A degree associated to linear eigenvalue problems in Hilbert spaces and applications to nonlinear spectral theory
Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera
aa r X i v : . [ m a t h . SP ] J un A DEGREE ASSOCIATED TO LINEAR EIGENVALUEPROBLEMS IN HILBERT SPACES AND APPLICATIONS TONONLINEAR SPECTRAL THEORY
PIERLUIGI BENEVIERI, ALESSANDRO CALAMAI, MASSIMO FURI,AND MARIA PATRIZIA PERA
Dedicated to the memory of our friend andoutstanding mathematician Russell Johnson
Abstract.
We extend to the infinite dimensional context the link between twocompletely different topics recently highlighted by the authors: the classicaleigenvalue problem for real square matrices and the Brouwer degree for mapsbetween oriented finite dimensional real manifolds. Thanks to this extension,we solve a conjecture regarding global continuation in nonlinear spectral theorythat we have formulated in a recent article. Our result (the ex conjecture)is applied to prove a Rabinowitz type global continuation property of thesolutions to a perturbed motion equation containing an air resistance frictionalforce. Introduction
Given a linear operator L : R k → R k , consider the classical eigenvalue problem(1.1) ( Lx = λx,x ∈ S , where S is the unit sphere of R k and λ ∈ R . The solutions of (1.1) are pairs( λ, x ) ∈ R × S, hereafter called eigenpoints , where λ is a real eigenvalue of L and x is one of the corresponding unit eigenvectors. Since the eigenpoints of (1.1) are thezeros of the C ∞ -map Ψ : R × S → R k , ( λ, x ) Lx − λx, in [8] we have shown that there is a link between the above purely algebraic problemand the Brouwer degree, deg(Ψ , U, ∈ R k on convenient opensubsets U of the cylinder R × S, which is a smooth k -dimensional real manifold witha natural orientation.Roughly speaking, in [8] we have shown that Date : 30th June 2020.1991
Mathematics Subject Classification.
Key words and phrases. eigenvalues, eigenvectors, nonlinear spectral theory, topological de-gree, bifurcation, differential equations.The first, second and fourth authors are members of the Gruppo Nazionale per l’Analisi Matem-atica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matem-atica (INdAM).A. Calamai is partially supported by GNAMPA - INdAM (Italy). • if λ ∗ ∈ R is a simple eigenvalue of L , and x ∗ and − x ∗ are the two cor-responding unit eigenvectors, then the “twin” eigenpoints p ∗ = ( λ ∗ , x ∗ ) and ¯ p ∗ = ( λ ∗ , − x ∗ ) are isolated zeros of Ψ and give the same contribution tothe Brouwer degree, which is either or − , depending on the sign jump at λ ∗ of the (real) characteristic polynomial of L . Still roughly speaking, here we extend this fact to the infinite dimensional caseby considering a problem of the type(1.2) ( Lx = λCx,x ∈ S , in which λ ∈ R , L and C are bounded linear operators acting between two realHilbert spaces G and H , C is compact, L − λC is invertible for some λ ∈ R , and Sis the unit sphere of the source space G . In this infinite dimensional context, thedegree is the one introduced in [9] for oriented C Fredholm maps of index zerobetween real differentiable Banach manifolds, which extends the Brouwer degree formaps between oriented finite dimensional manifolds, as well as the Leray-Schauderdegree for C compact vector fields. In this case the degree regards the mapΨ : R × S → H, ( λ, x ) Lx − λCx, acting between the 1-codimensional submanifold R × S of the Hilbert space R × G and the target space H , whose zeros are called the eigenpoints of (1.2). Theresult obtained here, Theorem 3.6 below, is similar to the one in [8] for the finitedimensional case, provided that one calls λ ∗ ∈ R a simple eigenvalue of (1.2) ifthere exists x ∗ ∈ S such that Ker( L − λ ∗ C ) = R x ∗ and H = R x ∗ ⊕ Img( L − λ ∗ C ).In fact, we obtain that • if λ ∗ is a simple eigenvalue of (1.2) and x ∗ and − x ∗ are the two corres-ponding unit eigenvectors, then the “twin” eigenpoints p ∗ = ( λ ∗ , x ∗ ) and ¯ p ∗ = ( λ ∗ , − x ∗ ) are isolated zeros of Ψ and give the same contribution tothe degree, which is either or − , depending on the orientation of Ψ . As in [8], this crucial result regarding the “fair contribution to the degree of thetwin eigenpoints” is applied to the study of the behaviour of the solution triples( s, λ, x ) of the following perturbation of (1.2):(1.3) ( Lx + sN ( x ) = λCx,x ∈ S , in which N : S → H is a compact C -map. Precisely, if we denote by Σ the subsetof R × R × S of the solutions ( s, λ, x ) of (1.3) and we call trivial those having s = 0,our main result, Theorem 4.5 below, yields the following Rabinowitz type globalcontinuation result that was conjectured in [7]. • If q ∗ = (0 , λ ∗ , x ∗ ) is a trivial solution of (1.3) corresponding to a simpleeigenvalue λ ∗ of the unperturbed problem (1.2) , then the connected compon-ent of Σ containing q ∗ is either unbounded or encounters a trivial solution q ∗ = (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ . Our global continuation result, Theorem 4.5, falls into the subject of nonlinearspectral theory , which finds applications to differential equations (see e.g. [3,16] andreferences therein).
DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 3
Here we mention the work of R. Chiappinelli [14], which inspired some of our re-cent articles. In [14] a sort of “local persistence property” for a perturbed eigenvalueproblem similar to (1.3) was proved. Precisely, under the assumptions • L : G → G is a self-adjoint operator, • C = I is the identity of G , • N : S → G is Lipschitz continuous, • λ ∗ ∈ R is an isolated simple eigenvalue of L , • x ∗ ∈ S is an eigenvector corresponding to λ ∗ ,it was shown that • in a neighborhood V of ∈ R it is defined a Lipschitz continuous function ε ( λ ε , x ε ) ∈ R × S with the properties that ( λ , x ) = ( λ ∗ , x ∗ ) and that Lx ε + εN ( x ε ) = λ ε x ε for any ε ∈ V . When G is infinite dimensional, the hypotheses of our global continuation res-ult seem incompatible with the assumptions of Chiappinelli, due to the fact thatthe identity is not a compact operator. However, Theorem 4.5 does apply to theequation Lx + εN ( x ) = λx, provided that N is C and compact, and L is of the type λ ∗ I + C , with λ ∗ ∈ R and C compact. In fact, putting ε = − σ/µ and λ = λ ∗ + 1 /µ , the above equationbecomes x + σN ( x ) = µCx , which is as in our problem (1.3) with L = I .For results regarding the local persistence property when the eigenvalue λ ∗ isnot necessarily simple we mention [4–8, 15, 17–21].The last section of the paper contains three examples illustrating our main result,as well as an application to the study of the solutions ( s, λ, x ) of the boundary valueproblem(1.4) (cid:26) x ′′ ( t ) + sg ( x ′ ( t )) + λx ( t ) = 0 ,x (0) = 0 = x ( π ) , x ∈ S , in which S is the unit sphere of the Hilbert space H (0 , π ), and g : R → R is anodd increasing C -function (such as the air resistance force g ( v ) = v | v | ). From ourresult, with the help of the well-known notion of winding number of a self-map ofthe circle S , we deduce that, given any trivial solution q ∗ of (1.4), the connectedcomponent of Σ containing q ∗ is unbounded and does not encounter other trivialsolutions.For pioneering articles regarding the use of the winding number in order to studythe behavior of solutions to ordinary differential equations we mention [13, 22, 23].2. Notation and preliminaries
We introduce some notation, preliminaries, and known or unknown conceptsthat we will need in subsequent sections. In particular, we will outline the mainnotions related to the topological degree for oriented C Fredholm maps of indexzero between real differentiable Banach manifolds introduced in [9] (see also [10,11]for additional details). Actually, some notions and results are new: we considerthem necessary for a better understanding of the topics in Sections 3 and 4, theproof of Theorem 3.6 in particular.
P. BENEVIERI, A. CALAMAI, M. FURI, AND M.P. PERA
Algebraic preliminaries.
Let, hereafter, E and F be two real vector spaces.By L ( E, F ) we shall denote the vector space of the linear operators from E into F .The same notation will be used if, in addition, E and F are normed. In this case,however, we will tacitly assume that all the operators of L ( E, F ) are bounded, andthat this space is endowed with the usual operator norm. If E = F , we will write L ( E ) instead of L ( E, E ). By Iso(
E, F ) we shall mean the subset of L ( E, F ) of theinvertible operators, and we will write GL( E ) instead of Iso( E, E ). The subspaceof L ( E, F ) of the operators having finite dimensional image will be denoted by F ( E, F ), or simply by F ( E ) when E = F .Let I ∈ L ( E ) indicate the identity of E . If T ∈ L ( E ) has the property that I − T ∈ F ( E ), we shall say that T is an admissible operator (for the determinant) .The symbol A ( E ) will stand for the affine subspace of L ( E ) of the admissibleoperators.It is known (see [25]) that the determinant of an operator T ∈ A ( E ) is welldefined as follows: det T := det T | ˆ E , where T | ˆ E is the restriction (as domain and ascodomain) to any finite dimensional subspace ˆ E of E containing Img( I − T ), withthe understanding that det T | ˆ E = 1 if ˆ E = { } .As one can easily check, the function det : A ( E ) → R inherits most of the prop-erties of the classical determinant. Some of them are stated in the following Remark 2.1.
Let
T, T , T ∈ A ( E ) . Then • det T = 0 if and only if T is invertible; • R ∈ Iso(
E, F ) implies RT R − ∈ A ( F ) and det( RT R − ) = det T ; • T T ∈ A ( E ) and det( T T ) = det( T ) det( T ) . See, for example, [12] for a discussion about other properties of the determinant.We will need the following remark, whose easy proof is left to the reader:
Remark 2.2.
