What Do You Mean by “Nonlinear Eigenvalue Problems”?

Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, I-53100 Siena, Italy

Axioms2018,7(2), 39;https://doi.org/10.3390/axioms7020039

Submission received: 30 March 2018 /Revised: 5 June 2018 /Accepted: 6 June 2018 /Published: 9 June 2018

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Abstract

A nonlinear eigenvalue problem is generally described by an equation of the form

F (λ, x) = 0

, where

F (λ, 0) = 0

for all

λ

, and contains by definition two unknowns: the eigenvalue parameter

λ

and the “nontrivial” vector(s)x corresponding to it. The nonlinear dependence ofF can be in either of them (and of course in both), and also the research in this area seems to follow two quite different directions. In this review paper, we try to collect some points of possible common interest for both fields.

Keywords:

eigenspace;algebraic multiplicity;bifurcation;bifurcation

MSC:

Primary 47J10; Secondary 47A56

1. Introduction

Nonlinear eigenvalue problems are generally described by equations of the form

F (λ, x) = 0 (λ \in K, x \in E)

(1)

where

K (= R

C

) is the field of real or complex numbers, andE is a real or complex Banach space that can in particular be then-space

R^{n}

C^{n}

. In Equation (1),F is a continuous map of

K \times E

intoE, and it is assumed that

F (λ, 0) = 0

for all scalars

λ

. That is to say,

x = 0

solves trivially Equation (1) for all

λ

; and one looks therefore for those

λ

’s (theeigenvalues ofF) such that Equation (1) has a solution

x \neq 0

(aneigenvector ofF corresponding to

λ

Of course, Equation (1) contains as a (very) special case the proper eigenvalue-eigenvector equation of Linear Algebra and Linear Functional Analysis,

F (λ, x) = A x - λ x = (A - λ I) x = 0

(2)

in which

A \in L (E)

, the space af all bounded linear operators acting inE andI is the identity map; to stress the linearity ofA, we write as usual

A x

rather than

A (x)

. In addition to Equation (2), consider now the following special forms of Equation (1):

F (λ, x) = G (λ) x = 0,

(3)

and

F (λ, x) = A (x) - λ C (x) = 0 .

(4)

Evidently, both Equations (3) and (4) encompass the classical case Equation (2). However, there is quite a difference between them: in Equation (3),F depends linearly onx and arbitrarily in

λ

, the latter dependence being driven by a map

G : K \to L (E)

, while, in Equation (4), it is rather the opposite, for here it is the dependence onx that is (possibly) nonlinear as dictated by the continuous maps

A, C : E \to E

. One first consequence is that the termseigenvalue/eigenvector/eigenspace retain their usual significance in the case of Equation (3), while on the contrary they have in general a poor meaning in the case of Equation (4). On the other hand, in the latter case, assuming that

C (x) \neq 0

for

x \neq 0

, the eigenvalue associated with an eigenvector is uniquely determined, for

A (\hat{x}) - λ_{1} C (\hat{x}) = A (\hat{x}) - λ_{2} C (\hat{x})

with

\hat{x} \neq 0

implies that

λ_{1} = λ_{2}

, while in the former case this is not necessarily true.

In fact, in the past decades both Equations (3) and (4) have been usually referred to asNonlinear Eigenvalue Problems. Those of the type in Equation (3), especially with

E = K^{n}

, have been extensively studied in Numerical Analysis and Matrix Analysis (see, for instance, the review paper [1], where the abbreviation NLEVP is used to designate them), while problems of the type in Equation (4) have formed a main subject in Nonlinear Functional Analysis and its applications to differential equations, and are at the basis of, among others, Bifurcation Theory; see, for instance, the nowadays classic book [2] or the most recent [3].

Historically, the study of Nonlinear Eigenvalue Problems can be dated back to nearly one century ago, if we look in particular to the work—inspired by D. Hilbert and E. Landau—of E. Schmidt and A. Hammerstein, and subsequently of M. Golomb, on parameter-dependent nonlinear integral equations of the form

λ u (x) = \int_{Ω} k (x, y) f (y, u (y)) d y .

On the eastern side of Europe, the investigation of this kind of problems received a strong impulse in the former Soviet Union on behalf of M.A. Krasnosel’skii and I.T. Gohberg. They were both pupils of M.G. Krein, and their subsequent work during many decades of the second half of the last century seems to have developed mainly on problems of the type in Equation (3) by Gohberg, and mainly on problems of the type in Equation (4) by Krasnosel’skii. For this reason, and in honor of these two true giants of Nonlinear Functional Analysis, I will often refer in the sequel to Equation (3) as describing problemsof type G, and to Equation (4) as describing problemsof type K.

The present paper does not contain new results in either field. It is rather a tentative review, having as a prominent scope that of indicating some problems and methods followed in each of the two classes, with a look for possible future interactions between them. This is done in the main part of the paper (Section 2). In fact, as to problems of type G—of which I became aware only a short time ago—my presentation (Section 2.1) will be that of a beginner, and limited to a few historical remarks, accompanied by some indication for further study and some motivation from ODEs.

