Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case
Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera
aa r X i v : . [ m a t h . SP ] J a n GLOBAL PERSISTENCE OF THE UNIT EIGENVECTORS OFPERTURBED EIGENVALUE PROBLEMS IN HILBERT SPACES:THE ODD MULTIPLICITY CASE
PIERLUIGI BENEVIERI, ALESSANDRO CALAMAI, MASSIMO FURI,AND MARIA PATRIZIA PERA
Abstract.
We study the persistence of eigenvalues and eigenvectors of per-turbed eigenvalue problems in Hilbert spaces. We assume that the unperturbedproblem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result.The approach is topological, based on a notion of degree for oriented Fred-holm maps of index zero between real differentiable Banach manifolds. Introduction
Nonlinear spectral theory is a research field of increasing interest, which findsapplication to properties of the structure of the solution set of differential equations,see e.g. [1, 14]. In this context a nontrivial question consists in studying nonlinearperturbations of linear problems and in investigating the so-called “persistence” ofeigenvalues and eigenvectors.More precisely, let G and H denote two real Hilbert spaces. By a “perturbedeigenvalue problem” we mean a system of the following type:(1.1) (cid:26) Lx + sN ( x ) = λCxx ∈ S , where s, λ are real parameters, L, C : G → H are bounded linear operators, Sdenotes the unit sphere of G , and N : S → H is a nonlinear map. We call solution of (1.1) a triple ( s, λ, x ) ∈ R × R × S satisfying the above system. The element x ∈ Sis then said a unit eigenvector corresponding to the eigenpair ( s, λ ) of (4.1), andthe set of solutions of (1.1) will be denoted by Σ ⊆ R × R × S.To investigate the topological properties of Σ, we consider (1.1) as a (nonlinear)perturbation of the eigenvalue problem(1.2) (cid:26) Lx = λCxx ∈ S , Date : 11th January 2021.2010
Mathematics Subject Classification.
Key words and phrases. eigenvalues, eigenvectors, nonlinear spectral theory, topological de-gree, bifurcation.The first, second and fourth authors are members of the Gruppo Nazionale per l’Analisi Mate-matica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Mate-matica (INdAM).A. Calamai is partially supported by GNAMPA - INdAM (Italy). where we assume that the operator L − λC ∈ L ( G, H ) is invertible for some λ ∈ R .When λ ∈ R is such that Ker( L − λC ) is nontrivial, we call λ an eigenvalue of theequation L = λC or, equivalently, of problem (1.2). A solution ( λ, x ) of (1.2) willbe called an eigenpoint ; in this case λ and x are, respectively, an eigenvalue and a unit eigenvector of the equation Lx = λCx .Let ( λ ∗ , x ∗ ) be an eigenpoint of (1.2) and suppose that the following conditionshold:(H1) C is a compact operator,(H2) Ker( L − λ ∗ C ) is odd dimensional,(H3) Img( L − λ ∗ C ) ∩ C (Ker( L − λ ∗ C )) = { } .Under assumptions (H1)–(H3) our main result, Theorem 4.4 below, asserts that • in the set Σ of the solutions of (1.1) , the connected component containing (0 , λ ∗ , x ∗ ) is either unbounded or includes a trivial solution (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ . The proof of Theorem 4.4, which can be thought of as a Rabinowitz-type globalcontinuation result [21], is based on a preliminary study of the “unperturbed”problem (1.2). In particular, notice that the eigenpoints of (1.2) coincide with thesolutions of the equation ψ ( λ, x ) = 0 , where ψ is the H -valued function ( λ, x ) Lx − λCx defined on the cylinder R × S,which is a smooth 1-codimensional submanifold of the Hilbert space R × G . A crucialpoint is then to evaluate the topological degree of the map ψ . Since the domain of ψ is a manifold, we cannot apply the classical Leary–Schauder degree. Instead, weuse a notion of topological degree for oriented Fredholm maps of index zero betweenreal differentiable Banach manifolds, developed by two authors of this paper, andwhose construction and properties are summarized in Section 3 for the reader’sconvenience. Such a notion of degree has been introduced in [8] (see also [7, 9, 10]for additional details).Taking advantage of the odd multiplicity assumption (H2), of condition (H1) onthe compactness of C , and of the transversality condition (H3), we are then able toapply a result of [6] concerning the case of simple eigenvalues . Precisely, call λ ∗ ∈ R a simple eigenvalue of (1.2) if there exists x ∗ ∈ S such that Ker( L − λ ∗ C ) = R x ∗ and H = Img( L − λ ∗ C ) ⊕ R Cx ∗ . In [6] we proved that • if λ ∗ is a simple eigenvalue of (1.2) 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 . Moreover, under the assumption thatthe operator C is compact, they give the same contribution to the bf -degree,which is either or − , depending on the orientation of ψ . Such an assertion generalizes, to the infinite dimensional case, an analogous resultin [5] concerning a “classical eigenvalue problem” in R k . Let us point out thatthe result in [5] is based on the notion of Brouwer degree for maps between finitedimensional oriented manifolds, whereas, as already stressed, the extension to theinfinite-dimensional setting of [6] requires a degree for Fredholm maps of index zeroacting between Banach manifolds, as the one introduced in [8]. To apply this degreewe need the unit sphere S to be a smooth manifold: for this reason, we restrict ourstudy to Hilbert spaces instead of the more general Banach environment. LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 3
The study of the local [2, 13, 15–19] as well as global [3–7] persistence propertywhen the eigenvalue λ ∗ is not necessarily simple has been performed in recentpapers by the authors, also in collaboration with R. Chiappinelli. In particulara first pioneering result in this sense is due to Chiappinelli [12], who proved theexistence of the local persistence of eigenvalues and eigenvectors, in Hilbert spaces,in the case of a simple isolated eigenvalue.Among others, let us quote our paper [3] in which we tackled a problem verysimilar to the one we consider here. The main result of [3] regards, roughly speaking,the global persistence property of the eigenpairs ( s, λ ) of (1.1), in the sλ -plane,under the odd multiplicity assumption. Thus, the result we obtain here on theglobal persistence of the solutions ( s, λ, x ) of (1.1) was, in some sense, implicitlyconjectured in [3].The present paper generalizes the “global persistence” property of solution tripleswhich, either in finite-dimensional or infinite-dimensional case, has been studiedin [4–7] in the case of a simple eigenvalue. Since it is known that the persistenceproperty need not hold if λ ∗ is an eigenvalue of even multiplicity, it is natural toinvestigate the odd-multiplicity case. However such an extension is not trivial andis based on advanced degree-theoretical tools.We close the paper with some illustrating examples showing, in particular, thatthe odd dimensionality of Ker( L − λ ∗ C ) cannot be removed, the other assumptionsremaining valid. 2. Preliminaries
