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DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SUPPLEMENT 2007

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EIGENVALUES OF HOMOGENEOUS GRADIENT MAPPINGS IN HILBERT SPACE AND THE BIRKHOFF-KELLOGG THEOREM

Raffaele Chiappinelli Dipartimento di Scienze Matematiche ed Informatiche Pian dei Mantellini 44 53100 Siena, Italia

Abstract. It is well known that any (nontrivial) linear compact self-adjoint operator acting in a Hilbert space possesses at least one non-zero eigenvalue. We present a generalization of this to nonlinear mappings as in the title, and discuss the relations of our results with the Birkhoff-Kellogg Theorem on one side, and with the spectral properties of self-adjoint operators on the other.

1. Introduction and statement of the results. Let E be a real, infinitedimensional Banach space and let F : E → E be a (possibly) nonlinear operator such that F (0) = 0. We suppose moreover that F is continuous and bounded (in the sense that it maps bounded sets onto bounded sets), and shall keep these hypotheses fixed through the paper without further mention. As for linear operators, we say that λ0 ∈ R is an eigenvalue of F if there exists x ∈ E, x 6= 0 such that F (x) = λ0 x; (1) in this case x is said to be an eigenvector associated with λ0 . This definition has perhaps little meaning in general, but is quite sensible when F is positively homogeneous , i.e., such that F (tx) = tF (x) (t > 0, x ∈ E); indeed in this case, if x0 is an eigenvector corresponding to λ0 , then so does every point on the ray {tx0 : t > 0}. It is thus clear that, for a positively homogeneous operator F acting in E, it is enough to consider eigenvectors of F belonging to the unit sphere S of E, S = {x ∈ E : kxk = 1} and that more generally, the ”global” properties of F are determined by its behaviour on S. Conversely, if f is a (continuous, bounded) map defined on S, then the map F : E → E defined by ( kxkf (x/kxk), x 6= 0 F (x) = (2) 0, x=0 is (continuous, bounded and) positively homogeneous. Thus, positively homogeneous mappings can be identified with mappings defined on the unit sphere, and 2000 Mathematics Subject Classification. Primary: 47J10; Secondary: 47J10. Key words and phrases. self-adjoint operators, Palais-Smale condition.

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statements which are usually given for the latter - such as the Birkhoff-Kellogg Theorem mentioned below - can be equivalently formulated for their positively homogeneous extensions defined in (2): see Theorem 1 below for example. The study of the eigenvalues and, more generally, of the spectral properties of positively homogeneous operators is of interest both in the theory (see, for instance, [2]) and in the applications to boundary value problems for semilinear elliptic and Sturm-Liouville operators: see for instance [4], [18] and the references therein, also in connection with the renowned ”Fuˇcik spectrum” of such operators. In this context, the eigenvalues are often called half-eigenvalues (see the Introduction to [4] for a historical account of this terminology), while our definition corresponds to that of classical eigenvalue - and, due to positive homogeneity, to that of connected eigenvalue - given in [1], Ch. 10. The scope of the present paper is to prove a general result on the existence of a (maximal) eigenvalue for positively homogeneous gradient operators acting in a Hilbert space: this is stated as Theorem 2 in this Section and is proved in Section 2. As we shall explain in detail, it generalizes a well known property of linear selfadjoint operators; on the other hand, it seems to have interesting connections with the famous Birkhoff-Kellogg Theorem [5]. The latter is probably one of the oldest results concerning the existence of eigenvalues for a nonlinear operator (see Remark 1.1), and in particular of eigenvalues different from 0 . To motivate the interest in such eigenvalues, just think of the distiguished role played by them when F is linear and compact, i.e., such that F (A) is relatively compact whenever A ⊂ E is bounded. Surprisingly, this special role is to some extent preserved in the spectral theory for nonlinear operators [1]: for instance, if F is positively homogeneous, compact and odd - i.e., such that F (−x) = −F (x) for all x ∈ E - then we have (precisely as for a linear F ) σ(F ) = {0} ∪ σp (F ) \ {0} (3) where σ(F ) denotes the spectrum of F and σp (F ) its point spectrum (that is, σp (F ) = {λ ∈ R : λ is an eigenvalue of F }): see for instance [1], Th. 9.12 for the equality (3) under the various definitions of σ(F ) existing in the literature. Now the question is: does σp (F ) \ {0} contain at least one point? Before stating some precise results about that, we recall one more definition. If A is a bounded subset of E, let α(A) denote the (Kuratowski) measure of noncompactness of A, that is, α(A) = inf{ > 0 : A can be covered by finitely many subsets of diameter ≤ }. We say that F : E → E is α−Lipschitz if α(F (A)) ≤ kα(A) for some k ≥ 0 and all bounded subsets A of E; in this case we put α(F ) = inf{k ≥ 0 : α(F (A)) ≤ kα(A) for all bounded A ⊂ E} (4) and note at once that α(F ) = 0 if and only if F is compact. W. Feng proved in [10], Th. 6.3, a generalization of the classical Birkhoff-Kellogg Theorem [5]. In our notations and terminology, her result reads as follows: Theorem 1. Let F be a positively homogeneous, α-Lipschitz mapping of E into itself, and suppose that m(F ) ≡ inf kF (x)k > α(F ). x∈S