Something more the reader will find about problems of type K, for which I have focused on a short account of some basic results and methods from Bifurcation Theory on the one hand (Section 2.2), and to a brief description of a very special—maybe the “closest to linear”—nonlinear eigenvalue problem on the other (Section 2.3). The latter is thep-Laplace equation

\{\begin{matrix} - div ({| \nabla u |}^{p - 2} \nabla u) & = μ {| u |}^{p - 2} u & in Ω \\ u & = 0 & on \partial Ω \end{matrix}

(5)

where

p > 1

and

Ω

is a bounded domain in

R^{n}

. Here, one can prove the existence—exactly as for the classical Laplace operator,

p = 2

—of countably many eigenvalues which can be naturally arranged in an increasing sequence

μ_{1} < μ_{2} \leq \dots μ_{k} \leq \dots, μ_{k} \to + \infty

(6)

The importance of this example is also because it shows—via the Lusternik–Schnirelmann theory—the full strength of the variational methods and of Critical Point Theory in particular. As is well known, these consist in searching a solution of a given equation as acritical point of a functional (i.e., a point where the derivative of the functional vanishes), and are of the utmost importance both for equations of the form

A (x) = 0

(7)

and for equations of the form

A (x) = λ C (x)

(8)

the latter being in fact the nonlinear eigenvalue problem in Equation (4) for the pair

(A, C)

. Indeed, if

A = \nabla f

and

C = \nabla g

, then solutions of Equation (7) are thefree critical points off, while solutions of Equation (8) are—modulo technicalities—theconstrained critical points off on the manifold

M = {x : g (x) = c o n s t .}

. As explained for instance in [4], the Lusternik–Schnirelmann theory not only guarantees, under appropriate assumptions on the nonlinear operatorsA andC, the existence of infinitely many distinct eigenvalues of Equation (8), and thus in particular of Equation (5), but also provides for them a “minimax" characterization of the form (when

C = I

)

λ_{n} = \sup_{K_{n}} \inf_{K} 〈 A (u), u 〉,

obtained via suitable families

K_{n}

of subsetsK of the unit sphere. This realizes a conceptually beautiful (and also practically useful, see for instance [5]) extension of the Courant–Weyl principle for the eigenvalues of a linear compact self-adjoint operator. The variational characterization of the eigenvalues seems to be a main point of common interest for either type of nonlinear eigenvalue problems, see, e.g., [6] and the references quoted inSection 2.1 andSection 2.3.

In the second part of the paper (Section 3), we return to Equation (1) and look at the case in which a small “perturbation parameter”

ϵ

enters in the problem, originating an equation of the form

F (ϵ, λ, x) = 0

(9)

of which Equation (1) is seen as the unperturbed form for

ϵ = 0

. We consider parameter-dependent forms of both Equations (3) and (4), precisely

G (ϵ, λ) x = 0

(10)

and, taking in Equation (4)

C = I

and adding to a linearA a nonlinear perturbation

ϵ B

A x + ϵ B (x) = λ x .

(11)

In both Equations (10) and (11), one common problem is—in the light of what is done for linear operators [7]—to see how the perturbed eigenvalues

λ (ϵ)

(provided that they exist) will depend on

ϵ

near a given unperturbed eigenvalue

λ_{0}

. To this purpose, we review the main points of the recent contributions [8,9], respectively, to Equation (10) and to Equation (11).

Two more points deserve to be mentioned before closing this Introduction. The first is that, for a better understanding, Nonlinear Eigenvalue Problems—both of type G and of type K—should be set in the more general respective context of Nonlinear Spectral Theory. References for this are [10,11], respectively. The interested reader might look at [12] for a recent contribution to the latter. The second fact, clear enough from this Introduction, is that we have not even attempted to mention the various numerical methods used for the practical solution of Equation (3) in the case

E = K^{n}

. The reader interested in this rich and fundamental research field might look into the excellent and very recent survey paper [13].

Let me repeat in conclusion that the only reasonable scope of this paper is to possibly arouse the curiosity of some expert in either field towards the problems treated in the other, and to give a chance of possible inspiration for further study.

2. The Two Types of NLEVP

2.1. Problems of Type G: (Linear) Operator- and Matrix-Valued Functions

A good point to start a presentation of nonlinear eigenvalue problems of the type in Equation (3) is perhaps R.E.L. Turner’s paper [6] of 1968. Given a complex Hilbert spaceH, rather than considering the spectrum of a single bounded linear operatorA acting inH, he considers for

λ \in C

operators of the form

A - B (λ) \equiv A - \sum_{1}^{N} λ^{k} B_{k}

(12)

where

B_{k} \in L (H)

are given

(k = 1, \dots, N)

, with

B_{1} = I

; thus if

N = 1

we are back to the familiar

A - λ I

considered in linear spectral theory. Thespectrum of

A - B (λ)

is defined as the set of those

λ \in C

for which

A - B (λ)

fails to be a homeomorphism ofH onto itself. In particular, a point

λ_{0} \in C

such that

A - B (λ_{0})

is not injective, i.e., such that the nullspace

Ker (A - B (λ_{0})) \neq {0}

, is aneigenvalue of

A - B (λ)

. Note that, in the case

N = 1

, these definitions of spectrum and eigenvalues of

A - λ I

yield what we usually call the spectrum and eigenvaluesof A. The new point of view is that the spectrum is now attributed to the (polynomial)function of

C

into

L (H)

defined putting

G (λ) = A - B (λ) .