In this section we recall some notions that will be used in the sequel. We mainlysummarize some concepts which are needed for the construction of the topologicaldegree for oriented Fredholm maps of index zero between real differentiable Banachmanifolds introduced in [8], here called bf -degree to distinguish it from the Leray–Schauder degree, called LS -degree (see [7, 9, 10] for additional details).It is necessary to begin by focusing on the preliminary concept of orientation forFredholm maps of index zero between manifolds. The starting point is an algebraicnotion of orientation for Fredholm linear operators of index zero.Consider two real Banach spaces E and F and denote by L ( E, F ) the space of thebounded linear operators from E into F with the usual operator norm. If E = F ,we write L ( E ) instead of L ( E, E ). By Iso(
E, F ) we mean the subset of L ( E, F ) ofthe invertible operators, and we write GL( E ) instead of Iso( E, E ). The subspace of L ( E, F ) of the compact operators will be denoted by K ( E, F ), or simply by K ( E )when F = E . Finally, F ( E, F ) will stand for the vector subspace of L ( E, F ) of theoperators having finite dimensional image (recall that, in the infinite dimensionalcontext, F ( E, F ) is not closed in L ( E, F )). We shall write F ( E ) when F = E .Recall that an operator T ∈ L ( E, F ) is said to be
Fredholm (see e.g. [23]) 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 . P. BENEVIERI, A. CALAMAI, M. FURI, AND M.P. PERA
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 } . By Φ( E ) andΦ n ( E ) we will designate, respectively, Φ( E, E ) and Φ n ( E, E ).We recall some important properties of Fredholm operators.(F1) If T ∈ Φ( E, F ) , then Img T is closed in F . (F2) The composition of Fredholm operators is Fredholm and its index is the sumof the indices of all the composite operators. (F3) If T ∈ Φ n ( E, F ) and K ∈ K ( E, F ) , then T + K ∈ Φ n ( E, F ) . (F4) For any n ∈ Z , the set Φ n ( E, F ) is open in L ( E, F ) . Let T ∈ L ( E ) be given. If I − T ∈ F ( E ), where I ∈ L ( E ) is the identity, wesay that T is an admissible operator (for the determinant) . The symbol A ( E ) willstand for the affine subspace of L ( E ) of the admissible operators.It is known (see [20]) 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 check, the functiondet : A ( E ) → R inherits most of the properties of the classical determinant. Formore details, see e.g. [11].Let T ∈ Φ ( E, F ) be given. As in [7], we will say that an operator K ∈ F ( E, F )is a companion of T if T + K is invertible.Observe in particular that any T ∈ Iso(
E, F ) has a natural companion : that is,the zero operator 0 ∈ L ( E, F ). This fact was crucial in [8] for the construction ofthe bf -degree.Given T ∈ Φ ( E, F ), we denote by C ( T ) the (nonempty) subset of F ( E, F ) ofall the companions of T . The following definition establishes a partition of C ( T )in two equivalence classes and is a key step for the definition of orientation givenin [8]. Definition 2.1 (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. Definition 2.2 (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 2.3 (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 2.4 (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 whenever K and K are positive companions of T and T , respectively. In previous papers, e.g. in [8], it was used the word corrector instead of companion
LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 5
Observe that the oriented composition is associative and, consequently, this no-tion can be extended to the composition of three (or more) oriented operators.
Definition 2.5 (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.A crucial fact in the definition of oriented map and the consequent constructionof the bf -degree is that • the orientation of any operator T ∗ ∈ Φ ( E, F ) induces an orientation of theoperators in a neighborhood of T ∗ . In fact, since Iso(
E, F ) is open in L ( E, F ), for any companion K of T ∗ we have that T + K is invertible when T is sufficiently close to T ∗ . Thus, because of property(F3) of the Fredholm operators, any such T belongs to Φ ( E, F ). Consequently, K is as well a companion of T . Definition 2.6.
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 [9] for details). Therefore, an orientation of a map Γ as in Definition 2.6could be regarded as a lifting ˆΓ of Γ. This implies that, if the domain X of Γ issimply 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 , called Φ n -map and hereafteralso denoted by f ∈ Φ n , if df x ∈ Φ n ( E, F ) for all x ∈ U . Therefore, if f ∈ Φ ,Definition 2.6 and the continuity of the differential map df : U → Φ ( E, F ) suggestthe following
Definition 2.7 (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 , accordingto Definition 2.6. The map f is said to be orientable if it admits an orientation,and oriented if an orientation has been chosen. Remark 2.8.
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 and that inwhich T is seen as a C -map, according to Definitions 2.2 and 2.7, respectively.In each case T admits exactly two orientations (in the second one this is due tothe connectedness of the domain E ). Hereafter, we shall tacitly assume that the P. BENEVIERI, A. CALAMAI, M. FURI, AND M.P. PERA 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 . Let us summarize how the notion of orientation can be given for maps actingbetween real Banach manifolds. In the sequel, by manifold we shall mean, for short,a smooth Banach manifold embedded in a 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 (see [22]).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 2.10). Definition 2.9.
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 2.4) of thedifferentials in the chain representing the differential at x of the composite map.Assume that f : M → N is a C -diffeomorphism. Thus, for any x ∈ M , wemay take as ω ( x ) the natural orientation of df x (recall Definition 2.3). This pre-orientation of f turns out to be continuous according to Definition 2.10 below (itis, in some sense, constant). From now on, unless otherwise stated, • any diffeomorphism will be considered oriented with the naturalorientation .In particular, in a composition of pre-oriented maps, all charts and parametrizationsof a manifold will be tacitly assumed to be naturally oriented. Definition 2.10 (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 ζ : V → F of N , such that f ( U ) ⊆ V , the pre-orientedcomposition ζ ◦ f ◦ ϕ − : U → V is an oriented map according to Definition 2.7.The map f is said to be orientable if it admits an orientation, and oriented if anorientation has been chosen.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 2.3).In contrast, a very simple example of non-orientable Φ -map is given by a con-stant map from the 2-dimensional projective space into R (see [9]). LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 7
Notation 2.11.