Let T ∈ L ( E ) and let E = E ⊕ E with dim E < + ∞ . Assumethat, with respect to the above decomposition, T can be represented in a block matrixform T = I T T ! , where I is the identity of E . Then T ∈ A ( E ) and det T = det T . Recall that an operator T ∈ L ( E, F ) is said to be (algebraic) Fredholm if itskernel, Ker T , and its cokernel, coKer T = F/T ( E ), are both finite dimensional.The index of a Fredholm operator T is the integerind T = dim(Ker T ) − dim(coKer T ) . In particular, any invertible linear operator is Fredholm of index zero. Observe alsothat, if T ∈ L ( R k , R s ), then ind T = k − s .The subset of L ( E, F ) of the Fredholm operators will be denoted by Φ(
E, F );while Φ n ( E, F ) will stand for the set { T ∈ Φ( E, F ) : ind T = n } . Obviously, Φ( E )and Φ n ( E ) designate, respectively, Φ( E, E ) and Φ n ( E, E ).One can easily check that A ( E ) is a subset of Φ ( E ). This is also a consequenceof a well known property regarding Fredholm operators. Namely,(1) if T ∈ Φ n ( E, F ) and K ∈ F ( E, F ) , then T + K ∈ Φ n ( E, F ) . DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 5
Another fundamental property states that(2) the composition of Fredholm operators is Fredholm and its index is the sumof the indices of all the composite operators.
An useful consequence of property (2) is the following: • If T ∈ Φ n ( E, F ) and k ∈ N , then the restriction of T to a k -codimensionalsubspace of E is Fredholm of index n − k .Let T ∈ Φ ( E, F ). In [9], an operator K ∈ F ( E, F ) was called a corrector of T if T + K is invertible. Since, during a conference, someone has critically observedthat it is not necessary to correct an invertible operator, hereafter we will use themore appropriate word companion instead of corrector .Notice that any T ∈ Iso(
E, F ) has a natural companion : the trivial element of L ( E, F ). This fact was crucial in [9] for the construction of the degree theory thatwe will apply here.Given T ∈ Φ ( E, F ), let us denote by C ( T ) the subset of F ( E, F ) of all thecompanions of T . As one can easily check, this set is nonempty. Moreover, C ( T )admits a partition in two equivalence classes according to the following Definition 2.3 (Equivalence relation) . Two companions K and K of an oper-ator T ∈ Φ ( E, F ) are equivalent (more precisely, T -equivalent) if the admissibleoperator ( T + K ) − ( T + K ) has positive determinant.Given two companions K and K of T ∈ Φ ( E, F ), the admissible automorph-ism ( T + K ) − ( T + K ) is not the unique one that can be used to check whetheror not K and K are equivalent. In fact, one has the following Remark 2.4.
Let T ∈ Φ ( E, F ) and K , K ∈ C ( T ) . Then, the determinants ofthe invertible operators ( T + K ) − ( T + K ) , ( T + K )( T + K ) − , ( T + K ) − ( T + K ) , ( T + K )( T + K ) − have the same sign. In fact, from the second property of Remark 2.1 one gets thatthe first two operators have the same determinant, while the third property impliesthe statement regarding the last two operators, being the inverses of the first two. Thanks to the above equivalence relation, in [9] it was introduced the following
Definition 2.5 (Orientation) . An orientation of T ∈ Φ ( E, F ) is one of the twoequivalence classes of C ( T ), denoted by C + ( T ) and called the class of positive com-panions of the oriented operator T . The set C − ( T ) = C ( T ) \ C + ( T ) of the negativecompanions is the opposite orientation of T .Some further definitions are in order. Definition 2.6 (Natural orientation) . Any T ∈ Iso(
E, F ) admits the natural ori-entation : the one given by considering the trivial operator of L ( E, F ) as a positivecompanion.
Definition 2.7 (Canonical orientation) . Any admissible operator T ∈ A ( E ) admitsthe canonical orientation : the one given by choosing as a positive companion any K ∈ F ( E ) such that det( T + K ) >
0. In particular, this applies for any T ∈ L ( E ),with dim E < ∞ . P. BENEVIERI, A. CALAMAI, M. FURI, AND M.P. PERA
Definition 2.8 (Associated orientation) . Let E and F have the same finite dimen-sion. Assume that they are oriented up to an inversion of both the orientations or,equivalently, assume that E × F has an orientation, say O . Then any T ∈ L ( E, F )admits the orientation associated with O : the one given by choosing as a positivecompanion any K ∈ F ( E, F ) such that T + K is orientation preserving. Definition 2.9 (Oriented composition) . The oriented composition of two orientedoperators, T ∈ Φ ( E , E ) and T ∈ Φ ( E , E ), is the operator T T with theorientation given by considering K = ( T + K )( T + K ) − T T as a positivecompanion, where K and K are positive companions of T and T , respectively.Observe that the oriented composition is associative. Indeed, if T ∈ Φ ( E , E ), T ∈ Φ ( E , E ) and T ∈ Φ ( E , E ), and K , K and K are, respectively, com-panions of T , T and T , one has (cid:0) ( T + K )( T + K ) (cid:1) ( T + K ) − (cid:0) T T (cid:1) T = ( T + K ) (cid:0) ( T + K )( T + K ) (cid:1) − T (cid:0) T T (cid:1) . The following result implies an important property of the oriented composition(see Corollary 2.13 below). Moreover, it shows that Definition 2.9 is well-posed.
Lemma 2.10.
Given T ∈ Φ ( E , E ) , T ∈ Φ ( E , E ) , K , K ′ ∈ C ( T ) and K , K ′ ∈ C ( T ) , consider the following companions of T T : K = ( T + K )( T + K ) − T T and K ′ = ( T + K ′ )( T + K ′ ) − T T . Then K is equivalent to K ′ if and only if K and K are both equivalent or bothnot equivalent to K ′ and K ′ , respectively.Proof. According to Definition 2.5, we need to compute the sign ofdet (cid:0) ( T T + K ) − ( T T + K ′ ) (cid:1) . We have( T T + K ) − ( T T + K ′ ) = (cid:0) ( T + K )( T + K ) (cid:1) − (cid:0) ( T + K ′ )( T + K ′ ) (cid:1) . Thus, because of the second property of Remark 2.1, we obtaindet (cid:0) ( T T + K ) − ( T T + K ′ ) (cid:1) = det (cid:16) ( T + K ′ ) (cid:0) ( T + K )( T + K ) (cid:1) − (cid:0) ( T + K ′ )( T + K ′ ) (cid:1) ( T + K ′ ) − (cid:17) = det (cid:0) ( T + K ′ )( T + K ) − ( T + K ) − ( T + K ′ ) (cid:1) . Therefore, applying the third property of the same remark, we getdet (cid:0) ( T T + K ) − ( T T + K ′ ) (cid:1) = det (cid:0) ( T + K ′ )( T + K ) − ) (cid:1) det (cid:0) ( T + K ) − ( T + K ′ ) (cid:1) , and the assertion follows. (cid:3) Definition 2.11 (Sign of an oriented operator) . Let T ∈ Φ ( E, F ) be an orientedoperator. Its sign is the integersign T = +1 if T is invertible and naturally oriented, − T is invertible and not naturally oriented,0 if T is not invertible. DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 7
As a straightforward consequence of Remark 2.4, and taking into account ofdefinitions 2.3, 2.5, 2.6, 2.11, we get the following
Remark 2.12.
Let T ∈ Iso(
E, F ) be oriented. Then, sign T = sign det (cid:0) ( T + K ) − T (cid:1) = sign det (cid:0) T ( T + K ) − (cid:1) = sign det (cid:0) T − ( T + K ) (cid:1) = sign det (cid:0) ( T + K ) T − (cid:1) , where K is a positive companion of T . Lemma 2.10 shows that, in the oriented composition, the inversion of the orient-ation of one (and only one) of the operators yields the inversion of the orientationof the composition. Hence, one gets the following
Corollary 2.13.
Let T ∈ Φ ( E , E ) and T ∈ Φ ( E , E ) be oriented. Then, sign( T T ) = sign T sign T , where T T is the oriented composition of T and T .Proof. If one of the two operators is not invertible, then the assertion is obvious.Assume, therefore, that T and T are isomorphisms. If both the operators arenaturally oriented, then the assertion follows from the definition of oriented com-position. The other cases are a consequence of Lemma 2.10. (cid:3) Given R ∈ Iso( E , F ) and R ∈ Iso( E , F ), observe that the functionΛ : L ( E , E ) → L ( F , F ) , T R T R − is a linear isomorphism (whose inverse is given by e T R − e T R ). One can seethat under Λ, some distinguished pair of subsets, one of L ( E , E ) and the otherof L ( F , F ), correspond. For example, Iso( E , E ) and Iso( F , F ), F ( E , E ) and F ( F , F ), Φ ( E , E ) and Φ ( F , F ). Moreover if, in particular, T ∈ Φ ( E , E ),then Λ sends the set C ( T ) onto the set C (Λ( T )), and if K , K ∈ C ( T ) are equivalent(according to Definition 2.3), so are Λ( K ) , Λ( K ) ∈ C (Λ( T )).Since the oriented composition is associative, this notion can be extended to thecomposition of three (or more) oriented operators.2.2. Topological preliminaries.
Let, hereafter, X denote a metric space. Werecall that a subset A of X is locally compact if any point of A admits a neigh-borhood, in A , which is compact. Therefore, any compact subset of X is locallycompact, as it is any relatively open subset of a locally compact set. However, theunion of two locally compact subsets of X may not be locally compact.We recall also that a continuous map between metric spaces is said to be proper if the inverse image of any compact set is a compact set, while it is called locallyproper if it is proper its restriction to a convenient closed neighborhood of any pointof its domain. Thus, level sets of locally proper maps are locally compact.One can check that proper maps are closed , in the sense that the image of anyclosed set is a closed set. Notation 2.14.