(13)

In the case

H = C

, the spectrum so defined consists very simply of the zeroes of the polynomialG itself. Now recall (see e.g., [14] or [15]) that, ifA is compact, self-adjoint and nonnegative, then:

The spectrum ofA is at most countable and consists of a finite or infinite decreasing sequence of non-negative eigenvalues $(λ_{n})$ :
$λ_{1} \geq λ_{2} \geq \dots λ_{k} \geq \dots \dots$
If the sequence is infinite, then $λ_{n} \to 0$ .
The eigenvectors $(u_{n})$ associated with the eigenvalues $(λ_{n})$ form an orthonormal basis ofH.

Turner first generalizes this to operators as in Equation (12) whereA is compact, self-adjoint and nonegative,

B_{k}

is self-adjoint and non-negative for

k = 1, \dots, N

andA belongs to the Schatten class

C_{r}

(i.e., its eigenvalues

(α_{i})

satisfy the condition

\sum_{i} {(α_{i})}^{r} < \infty

) for some

r < \frac{1}{2}

. Another basic fact concerning the spectrum of an operatorA as above is thevariational characterization of its positive eigenvalues

(λ_{n})

: indeed,

λ_{n} = \max_{x ⊥ u_{1}, \dots, u_{n - 1}} \frac{〈 A x, x 〉}{〈 x, x 〉} = \min_{x \in [u_{1}, \dots, u_{n}]} \frac{〈 A x, x 〉}{〈 x, x 〉}

(14)

= \min_{V \in V_{n - 1}} \max_{x ⊥ V} \frac{〈 A x, x 〉}{〈 x, x 〉} = \max_{V \in V_{n}} \min_{x \in V} \frac{〈 A x, x 〉}{〈 x, x 〉}

(15)

whereV is a vector subspace ofH, and

V_{n}

denotes the family of all vector subspaces of dimensionn.

Turner generalizes the variational principle as follows. For

x \in H, x \neq 0

, let

Z (x)

be the unique non-negative zero of the polynomial

λ \to 〈 G (λ) x, x 〉

[6]. Note that, in the case

N = 1

, as

〈 G (λ) x, x 〉 = 〈 A x, x 〉 - λ 〈 x, x 〉

, we have

Z (x) = \frac{〈 A x, x 〉}{〈 x, x 〉}

(16)

so that the functionZ is the usualRayleigh quotient of A, of which the eigenvalues areextremal values as shown by Equation (14). Then, under the stated assumptions onA and

B_{k}

, if moreover the eigenvectors of

A - B (λ)

, corresponding to non-negative eigenvalues, form a basis forH, then the variational principles in Equations (14) and (15) hold replacing

〈 A x, x 〉 / 〈 x, x 〉

with

Z (x)

Finally, we have by definition of

Z (x)

that

〈 G (Z (x)) x, x 〉 = 0 for all x \in H, x \neq 0

(17)

Results similar to those of Turner, and practically at the same time, were obtained by K.P. Hadeler in [16,17]. He considered several-parameter dependent operators of the form

A - (λ_{1} B_{1} + λ_{2} B_{2} + \dots λ_{N} B_{N})

with

B_{j}

bounded self-adjoint for

j = 1, \dots, N

, and in connection with the variational property of their eigenvalues introduced the general concept ofRayleigh functional of a matrix function as follows. Let

α \to T (α)

be a differentiable mapping of the real interval

(a, b)

to the set

S_{n}

of real symmetric matrices of ordern. Then, a Rayleigh functional ofT is a continuous real-valued functionp on

R^{n} \ {0}

such that

p (x) \in (a, b)

for all

x \in R^{n} \ {0}

and

$p (c x) = p (x) if c \neq 0$
$〈 T (p (x) x, x 〉 = 0$
$〈 T^{'} (p (x) x, x 〉 > 0$ .

The last is a definiteness condition that can be replaced by

〈 T^{'} (p (x) x, x 〉 < 0

, and is plainly satisfied in the basic case

T (α) = A - α I

, where

T^{'} (α) = - I

. Thus, looking at Equations (16) and (17), we see that this is a sensible extension of the definition and properties of the Rayleigh quotient.

The results of Turner and Hadeler indicated above were developed and improved by, among others, H. Langer. For instance, in [18], studying combinations

T (λ)

of bounded self-adjoint operators of the form of Equation (12) considered by Turner, he assumed that, for each nonzero vectorx, the polynomial

p_{x} (L) \equiv 〈 T (λ) x, x 〉

has only real roots

λ_{1} (x) \geq λ_{2} (x) \geq \dots \geq λ_{n} (x)

Under this assumption he showed that the ranges

Λ_{i}

of the functions

λ_{i}

are intervals, calledspectral zones, whose interiors do not overlap.

A systematization of the spectral theory (that is, of the properties of eigenvalues and eigenvectors) ofpolynomial operator pencils, as had been named the families

A (λ) = A_{0} + λ A_{1} + \dots λ^{n} A_{n}

(18)

where

λ \in C

is a spectral parameter, and

A_{i}, i = 1, \dots, n,

are linear operators in a Hilbert space, was given by A.S. Markus in his book [19]. Among others, he considered in depth the problem of thefactorization of pencils, which in the simplest case consists in representing a quadratic pencil

A (λ) = λ^{2} I + λ B + C

in the form

A (λ) = (λ I - Y) (λ I - Z) .