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 } . Moreover, if f : D → Z is a map into a metric space Z , we denote by f x : D x → Z the partialmap of f defined by f x = f ( 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 2.12 (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 , ζ : 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 ( ζ ◦ h s ◦ ϕ − ) x = dζ h ( s,x ) d ( h s ) x ( dϕ x ) − is an orientation, according to Definition 2.6.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, one has the following Proposition 2.13 ( [8, 9]) . 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. As a consequence of Proposition 2.13, 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 bf -degree, introduced in [8], satisfies the three fundamental properties listedbelow: Normalization, Additivity and Homotopy Invariance . In [10], by means ofan axiomatic approach, it is proved that the bf -degree is the only possible integer-valued function that satisfies these three properties.More in detail, the bf -degree is defined in a class of admissible triples . Givenan oriented Φ -map f : M → N , an open (possibly empty) subset U of M , and atarget value y ∈ N , the triple ( f, U, y ) is said to be admissible for the bf -degreeprovided that U ∩ f − ( y ) is compact. From the axiomatic point of view, the bf -degree is an integer-valued function, deg bf , defined on the class of all the admissibletriples, that satisfies the following three fundamental properties . • (Normalization) If f : M → N is a naturally oriented diffeomorphism ontoan open subset of N , then deg bf ( f, M , y ) = 1 , ∀ y ∈ f ( M ) . P. BENEVIERI, A. CALAMAI, M. FURI, AND M.P. PERA • (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 bf ( f, U, y ) = deg bf ( f | U , U , y ) + deg bf ( 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 bf ( h ( s, · ) , M , γ ( s )) does not depend on s ∈ [0 , . Other useful properties are deduced from the fundamental ones (see [10] fordetails). Here we mention some of them. • (Localization) If ( f, U, y ) is an admissible triple, then deg( f, U, y ) = deg( f | U , U, y ) . • (Existence) If ( f, U, y ) is admissible and deg bf ( f, U, y ) = 0 , then the equa-tion f ( x ) = y admits at least one solution in U . • (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 ) . In some sense, given an admissible triple ( f, U, y ), the integer deg bf ( f, U, y ) isan algebraic count of the solutions in U of the equation f ( x ) = y . In fact, from thefundamental 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 bf ( f, U, y ) = X x ∈ U ∩ f − ( y ) sign( df x ) . Another useful property that can be deduced from the fundamental ones is the • (Topological Invariance) If ( f, U, y ) is admissible and g : N → O is a nat-urally oriented diffeomorphism onto a manifold O , then deg bf ( f, U, y ) = deg bf ( g ◦ f, U, g ( y )) . Some further notation and definitions are in order.
Notation 2.14.
Hereafter we will use the shorthand notation deg bf ( f, U ) insteadof deg bf ( f, U, f : M → F is an oriented Φ -map from a manifold into aBanach space, U is an open subset of M , and 0 is the null vector of F . Analogously,deg LS ( f, U ) means the Leray–Schauder degree deg LS ( f, U, U is an openbounded subset of a Banach space E , f : U → E is a compact vector field definedon the closure of U , and 0 is the null vector of E . Definition 2.15.
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 . The set U is called an isolatingneighborhood of K among (the elements of ) A . LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 9
Definition 2.16.
Let f : M → F be an oriented Φ -map from a manifold intoa Banach space. If K is an isolated subset of f − (0), we shall call contributionof K to the bf -degree of f the integer deg bf ( f, U ), where U ⊆ M is any isolatingneighborhood of K among f − (0). The excision property of the degree implies thatdeg bf ( f, U ) does not depend on the isolating neighborhood U .Regarding Definition 2.16, we observe that the finite union of isolated subsets of f − (0) is still an isolated subset. Moreover, from the excision and the additivityproperties of the bf -degree one gets that the contribution to the bf -degree of thisunion is the sum of the single contributions of these subsets.3. The eigenvalue problem and the associated topological degree
Let, hereafter, G and H denote two real Hilbert spaces and consider the eigen-value problem(3.1) (cid:26) Lx = λCxx ∈ S , where λ is a real parameter, L, C : G → H are bounded linear operators, and Sdenotes the unit sphere of G . To prevent the problem from being meaningless, • we will always assume that the operator L − λC ∈ L ( G, H ) is in-vertible for some λ ∈ R . When λ ∈ R is such that Ker( L − λC ) is nontrivial, then λ is called an eigenvalue of the equation L = λC or, equivalently, of problem (3.1).A solution ( λ, x ) of (3.1) will also be called an eigenpoint . In this case λ and x are, respectively, an eigenvalue and a unit eigenvector of the equation Lx = λCx .Notice that the eigenpoints are the solutions of the equation ψ ( λ, x ) = 0 , where ψ is the H -valued function ( λ, x ) Lx − λCx defined on the cylinder R × S,which is a smooth 1-codimensional submanifold of the Hilbert space R × G .By S we will denote the set of the eigenpoints of (3.1). Therefore, given any λ ∈ R , the λ -slice S λ = { x ∈ S : ( λ, x ) ∈ S} of S coincides with S ∩ Ker( L − λC ).Thus, S λ is nonempty if and only if λ is an eigenvalue of problem (3.1). Inthis case S λ will be called the eigensphere of (3.1) corresponding to λ or, simply,the λ -eigensphere . Observe that S λ is a sphere whose dimension equals that ofKer( L − λC ) minus one. The nonempty subset { λ }×S λ of the cylinder R × S willbe called an eigenset of (3.1).
Remark 3.1.
The assumption that L − λC is invertible for some λ ∈ R impliesthat, for any λ ∈ R , the restriction of C to the (possibly trivial) kernel of L − λC is injective. Remark 3.1 can be proved arguing by contradiction. In fact, assume that theassertion is false. Then, there are λ ∗ ∈ R and a nonzero vector x ∗ ∈ Ker( L − λ ∗ C ) ∩ Ker C. This implies that, for any λ , the operator L − λC is non-injective and, consequently,non-invertible, in contrast to the assumption. In fact, for any λ , one has( L − λC ) x ∗ = ( L − λ ∗ C ) x ∗ − ( λ − λ ∗ ) Cx ∗ = 0 . Remark 3.2.