(5)

Then F has a positive and a negative eigenvalue. Remark 1.1. The Birkhoff-Kellogg Theorem is usually formulated for mappings defined on the unit sphere S (see the above remarks). Its classical version (see, for

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instance, Riedrich [17] also for the terminology) states the existence of ”invariant directions” on S for a compact map - i.e., for the case that α(F ) = 0 in (5) - and is related to fundamental topological tools such as the Schauder Fixed Point Theorem [19]. The original work [5] has stimulated all along the past century a large amount of research on the subject, see for instance the Introduction to the paper by Riedrich [17] for the state of the art until the Sixties and the Notes to Ch. 10 of the book by Appell, De Pascale and Vignoli [1] for more updated information. In particular, for compact maps Furi, Martelli and Vignoli [11] gave a proof of Theorem 1 based on their spectral theory for nonlinear operators; Feng [10] used similar arguments to include α−Lipschitz operators. A different proof of Theorem 1, based on homotopy theory for k-epi operators, can be found in [1], Th. 10.3. A version of Theorem 1 for A−proper mappings was given by Infante and Webb in [12]. Finally, it has to be said that the Birkhoff-Kellogg Theorem holds more generally with the the sphere S replaced by the boundary of an open bounded subset Ω of E with 0 ∈ Ω: see for instance [13], Ch. 4, Th. 1.1. Remark 1.2. Evidently, (5) is a rather strong condition, implying in particular that F vanishes nowhere on S. Moreover, Theorem 1 is a ”strongly nonlinear” result, in the sense that that (5) cannot be satisfied by a linear operator: indeed for a linear F , the opposite inequality m(F ) ≤ α(F )

(6)

takes place, see for instance Prop. 3.2.1 in [11]. This ”linear obstruction” to the Birkhoff-Kellogg Theorem has been already observed by Appell, De Pascale and Vignoli for the special case that F be compact (see the comments following Th. 10.2 in [1]): indeed for such an operator, condition (5) becomes m(F ) > 0 and would imply that F has a bounded inverse, which is impossible in infinite-dimensional spaces. It is our aim in this note to complement Theorem 1 with a different statement, which holds in the restricted context of Hilbert spaces and for a more restricted class of mappings, but: i) on the one hand, it weakens assumption (5) and in particular the requirement that F should vanish nowhere on S; ii) on the other hand, the class of mappings to which it applies is not too restricted, as it includes the linear self-adjoint operators. Thus let H be a (real, infinite-dimensional) Hilbert space with scalar product denoted (.,.). If f is a (Fr´echet) differentiable functional defined on H, we let f 0 (x) denote the derivative of f at the point x ∈ H, which is a bounded linear functional on H, and with ∇f (x) the unique vector of H corresponding to f 0 (x) via the Riesz representation Theorem: thus by definition, f 0 (x)y = (∇f (x), y) for all x, y ∈ H. A mapping F : H → H is said to be a gradient (or potential ) operator if there exists a differentiable functional f on H (the potential of F ) such that F = ∇f , i.e., (F (x), y) = f 0 (x)y

∀x, y ∈ H.