The importance of many results in [19] is due to the fact that they hold for the more general case ofholomorphic(i.e., analytic) operator-valued functions, namely operators

A (λ)

expressed as the sum of convergent power series in

L (E)

A (λ) = \sum_{0}^{\infty} λ^{n} A_{n} .

(19)

For an updated reference reviewing the spectral properties of self-adjoint analytic operator functions, and in particular the factorization problem, see [10]. On the other hand, for further work on the variational characterization of eigenvalues as well as for the development of the theory of Rayleigh functionals, the interested reader can look for instance into the quite recent papers by Binding, Eschwé and Langer [20], Hasanov [21], Voss [22], and Schwetlick and Schreiber [23], and the references therein.

Let us now add some more specific indication for the case in which

E = K^{m}

, so that the functionG appearing in Equation (3) takes its values in the space

K^{m \times m}

m \times m

real or complex matrices. We shall stress the finite-dimensionality of the ambient spaceE using the letterM rather thanG, and often the letterv rather thanx for the vectors ofE. A well known reference book for the matter is the one by Gohberg, Lancaster and Rodman [24], and the very Introduction to this book explains to us that problems of the form

M (λ) v = 0, λ \in C, v \in C^{m}, v \neq 0

(20)

where

M (λ) \in C^{m \times m}

appear naturally when dealing with linear systems of higher order ordinary differential equations (ODE) with constant coefficients:

\frac{d^{n} u}{d t^{n}} + A_{n - 1} \frac{d^{n - 1} u}{d t^{n - 1}} + \dots + A_{1} \frac{d u}{d t} + A_{0} u = 0

(21)

where

A_{i} \in C^{m \times m}

for

i = 0, 1, \dots, n - 1

. Indeed, looking for solutions of the form

u (t) = e^{λ t} v (λ \in C, v \in C^{m})

of Equation (21) leads to the equation

e^{λ t} {λ^{n} + λ^{n - 1} A_{n - 1} + \dots + λ A_{1} + A_{0}} v = 0

(22)

which—as long as

v \neq 0

, and putting

A_{n} = I

—is equivalent to Equation (20) with

M (λ) = \sum_{i = 0}^{n} λ^{i} A_{i} .

(23)

Thus,

e^{λ_{0} t} v_{0}

is a nontrivial solution of Equation (21) if and only if

λ_{0}

is an eigenvalue of Equation (20), i.e., it is a zero of thecharacteristic equation

\det M (λ) = 0

(24)

and

v_{0} \in Ker M (λ_{0})

. More generally, the function

u (t) = e^{λ_{0} t} {\frac{t^{k}}{k!} v_{0} + \dots + \frac{t}{1!} v_{k - 1} + v_{k}}

(25)

is a solution of Equation (21) if and only if the vectors

v_{0}, v_{1}, \dots, v_{k}

satisfy the relations

\sum_{j = 0}^{l} \frac{1}{j!} \frac{d^{j} M}{d λ^{j}} (λ_{0}) v_{l - j} = 0, l = 0, 1, \dots, k .

(26)

Such a set of vectors

v_{0}, v_{1}, \dots v_{k}

is called aJordan chain of length

k + 1

for the matrix function

M (λ)

, corresponding to the eigenvalue

λ_{0}

andstarting with the eigenvector

v_{0}

. The above definitions extend from matrix polynomials as in Equation (23) to any analytic matrix function

M (λ)

. It is good to see the explicit form of Equation (26), which is

\{\begin{matrix} l = 0 : & M (λ_{0}) v_{0} = 0 \\ l = 1 : & M (λ_{0}) v_{1} + \frac{1}{1!} \frac{d M}{d λ} (λ_{0}) v_{0} = 0 \\ l = 2 : & M (λ_{0}) v_{2} + \frac{1}{1!} \frac{d M}{d λ} (λ_{0}) v_{1} + \frac{1}{2!} \frac{d^{2} M}{d λ^{2}} (λ_{0}) v_{0} = 0 \\ \dots \\ l = k : & M (λ_{0}) v_{k} + \frac{1}{1!} \frac{d M}{d λ} (λ_{0}) v_{k - 1} + \dots + \frac{1}{k!} \frac{d^{k} M}{d λ^{k}} (λ_{0}) v_{0} = 0 . \end{matrix}

(27)

n = 1

in Equation (23), we have

M (λ) = λ A_{1} + A_{0}

; and if moreover

A_{1} = - I

, then

M (λ) = A_{0} - λ I

. In this case,

\frac{d M}{d λ} (λ_{0}) = - I

, while

\frac{d^{j} M}{d λ^{j}} (λ_{0}) = 0

for all

j > 1

, so that the above equalities reduce to (putting

A_{0} = A

)

\{\begin{matrix} (A - λ_{0} I) v_{0} = 0 \\ (A - λ_{0} I) v_{1} = v_{0} \\ \dots \\ (A - λ_{0} I) v_{k} = v_{k - 1} \end{matrix}

(28)

and are those defining anordinary Jordan chain for the matrixA corresponding to

λ_{0}

and

v_{0}

, used to representA in its Jordan canonical form and in particular to construct a basis of thegeneralized eigenspace

E_{λ_{0}} (A)

associated with

λ_{0}

. We recall that this is defined as

E_{λ_{0}} (A) = Ker ({(A - λ_{0} I)}^{p})

(29)

wherep is the least integer such that

Ker ({(A - λ_{0} I)}^{p}) = Ker ({(A - λ_{0} I)}^{p + 1})

, and that the dimension

\dim E_{λ_{0}} (A)

E_{λ_{0}} (A)

is equal to thealgebraic multiplicity of

λ_{0}

, that is, the multiplicity of the eigenvalue as a zero of the characteristic polynomial

\det (A - λ I)

. We say that

λ_{0}

issemisimple if

p = 1

in Equation (29)—that is, if the algebraic multiplicity coincides with thegeometric multiplicity of

λ_{0}

, defined as

\dim Ker (A - λ_{0} I)

—and that

λ_{0}

issimple if they are both equal to 1.

