipartimento di 4. FURI M. & V~GNOLI A., On surjectivity for non linear maps in Banach spaces, Pubblicazioni Matematic, Universita della Calabria, Cosenza (1975). operators with 5. PETRYSHY~ W. V. & FITZPATRICK P. M., Positive eigenvalues for non-linear multivalued noncompact application to differential operators, .I. dif$ Eqns 22, 428441 (1976). Pacif: .I. Math. 9, 847-860 (1959). 6. SCHAEFER H. H.. On non linear positive operators, 7. RABINOWITZ P. k., Some global results fdr non linear e”igenvalue problems, .I. >inc; Analysis 7, 487-5 13 (1971). 8. DANCER E. N., Global solution branches for positive maps, Archs ration. mech. Analysis 52, 181-192 (1973). 9. AMANN H., Fixed point equations and non linear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (4), 621-709 (1976). 10. STUART C. A., The principal component of solutions for non-linear eigenvalue problems, Boll. (in. mat. Ital. 13B (5), 726-742 (1976). elliptic eigenvalue problems, R. Circ. mat. Palermo 20, 113-138 11. KUIPER H. J., On positive solutions of non-linear (1971). equations in ordered Banach spaces, .I. funct. Analysis 11, 12. AMANN H., On the number of solutions of non-linear 346-384 (1972). 13. WHYBURN G. T., Topological Analysis. Princeton Univ. Press, Princeton, N.Y. (1958). theory for k-set contractions, Proc. Land. math. Sot. 27 (3), 531-550 (1973) 14. STUART C. A., Some bifurcation Berlin, Heidelberg, N.Y. (1966). 15. KATO T.. Perturbation Theorv fbr Linear Operators. Springer-Verlaa. theory of elliptic differential operators 1, Math. Annln-142, 22-130 (1961). 16. BROWDER F. E., On the spe&l 17. MARTIN R. H., Non-linear operators and dqjbrential equations in Banach spaces. Wiley, New York (1976). 18. KRIEN M. G. & RUTMAN M. A., Linear operators leaving a cone invariant in a Banach space, Usp. mat. Nauk 23, 3 (1948). positive operators, Proc. R. Sot. Lond .4 328, 19. EDMUNDS D. A., POTTER A. J. B. & STUART C. A., Non-compact 67-81 (1972).