(7)

We then have: Theorem 2. Let F be a positively homogeneous, α- Lipschitz gradient mapping of H into itself, and suppose that r(F ) ≡ sup |(F (x), x)| > α(F ). x∈S

(8)

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Then F has an eigenvalue λ0 with |λ0 | = r(F ). Moreover, |λ0 | ≥ |λ| for any possible eigenvalue λ of F . Remark 1.3. The maximality property of λ0 follows immediately from its existence, because if F (x) = λx for some λ ∈ R and some x ∈ S, then λ = (F (x), x) and so |λ| ≤ r(F ). The proof of Theorem 2 is based on variational methods, and consists in showing that under the given assumptions either the supremum or the infimum on S of the functional (F (x), x) must be attained, thus giving rise to an eigenvalue of F by the Lagrange multiplier’s rule. It exploits the crucial fact that for |c| > α(F ), F − cI (I the identity map) is proper on closed bounded sets : that is, given any closed bounded set M of E, M ∩ (F − cI)−1 (K) = {x ∈ M : F (x) − cx ∈ K} is compact whenever K ⊂ E is compact. We postpone the proof of Theorem 2 to Section 2 (after the presentation of some preliminaries on Critical Point Theory) and give here some comments, illustrating the points i) and ii) listed above. First, our condition (8) weakens (5). Indeed, (5) ensures - by Theorem 1 - that there exists some λ ∈ R and some x ∈ S so that F (x) = λx. Therefore, |(F (x), x)| = kF (x)k(= λ), which implies (by the definitions of r(F ) and m(F )) that r(F ) ≥ m(F )

(9)

and shows that (5) is a stronger condition that (8). Open problem 1: Does any gradient operator F satisfy the inequality (9) independently from the condition (5)? (This is true if in addition F is linear, see equation (12) below). As we have just seen, for gradient operators Theorem 2 weakens the assumptions of Theorem 1. On the other hand, the latter yields the existence only of one eigenvalue (rather than two as in the former). Open problem 2: Is there any relation between these eigenvalues, other than the bound given in Remark 1.3? The weakening of condition (5) becomes more apparent for compact operators. Indeed as already noted, Theorem 1 requires in particular that F (x) 6= 0 at each point x ∈ S. Condition (8) instead - which becomes r(F ) > 0 for F compact - is satisfied as soon as (F (x), x) 6= 0 at just one point of S, and this in turn is the same as requiring that F 6= 0. In fact, if a positively homogeneous gradient mapping F satisfies the condition Q(x) ≡ (F (x), x) = 0 for all x ∈ S (and thus for all x ∈ H by homogeneity), then it is identically zero: see Section 2 and in particular equation (19), which roughly speaking says that 2F is the derivative of Q (whence F is identically zero if so is Q). Let us give a formal statement for these last remarks: Proposition 1. Let F be a positively homogeneous, compact, gradient operator acting in a Hilbert space. Then if F 6= 0, F has at least one non-zero eigenvalue.

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Example 1.1. Let Ω be a bounded open set in RN (N ≥ 1) and let H = H01 (Ω) be the first Sobolev space on Ω. Consider the semilinear elliptic eigenvalue problem u ∈ H01 (Ω)

Lu = µg(x, u),

(10)

where L is a uniformly elliptic formally selfadjoint operator, Lu := −

N X

∂u ∂ (aij (x) ) + a0 (x)u ∂x ∂x j i i,j=1

with L∞ coefficients aij = aji (i, j = 1, . . . , N ) and a0 , with a0 ≥ 0 a.e. in Ω. The nonlinearity is given by the ”piecewise linear” function g(x, s) = α(x)s+ − β(x)s− where s+ = max {s, 0}, s = s+ − s− and α, β ∈ L∞ (Ω) with α, β ≥ 0 a.e. in Ω. If we take as scalar product in H01 (Ω) the Dirichlet form of L, then (10) consists in finding u ∈ H, u 6= 0 and µ ∈ R such that u = µF (u), where F : H → H is defined by the rule Z (F (u), v) = (α(x)u+ v − β(x)u− v) dx Ω

for u, v ∈ H. It is straightforward to check (see for instance Chapter 2 of [3]) that F is a positively homogeneous, compact, gradient operator in H. Therefore it follows from Proposition 1 that - provided that α and β are not both zero a.e. in Ω - the problem (10) has a smallest positive eigenvalue µ1 , with Z (µ1 )−1 = sup(F (u), u) = sup [α(x)u+2 + β(x)u−2 ] dx. u∈S

u∈S

Ω

The Example just discussed generalizes a property of (10) which is well known (see for instance [7]) when α ≡ β (a.e. in Ω), in which case the problem is linear. Let us now show in general that our results extend similar ones holding for linear operators. To this scope, we need first recall (see, e.g., [3], Ch. 2, Sect. 5) that a linear bounded operator L acting in H is a gradient if and only if it is self-adjoint, that is, such that ∀x, y ∈ H.