These familiar concepts from Linear Algebra, concerning the basic case

M (λ) = A_{0} - λ I

, need to be extended to analytic matrix functions

M (λ)

. To this purpose, we quote from [25]; see also ([26], Chapter 7).

Let $x_{0}$ be an eigenvector corresponding to an eigenvalue $λ_{0}$ . The maximal length of a Jordan chain starting at $x_{0}$ is called themultiplicity of $x_{0}$ and denoted by $m (x_{0})$ . An eigenvalue $λ_{0}$ is said to benormal if it is an isolated eigenvalue and the multiplicity of each corresponding eigenvector is finite.
Suppose that $λ_{0}$ is a normal eigenvalue. Then, a correspondingcanonical system of Jordan chains
$x_{0}^{k}, x_{1}^{k}, \dots, x_{m_{k} - 1}^{k} (k = 1, \dots, N)$
is defined by the following rules:
(1)
The vectors $x_{0}^{1}, \dots, x_{0}^{N}$ form a basis of $Ker M (λ_{0})$ (and so $N = \dim Ker M (λ_{0})$ ).
(2)
$x_{0}^{1}, x_{1}^{1}, \dots, x_{m_{1} - 1}^{1}$ is a Jordan chain of the maximal length $m_{1} \equiv m (x_{0}^{1})$ .
(3)
Once that the vectors $x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{k - 1}$ ( $1 \leq k < N$ ) have been chosen, then pick an eigenvector $x_{0}^{k}$ linearly independent from $x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{k - 1}$ and form a Jordan chain $x_{0}^{k}, x_{1}^{k}, \dots, x_{m_{k} - 1}^{k}$ of the maximal length $m_{k} \equiv m (x_{0}^{k})$ .
A canonical system is not defined uniquely; however, the numbers $m_{1}, m_{2}, \dots, m_{N}$ do not depend on the choice of Jordan chains and are calledpartial multiplicities of the eigenvalue $λ_{0}$ . The number $Q (λ_{0}) \equiv m_{1} + \dots + m_{N}$ is thealgebraic multiplicity of the eigenvalue $λ_{0}$ .

The next statement—which is based on results found in [27]—proves that these definitions are a coherent generalization of the usual ones.

Proposition 1.

An eigenvalue

λ_{0}

is a zero of

\det M (λ)

of multiplicity

Q (λ_{0})

Based on Proposition 1, the definitions of simple and semisimple eigenvalue carry over to the case of matrix polynomials and more generally to analytic matrix functions. For instance, one may check that the matrix function

M_{2} (λ) = \begin{matrix} (\begin{matrix} λ - 1 + e^{- λ} & 0 \\ 0 & λ + 1 \end{matrix}) \end{matrix}

(30)

considered in [8] has

λ_{0} = 0

as a double (i.e., of algebraic multiplicity 2), nonsemisimple (i.e., of geometric multiplicity 1) eigenvalue, with Jordan chain

H_{0} = \begin{matrix} (\begin{matrix} 1 \\ 0 \end{matrix}), \end{matrix} H_{1} = \begin{matrix} (\begin{matrix} α \\ 0 \end{matrix}) \end{matrix}

(31)

for any

α \in R

. This example also shows that in the nonlinear case, generalized eigenvectors do not need to be linearly independent. Indeed, in the construction (and notation) recalled above, the generating vectors

x_{0}^{1}, \dots, x_{0}^{N}

of the system of Jordan chain are chosen to be linearly independent, but it is not necessarily so for the vectors in each corresponding chain, generated by the rules given by the system in Equation (27).

An especially important source for the study of NLEVP are the Delay Differential Equations (DDE), or systems of them. For instance, in [26] is considered the so-called Wright equation

x^{'} (t) = - α x (t - 1) [1 + x (t)]

(32)

where

α > 0

. The objective is to determine the periodic orbits (if any) of Equation (32). To do this, one must first look at the linearized equation of Equation (32) near

x \equiv 0

, which is

x^{'} (t) = - α x (t - 1)

(33)

Solutions

e^{λ t}

of this exist iff

λ

satisfies the characteristic equation

λ e^{λ} + α = 0

(34)

For

α = π / 2

, this has

λ = i \frac{π}{2}

as a simple purely imaginary root, corresponding to the periodic solution

e^{i \frac{π}{2} t}

. Studying the properties of these nonlinear eigenvalues, that is of the roots

λ (α)

of Equation (34) as a function of

α

, and using deep topological and functional-analytic results from [26], it is possible to demonstrate that Equation (32) has a Hopf bifurcation at

α = π / 2

, and that forevery

α > π / 2

Equation (32) has a nonconstant periodic solution with period close to 4. Finally, the authors show that for

p > 4

, there is a periodic solution of Equation (32) of periodp.