Let D be a subset of the product X × Y of two metric spaces.Given x ∈ X , we call x -slice of D the set D x = { y ∈ Y : ( x, y ) ∈ D } .Assume, from now on, that the vector spaces E and F are actually Banach. Inthis framework, any Fredholm operator is assumed to be bounded. Therefore, inaddition to the algebraic properties (1) and (2) stated in Subsection 2.1, one hasthe following topological ones (see e.g. [29]): P. BENEVIERI, A. CALAMAI, M. FURI, AND M.P. PERA (3) if T ∈ Φ( E, F ) , then Img E is closed in F ; (4) if T ∈ Φ( E, F ) , then T is proper on any bounded closed subsets of E ; (5) for any n ∈ Z , the set Φ n ( E, F ) is open in L ( E, F ) ; (6) if T ∈ Φ n ( E, F ) and K ∈ L ( E, F ) is compact, then T + K ∈ Φ n ( E, F ) . Let us now sketch the construction and summarize the main properties of thedegree introduced in [9].The basic fact is that, in the context of Banach spaces, the orientation of anoperator T ∗ ∈ Φ ( E, F ) induces an orientation to the operators in a neighborhoodof T ∗ . Indeed, due to the fact that Iso( E, F ) and Φ ( E, F ) are open in L ( E, F ),any companion of T ∗ remains a companion of all T sufficiently close to T ∗ .Therefore, it makes sense the following Definition 2.15.
Let Γ : X → Φ ( E, F ) be a continuous map defined on a metricspace X . A pre-orientation of Γ is a function that to any x ∈ X assigns an orient-ation ω ( x ) of Γ( x ). A pre-orientation (of Γ) is an orientation if it is continuous ,in the sense that, given any x ∗ ∈ X , there exist K ∈ ω ( x ∗ ) and a neighborhood W of x ∗ such that K ∈ ω ( x ) for all x ∈ W . The map Γ is said to be orientable ifit admits an orientation, and oriented if an orientation has been chosen. In par-ticular, a subset Y of Φ ( E, F ) is orientable or oriented if so is the inclusion map
Y ֒ → Φ ( E, F ).Observe that the set ˆΦ ( E, F ) of the oriented operators of Φ ( E, F ) has a naturaltopology, and the natural projection π : ˆΦ ( E, F ) → Φ ( E, F ) is a 2-fold coveringspace (see [10] for details). Therefore, an orientation of a map Γ as in Definition2.15 could be regarded as a lifting ˆΓ of Γ. This implies that, if the domain X of Γis simply connected and locally path connected, then Γ is orientable.Let f : U → F be a C -map defined on an open subset of E , and denote by df x ∈ L ( E, F ) the Fr´echet differential of f at a point x ∈ U .We recall that f is said to be Fredholm of index n , from now on written f ∈ Φ n ,if df x ∈ Φ n ( E, F ) for all x ∈ U . Therefore, if f ∈ Φ , Definition 2.15 and thecontinuity of the differential map df : U → Φ ( E, F ) suggest the following
Definition 2.16 (Orientation of a Φ -map in Banach spaces) . Let U be an opensubset of E and f : U → F a Fredholm map of index zero. A pre-orientation or an orientation of f are, respectively, a pre-orientation or an orientation of df ,according to Definition 2.15. The map f is said to be orientable if it admits anorientation, and oriented if an orientation has been chosen. Remark 2.17.
A very special Φ -map is given by an operator T ∈ Φ ( E, F ) . Thus,for T there are two different notions of orientations: the algebraic one, accordingto Definition 2.5; and the one regarding T as a C -map (according to Definition2.16). In each case T admits exactly two orientations (in the second one this isdue to the connectedness of the domain E ). Hereafter, we shall tacitly assume thatthe two notions agree. Namely, T has an algebraic orientation ω if and only if itsdifferential dT x : ˙ x T ˙ x has the ω orientation for all x ∈ E . We will show how the notion of orientation in Definition 2.16 can be extended tothe case of maps acting between real Banach manifolds. To this purpose, we needsome further notation.
DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 9
For short, by a manifold we shall mean a smooth Banach manifold embedded ina real Banach space.Given a manifold M and a point x ∈ M , the tangent space of M at x will bedenoted by T x M . If M is embedded in a Banach space e E , T x M will be identifiedwith a closed subspace of e E , for example by regarding any tangent vector of T x M as the derivative γ ′ (0) of a smooth curve γ : ( − , → M such that γ (0) = x .Assume that f : M → N is a C -map between two manifolds, respectively em-bedded in e E and e F and modelled on E and F . As in the flat case, f is said to be Fredholm of index n (written f ∈ Φ n ) if so is the differential df x : T x M → T f ( x ) N ,for any x ∈ M .Given f ∈ Φ , suppose that to any x ∈ M it is assigned an orientation ω ( x ) of df x (also called orientation of f at x ). As above, the function ω is called a pre-orientation of f , and an orientation if it is continuous, in a sense to be specified(see Definition 2.19). Definition 2.18.
The pre-oriented composition of two (or more) pre-oriented mapsbetween manifolds is given by assigning, at any point x of the domain of the com-posite map, the composition of the orientations (according to Definition 2.9) of thedifferentials in the chain representing the differential at x of the composite map.Assume that f : M → N is a C -diffeomorphism. Then, in particular, given any x ∈ M , the differential df x is an isomorphism. Thus, for any x ∈ M , we may takeas ω ( x ) the natural orientation of df x (recall Definition 2.6). This pre-orientationof f turns out to be continuous according to Definition 2.19 below (it is, in somesense, constant). From now on, unless otherwise stated, any diffeomorphismwill be considered oriented with the natural orientation . In particular, ina composition of pre-oriented maps, all charts and parametrizations of a manifoldwill be tacitly assumed to be naturally oriented. Definition 2.19 (Orientation of a Φ -map between manifolds) . Let f : M → N be a Φ -map between two manifolds modelled on E and F , respectively. A pre-orientation of f is an orientation if it is continuous in the sense that, given any twocharts, ϕ : U → E of M and ψ : V → F of N , such that f ( U ) ⊆ V , the pre-orientedcomposition ψ ◦ f ◦ ϕ − : U → V is an oriented map according to Definition 2.16.The map f is said to be orientable if it admits an orientation, and oriented if anorientation has been chosen.Perhaps, the simplest example of non-orientable Φ -map is given by a constantmap from the 2-dimensional projective space into R (see [10]). Remark 2.20.
One can check that the pre-oriented composition of orientations isan orientation.
Remark 2.21.
Regarding the attribute that we will assign to some particular ori-entations of Φ -maps between manifolds, whenever it makes sense, we will adaptthe terminology for Φ -operators, such as “natural orientation”, “associated orient-ation”, “canonical orientation”. For example any local diffeomorphism f : M → N admits the natural orienta-tion , given by assigning the natural orientation to the operator df x , for any x ∈ M (see Definition 2.6). As another example, assume the manifolds M and N have thesame finite dimension and are oriented, then any C -map between them admits the associated orientation (see Definition 2.8). A third example is given by a C -map f : R k → R k : it can be given the canonical orientation (see Definition 2.7).The concept of canonical orientation of a C -map f : R k → R k can be extendedto a more general situation that we shall need in the next section. In fact, if E isa real Banach space, in spite of the fact that the function det : A ( E ) → R can bediscontinuous (see e.g. [12]), one has the following Remark 2.22.