(Lx, y) = (x, Ly)

(11)

Therefore, considering first Proposition 1, we see that it generalizes the well known property of (nontrivial) compact self-adjoint operators of having at least one non-zero eigenvalue (see, e.g., [23], Th. 6.4-A). Also, it is useful to recall here that for any linear bounded self-adjoint operator L, we have the equality (see, for instance, Th. 6.11-C in [23]) r(L) = kLk ≡ sup kF (x)k.

(12)

x∈S

Thus for linear operators, our Theorem 2 amounts to replacing the ”inf” on S in condition (5) of the Birkhoff-Kellogg Theorem with the ”sup” on S. Also note that in particular, (12) reminds us that L ≡ 0 if its quadratic form Q0 (x) ≡ (Lx, x) is identically zero, and thus yields the linear version of what we have seen above to be a property of any gradient operator.

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265

Let us remain into the larger class of linear bounded self-adjoint (not necessarily compact) operators and let us consider the significance of condition (8) in this case. It is known from papers of Nussbaum [15] and Stuart [21] that for such an operator L, its measure of noncompactness α(L) coincides with re (L), the radius of the essential spectrum, σe (L), of L: σe (L) = {λ ∈ R : L − λI is not a Fredholm operator}. (13) Theorem 2 thus says that if |Q0 (x)| = |(Lx, x)| > re (L) at some x ∈ S, then L has at least one eigenvalue outside the essential spectrum (and therefore necessarily isolated and of finite multiplicity: see, e.g., [9], Ch. IX, Th. 1.6). This is another known fact from linear spectral theory (see, e.g., [9], Ch. XI, Th. 1.2) which in fact generalizes what was said before for compact operators, since re (L) = 0 if L is compact. 2. Proof of the main result. To prove Theorem 2, we start noting that r(F ) = max{|p(F )|, |q(F )|}, where p(F ) = inf (F (x), x)

q(F ) = sup(F (x), x)

x∈S

(14)

x∈S

and the assumption r(F ) > α(F ) is equivalent to either q > α(F ) or p < −α(F ) (or both); we put here and in the sequel p = p(F ), q = q(F ). Suppose for instance q > α(F ). Our claim is that there exists a normalized eigenvector of F corresponding to q, that is, an x0 ∈ S such that F (x0 ) = qx0 . As already remarked, (15) implies that

(15)

(F (x0 ), x0 ) = q. Now if we let Q denote the pseudo-quadratic form of F , that is,

(16)

Q(x) = (F (x), x) (x ∈ H) (17) then, in the language of Critical Point Theory, conditions (15),(16) express the fact that q is a critical value of Q on S, corresponding to the critical point x0 . To make this clear, we postpone for a while the proof of Theorem 2 and recall some definitions and concepts which are basic in this context (standard references are, for instance, [3], [8], [13], [14], [16], [20]); the reader experienced with Critical Point Theory is invited to jump directly to Theorem 3 below. Let f be a C 1 functional defined on H. Then x0 ∈ H is a (”free”) critical point of f if f 0 (x0 ) = 0, while given a manifold M = {x ∈ H : g(x) = R}, i.e. the level set of another C 1 functional g : H → R, x0 ∈ M is a (”constrained”) critical point of f on M if f 0 (x0 ) = λg 0 (x0 ) for some ”Lagrange multiplier” λ ∈ R. (This means that f 0 (x0 ) vanishes on the tangent space to M at x0 , or in other words that x0 is a critical point of the restriction fM of f to M ). Finally a real number c is said to be a critical value of f (resp. of f on M ) if c = f (x0 ) for some critical point x0 of f (resp. of f on M ). Now suppose F = ∇f . Assuming - as we do - F continuous, i.e. f of class C 1 , F and f are related by the formula (see, e.g., [3], Ch. II, Section 2) Z f (x) =

1

(F (tx), x) dt 0

(18)