One can also consider systems of DDE, for instance

x^{'} (t) = \begin{matrix} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) \end{matrix} x (t) + \begin{matrix} (\begin{matrix} - 1 & 0 \\ 0 & 0 \end{matrix}) \end{matrix} x (t - 1)

whosecharacteristic matrix is precisely that displayed in Equation (30). The general form of a system ofN delay differential equations, with delays

τ_{1}, \dots, τ_{N}

x^{'} (t) = A_{0} x (t) + \sum_{i = 1}^{N} A_{i} x (t - τ_{i})

(35)

with

A_{0}, A_{i} \in C^{N \times N}

, and the corresponding characteristic matrix is

M (λ) = λ I - A_{0} - \sum_{i = 1}^{N} A_{i} e^{- λ τ_{i}} .

More general forms of Equation (35) are considered inSection 3.

2.2. Problems of Type K: Nonlinear Operators and Bifurcation

Throughout this SectionE will be a real Banach space, of finite or infinite dimension. Originally, bifurcation theory deals with the local study of Equation (1) near a point

(λ_{0}, 0) \in R \times E

, and studies precisely the conditions under which from the given point

(λ_{0}, 0)

of the line

R \times {0} \subset R \times E

of thetrivial solutions of Equation (1), therebifurcates a branch of nontrivial solutions, that is, of solutions

(λ, x)

with

x \neq 0

. Of course, the basic situation that comes to one’s mind is the case

F (λ, x) = A x - λ x

, with

λ_{0}

an eigenvalue of the linear operatorA, the “branch” being here the special subset

{λ_{0}} \times (Ker (A - λ_{0} I) \ {0})

R \times E

. The interesting case is whenF depends in a less obvious way from

λ

andx; an easy example of what we mean is given for instance by the equation

F (λ, x) \equiv a x + b x^{3} - λ x = 0, (λ, x) \in R^{2}

in which the parabola

λ = a + b x^{2}

bifurcates at the point

(a, 0)

from the line of the trivial solutions. For a motivating introduction to the theory, and a discussion of some important physical problems that fall in this context, an excellent source is the old review paper by Stackgold [28].

The previous “naif” idea of bifurcation needs to be made both more precise and more general, and this is done by saying that

(λ_{0}, 0)

is a bifurcation point for Equation (1) if any neighborhood of

(λ_{0}, 0)

R \times E

contains nontrivial solutions of Equation (1). For this definition to make sense, it is enough thatF be defined in an open set

U \subset R \times E

with

(λ_{0}, 0) \in U

, and this is what we assume from now on. For the next step, we further assume thatF is differentiable at the point

(λ_{0}, 0)

, so thatF can be linearized near that point as

F (λ, x) = F (λ_{0}, 0) + D_{λ} F (λ_{0}, 0) (λ - λ_{0}) + D_{x} F (λ_{0}, 0) x + R (λ, x) = D_{x} F (λ_{0}, 0) x + R (λ, x)

(36)

where the remainder termR satisfies

R (λ, x) = o (∥ (λ, x) ∥) as (λ, x) \to (λ_{0}, 0) .

Some more regularity onF yields immediately a necessary condition for bifurcation:

Theorem 1.

Suppose that F is of class

C^{1}

in a neighborhood of

(λ_{0}, 0)

. If

D_{x} F (λ_{0}, 0)

is a homeomorphism of E onto itself, then

(λ_{0}, 0)

cannot be a bifurcation point for Equation (1).

Proof.

The assumption implies, via the Implicit Function Theorem, that there is a neighborhood

I \times V

(λ_{0}, 0)

such that, for any

λ \in I

, there is aunique

x = x (λ) \in V

such that

F (λ, x) = 0

. As by assumption

F (λ, 0) = 0

for any

λ

, we must have

x (λ) = 0

for

λ \in I

, so that there is no nontrivial solution to Equation (1) in the neighborhood

I \times V

(λ_{0}, 0)

. ☐

For simplicity, we shall henceforth consider only the special case

F (λ, x) = A (x) - λ x

(37)

where

A (0) = 0

andA is of class

C^{1}

near

x = 0

. Here,

D_{x} F (λ_{0}, 0) = A^{'} (0) - λ_{0} I

, and we have a more explicit form of the remainder term in the linearized form in Equation (36) ofF: for we can write

A (x) = A^{'} (0) x + B (x)

with

B (x) = o (∥ x ∥)

∥ x ∥ \to 0

, so that Equation (37) yields

F (λ, x) = A^{'} (0) x + B (x) - λ x

= (A^{'} (0) - λ_{0} I) x + B (x) - (λ - λ_{0}) x

and comparing this with Equation (36) we see that

R (λ, x) = B (x) - (λ - λ_{0}) x

in this special case. Resuming, the equation we want to study is

A (x) - λ x = 0

(38)

with

A (0) = 0

andA of class

C^{1}

near

x = 0

, and can be written as

T x - λ_{0} x + B (x) = (λ - λ_{0}) x

(39)

where

T \equiv A^{'} (0)

and

B (x) = o (∥ x ∥)

∥ x ∥ \to 0

. The necessary condition for bifurcation implicitly stated in Theorem 1 can now be rephrased as follows:

(λ_{0}, 0) bifurcation point of A (x) - λ x = 0 \Rightarrow λ_{0} \in σ (A^{'} (0)) .