Let X be a metric space and E = E × E a real Banach space,with dim E < + ∞ . Assume that Γ : X → A ( E ) is a continuous map that can berepresented in a block matrix form as follows: Γ = I Γ ! , where I is the identity of E , Γ : X → L ( E , E ) , and Γ : X → L ( E ) . Then,according to Remark 2.2, one has det Γ( x ) = det Γ ( x ) , for all x ∈ X . Moreover,the pre-orientation of Γ given by assigning, to any x ∈ X , the canonical orientationof the operator Γ( x ) is actually an orientation, and has the property that sign Γ( x ) =sign det Γ( x ) for all x ∈ X . Similarly to the case of a single map, one can define a notion of orientation of acontinuous family of Φ -maps depending on a parameter s ∈ [0 , Definition 2.23 (Oriented Φ -homotopy) . A Φ -homotopy between two Banachmanifolds M and N is a C -map h : [0 , ×M → N such that, for any s ∈ [0 , h s = h ( s, · ) is Fredholm of index zero. An orientation of h is a continuous function ω that to any ( s, x ) ∈ [0 , ×M assigns an orientation ω ( s, x )to the differential d ( h s ) x ∈ Φ ( T x M , T h ( s,x ) N ). Where “continuous” means that,given any chart ϕ : U → E of M , a subinterval J of [0 , ψ : V → F of N such that h ( J × U ) ⊆ V , the pre-orientation of the map Γ : J × U → Φ ( E, F )that to any ( s, x ) ∈ J × U assigns the pre-oriented composition d ( ψ ◦ h s ◦ ϕ − ) x = dψ h ( s,x ) d ( h s ) x ( dϕ x ) − is an orientation, according to Definition 2.15.The homotopy h is said to be orientable if it admits an orientation, and oriented if an orientation has been chosen.If a Φ -homotopy h has an orientation ω , then any partial map h s = h ( s, · ) hasa compatible orientation ω ( s, · ). Conversely, on has the following Proposition 2.24 ( [9, 10]) . Let h : [0 , ×M → N be a Φ -homotopy, and assumethat one of its partial maps, say h s , has an orientation. Then, there exists and isunique an orientation of h which is compatible with that of h s . In particular, iftwo maps from M to N are Φ -homotopic, then they are both orientable or bothnon-orientable. DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 11
As a consequence of Proposition 2.24, one gets that any C -map f : M → M which is Φ -homotopic to the identity is orientable, since so is the identity (evenwhen M is finite dimensional and not orientable).The degree for oriented Φ -maps defined in [9] satisfies the three fundamentalproperties stated below and called Normalization, Additivity and Homotopy Invari-ance . By an axiomatic approach similar to the one due to Amann-Weiss in [2] forthe Leray–Schauder degree, in [11] it is shown that the degree constructed in [9] isthe only possible integer valued function that satisfies these theree properties.To be more explicit, let us define, first, the domain of this degree function. Givenan oriented Φ -map f : M → N , an open (possibly empty) subset U of M , anda target value y ∈ N , the triple ( f, U, y ) is said to be admissible for the degreeprovided that U ∩ f − ( y ) is compact. From the axiomatic point of view, the degreeis an integer valued function, deg, defined on the class of all the admissible triples,that satisfies the following three fundamental properties : • (Normalization) If f : M → N is a naturally oriented diffeomorphism ontoan open subset of N , then deg( f, M , y ) = 1 , ∀ y ∈ f ( M ) . • (Additivity) Let ( f, U, y ) be an admissible triple. If U and U are twodisjoint open subsets of U such that U ∩ f − ( y ) ⊆ U ∪ U , then deg( f, U, y ) = deg( f | U , U , y ) + deg( f | U , U , y ) . • (Homotopy Invariance) Let h : [0 , ×M → N be an oriented Φ -homotopy,and γ : [0 , → N a continuous path. If the set (cid:8) ( s, x ) ∈ [0 , ×M : h ( s, x ) = γ ( s ) (cid:9) is compact, then deg( h ( s, · ) , M , γ ( s )) does not depend on s ∈ [0 , . Other properties can be deduced from the fundamental ones (see [11] for details).We mention only some of them. One of these is the • (Localization) If ( f, U, y ) is an admissible triple, then deg( f, U, y ) = deg( f | U , U, y ) . Another one is the • (Excision) If ( f, U, y ) is admissible and V is an open subset of U such that f − ( y ) ∩ U ⊆ V , then deg( f, U, y ) = deg( f, V, y ) . A significative one is the • (Existence) If ( f, U, y ) is admissible and deg( f, U, y ) = 0 , then the equation f ( x ) = y admits at least one solution in U . Roughly speaking, given an admissible triple ( f, U, y ), the integer deg( f, U, y ) isan algebraic count of the solutions in U of the equation f ( x ) = y . More precisely,as a consequence of the fundamental properties, one gets the following • (Computation Formula) If ( f, U, y ) is admissible and y is a regular valuefor f in U , then the set U ∩ f − ( y ) is finite and deg( f, U, y ) = X x ∈ U ∩ f − ( y ) sign( df x ) . Another property that can be deduced from the fundamental ones is a general-ization of the Homotopy Invariance Property, that we will need in Section 4. Thisrequires the following extension of the concept of Φ -homotopy: Definition 2.25 (Extended Φ -homotopy) . An extended Φ -homotopy from M to N is a C -map h : I ×M → N , where I is an arbitrary (nontrivial) real interval,such that any partial map h s = h ( s, · ) of h is a Φ -map.The notion of orientation for an extended Φ -homotopy is practically identicalto the one in Definition 2.23 and its formulation is left to the reader.As a consequence of the Excision and the Homotopy Invariance properties of thedegree we get the following • (Generalized Homotopy Invariance) Let h : I × M → N be an orientedextended Φ -homotopy, γ : I → N a continuous path, and W an open subsetof I × M . Given any s ∈ I , denote by W s = { x ∈ M : ( s, x ) ∈ W } the s -slice of W . If the set (cid:8) ( s, x ) ∈ W : h ( s, x ) = γ ( s ) (cid:9) is compact, then deg( h s , W s , γ ( s )) does not depend on s ∈ I . The easy proof of this property can be performed by showing that the integervalued function s ∈ I 7→ ν ( s ) := deg( h s , W s , γ ( s )) is locally constant. In fact, givenany s ∗ ∈ I , because of the compactness of the set (cid:8) ( s, x ) ∈ W : h ( s, x ) = γ ( s ) (cid:9) , one can find a box J × V ⊆ W , with V open in M and J an open interval containing s ∗ , such that W s ∩ h − s ( γ ( s )) ⊆ V for all s ∈ J ∩ I . Thus, from the ExcisionProperty, one gets ν ( s ) = deg( h s , V, γ ( s )) for all s ∈ J ∩ I . Moreover, because ofthe Homotopy Invariance Property, ν ( s ) does not depend on s ∈ J ∩ I . Hence,since I is connected and s ∗ ∈ I is arbitrary, one gets the assertion.3. The eigenvalue problem and the associated topological degree
Hereafter, G and H will be two real Hilbert spaces, with inner product and normdenoted by h· , ·i and k · k , respectively.Consider the eigenvalue problem(3.1) (cid:26) Lx = λCxx ∈ S , where λ ∈ R , L, C ∈ L ( G, H ), C is compact, and S is the unit sphere of G .We assume that the operator L − λC ∈ L ( G, H ) is invertible for some λ ∈ R .Therefore, because of the compactness of C , L − λC is Fredholm of index zerofor any λ ∈ R (recall property (6) of Fredholm operators in Subsection 2.2). Inparticular, Ker( L − λC ) is always finite dimensional, and nontrivial if and only if λ ∈ R is an eigenvalue of (3.1). Moreover, the set of all the real eigenvalues of (3.1)is discrete. DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 13
A pair ( λ, x ) belonging to the cylinder R × S will be called an eigenpoint of (3.1)if it satisfies the equation Lx = λCx . In this case, x is a unit eigenvector of (3.1)corresponding to the eigenvalue λ .The set of the eigenpoints of (3.1) will be denoted by S . Hence, given any λ ∈ R ,the λ -slice S λ = { x ∈ S : ( λ, x ) ∈ S} of S coincides with S ∩ Ker( L − λC ).Observe that S λ is nonempty if and only if λ is an eigenvalue of (3.1). In thiscase S λ will be called the eigensphere of (3.1) corresponding to λ . In fact, it is asphere whose (finite) dimension equals that of Ker( L − λC ) minus one.If λ ∈ R is an eigenvalue of (3.1), the nonempty subset { λ } × S λ of S will becalled the λ -eigenset of (3.1).It is convenient to regard R × S as the subset of the Hilbert space R × G satisfyingthe equation g ( λ, x ) = 1, where g : R × G → R is defined by g ( λ, x ) = h x, x i . Thedifferential dg p ∈ L ( R × G, R ) of g at a point p = ( λ, x ) is given by ( ˙ λ, ˙ x ) h x, ˙ x i .Therefore, the set of the critical points of g is the λ -axis x = 0 and, consequently,the number 1 is a regular value for g . This shows that R × S is a smooth manifoldof codimension one in R × G and, given any p = ( λ, x ) ∈ g − (1), the tangent spaceof R × S at p is the kernel of dg p , namely T ( λ,x ) ( R × S) = (cid:8) ( ˙ λ, ˙ x ) ∈ R × G : h x, ˙ x i = 0 (cid:9) = R × x ⊥ . Observe that, if dim G = 1 and ( λ, x ) ∈ R × S, then x ⊥ = { } and the tangentspace T ( λ,x ) ( R × S) is the subspace R ×{ } of R × R . Moreover, the cylinder R × S isdisconnected: it is the union of two horizontal lines, R ×{− } and R ×{ } . Due tothis fact, in order to write some statements in a simpler form, hereafter, unlessotherwise stated, we will assume that the dimensions of the Hilbertspaces G and H are bigger than
1, so that the cylinder R × S is connected.Define the smooth mapΨ : R × G → H by ( λ, x ) Lx − λCx and observe that it is Fredholm of index one. Therefore, its restrictionΨ : R × S → H to the 1-codimensional submanifold R × S of R × G is Fredholm of index zero. Tosee this, recall that the differential of Ψ at p ∈ R × S is the restriction of d Ψ p to the1-codimensional subspace T p ( R × S) of R × G .The map Ψ will play a fundamental role in this paper. Notice that its zeros arethe eigenpoints of (3.1). That is, S = Ψ − (0).We point out that Ψ is orientable and, because of the connectedness of themanifold R × S, admits exactly two orientations. In fact, in the finite dimensionalcase, an orientation of Ψ is equivalent to a pair of orientations, one of the domainand one of the codomain, up to an inversion of both of them (see Definition 2.8);while, if dim G = + ∞ , the orientability of Ψ is a consequence of the fact that thecylinder R × S is simply connected (it is actually contractible). Therefore, fromnow on, we shall assume that Ψ is oriented . No matter which one of the twoorientations one selects, all the statements in this paper hold true. Definition 3.1.
Let X be a metric space and K ⊆ A ⊆ X . We shall say that K is an isolated subset of A if it is compact and relatively open in A . Thus, thereexists an open subset U of X such that U ∩ A = K , which will be called an isolatingneighborhood of K among (the elements of ) A . Definition 3.2.
Let
K ⊂ R × S be an isolated subset of Ψ − (0). By the Ψ -degree of K we mean the integer Ψ-deg( K ) := deg(Ψ , U, U ⊆ R × S is any isolatingneighborhood of K among Ψ − (0). If p is an isolated zero of Ψ, we shall simplywrite Ψ-deg( p ) instead of Ψ-deg( { p } ).Notice that this definition is well-posed, thanks to the Excision Property of thedegree. Remark 3.3. If p ∈ Ψ − (0) is such that the differential d Ψ p is an isomorphism,then, as a consequence of the Local Inverse Function Theorem, Ψ maps diffeomorph-ically a neighborhood of p in R × S onto a neighborhood of in H . Thus, { p } isisolated among Ψ − (0) and, because of the Computation Formula of the degree (seeSection 2), Ψ - deg( p ) = sign( d Ψ p ) , which is either or − , depending on whetheror not the orientation of d Ψ p is the natural one. Definition 3.4.