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which obviously reduces to f (x) = (Lx, x)/2 when F = L is linear. The key point is that the same simple expression for the potential f of F still holds if F is merely assumed to be positively homogeneous. Thus if Q is the pseudo-quadratic form of F defined in (17), we have Q(x) = 2f (x) and therefore Q0 (x)y = 2f 0 (x)y = 2(F (x), y) ∀x, y ∈ H. (19) On the basis of these facts, we can now state precisely the criterion mentioned above for the existence of normalized eigenvectors of a gradient operator which is positively homogeneous. Proposition 2. Let F : H → H be gradient and positively homogeneous, and let Q(x) = (F (x), x). Let x0 ∈ S = {x ∈ H : kxk = 1} and λ0 ∈ R. Then F (x0 ) = λ0 x0 if and only if x0 is a critical point of Q on S and λ0 is the corresponding critical value. Proof. Suppose first that x0 is a critical point of Q on S, and let λ0 = Q(x0 ). As S is a level set of the functional N (x) = kxk2 , we have by definition Q0 (x0 )y = λN 0 (x0 )y for some λ ∈ R. Using (19), this yields

∀y ∈ H

2(F (x0 ), y) = 2λ(x0 , y) ∀y ∈ H, whence F (x0 ) = λx0 . In turn, this yields λ = (F (x0 ), x0 ) = Q(x0 ) = λ0 , and thus proves the reverse implication in the above statement. The proof of the direct implication is similar. Proposition 2 characterizes the eigenvalues of F as the critical values of the restriction QS of Q to S (and moreover, the eigenvectors associated with a given eigenvalue as the critical points where that value is attained). Now if we look at a general C 1 functional f defined on H, its simplest critical values on S are the absolute minimum and maximum on S. Of course, one has to make sure that these are attained: this requires some further assumption - compensating for the lack of compactness of S due to the infinite-dimensional character of H - such as the famous Palais-Smale condition that we now recall (see, for instance, [16], Ch. 1). For x0 ∈ S, let Tx0 (S) = {y ∈ H : (x0 , y) = 0} denote the tangent space to S at x0 ; then for y ∈ H, y−(x0 , y)x0 is the orthogonal projection of y onto Tx0 (S), that is, the tangential component of y to S at x0 . Now define a bounded linear form fS0 (x0 ) on H via the formula fS0 (x0 )y = f 0 (x0 )[y − (x0 , y)x0 ] (y ∈ H). Thus represents (for each x0 ∈ S) the restriction of f 0 (x0 ) to Tx0 (S); essentially, it is the derivative at x0 of fS viewed as a differentiable map on the manifold S, which justifies the notation. fS0 (x0 )

Definition 1. Let f be a C 1 functional on H and let fS denote its restriction to S. fS is said to satisfy the Palais-Smale ((P S)) condition at level c ∈ R ((P S)c for short) if any sequence (xn ) ⊂ S such that f (xn ) → c and fS0 (xn ) → 0 contains a convergent subsequence. We say that fS satisfies the Palais-Smale condition if it satisfies (P S)c for any c ∈ R.

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Before going further, it is useful to note that fS0 (x0 )y =f 0 (x0 )y − (x0 , y)f 0 (x0 )x0 =(∇f (x0 ), y) − (∇f (x0 ), x0 )(x0 , y) ≡ (∇fS (x0 ), y) with ∇fS (x0 ) = ∇f (x0 ) − (∇f (x0 ), x0 )x0 .

(20)

The above equations tell us that the vector ∇fS (x0 ) corresponding to the linear form fS0 (x0 ) is precisely the tangential component to S of ∇f (x0 ). The importance of the (P S) condition can be seen from the following simple result about the minimization (or maximization) of a functional on the sphere. Theorem 3. Let f : H → R be a C 1 functional which is bounded from below on the sphere S, and let m = inf f (x). x∈S

Then if fS satisfies (P S)m , m (is attained and) is a critical value of f . A similar statement holds if f is bounded from above on S, with M = supx∈S f (x). Proof. For the proof of Theorem 3, see e.g. [8], Th. 4.7 or [16], Th. 2.7. We are now in a position to prove Theorem 2. To do this, we merely apply Theorem 3 to the functional Q related to F via (17). Indeed recall that the hypotheses of Theorem 2 allow us to assume - as explained at the beginning of this Section that q ≡ supx∈S Q(x) > α(F ). Now, by virtue of Theorem 3, to guarantee that q is attained - and so is an eigenvalue of F by virtue of Proposition 2 - we need only verify that QS satisfies the Palais- Smale condition at level q. Indeed, our final result shows that a much more general property holds. Proposition 3. Let F satisfy the assumptions of Theorem 2. Then QS satisfies (P S)c for each c ∈ R with |c| > α(F ). Proof. Let (xn ) ⊂ S be such that Q(xn ) → c and Q0S (xn ) → 0. We have ∇Q = 2F by (19) and therefore by (20) ∇QS (x) = 2[F (x) − Q(x)x]

(x ∈ S).