The standard case considered in the literature is when

λ_{0}

is in thepoint spectrum of

A^{'} (0)

, and we formalize this more precisely under the form of a basic assumption, which is plainly satisfied if

\dim E < \infty

H0.

λ_{0}

is an isolated eigenvalue of

T \equiv A^{'} (0)

and

T - λ_{0} I

is a Fredholm operator of index zero.

Let us recall (see, e.g., [14]) that a bounded linear operatorL between two Banach spacesE andF is said to be aFredholm operator if its nullspace

Ker L

has finite dimension and its range

Im L

is closed and has finite codimension; in this case, theindex ofL,

ind L

, is defined as

ind L = \dim Ker L - codim Im L .

Thus, if

\dim E = \dim F < \infty

, then any linear operator is Fredholm of index zero. From the Riesz–Schauder theory of such operators (see e.g., [15]), it is known that also the nullspaces

Ker L^{j}

(

j > 1

) are finitedimensional, and that they stabilize forj sufficiently large; with reference to the case

L = T - λ_{0} I

, this means that there exists a least integerp such that

Ker {(T - λ_{0} I)}^{p} = Ker {(T - λ_{0} I)}^{p + 1}

, and moreover one has

E = Ker {(T - λ_{0} I)}^{p} \oplus Im {(T - λ_{0} I)}^{p} .

It follows in particular that thealgebraic multiplicity of

λ_{0}

is finite, where in general this is is defined—consistently with the definition recalled inSection 2.1 for the case

\dim E < \infty

—as the dimension of the subspace

⋃_{j = 1}^{\infty} Ker {(T - λ_{0} I)}^{j} .

In the following, when speaking ofmultiplicity of an eigenvalue, we refer to the algebraic multiplicity. We recall that this coincides with thegeometric multiplicity

\dim Ker (T - λ_{0} I)

whenT is a self-adjoint operator in a Hilbert space.

Remark 1.

The assumptionH0 alone is not sufficient to guarantee that an eigenvalue of the “linear part” T of A at 0 is a bifurcation point for A. To see this, consider the example ([29], Chapter 11) given by the system

\{\begin{matrix} x + y^{3} = λ x \\ y - x^{3} = λ y . \end{matrix}

(40)

Here,

E = R^{2}

and we have (in our notations)

T = I

λ_{0} = 1

and

B (x, y) = (y^{3}, - x^{3})

. Multiplying the first equation by y, the second by x and subtracting the second from the first, we obtain

x^{4} + y^{4} = 0

, showing that Equation (40)has no nontrivial solution whatsoever. One way of seeing this is that the two-dimensional eigenspace associated with

λ_{0}

is completely destroyed by the addition of the perturbing term B.

Three typical situations are then considered, each of them guaranteeing bifurcation from

λ_{0}

, and described by the following assumptions, respectively:

H1.

λ_{0}

is asimple eigenvalue of

A^{'} (0)

H2.

A iscompact and

λ_{0} \neq 0

is an eigenvalue ofodd multiplicity of

A^{'} (0)

H3.

A is agradient operator in a Hilbert space and

λ_{0}

is an isolated eigenvalue of finite multiplicity of

A^{'} (0)

These assumptions call immediately for some explanation. In fact, it could be noted at once that bothH2 andH3 are a strengthening ofH0. However, to proceed with some order, in the remaining part of this subsection, we shall give a precise statement for each of the three bifurcation results roughly indicated above, preceded by a comment on the respective assumption, and followed by an indication of the proof.

Thus, starting withH2, we recall that, ifA is compact, then the linear operator

A^{'} (0)

is a compact, too [30]. ThereforeH0 is redundant in this case, as it is a basic spectral property of any such operator [14].

Theorem 2.

IfH2 is satisfied, then

λ_{0}

is a bifurcation point for Equation (38).Moreover, it is a globalbifurcation point in the following sense: if S denotes the closure in

R \times E

of the set of nontrivial solutions of Equation (38),then

S \cup {(λ_{0}, 0)}

has a connectedsubset

S_{λ_{0}}

containing

{(λ_{0}, 0)}

, and which is either unboundedin

R \times E

or contains a point

{(λ_{1}, 0)}

with

λ_{1}

an eigenvalue of odd multiplicity of T.

Proof.

The proof relies on theLeray–Schauder degree. Roughly speaking, this is a topological tool to detect the fixed points of a compact map and can be briefly introduced as follows (see, for instance, [3] (Part I) for a complete presentation). Suppose we have a continuous compact mapC ofE into itself, a bounded open set

Ω \subset E

with

0 \in Ω

, and suppose that

C (x) \neq x

for

x \in \partial Ω

. Then, there exists an integer, denoted

d (I - C, Ω, 0)

and called the (Leray–Schauder)degree of

I - C

relative to the set Ωand to the point 0, having the following properties:

(i): If $d (I - C, Ω, 0) \neq 0$ , then there exists an $x \in Ω$ such that $x = C (x)$ .
(ii): $d (I, Ω, 0) = 1$ .
(iii): If $Ω_{0} \subset Ω$ and $I - C$ has no zeroes in $Ω \ {\bar{Ω}}_{0}$ , then $d (I - C, Ω, 0) = d (I - C, Ω_{0}, 0) .$
(iv): Suppose $C_{1}, C_{2} : E \to E$ are compact maps, and put
$H (t, x) = x - [C_{1} (x) - t (C_{2} (x) - C_{1} (x)], t \in [0, 1], x \in E$
If $H (t, x) \neq 0$ for $t \in [0, 1]$ and $x \in \partial Ω$ , then
$d (I - C_{1}, Ω, 0) = d (I - C_{2}, Ω, 0) .$
(v): IfC is alinear compact map and $I - C$ is injective, then
$d (I - C, Ω, 0) = {(- 1)}^{ν}$
where $ν$ is the number of eigenvalues >1 ofC, each counted with its algebraic multiplicity.