If the operator C is compact, then, from the assumption that L − λC is invertible for some ˆ λ ∈ R , it follows that L − λC is Fredholm of index zero forany λ ∈ R and, consequently, the set of the eigenvalues of problem (3.1) is discrete.Moreover, Ker( L − λC ) is always finite dimensional, and so is the intersection Img( L − λC ) ∩ C (Ker( L − λC )) . Consequently, if this intersection is the singleton { } , taking into account Remark3.1 and the fact that L − λC ∈ Φ ( G, H ) , one has H = Img( L − λC ) ⊕ C (Ker( L − λC )) . To prove Remark 3.2 notice that, if L − ˆ λC is invertible, then it is triviallyFredholm of index zero. Now, given any λ ∈ R , one has( L − λC ) = ( L − ˆ λC ) − ( λ − ˆ λ ) C. Thus, because of the compactness of C , from property (F3) of Fredholm operators,one gets that L − λC is also Fredholm of index zero. Finally, the set of the eigen-values of problem (3.1) is discrete since so is, according to the spectral theory oflinear operators, the set of the characteristic values of ( L − ˆ λC ) − C .Because of Remark 3.2, • from now until the end of this section we assume that the operator C is compact .Observe that the function ψ defined above is the restriction to R × S of thenonlinear smooth map ψ : R × G → H, ( λ, x ) Lx − λCx. According to Remark 3.2, any partial map ψ λ : G → H of ψ is Fredholm of indexzero. Since the map σ : R × G → G given by σ ( λ, x ) = x is clearly Φ , the sameholds true, because of the property (F2) of Fredholm operators, for the composition ψ = ψ λ ◦ σ . Consequently, again because of property (F2), one has that therestriction ψ of ψ to the 1-codimensional submanifold R × S of R × G is Φ .Notice that, if dim G = 1, the cylinder R × S is disconnected: it is the union oftwo horizontal lines, R ×{− } and R ×{ } . Because of this, to make some statementssimpler, • from now on, unless otherwise stated, we assume that the dimen-sion of the space G is greater than R × S is connected, and simply connected if dim
G > G is infinite dimensional. Therefore, the Φ -map ψ ,defined above, is orientable and admits exactly two orientations. We choose one ofthem and • hereafter we assume that ψ is oriented . Remark 3.3.
Let ˆ λ ∈ R be such that L − ˆ λC is invertible and let Z : H → G denote its inverse. Then, given any λ ∈ R , the two equations • ψ λ ( x ) = ( L − λC ) x = 0 ∈ H , • η λ ( x ) = Zψ λ ( x ) = ( I − ( λ − ˆ λ ) ZC ) x = 0 ∈ G LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 11 are equivalent ( I being the identity on G ). Therefore, if B denotes the unit ball of G ,the Leray–Schauder degree with target ∈ G , deg LS ( η λ , B ) , of the compact vectorfield η λ is well defined whenever λ is not an eigenvalue of the equation Lx = λCx . Observe that, as a consequence of the homotopy invariance property of theLeray–Schauder degree, the function λ deg LS ( η λ , B ) is constant on any intervalin which it is defined. Moreover, in these intervals, deg LS ( η λ , B ) is either 1 or − η λ ( x ) = 0 has only one solution: the regular point 0 ∈ G . Remark 3.4.
Let U be an isolating neighborhood of a compact subset of the set S of the eigenpoints of (3.1) , and let Z : H → G be as in Remark 3.3. Then deg bf ( ψ, U ) = deg bf ( η, U ) , provided that the map η = Zψ is the oriented composi-tion obtained by considering Z as a naturally oriented diffeomorphism. Concerning possible relations between the LS -degree of η λ and the bf -degree of ψ (or, equivalently, of η = Zψ ), we believe that the following is true (but up tonow we were unable to prove or disprove). Conjecture 3.5.
Let [ α, β ] be a compact (nontrivial) real interval such that theextremes are not eigenvalues of Lx = λCx . Then the bf -degree of ψ (or, equival-ently, of η = Zψ ) on the open subset U = ( α, β ) × S of R × S is different from zeroif and only if deg LS ( η α , B ) = deg LS ( η β , B ) . In support of the above conjecture we observe that both the conditionsdeg bf ( ψ, U ) = 0 and deg LS ( η α , B ) = deg LS ( η β , B )imply the existence of at least one eigenpoint p ∗ = ( λ ∗ , x ∗ ) ∈ U . The first onebecause of the existence property of the bf -degree and the last one due to thehomotopy invariance property of the LS -degree. Definition 3.6.
An eigenpoint ( λ ∗ , x ∗ ) of (3.1) is said to be simple provided thatthe operator T = L − λ ∗ C is Fredholm of index zero and satisfies the conditions:(1) Ker T = R x ∗ ,(2) Cx ∗ / ∈ Img T .We point out that, if an eigenpoint p ∗ = ( λ ∗ , x ∗ ) is simple, then the correspond-ing eigenset { λ ∗ }×S λ ∗ is disconnected. In fact, it has only two elements: p ∗ andits twin eigenpoint ¯ p ∗ = ( λ ∗ , − x ∗ ), which is as well simple.The following theorem obtained in [7] was essential in the proofs of some resultsin [7] concerning perturbations of (3.1), as problem (4.1) in the next section. Theorem 3.7.
In addition to the compactness of C , assume that p ∗ = ( λ ∗ , x ∗ ) and ¯ p ∗ = ( λ ∗ , − x ∗ ) are two simple twin eigenpoints of (3.1) . Then, the contributions of p and ¯ p to the bf -degree of ψ are equal: they are both either or − depending onthe orientation of ψ . Consequently, if U is an isolating neighborhood of the eigenset { λ ∗ }×S λ ∗ , one has deg bf ( ψ, U ) = ± . We close this section strictly devoted to the unperturbed eigenvalue problem(3.1) with a consequence of Theorem 3.7, which will be crucial in the proof of ourmain result (Theorem 4.4 in Section 4).
Theorem 3.8.