The requirement that Q0S (xn ) → 0 is equivalent to ∇QS (xn ) → 0, that is, F (xn ) − Q(xn )xn → 0.

(21)

Rewrite (21) as F (xn ) − cxn − [Q(xn )xn − cxn ] → 0, which implies (since Q(xn ) → c by assumption) that F (xn ) − cxn → 0.

(22)

Now if |c| > α(F ), F − cI is proper on closed bounded sets (see e.g. [11], Prop. 3.1.3) and therefore (xn ) contains a convergent subsequence, as required.

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Remark 2.1. For a (bounded or unbounded) linear self-adjoint operator, the relation between its essential spectrum and the Palais-Smale condition of its quadratic form has been thoroughly investigated by Stuart [22]. In particular, for a bounded linear self-adjoint L he obtained a beautiful ”variational characterization” of σe (L), namely (in our notations) σe (L) = {c ∈ R : QS does not satisfy (P S)c }. (23) Some information about the Palais-Smale condition for the pseudo-quadratic form of nonlinear homogeneous perturbations of bounded linear self-adjoint operators can be gathered from the recent paper [6] of this author. REFERENCES [1] J. Appell, E. De Pascale and A. Vignoli, “Nonlinear Spectral Theory,” de Gruyter, 2004. [2] J. Appell, E. Giorgieri and M. V¨ ath, Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589–618. [3] M. S. Berger, “Nonlinearity and Functional Analysis,” Academic Press, 1977. [4] P. A. Binding and B. P. Rynne, Half-eigenvalues of periodic Sturm-Liouville problems, J. Differential Equations, 206 (2004), 280–305. [5] G. D. Birkhoff and O. D. Kellogg, Invariant points in function space, Trans. Amer. Math. Soc., 23 (1922), 96–115. [6] R. Chiappinelli, Nonlinear homogeneous perturbation of the discrete spectrum of a self-adjoint operator and a new Constrained Saddle Point Theorem, J. Math. Anal. Appl., 318 (2006), 323–332. [7] D. G. De Figueiredo, Positive solutions of semilinear elliptic problems, Differential Equations, Proceedings of the 1st Latin American School, D.G. De Figueiredo and C.S. H¨ onig Eds., Lecture Notes in Mathematics, Springer, 957 (1982), 34–87. [8] D. G. De Figueiredo, “Lectures on the Ekeland variational principle with applications and detours,” Tata Institute of Fundamental Research, Bombay, 1989. [9] D. E. Edmunds and W.D. Evans, “Spectral Theory and Differential Operators,” Oxford University Press, 1987. [10] W. Feng, A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163–183. [11] M. Furi, M. Martelli and A. Vignoli, Contributions to the spectral theory for nonlinear operators in Banach spaces, Ann. Mat. Pura Appl., 118 (1978), 229–294. [12] G. Infante and J. R. L. Webb, A finite dimensional approach to nonlinear spectral theory, Nonlinear Anal. TMA, 51 (2002), 171–188. [13] M. A. Krasnoselskii, “Topological Methods in the Theory of Nonlinear Integral Equations,” Macmillan, 1964. [14] J. Mawhin and M. Willem, “Critical Point Theory and Hamiltonian Systems,” Springer, 1989. [15] R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473–478. [16] P. Rabinowitz, Minimax Methods in Critical point Theory with applications to Differential Equations, CBMS Regional Conference Series Math. Vol. 65, Amer. Math. Soc., 1986. [17] T. Riedrich, Das Birkhoff-Kellogg-Theorem f¨ ur lokal radial beschr¨ ankte R¨ aume, Math. Ann., 166 (1966), 264–276. [18] B. P. Rynne, Half-eigenvalues of elliptic operators, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1439–1451. [19] J. Schauder, Der Fixpunktsatz in Funktionalra¨ umen, Studia Math., 2 (1930), 171–180. [20] M. Struwe, “Variational Methods,” Springer, 1996. [21] C. A. Stuart, Some bifurcation theory for k−set contractions, Proc. London Math. Soc., 27 (1973), 532–550. [22] C. A. Stuart, Spectrum of a self-adjoint operator and Palais-Smale conditions, J. London Math. Soc., 61 (2000), 581–592. [23] A. E. Taylor, “Introduction to Functional Analysis,” J. Wiley, New York, 1958.

Received July 2006; revised April 2007. E-mail address: [email protected]