An eigenpoint ( λ ∗ , x ∗ ) of (3.1) will be called simple if the associ-ated Φ -operator T = L − λ ∗ C satisfies the following conditions:(1) Ker T = R x ∗ ,(2) Cx ∗ / ∈ Img T .Notice that, if p ∗ = ( λ ∗ , x ∗ ) is a simple eigenpoint, then the eigenset { λ ∗ }×S λ ∗ has only two elements: p ∗ and its twin eigenpoint ¯ p ∗ = ( λ ∗ , − x ∗ ), which is as wellsimple. Moreover, since T = L − λ ∗ C is Fredholm of index zero, its image hascodimension one in H . Therefore one has the following Remark 3.5. If ( λ ∗ , x ∗ ) is a simple eigenpoint of (3.1) , then H = Img T ⊕ R Cx ∗ .Thus, λ ∗ is a simple eigenvalue of the equation Lx = λCx . The following result is the key that brings to the proof of Theorem 4.5. Despitethe fact that we have assumed dim
G >
1, Theorem 4.5 holds true in any dimension:its assertion in the 1-dimensional case will be verified in Example 5.1.
Theorem 3.6.
Let p ∗ = ( λ ∗ , x ∗ ) and ¯ p ∗ = ( λ ∗ , − x ∗ ) be two simple twin eigenpointsof (3.1) . Then, Ψ - deg( p ) = Ψ - deg(¯ p ) = ± . Consequently, the Ψ - degree of the λ ∗ -eigenset { λ ∗ }×S λ ∗ is non-zero.Proof. It is enough to prove that Ψ-deg( p ) = Ψ-deg(¯ p ) = ±
1: the last assertionfollows from the Additivity Property of the degree.Since p ∗ is simple, the λ ∗ -eigensphere S λ ∗ of (3.1) consists of two antipodalpoints: x ∗ and − x ∗ . Both the tangent spaces of S at these points coincide with the1-codimensional subspace x ⊥∗ of G . Thus, the tangent spaces of the cylinder R × Sat the twin eigenpoints p ∗ and ¯ p ∗ are equal to the 1-codimensional subspace R × x ⊥∗ of the Hilbert space R × G . The differentials d Ψ p ∗ and d Ψ ¯ p ∗ (acting from R × x ⊥∗ to H ) are given, respectively, by( ˙ λ, ˙ x ) T ˙ x − ˙ λCx ∗ and ( ˙ λ, ˙ x ) T ˙ x + ˙ λCx ∗ , where T , as in Definition 3.4, denotes the operator L − λ ∗ C .As one can check (see, for example, [7, Lemma 3.2])), the fact that the eigen-points p ∗ and ¯ p ∗ are simple implies that the differentials d Ψ p ∗ and d Ψ ¯ p ∗ are invert-ible. Consequently, according to Remark 3.3, the Ψ-degrees of p ∗ and ¯ p ∗ coincide,respectively, with sign( d Ψ p ∗ ) and sign( d Ψ ¯ p ∗ ), which are both non-zero.Thus, it remains to prove that these two signs are equal, which means that theorientations of Ψ at these points are both natural or both not natural. DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 15
To this purpose, it is convenient to fix an orientation of Ψ at one of the twoeigenpoints p ∗ and ¯ p ∗ (for example by choosing the natural orientation of d Ψ ¯ p ∗ )and to transport it, continuously, along a curve, up to the other eigenpoint .A suitable curve is a λ ∗ -meridian . That is, a geodesic in R × S of the type G = (cid:8) ( λ ∗ , x ) ∈ R × S : x = sin θ x ∗ + cos θ x e , θ ∈ [ − π/ , π/ (cid:9) , where x e is an element of the equator S ∩ x ⊥∗ of S (recall that dim G >
1) and θ may be regarded as a latitude .Having chosen x e , and the consequent λ ∗ -meridian, we will “observe” the differ-ential of Ψ, along G , from the point of view of a self-map defined on a convenient“flat space”; namely, the Hilbert space G ∗ × R × R , where G ∗ is the 2-codimensionalsubspace x ⊥∗ ∩ x ⊥ e of G , which is nonempty because of the assumption dim G > G = 2.We will “observe” the map Ψ by means of a convenient composition e Ψ = σ − ◦ Ψ ◦ η , where η : W → R × S is a parametrization (i.e. the inverse of a chart)defined on an open subset W of G ∗ × R × R and σ : G ∗ × R × R → H is an invertiblebounded linear operator (a global parametrization of H ).Let us define η , first. Consider the open subset W = (cid:8) ( y, θ, λ ) ∈ B × ( − π, π ) × R (cid:9) of G ∗ × R × R , where B stands for the open unit ball of G ∗ , and let η : W → R × Sbe the map given by η ( y, θ, λ ) = (cid:0) λ, y + p − k y k (sin θ x ∗ + cos θ x e ) (cid:1) . Notice that, under η , the eigenpoints ¯ p ∗ and p ∗ correspond to ¯ u ∗ = (0 , − π/ , λ ∗ )and u ∗ = (0 , π/ , λ ∗ ), respectively. One can check that η is a diffeomorphism ontoan open subset of R × S containing the meridian G .We now define σ . From Remark 3.5 we get the splitting(3.2) H = T ( x ⊥∗ ) ⊕ R Cx ∗ = ( T ( G ∗ ) ⊕ R T x e ) ⊕ R Cx ∗ . Thus, H can be identified with G ∗ × R × R by means of the isomorphism σ : G ∗ × R × R → H, ( y, a, b ) T y + aT x e + bCx ∗ . We assume that η and σ − are naturally oriented and that e Ψ has the compositeorientation. Therefore, recalling Corollary 2.13,(3.3) sign( d e Ψ ¯ u ∗ ) = sign( d Ψ ¯ p ∗ ) and sign( d e Ψ u ∗ ) = sign( d Ψ p ∗ ) , whatever the orientation of Ψ.Hence, it remains to prove that sign( d e Ψ ¯ u ∗ ) = sign( d e Ψ u ∗ ), no matter what is theorientation of e Ψ.To this purpose, consider the straight path γ : [ − π/ , π/ → W defined by γ ( θ ) = (0 , θ, λ ∗ ). This path joins γ ( − π/
2) = ¯ u ∗ with γ ( π/
2) = u ∗ , therefore it issuitable for the continuous transport of the orientation of e Ψ from ¯ u ∗ to u ∗ . Noticethat the image of the simple arc θ η ( γ ( θ )) is the λ ∗ -meridian G .Taking into account that Ψ( λ, x ) = T x − ( λ − λ ∗ ) Cx and that T x ∗ = 0, givenany θ ∈ [ − π/ , π/ d (Ψ ◦ η ) γ ( θ ) ( ˙ y, ˙ θ, ˙ λ ) = T ˙ y − ˙ θ sin θ T x e − ˙ λ sin θ Cx ∗ − ˙ λ cos θ Cx e . Therefore, recalling that σ − , being linear, coincides with its differential, we obtain d ( e Ψ) γ ( θ ) ( ˙ y, ˙ θ, ˙ λ ) = σ − (cid:0) T ˙ y − ˙ θ sin θ T x e − ˙ λ sin θ Cx ∗ − ˙ λ cos θ Cx e (cid:1) . Since, according to the splitting (3.2), Cx e can be written as T y ∗ + αT x e + βCx ∗ for some y ∗ ∈ G ∗ and α, β ∈ R , we have d ( e Ψ) γ ( θ ) ( ˙ y, ˙ θ, ˙ λ ) = (cid:0) ˙ y, − ˙ θ sin θ, − ˙ λ sin θ (cid:1) − ˙ λ cos θ (cid:0) y ∗ , α, β (cid:1) , that can be represented as d e Ψ γ ( θ ) ( ˙ y, ˙ θ, ˙ λ ) = I − cos θ y ∗ − sin θ − α cos θ − (sin θ + β cos θ ) ˙ y ˙ θ ˙ λ , where I is the identity of G ∗ .Thus, the continuous map Γ : [ − π/ , π/ → A ( G ∗ × R × R ), given by θ d e Ψ γ ( θ ) ,is in block matrix form as in Remark 2.22. Consequently, up to an inversion ofthe orientation of Ψ, we may assume that Γ has the canonical orientation, which issuch that sign( d e Ψ γ ( θ ) ) = sign det( d e Ψ γ ( θ ) ) = sign (cid:0) sin θ (sin θ + β cos θ ) (cid:1) . Recalling that ¯ u ∗ = γ ( − π/
2) and u ∗ = γ ( π/ d e Ψ ¯ u ∗ ) =sign( d e Ψ u ∗ ), and the assertion “Ψ-deg( p ) = Ψ-deg(¯ p ) = ±
1” follows from (3.3). (cid:3) The perturbed eigenvalue problem and global continuation
Given, as before, two real Hilbert spaces G and H , consider the problem(4.1) (cid:26) Lx + sN ( x ) = λCxx ∈ S , where s, λ ∈ R , L, C ∈ L ( G, H ), C is compact, and N : S → H is a C compactmap defined on the unit sphere of G . As in the unperturbed problem (3.1), weassume that L − λC is invertible for some λ ∈ R .A triple ( s, λ, x ) ∈ R × R × S is a solution of (4.1) if it satisfies the equation Lx + sN ( x ) = λCx . The third element x ∈ S is said to be a unit eigenvector corresponding to the eigenpair ( s, λ ). The set of all the solutions of (4.1) is denotedby Σ, while E stands for the subset of R of the eigenpairs of (4.1).Observe that E is the projection of Σ into the sλ -plane, and that the s = 0 sliceΣ of Σ coincides with the set S = Ψ − (0) of the eigenpoints of (3.1).A solution of (4.1) is said to be trivial if it is of the type (0 , λ, x ). In thiscase p = ( λ, x ) is the corresponding eigenpoint (of the unperturbed problem (3.1)).When p is simple, the solution (0 , λ, x ) will be as well said to be simple . Therefore,any simple solution is trivial, but not viceversa.Consider the C -mapΨ + : R × R × S → H, ( s, λ, x ) Ψ( λ, x ) + sN ( x ) , where, we recall, Ψ : R × S → H is defined by Ψ( λ, x ) = Lx − λCx . Notice that thezeros of Ψ + are the solutions of (4.1); that is, Σ = (Ψ + ) − (0).Since Ψ is Fredholm of index zero, because of the compactness of N , any partialmap Ψ + s = Ψ + ( s, · , · ) of Ψ + is a Φ -map from R × S to H (recall that the differential, DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 17 at any point, of a compact C -map is a compact operator). Therefore, accordingto Definition 2.25, Ψ + is an extended Φ -homotopy from R × S into H .Due to the fact that the partial map Ψ +0 of Ψ + coincides with the oriented mapΨ, thanks to Proposition 2.24, the extended Φ -homotopy Ψ + admits an orientation(a unique one) which is compatible with that of Ψ. Therefore, from now on, Ψ + will be considered an oriented extended Φ -homotopy.Since the set Σ = (Ψ + ) − (0) has a distinguished (trivial) subset, namely { }× Σ ,it makes sense