Let λ ∗ ∈ R , put T = L − λ ∗ C , and suppose that (H1) C is a compact operator, (H2) Ker T is odd dimensional, (H3) Img T ∩ C (Ker T ) = { } .Then, given (in R × S ) an isolating neighborhood U of the eigenset { λ ∗ }×S λ ∗ , onehas deg bf ( ψ, U ) = 0 .Proof. Because of the assumption Img T ∩ C (Ker T ) = { } , as well as the fact that T is Fredholm of index zero, we can split the spaces G and H as follows: G = G ⊕ G with G = (Ker T ) ⊥ and G = Ker T ; H = H ⊕ H with H = Img T and H = C (Ker T ) . With these splittings, T and C can be represented in block matrix form as follows: T = T
00 0 ! , C = C C C ! . The operators T : G → H and C : G → H are isomorphisms (the second onebecause of Remark 3.1), while C : G → H and C : G → H are, respectively,compact and finite dimensional.We can equivalently regard the equation ψ ( λ, x ) = 0 as Zψ ( λ, x ) = 0, where Z : H → G is an isomorphism. We choose Z as follows: Z = T − C − . Given any λ ∈ R , the operator ψ λ = L − λC ∈ L ( G, H ) can be written as T − ( λ − λ ∗ ) C . Therefore, putting η = Zψ : R × G → G , the partial map η λ : G → G (see Notation 2.11) can be represented as η λ = I − ( λ − λ ∗ ) b C − ( λ − λ ∗ ) b C λ ∗ I − λI , where I is the identity on G = G ⊕ G and b C = ZC (observe that b C coincideswith the identity I ∈ L ( G )).This shows that, given any λ ∈ R , the endomorphism η λ : G → G is a compactvector field. Therefore, its Leray–Schauder degree on the unit ball B of G is welldefined whenever λ is not an eigenvalue of the equation Lx = λCx , and this happenswhen λ is close to, but different from, λ ∗ . Since G is odd dimensional and, becauseof assumption (H3), the geometric and algebraic multiplicities of λ ∗ coincide, thefunction λ deg LS ( η λ , B ) has a sign-jump crossing λ ∗ . Therefore, if Conjecture3.5 were true, we would have done. So we need to proceed differently.We consider an isolating neighborhood of the eigenset { λ ∗ } × S λ ∗ of the type U = ( α, β ) × S and we approximate the family of operators η λ , λ ∈ [ α, β ], with afamily η ελ ∈ L ( G ), λ ∈ [ α, β ], having in ( α, β ) only simple eigenvalues; the numberof them equal to the dimension of G = Ker T .First of all we point out that • the operator I − ( λ − λ ∗ ) b C ∈ L ( G ) is invertible for all λ ∈ [ α, β ],since otherwise the equation Lx = λCx would have eigenvalues different from λ ∗ in the interval [ α, β ]. LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 13
Now, given ε > λ ∗ − ε, λ ∗ + ε ) ⊂ ( α, β ), we choose a linear operator A ε ∈ L ( G ) with the following properties: • in the operator norm, the distance between A ε and λ ∗ I is less than ε , • the eigenvalues of A ε are real and simple, • any eigenvalue λ of A ε is such that | λ − λ ∗ | < ε .For any λ ∈ R we define η ελ ∈ L ( G ⊕ G ) by η ελ = I − ( λ − λ ∗ ) b C − ( λ − λ ∗ ) b C A ε − λI . Then, any eigenvalue λ of A ε is as well an eigenvalue of the equation η ελ ( x ) = 0, andviceversa provided that λ ∈ [ α, β ]. Therefore, η ελ ( x ) = 0 has exactly n = dim( G )simple eigenvalues in the interval ( α, β ). Consequently, the function η ε : R × S → G, ( λ, x ) η ελ ( x )has exactly n eigensets in the open subset U = ( α, β ) × S of the cylinder R × S, all ofthem corresponding to a simple eigenvalue. Therefore, according to Theorem 3.7,the contribution of each of them to deg bf ( η ε , U ) is either 2 or −
2. Consequently,taking into account that n is odd, one gets deg bf ( η ε , U ) = 0.Let the isomorphism Z be naturally oriented and let the restriction η of η to themanifold R × S be oriented according to the composition Zψ . Thus, because of thetopological invariance property of the bf -degree, we getdeg bf ( η, U ) = deg bf ( ψ, U ) . Hence, it remains to show that, if ε > bf ( η ε , U ) = deg bf ( η, U ) . In fact, this is a consequence of the homotopy invariance property of the bf -degree.To see this it is sufficient to show that (if ε is small) the homotopy h : [0 , × U → G ,defined by h ( t, λ, x ) = tη ε ( λ, x ) + (1 − t ) η ( λ, x ), is admissible. That is, h ( t, λ, x ) = 0 for t ∈ [0 ,
1] and ( λ, x ) ∈ ∂U = { ( λ, x ) ∈ [ α, β ] × S : λ = α or λ = β } . Let us prove that this is true for the left boundary of U ; that is, for λ = α . Theargument for λ = β will be the same.We need to show that (if ε is small) the linear operator A t = tη εα + (1 − t ) η α of L ( G ) is invertible for any t ∈ [0 , A = η α is invertible, and the setof the invertible operators of L ( G ) is open, this holds true for all A t provided that ε is sufficiently small. (cid:3) The perturbed eigenvalue problem and global continuation
Here, as in Section 3, G and H denote two real Hilbert spaces, L, C : G → H arebounded linear operators, S is the unit sphere of G and, as in problem (3.1), theoperator L − λC is invertible for some λ ∈ R .Consider the perturbed eigenvalue problem(4.1) (cid:26) Lx + sN ( x ) = λCxx ∈ S , where N : S → H is a C compact map and s is a real parameter. A solution of (4.1) is a triple ( s, λ, x ) ∈ R × R × S satisfying (4.1). The element x ∈ S is a unit eigenvector corresponding to the eigenpair ( s, λ ).The set of solutions of (4.1) will be denoted by Σ and E is the subset of R ofthe eigenpairs. Notice that E is the projection of Σ into the sλ -plane and the s = 0slice Σ of Σ is the same as the set S = ψ − (0) of the eigenpoints of (3.1), where ψ has been defined in the previous section.A solution ( s, λ, x ) of (4.1) is regarded as trivial if s = 0. In this case p = ( λ, x )is the corresponding eigenpoint of problem (3.1). When p is simple, the triple(0 , λ, x ) ∈ Σ will be as well said to be simple . A nonempty subset of Σ of the type { }×{ λ }×S λ will be called a solution-sphere .We consider the subset { ( s, λ, x ) ∈ Σ : s = 0 } = { }× Σ = { }×S of the trivialsolutions of Σ as a distinguished subset . Thus, it makes sense to call a solution q ∗ = (0 , λ ∗ , x ∗ ) of (4.1) a bifurcation point if any neighborhood of q ∗ in Σ containsnontrivial solutions.We say that a bifurcation point q ∗ = (0 , λ ∗ , x ∗ ) is global (in the sense of Ra-binowitz [21]) if in the set of nontrivial solutions there exists a connected com-ponent, called global (bifurcating) branch , whose closure in Σ contains q ∗ and it iseither unbounded or includes a trivial solution q ∗ = (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ . Inthe second case q ∗ is as well a global bifurcation point.A meaningful case is when a bifurcation point q ∗ = (0 , λ ∗ , x ∗ ) belongs to aconnected solution-sphere { } × { λ ∗ } × S λ ∗ . In this case the dimension of S λ ∗ ispositive and we will simply say that x ∗ is a bifurcation point. In fact, 0 and λ ∗ being known, x ∗ can be regarded as an alias of q ∗ .For a necessary condition as well as some sufficient conditions for a point x ∗ ofa connected eigensphere to be a bifurcation point see [15]. Other results regardingthe existence of bifurcation points belonging to even-dimensional eigenspheres canbe found in [2–7, 16, 17, 19].As already pointed out, if the operator C is compact, then ψ : R × S → H isFredholm of index zero, and this is crucial for the global results regarding theperturbed eigenvalue problem (4.1). Because of this, • from now on, unless otherwise stated, we will tacitly assume thatthe linear operator C is compact .We define the C -map ψ + : R × R × S → H, ( s, λ, x ) ψ ( λ, x ) + sN ( x ) , in which ψ : R × S → H , as in Section 3, is given by ψ ( λ, x ) = Lx − λCx . Thereforethe set ( ψ + ) − (0) of the zeros of ψ + coincides with Σ.As shown in [7], because of the compactness of C and N , one gets that • ψ + is proper on any bounded and closed subset of its domain. Consequently, any bounded connected component of Σ is compact. This fact willbe useful later.Notice that ψ + is the restriction to the manifold R × R × S of the nonlinear map ψ + : R × R × G → H, ( s, λ, x ) ψ ( λ, x ) + sN ( x ) , where ψ is as in Section 3 and N is the positively homogeneous extension of N .The following result of [7] is crucial for proving the existence of global bifurcationpoints. LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 15
Theorem 4.1.
Given an open subset Ω of R × R × S , let Ω = (cid:8) ( λ, x ) ∈ R × S : (0 , λ, x ) ∈ Ω (cid:9) be its -slice. If deg bf ( ψ, Ω ) is well defined and nonzero, then Ω contains a con-nected set of nontrivial solutions whose closure in Ω is non-compact and meets atleast one trivial solution of (4.1) . Corollary 4.2 below, which was deduced in [7] from Theorem 4.1, asserts that thecontribution to the bf -degree of the 0-slice of any compact (connected) componentof Σ is null. We will need this basic property later. Corollary 4.2.
Let D be a compact component of Σ , and let D ⊂ R × S be its(possibly empty) -slice. Then, if U ⊂ R × S is an isolating neighborhood of D , onehas deg bf ( ψ, U ) = 0 . The following result, obtained in [7, Theorem 4.5], regards the existence of aglobal branch of solutions emanating from a trivial solution of problem (4.1) whichcorresponds to a simple eigenpoint of (3.1).
Theorem 4.3. If ( λ ∗ , x ∗ ) is a simple eigenpoint of problem (3.1) , then, in the set Σ of the solutions of (4.1) , the connected component containing (0 , λ ∗ , x ∗ ) is eitherunbounded or includes a trivial solution (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ . We are now ready to prove our main result, which extends Theorem 4.3 andprovides a global version of Theorem 3.9 in [19], the latter concerning the existenceof local bifurcation points belonging to even dimensional eigenspheres.
Theorem 4.4.
In addition to the compactness of C , let ( λ ∗ , x ∗ ) be an eigenpointof (3.1) and denote by T the non-invertible operator L − λ ∗ C . Assume that • Ker T is odd dimensional, • Img T ∩ C (Ker T ) = { } .Then, in the set Σ of the solutions of (4.1) , the connected component containing (0 , λ ∗ , x ∗ ) is either unbounded or includes a trivial solution (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ .Proof. Because of the compactness of C , according to Remark 3.2, the operator L − λC is Fredholm of index zero for all λ ∈ R . Moreover, the set of the eigenvaluesof problem (3.1) is discrete. Consequently, the eigenset { λ ∗ }×S λ ∗ , which is compactand nonempty, is relatively open in the set S of the eigenpoints. Thus, it admitsan isolating neighborhood U ⊂ R × S and, therefore, deg bf ( ψ, U ) is well defined.Denote by D the connected component of Σ containing (0 , λ ∗ , x ∗ ). We mayassume that D is bounded. Thus, it is actually compact, since ψ + is proper on anybounded and closed subset of R × R × S. We need to prove that D contains a trivialsolution (0 , λ ∗ , x ∗ ) with λ ∗ = λ ∗ .Assume, by contradiction, that this is not the case. Then the 0-slice D of D iscontained in the eigenset { λ ∗ }×S λ ∗ . We will show that this contradicts Corollary4.2. We distinguish two cases: n = 1 and n >
1, where n is the dimension of Ker T . Case n = 1 . Because of the assumption Img T ∩ C (Ker T ) = { } , the eigenpoint p ∗ = ( λ ∗ , x ∗ ) is simple and { λ ∗ } × S λ ∗ has only two points: p ∗ = ( λ ∗ , x ∗ ) and¯ p ∗ = ( λ ∗ , − x ∗ ). In this case, according to Theorem 3.7, the contribution to the bf -degree of any subset of { λ ∗ }×S λ ∗ is different from zero, and this, having assumed D ⊆ { λ ∗ }×S λ ∗ , is incompatible with Corollary 4.2. Case n > . The solution-sphere { }×{ λ ∗ }×S λ ∗ is connected and, consequently,it is contained in the component D of Σ. Thus, the eigenset { λ ∗ }×S λ ∗ is containedin the slice D of D . Having assumed D ⊆ { λ ∗ } × S λ ∗ , we get D = { λ ∗ } × S λ ∗ .Hence, because of Theorem 3.8, given an isolating neighborhood U of D , one getsdeg bf ( ψ, U ) = 0, and we obtain a contradiction with Corollary 4.2. (cid:3) Remark 4.5.
Under the notation and assumptions of Theorem 4.4 suppose, inaddition, that dim(Ker T ) > . Then, the connected component D containing (0 , λ ∗ , x ∗ ) contains as well the connected solution-sphere D ∗ = { }×{ λ ∗ }×S λ ∗ .This implies that there exists at least one point ˆ q = (0 , λ ∗ , ˆ x ) ∈ D ∗ which isin the closure D \ D ∗ of the difference D \ D ∗ . Thus, ˆ q (or, equivalently, its alias ˆ x ∈ S λ ∗ ) is a global bifurcation point. Some illustrating examples
In this section we provide three examples in ℓ concerning Theorem 4.4. Thedimensions of Ker T (where T = L − λ ∗ C ) are, respectively, 3, 2, and 1. The secondexample, in which Ker T is two dimensional, shows that in Theorem 4.4, as wellas in Remark 4.5, the hypothesis of the odd dimensionality of Ker T cannot beremoved, the other assumptions remaining valid.Given a positive integer k , let T k ∈ L ( ℓ ) be the bounded linear operator thatto any x = ( ξ , ξ , ξ , . . . ) ∈ ℓ associates the element T k x = (0 , , . . . , , ξ k +1 , ξ k +2 , . . . ) , in which the first k components are 0. Notice that T k is Fredholm of index zeroand its kernel is the k -dimensional spaceKer T k = { x ∈ ℓ : x = ( ξ , ξ , . . . , ξ k , , , . . . ) } , which is orthogonal to Img T k .Hereafter, C will be the well-known compact linear operator defined by( ξ , ξ , ξ , . . . ) ( ξ / , ξ / , ξ / , . . . , ξ n /n, . . . ) . Given any compact (possibly nonlinear) map N : ℓ → ℓ of class C , considerthe perturbed eigenvalue problem(5.1) (cid:26) T k x + sN ( x ) = λCx,x ∈ S , where S is the unit sphere of ℓ . As before, we denote by Σ the set of solutions( s, λ, x ) of (5.1).Observe that, for any k ∈ N , k ≥ λ ∗ = 0 is an eigenvalue of the unperturbedequation T k x = λCx and the condition Img T k ∩ C (Ker T k ) = { } is satisfied.Therefore, according to Theorem 4.4, given any positive odd integer k , any compactperturbing map N : ℓ → ℓ of class C , and any x ∗ ∈ S ∩ Ker T k , the connectedcomponent of Σ containing (0 , , x ∗ ) is either unbounded or encounters a trivialsolution (0 , λ ∗ , x ∗ ) with λ ∗ = 0.In the three examples below we will check whether or not the assertions ofTheorem 4.4 and Remark 4.5 hold, by taking, for all of them, the same perturbingmap. Namely, N : ℓ → ℓ , ( ξ , ξ , ξ , ξ , . . . ) ( − ξ , ξ , − ξ , ξ , , , , . . . ) . LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 17
Example 5.1 ( k = 3) . The eigenvalues of the unperturbed equation T x = λCx are 0 , , , , . . . The first one, λ ∗ = 0, has geometric and algebraic multiplicity 3and all the other eigenvalues are simple.A standard computation shows that, in the sλ -plane, the set E of the eigenpairshas a connected subset E satisfying the equation 3 s +( λ − / , , ξ , ξ , , , ... ). The set E is an ellipse with center(0 ,
2) and half-axes 1 / √ s, λ ) =(0 , λ = 5, λ = 6, λ = 7, etc. Thus, the connected component in Σ containing any trivial solution(0 , λ, x ) with eigenvalue λ ≥ s = (1 / √
3) sin θ , λ = 2(1 − cos θ ), θ ∈ [0 , π ], and for any θ in the open interval (0 , π ), the kernel of the equation T x + (1 / √
3) sin θN x − − cos θ ) Cx = 0is 1-dimensional and spanned by the vector x ( θ ) = (0 , , (1 / √
3) sin θ, − (2 / − cos θ ) , , , . . . ) . Since E is bounded, so is the connected component D of Σ containing the2-dimensional solution-sphere D ∗ = { } × { λ ∗ } × S λ ∗ (recall that λ ∗ = 0). As weshall see, D includes the twin trivial solutions (0 , λ ∗ , ± x ∗ ), where λ ∗ = 4 and x ∗ = x ( π ) / k x ( π ) k = (0 , , , , , . . . ) . According to Remark 4.5, there exists at least one bifurcation point ˆ x ∈ S λ ∗ .Actually, in this case one gets exactly two (global) bifurcation points. This is dueto the fact that D \ D ∗ has two disjoint “twin” branches whose closures meet thesolution-sphere D ∗ . The branches can be parametrized with θ ∈ (0 , π ) as follows: q ( θ ) = (cid:0) (1 / √
3) sin θ, − cos θ ) , x ( θ ) / k x ( θ ) k (cid:1) , ¯ q ( θ ) = (cid:0) (1 / √
3) sin θ, − cos θ ) , − x ( θ ) / k x ( θ ) k (cid:1) . Then, if the following limits exist:lim θ → q ( θ ) and lim θ → ¯ q ( θ ) , we get the bifurcation points (as elements of D ∗ ). Equivalently, to find the aliasesof these points (that is, the corresponding elements in the eigensphere S λ ∗ ) wecompute ± lim θ → ( x ( θ ) / k x ( θ ) k )obtaining ± (0 , , , , , . . . ) ∈ S λ ∗ . In fact, to compute the limits, observe thatsin θ = u s ( θ ) θ and 2(1 − cos θ ) = u c ( θ ) θ , where u s and u c are continuous functionssuch that u s (0) = u c (0) = 1. Hence, one quickly obtainslim θ → (1 / √
3) sin θ q (1 /
3) sin θ + (4 / − cos θ ) = 1 , lim θ → − (2 / − cos θ ) q (1 /
3) sin θ + (4 / − cos θ ) = 0 . Example 5.2 ( k = 2) . The eigenvalues of the unperturbed equation T x = λCx are 0 , , , , , . . . The first one, λ ∗ = 0, has geometric and algebraic multiplicity2 and all the others are simple. As in Example 5.1, for any eigenvalue λ ≥
5, one gets an horizontal line ofeigenpairs containing (0 , λ ). Moreover, as one can check, the trivial eigenpairs(0 ,
3) and (0 ,
4) are vertices of an ellipse of eigenpairs with center (0 , /