to consider the notion of bifurcation point. A trivial solution q ∗ =(0 , λ ∗ , x ∗ ) of (4.1) is a bifurcation point provided that any neighborhood of q ∗ contains nontrivial solutions.A bifurcation point q ∗ is said to be global (in the sense of Rabinowitz [27]) ifthere exists a connected set of nontrivial solutions whose closure contains q ∗ and iseither unbounded or meets a bifurcation point q ∗ different from q ∗ .Particularly meaningful is the study of bifurcation points belonging to a set oftrivial solutions of the type { } × { λ ∗ } × S λ ∗ , whose eigensphere S λ ∗ is nontrivial (that is, with positive dimension). Since, in this case, 0 and λ ∗ are given, one cansimply say that a point x ∗ ∈ S λ ∗ , regarded as an alias of q ∗ = (0 , λ ∗ , x ∗ ), is a bifurcation point if so is q ∗ .For a necessary condition and some sufficient conditions for a point x ∗ of anontrivial eigensphere to be a bifurcation point see [17]. Results regarding theexistence of (global or non-global) bifurcation points belonging to even-dimensionaleigenspheres can be found in [4–8, 18, 19, 21].Theorem 4.2 below, which is crucial for our main result (Theorem 4.5), provides,in particular, a sufficient condition for an isolated subset of trivial solutions of(4.1) to contain at least one bifurcation point. To prove it, we need the followinglemma of point-set topology, which is particularly suitable to our purposes and isobtained from general results by C. Kuratowski (see [26], Chapter 5, Vol. 2). Foran interesting paper on connectivity theory we also recommend [1]. Lemma 4.1 ( [24] ) . Let K be a compact subset of a locally compact metric space X . If every compact subset of X containing K has nonempty boundary, then X \ K contains a connected set whose closure in X is non-compact and intersects K . Recall that, according to Notation 2.14, given a subset D of R × R × S and s ∈ R ,the symbol D s stands for the s -slice of D . Namely, D s = (cid:8) ( λ, x ) ∈ R × S : ( s, λ, x ) ∈ D (cid:9) . Theorem 4.2.
Let Ω be an open subset of R × R × S . If deg(Ψ , Ω , = 0 , then Ω has a connected set of nontrivial solutions whose closure in Ω is non-compact andcontains at least one bifurcation point.Proof. Since, by definition, a trivial solution of (4.1) is a bifurcation point if it isin the closure of the set of nontrivial solutions, the assertion is the same as that ofLemma 4.1 provided that X is the set of the solutions in Ω and K is its subset ofthe trivial ones. Namely, X = (Ψ + ) − (0) ∩ Ω and K = { }× X , where X is the s = 0 slice of X , which coincides with Ψ − (0) ∩ Ω = Σ ∩ Ω .Thus, it is enough to prove that the metric pair ( X, K ) satisfies the assumptionsof Lemma 4.1.Let us show first that X is locally compact. Recall that Ψ : R × S → H is a Fred-holm map of index zero. Therefore, its extension ( s, λ, x ) Ψ( λ, x ) is Fredholm of index one, being obtained by composing Ψ with the projection ( s, λ, x ) ( λ, x ),which is a Φ -map (recall the property about the index of the composition of Fred-holm operators in Subsection 2.1). Since Ψ + is obtained by adding to this extensionof Ψ the compact C -map ( s, λ, x ) sN ( x ), we get that Ψ + is as well Fredholmof index one. Therefore, Ψ + , being Fredholm, is a locally proper map (see [28]).This implies that the set Σ = (Ψ + ) − (0) is locally compact, and so is its relativelyopen subset X . Moreover, K = { }× X is compact, since X coincides with the setΨ − (0) ∩ Ω , whose compactness is implicit in the assumption that deg(Ψ , Ω ,
0) isdefined.It remains to prove that any compact subset of X containing K has nonemptyboundary in X . Assume, by contradiction, that this is not the case. Hence thereexists a compact subset D of X containing K whose boundary, in X , is empty.Therefore, D is relatively open in X and, consequently, there exists an open subset W of Ω such that X ∩ W = D . Incidentally we observe that, according to Definition3.1, the compact set D is isolated among the elements of X .Notice that, since W ⊆ Ω, one has D = (cid:8) ( λ, s, x ) ∈ W : Ψ + ( λ, s, x ) = 0 (cid:9) . Thus,according to the Generalized Homotopy Invariance Property, deg(Ψ + s , W s ,
0) doesnot depend on s ∈ R .Because of the Excision Property, for s = 0 one hasdeg(Ψ +0 , W ,
0) = deg(Ψ +0 , Ω , . Therefore, since the partial map Ψ +0 of Ψ + coincides with Ψ, one getsdeg(Ψ + s , W s ,
0) = deg(Ψ , Ω , = 0 , ∀ s ∈ R . Now, the compactness of D implies that there exists s ∗ ∈ R such that the set D s ∗ = (Ψ + s ∗ ) − (0) ∩ W s ∗ is empty. Consequently, because of the Existence Propertyone gets deg(Ψ + s ∗ , W s ∗ ,
0) = 0, and the assertion follows from the contradiction. (cid:3)
Corollary 4.3 below provides a sufficient condition for the existence of a globalbifurcation point “emanating” from an isolated subset of trivial solutions. In orderto deduce it from Theorem 4.2, we need to show that the map Ψ + is more thanlocally proper. Actually, • Ψ + is proper on any bounded and closed subset of its domain. To check this, observe that Ψ + is the sum of two maps: one is the restriction ˆ L tothe manifold R × R × S of the linear operator¯ L : R × R × G → F, ( s, λ, x ) Lx, which is Fredholm of index two; the other one is the map ( s, λ, x ) sN ( x ) − λCx ,which sends bounded sets into relatively compact sets. The linear operator ¯ L , beingFredholm, is proper on bounded and closed subsets of its domain. Therefore, thesame property is inherited by ˆ L to the closed subset R × R × S of R × R × G . One cancheck that this property is preserved by adding to ˆ L a compact map.From Theorem 4.2 we get a sufficient condition for the existence of a globalbifurcation point. Recall that the slice Σ of the set Σ = (Ψ + ) − (0) of the solutionsof (4.1) coincides with the set S = Ψ − (0) of the eigenpoints of (3.1). Corollary 4.3.
Let K be an isolated subset of Σ such that Ψ - deg( K ) = 0 . Then,there exists a connected set of nontrivial solutions of (4.1) whose closure contains DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 19 a bifurcation point q ∗ ∈ { }×K and is either unbounded or encounters a bifurcationpoint q ∗ / ∈ { }×K . Consequently, q ∗ is a global bifurcation point.Proof. Let Ω be the open subset of R × R × S obtained by removing the closed setof the elements of { } × Σ which are not in { } × K (recall that K , according toDefinition 3.1, is relatively open in Σ ). Thus, Ω is an isolating neighborhoodof K among Σ and, consequently, deg(Ψ , Ω ,
0) = Ψ-deg( K ) = 0. Because ofTheorem 4.2, there exists a connected set C of nontrivial solutions whose closure inΩ, call it C + , is non-compact and contains at least one bifurcation point q ∗ , which,necessarily, belongs to Ω.It is enough to prove that C satisfies the first assertion: the second one is aconsequence of the fact that the closure of a connected set is as well connected.To this purpose, we need to show, first, that q ∗ ∈ { }×K and, after this, we mayassume that C + is bounded.The point q ∗ belongs to { }×K , since, because of the definition of Ω, one hasΩ ∩ ( { }× Σ ) = { }×K .Assume now that C + is bounded. Then, so is the closure C of C (in R × R × S).It remains to show that C contains a trivial solution q ∗ which does not belong to { }×K .Recall that Ψ + is proper on bounded closed subsets of its domain. Therefore, thesubset C of (Ψ + ) − (0) is compact. Moreover, C contains C + , which is not compact.This implies that C has at least one point q ∗ which is not in C + . The fact that q ∗ is a bifurcation point not in { }×K follows from the definition of Ω. (cid:3) Corollary 4.4. If D is a compact component of Σ , then Ψ - deg( D ) = 0 .Proof. Observe that D is an isolated subset of Σ , due to the fact that the set ofall the eigenvalues of (3.1) is discrete.Suppose, by contradiction, that Ψ-deg( D ) = 0. Then, Corollary 4.3 appliesensuring the existence of a connected set C of nontrivial solutions of (4.1) whoseclosure C , which is as well connected, contains at least two trivial solutions: one,say q ∗ , belonging to { }× D , and one, call it q ∗ , outside { }× D .Since q ∗ belongs to both the connected set C and the component D , one gets C ⊆ D . Therefore, q ∗ belongs to D and, consequently, being trivial, belongs as wellto { }× D , which is a contradiction yielding the assertion. (cid:3) We are ready to prove our main achievement (Theorem 4.5). Its proof is basedon previous results requiring the notion of degree for the oriented map Ψ and theconvenient hypothesis dim