2) and half-axes 1 / √
48 and 1 /
2, corresponding, as in Example 5.1, to eigenvectors of the type(0 , , ξ , ξ , , , ... ). However, in a neighborhood of the origin of the sλ -plane thereare no eigenpairs, except the isolated one (0 , solution-circle D ∗ = { }×{ λ ∗ }×S λ ∗ is an isolated subset of Σ. Therefore, the assertions of Theorem4.4 and Remark 4.5 do not hold in this case. Moreover, according to Corollary 4.2,the contribution of D ∗ to the bf -degree of the map ψ is zero.In conclusion, in Theorem 4.4 and Remark 4.5, the assumption that Ker T isodd dimensional cannot be removed. Example 5.3 ( k = 1) . In this case the eigenvalues of the unperturbed problemare 0 , , , , , . . . All of them are simple. As in the previous two examples, the sλ -plane contains infinitely many horizontal lines of eigenpairs. Their equationsare λ = 5, λ = 6, λ = 7, . . .In addition to the horizontal lines, the set of the eigenpairs has two boundedcomponents: an ellipse with center (0 ,
1) and half-axes 1 / √ ,
0) and (0 , ,
3) with (0 , , /
2) and half-axes 1 / √
48 and 1 / { }×{ }×S , its connected componentin Σ is bounded and contains a point of { }×{ }×S . This agrees with Theorem 4.4. References [1] Appell J. - De Pascale E. - Vignoli A.,
Nonlinear spectral theory , de Gruyter, Berlin, 2004.[2] Benevieri P. - Calamai A. - Furi M. - Pera M.P.,
On the persistence of the eigenvalues of aperturbed Fredholm operator of index zero under nonsmooth perturbations , Z. Anal. Anwend. (2017), no. 1, 99–128.[3] Benevieri P. - Calamai A. - Furi M. - Pera M.P., Global continuation of the eigenvalues of aperturbed linear operator , Ann. Mat. Pura Appl. (2018), no. 4, 1131–1149.[4] Benevieri P. - Calamai A. - Furi M. - Pera M.P.,
Global continuation in Euclidean spacesof the perturbed unit eigenvectors corresponding to a simple eigenvalue , Topol. MethodsNonlinear Anal. (2020), no. 1, 169–184.[5] Benevieri P. - Calamai A. - Furi M. - Pera M.P., Global persistence of the unit eigenvectors ofperturbed eigenvalue problems in Hilbert spaces , Z. Anal. Anwend. (2020), no. 4, 475–497.DOI: 10.4171/ZAA/1669.[6] Benevieri P. - Calamai A. - Furi M. - Pera M.P., The Brouwer degree associated to classicaleigenvalue problems and applications to nonlinear spectral theory , preprint arXiv:1912.03182.[7] Benevieri P. - Calamai A. - Furi M. - Pera M.P.,
A degree associated to linear eigenvalueproblems in Hilbert spaces and applications to nonlinear spectral theory , to appear in J.Dynamics and Differential Equations.[8] Benevieri P. - Furi M.,
A simple notion of orientability for Fredholm maps of index zerobetween Banach manifolds and degree theory,
Ann. Sci. Math. Qu´ebec, (1998), 131–148.[9] Benevieri P. - Furi M., On the concept of orientability for Fredholm maps between real Banachmanifolds,
Topol. Methods Nonlinear Anal. (2000), 279–306.[10] Benevieri P. - Furi M., On the uniqueness of the degree for nonlinear Fredholm maps of indexzero between Banach manifolds,
Commun. Appl. Anal. (2011), 203–216.[11] Benevieri P. - Furi M. - Pera M.P. - Spadini M., About the sign of oriented Fredholm operatorsbetween Banach spaces , Z. Anal. Anwendungen (2003), no. 3, 619–645.[12] Chiappinelli R., Isolated Connected Eigenvalues in Nonlinear Spectral Theory , NonlinearFunct. Anal. Appl. (2003), no. 4, 557–579. LOBAL PERSISTENCE OF THE UNIT EIGENVECTORS 19 [13] Chiappinelli R.,
Approximation and convergence rate of nonlinear eigenvalues: Lipschitzperturbations of a bounded self-adjoint operator , J. Math. Anal. Appl. (2017), no. 2,1720–1732.[14] Chiappinelli R.,
What do you mean by “nonlinear eigenvalue problems”? , Axioms (2018),Paper no. 39, 30 pp.[15] Chiappinelli R. - Furi M. - Pera M.P., Normalized Eigenvectors of a Perturbed Linear Oper-ator via General Bifurcation , Glasg. Math. J. (2008), no. 2, 303–318.[16] Chiappinelli R. - Furi M. - Pera M.P., Topological persistence of the normalized eigenvectorsof a perturbed self-adjoint operator , Appl. Math. Lett. (2010), no. 2, 193–197.[17] Chiappinelli R. - Furi M. - Pera M.P., A new theme in nonlinear analysis: continuationand bifurcation of the unit eigenvectors of a perturbed linear operator , Communications inApplied Analysis (2011), no. 2, 3 and 4, 299–312.[18] Chiappinelli R. - Furi M. - Pera M.P., Persistence of the normalized eigenvectors of a per-turbed operator in the variational case , Glasg. Math. J. (2013), no. 3, 629–638.[19] Chiappinelli R. - Furi M. - Pera M.P., Topological Persistence of the Unit Eigenvectors of aPerturbed Fredholm Operator of Index Zero , Z. Anal. Anwend. (2014), no. 3, 347–367.[20] Kato T., Perturbation Theory for Linear Operators , Classics in Mathematics, Springer-Verlag, Berlin, 1995.[21] Rabinowitz P.H.,
Some global results for nonlinear eigenvalue problems , J. Funct. Anal. (1971), 487–513.[22] Smale S., An infinite dimensional version of Sard’s theorem,
Amer. J. Math. (1965),861–866.[23] Taylor A.E. - Lay D.C., Introduction to Functional Analysis , John Wiley & Sons, New York-Chichester-Brisbane, 1980.
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 -