G >
1. In spite of this, its assertion is still valid whenthe space G has dimension one, as Example 5.1 shows. Theorem 4.5.
Let ( λ ∗ , x ∗ ) be a simple eigenpoint of problem (3.1) . Then, in theset Σ of the solutions of (4.1) , the connected component containing (0 , λ ∗ , x ∗ ) iseither unbounded or includes a trivial solution (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ .Proof. We may assume that the connected component D of Σ containing (0 , λ ∗ , x ∗ )is bounded, and we need to show that its slice D is not contained in { λ ∗ }×S λ ∗ .Recalling that Σ = (Ψ + ) − (0) and that Ψ + is proper on bounded closed subsetsof its domain, we get that D is compact. Then, Corollary 4.4 implies Ψ-deg( D ) = 0and, consequently, because of Theorem 3.6, D
6⊆ { λ ∗ }×S λ ∗ . (cid:3) Some illustrating examples and an application
We give now three examples illustrating the assertion of Theorem 4.5. The lastone shows also that, in this theorem, the assumption that the solution (0 , λ ∗ , x ∗ ) issimple cannot be removed.After the examples, we will give an application of Theorem 4.5 to a motionequation containing a nonlinearity like an air resistance force.5.1. Examples.
The first example regards an exhaustive discussion about the solu-tions of problem (4.1) in the case when dim G = 1. As we shall see, the assertionof Theorem 4.5 holds true also in this minimal dimension. Example 5.1.
Let G = H = R and consider the problem(5.1) (cid:26) lx + sN ( x ) = λcx, | x | = 1 , in which l and c are two given real numbers, and N is an arbitrary real function.We assume c = 0, so that the unperturbed problem (obtained by putting s = 0)has a unique eigenvalue, λ ∗ = l/c , and two corresponding twin eigenpoints: p = ( λ ∗ , x ∗ ) = ( l/c, , ¯ p = ( λ ∗ , − x ∗ ) = ( l/c, − , both simple. We will interpret the assertion of Theorem 4.5 in this extreme situ-ation.For a solution ( s, λ, x ) of problem (5.1) we have two possibilities: x = 1 or x = − x = 1 one has λ = ( l + sN (1)) /c . Thus, the set of solutions of this type isgiven by the straight lineΣ + = (cid:8)(cid:0) s, ( l + sN (1)) /c, (cid:1) ∈ R : s ∈ R (cid:9) . Analogously, for x = − − = (cid:8)(cid:0) s, ( l − sN ( − /c, − (cid:1) ∈ R : s ∈ R (cid:9) . Therefore, the set Σ of all the solutions of (5.1) is Σ + ∪ Σ − , and the assertion ofTheorem 4.5 is satisfied for both the simple eigenpoints ( λ ∗ , x ∗ ) and ( λ ∗ , − x ∗ ).The following example regards a differential equation with an evident physicalmeaning, and the parameter 2 s , when positive, can be regarded as a frictionalcoefficient. Its abstract formulation has infinitely many eigenpoints, all of themsimple. The set Σ of the solution triples ( s, λ, x ) is the union of infinitely manyunbounded components, each of them corresponding to one and only one eigenpoint. Example 5.2.
Let us show how Theorem 4.5 agrees with the structure of thenon-zero solutions of the following boundary value problem:(5.2) (cid:26) x ′′ ( t ) + 2 sx ′ ( t ) + λx ( t ) = 0 ,x (0) = 0 = x ( π ) . To this purpose, we will interpret it as an abstract problem of the type (4.5) byspecifying what are here the spaces G and H , the linear operators L and C , andthe map N .Let H (0 , π ) be the Hilbert space of the absolutely continuous real functionsdefined in [0 , π ] with derivative in L (0 , π ), and denote by H (0 , π ) the Hilbertspace of the C real functions in [0 , π ] with derivative in H (0 , π ). DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 21
Clearly H (0 , π ) is a subset of the Banach space C [0 , π ], and the inclusion, werecall, is a compact operator. Therefore the injection of H (0 , π ) into L (0 , π ) is aswell compact, due to the bounded injection of C [0 , π ] into L (0 , π ). Analogously,the inclusion of H (0 , π ) into C [0 , π ] is compact and the inclusion of C [0 , π ] into H [0 , π ] is continuous.As a source space G we take the 2-codimensional closed subspace of H (0 , π )consisting of the functions x satisfying the boundary condition x (0) = 0 = x ( π ).The target space H is L (0 , π ).Observe that the second derivative x x ′′ , as a linear operator from H (0 , π ) to L (0 , π ), is bounded and Fredholm of index 2, being surjective with 2-dimensionalkernel. Therefore, its restriction L ∈ L ( G, H ) is a Φ -operator (recall the propertyabout the composition of Fredholm operators).Here C associates to any x ∈ G the element − x ∈ H . Thus, C is a compactlinear operator, since so is the injection of H (0 , π ) into L (0 , π ). The map N transforms x ∈ G in 2 x ′ ∈ H and, therefore, it is as well compact, as compositionof a bounded linear operator into H (0 , π ) with the compact injection into L (0 , π ).Among the infinitely many equivalent norms in H (0 , π ) and, consequently, in G we choose the one associated with the inner product h x, y i = 1 π Z π (cid:0) x ( t ) y ( t ) + x ′′ ( t ) y ′′ ( t ) (cid:1) dt. As in the previous sections, S denotes the unit sphere of G . Since we are inter-ested in the non-zero solutions of (5.2), the linearity of N justifies the condition x ∈ S in the following abstract formulation of our problem:(5.3) (cid:26) Lx + sN ( x ) = λCxx ∈ S . Elementary computations show that the eigenvalues of the unperturbed problem(obtained with s = 0) are λ = 1 , λ = 4 , . . . , λ n = n , . . . and to any λ n corres-ponds the 1-dimensional eigenspace R x n = Ker( L − λ n C ), with x n ∈ S given by x n ( t ) = r
21 + n sin( nt ) . Let us show that p n = ( λ n , x n ) and ¯ p n = ( λ n , − x n ) are simple eigenpoints,according to Definition 3.4. Since C is compact, the operator T n = L − λ n C is Fredholm of index zero. Therefore, we need only to prove that Cx n does notbelong to T n ( G ), which means that the equation T n ( x ) = Cx n has no solutions in G . In fact, there are no solutions of the resonant problem (cid:26) x ′′ ( t ) + n x ( t ) = sin( nt ) ,x (0) = 0 = x ( π ) . With standard computations one can prove that, given s ∈ R , the differentialequation x ′′ ( t ) + 2 sx ′ ( t ) + λx ( t ) = 0 has a non-zero solution verifying the boundarycondition x (0) = 0 = x ( π ) if and only if λ = n + s , with n ∈ N . Therefore,the subset E of the sλ -plane of the eigenpairs of (5.3) is composed by the disjointunion of infinitely many parabolas of equation λ = n + s , n ∈ N . Moreover, given( s, n + s ) ∈ E , any solution in G of the differential equation x ′′ ( t ) + 2 sx ′ ( t ) + ( s + n ) x ( t ) = 0belongs to the straight line R x s,n , where x s,n ( t ) = exp( − st ) sin( nt ). As a consequence of this, given any eigenpoint p n = ( λ n , x n ), the connectedcomponent, in Σ, containing the corresponding trivial solution (0 , λ n , x n ) is theunbounded curve (cid:8) ( s, s + n , x s,n / k x s,n k ) : s ∈ R (cid:9) . Obviously, for the twin eigenpoint ¯ p n = ( λ n , − x n ), one gets (cid:8) ( s, s + n , − x s,n / k x s,n k ) : s ∈ R (cid:9) . In conclusion, for any eigenpoint the assertion of Theorem 4.5 is verified.Example 5.3 below has already been considered in [7] in relation to the conjectureformulated there and solved by Theorem 4.5 above. It concerns a system of twoordinary differential equations with periodic boundary conditions, and the set Σof its solutions ( s, λ, x ) has a component which is diffeomorphic to a circle andcontains exactly four trivial solutions, all of them simple. These four solutionsare associated with two eigenvalues of the unperturbed problem: a pair of twinsfor each eigenvalue. The other components of Σ are infinitely many 1-dimensionalspheres (geometric circles). The projection of each of them into the sλ -plane is asingleton { (0 , λ ) } , with λ an eigenvalue of the unperturbed problem. Example 5.3.
We are interested in the non-zero solutions of the following systemof coupled differential equations with 2 π -periodic boundary conditions: x ′ ( t ) + x ( t ) − sx ( t ) = λx ( t ) ,x ′ ( t ) − x ( t ) − sx ( t ) = − λx ( t ) ,x (0) = x (2 π ) , x (0) = x (2 π ) . As in Example 5.2, we interpret our problem in the abstract form(5.4) (cid:26) Lx + sN ( x ) = λCxx ∈ S , where L , C and N are operators to be defined below, together with the source andthe target spaces G and H .Let H ((0 , π ) , R ) be the Hilbert space of the absolutely continuous functions x = ( x , x ) : [0 , π ] → R with derivative in L ((0 , π ) , R ).We take as G the closed subspace of H ((0 , π ) , R ) of the functions satisfyingthe periodic condition x (0) = x (2 π ), and as H the space L ((0 , π ) , R ). Observethat G has codimension 2 in H ((0 , π ) , R ). Therefore, the operator L : G → H ,given by ( x , x ) ( x ′ + x , x ′ − x ), is Fredholm of index zero.The operators N and C , given by ( x , x ) ( − x , − x ) and ( x , x ) ( x , − x )respectively, are compact, since so is the injection H ((0 , π ) , R ) ֒ → L ((0 , π ) , R ) . The norm in G is the one associated with the inner product h x, y i = 12 π Z π (cid:0) x ( t ) · y ( t ) + x ′ ( t ) · y ′ ( t ) (cid:1) dt, where, given two vectors a = ( a , a ) and b = ( b , b ) in R , a · b denotes thestandard dot product. As in the previous sections, S is the unit sphere in G . DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 23
The eigenvalues of the unperturbed problem (obtained by putting s = 0 in (5.4))are λ = ±√ n , n = 0 , , , . . . and the set E of the eigenpairs of (5.4) is thedisjoint union of two sets: the circle C = (cid:8) ( s, λ ) ∈ R : s + λ = 1 (cid:9) and the isolated eigenpairs (cid:8) (0 , λ ) ∈ R : λ = ± p n , n = 1 , , , . . . (cid:9) . The non-zero solutions of the periodic boundary value problem (5.3) are of twotypes: the constant ones, corresponding to the eigenpairs of the circle C , and theoscillating ones associated with the isolated eigenpairs.Let us see first the case ( s, λ ) ∈ C . One has ( s, λ ) = (cos θ, sin θ ), with θ ∈ [0 , π ],and the kernel of the linear operator L + (cos θ ) N − (sin θ ) C ∈ L ( G, H )is the straight line R x θ , where x θ ∈ S is the constant function x θ : [0 , π ] → R , t (cos( θ/ , sin( θ/ . Therefore, the connected component D of Σ containing the trivial solution q ∗ = (0 , λ ∗ , x ∗ ) = (0 , , x π/ )of (5.4) is diffeomorphic to a circle, as its parametrization θ ∈ [0 , π ] (cos θ, sin θ, x θ ) ∈ R × R × Sshows. This component contains four trivial solutions: two of them associated withthe eigenvalue λ ∗ = 1 and the others with λ ∗ = −
1. They are all simple solutions,and the component D agrees with the statement of Theorem 4.5.Incidentally, we observe that the projection of D onto the circle C is a doublecovering map, and the above parametrization of D is just a lifting of the map θ (cos θ, sin θ ) ∈ C , θ ∈ [0 , π ].Let us consider now the isolated eigenpairs. That is, the ones having | λ | >
1. Inthis case (5.3) admits non-zero solutions if and only if s = 0 and λ = ±√ n , with n = 1 , , , . . . More precisely, these solutions are oscillating and, given any isolatedeigenpair (0 , λ ∗ ), the corresponding solutions of (5.3), plus the zero one, form a two-dimensional subspace of H ((0 , π ) , R ). This implies that the eigensphere S λ ∗ ofthe unperturbed problem is the geometric circle Ker( L − λ ∗ C ) ∩ S. Therefore,if x ∗ is any element of this circle, the connected component in Σ containing thecorresponding trivial solution (0 , λ ∗ , x ∗ ) does not satisfy the assertion of Theorem4.5. Thus, the assumption that the eigenpoint ( λ ∗ , x ∗ ) is simple cannot be removed.5.2. An application.
We close by showing how both Theorem 4.5 and the well-known notion of winding number allow us to deduce theoretically, without explicitlysolving the differential equation, that the structure of set Σ of solutions ( s, λ, x ) ofthe nonlinear boundary value problem (cid:26) x ′′ ( t ) + sg ( x ′ ( t )) + λx ( t ) = 0 ,x (0) = 0 = x ( π ) , x ∈ Sis essentially the same as in Example 5.2. Here g : R → R is an increasing odd C -function, as it is the classical air resistance force g ( v ) = v | v | , and the sphere S is as in Example 5.2. The parameter s , when positive, may be regarded as a frictionalcoefficient.The problem can be rewritten in the abstract form as follows:(5.5) (cid:26) Lx + sN ( x ) = λCxx ∈ S . The spaces G and H are the same as in Example 5.2, and so are the operators L and C . The compact map N : G → H sends x into the function N ( x ) : t g ( x ′ ( t )).It is not difficult to prove that N is C and its Fr´echet differential at x ∈ G is givenby dN x ( h ) : t g ′ ( x ′ ( t )) h ′ ( t ).The unperturbed problem is the same as in Example 5.2. Therefore, its eigen-values are λ = 1 , λ = 4 , . . . , λ n = n , . . . They are all simple and, consequently,each of them corresponds to a pair of isolated unit eigenpoints.We will prove that the set Σ of the solutions ( s, λ, x ) of (5.5) contains infinitelymany unbounded components, each of them corresponding to one and only oneeigenpoint.Let S denote the unit circle of C and let w : C ( S ) → Z stand for the wind-ing number function, defined on the set of the continuous maps from S into it-self. Recall that, given γ ∈ C ( S ), w( γ ) is the same as the Brouwer degree of γ and, speaking loosely, denotes the number of times that γ travels counterclockwisearound the origin of C , and it is negative if the curve travels clockwise.Call wj the integer valued function that to any non-zero solution x of the para-metrized differential equation(5.6) x ′′ ( t ) + sg ( x ′ ( t )) + λx ( t ) = 0 , depending on s, λ ∈ R , assigns the winding number wj( x ) of the closed curvej( x ) ∈ C ( S ) defined by z = e iθ (cid:0) x ′ ( θ/
2) + ix ′ (0) x ( θ/ (cid:1) x ′ ( θ/ + x ′ (0) x ( θ/ , θ ∈ [0 , π ] . Notice that, given any non-zero solution x of (5.6), j( x ) is well defined, since x ( t )and x ′ ( t ) cannot be simultaneously zero, due to the uniqueness of the Cauchyproblem. Observe also that j( x ) is a closed curve, since both the endpoints coincidewith 1 ∈ C .It is convenient to extend the map x j( x ) to the symmetric set of all thefunctions x ∈ G having the property that x ( t ) + x ′ ( t ) > t ∈ [0 , π ].We denote this set by X and we observe that it is open, because of the boundedinclusions H (0 , π ) ֒ → C [0 , π ] and H (0 , π ) ֒ → C [0 , π ].One can check that, for example, if x ( t ) = sin( nt ) with n ∈ Z , then j( x ) is themap z z n , whose winding number is n .One can also check that, if a is a positive constant and x ∈ X , then j( x ) andj( ax ) are homotopic, therefore they have the same winding number. Moreover, if x ∈ X , then j( − x ) = 1 / j( x ), which is the same as the conjugate map j( x ) of j( x ).Therefore, the winding numbers of j( x ) and j( − x ) are opposite each other, and thishappens for the two unit eigenvectors corresponding to any eigenvalue λ n = n ofour problem. In fact, this number is n for the unit eigenfunction x n ( t ) = r
21 + n sin( nt ) , DEGREE ASSOCIATED TO LINEAR EIGENVALUE PROBLEMS IN HILBERT SPACES 25 and − n for the opposite one.Observe that, due to the fact that X is open in G , if two functions of X aresufficiently close, then the segment joining them lies in X . Therefore, the corres-ponding two images under j : X → C ( S ) are homotopic and, consequently, theyhave the same winding number. Thus, the integer valued function wj : X → Z islocally constant.Since the projection map p : Σ → X that to any solution ( s, λ, x ) of (5.5) assignsthe function x is continuous, we have the following Remark 5.4.
The map wjp : Σ → Z that to any solution ( s, λ, x ) of (5.5) assignsthe winding number of the closed curve j( x ) is locally constant. Let q ∗ = (0 , λ ∗ , x ∗ ) be any trivial solution of (5.5). We want to prove that theconnected component D ∗ of Σ containing q ∗ is unbounded and does not meet othertrivial solutions.To this purpose, observe first that, since D ∗ is connected, Remark 5.4 implieswjp( q ) = wjp( q ∗ ) for all q ∈ D ∗ . In particular, if λ ∗ = λ n = n , then wjp( q ∗ ) is n or − n , depending on whether x ∗ is the above function x n or its opposite. Thus, D ∗ does not contain trivial solutions different from q ∗ , consequence of the fact thatthe function that to any trivial solution q assigns the integer wjp( q ) is injective.Finally, from Theorem 4.5 we get that D ∗ is unbounded, since otherwise D ∗ would contain a trivial solution different from q ∗ . References [1] Alexander J.C.,
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Pierluigi Benevieri - Instituto de Matem´atica e Estat´ıstica, Universidade de S˜aoPaulo, Rua do Mat˜ao 1010, S˜ao Paulo - SP - Brasil - CEP 05508-090 -
E-mail address: [email protected]
Alessandro Calamai - Dipartimento di Ingegneria Civile, Edile e Architettura, Uni-versit`a Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy -
E-mailaddress: [email protected]
Massimo Furi - Dipartimento di Matematica e Informatica “Ulisse Dini”, Univer-sit`a degli Studi di Firenze, Via S. Marta 3, I-50139 Florence, Italy -
E-mail address: [email protected]
Maria Patrizia Pera - Dipartimento di Matematica e Informatica “Ulisse Dini”, Uni-versit`a degli Studi di Firenze, Via S. Marta 3, I-50139 Florence, Italy -