To prove that

λ_{0}

is a bifurcation point, it is enough to show that for any sufficiently small

r > 0

, there exists a solution

(λ_{r}, x_{r})

of Equation (38) with

λ_{r} \in [λ_{0} - r, λ_{0} + r]

and

∥ x_{r} ∥ = r

. Thus, let

B_{r} = {x \in E : ∥ x_{r} ∥ < r}

be the open ball centered at

x = 0

and with radiusr; we consider the degree of various maps with respect to this neighborhood of 0. Precisely, assume for instance

λ_{0} > 0

and write

A (x) - λ x = - λ (x - μ A (x))

μ = 1 / λ

, for

λ

near

λ_{0}

. Consider thus the equivalent equation

x - μ A (x) = 0

(41)

and let

μ

vary in an interval

[\underset{̲}{μ}, \bar{μ}]

containing as interior point

μ_{0} = 1 / λ_{0}

and no othercharacteristic values(as are named the reciprocals of the nonzero eigenvalues) of

T \equiv A^{'} (0)

except

μ_{0}

. Assume by way of contradiction that

x - μ A (x) \neq 0

for

∥ x ∥ = r

and

μ \in [\underset{̲}{μ}, \bar{μ}]

; then using theHomotopy invariance Property (iv) with

C_{1} = \underset{̲}{μ} A, C_{2} = \bar{μ} A

we would have

d (I - \underset{̲}{μ} A, B_{r}, 0) = d (I - \bar{μ} A, B_{r}, 0) .

(42)

On the other hand, for small

r > 0

, using again Property (iv) we have

d (I - \underset{̲}{μ} A, B_{r}, 0) = d (I - \underset{̲}{μ} T, B_{r}, 0)

(43)

because

I - \underset{̲}{μ} A

is homotopic to

I - \underset{̲}{μ} T

\partial B_{r}

; indeed, since the latter operator is a homeomorphism and since

B (x) = o (∥ x ∥)

∥ x ∥ \to 0

, we have (diminishingr if necessary)

∥ x - \underset{̲}{μ} (T x + t B (x)) ∥ \geq ∥ x - \underset{̲}{μ} T x ∥ - ∥ \underset{̲}{μ} t B (x) ∥ \geq k ∥ x ∥

for some

k > 0

and for all

(t, x) \in [0, 1] \times {\bar{B}}_{r}

. Similarly,

d (I - \bar{μ} A, B_{r}, 0) = d (I - \bar{μ} T, B_{r}, 0) .

(44)

However, using Property (v), we have

d (I - \underset{̲}{μ} T, B_{r}, 0) = {(- 1)}^{\underset{̲}{ν}}, d (I - \bar{μ} T, B_{r}, 0) = {(- 1)}^{\bar{ν}}

(45)

where

\bar{ν} = \underset{̲}{ν} + h

,h an odd integer (the algebraic multiplicity of

λ_{0}

); therefore the two degrees in Equation (45) are different, contradicting the previous equalities in Equations (42)–(44). This proves that

x_{r} - μ_{r} A (x_{r}) = 0

for some

x_{r} \in \partial B_{r}

and some

μ_{r} \in [\underset{̲}{μ}, \bar{μ}]

, and therefore that there is bifurcation from

(μ_{0}, 0)

for the Equation (41), or equivalently from

(λ_{0}, 0)

for Equation (38). The proof that under the stated assumptions the bifurcation has aglobal character, in the sense described by the statement of Theorem 2, requires the much deeper topological analysis performed by P.H. Rabinowitz in his famous paper [31]. ☐

We now go on to comment assumptionH3, and to briefly discuss the corresponding bifurcation result. For the next definition, and the statements following it, see for instance [2] or [30].

Definition 1.

Let H be a real Hilbert space with scalar product denoted

〈 . 〉

. An operator

A : H \to H

is said to be a gradient (or potential) operatorif there exists a differentiable functional

a : H \to R

such that

〈 A (x), y 〉 = a^{'} (x) y f o r a l l x, y \in H .

(46)

One then writes

A = \nabla a

; the functionala—thepotential ofA—is uniquely determined by the requirement that

a (0) = 0

, and is explicitly given by the formula

a (x) = \int_{0}^{1} 〈 A (t x), x 〉 d t .

(47)

A bounded linear operator is a gradient if and only if it is self-adjoint. Moreover, if a gradient operatorA is differentiable at a point

x_{0}

, then

A^{'} (x_{0})

is self-adjoint.

Theorem 3.

IfH3 is satisfied, then

λ_{0}

is a bifurcation point of Equation (38).Moreover, for each

r > 0

sufficiently small, Equation (38)has at least twodistinct solutions

(λ_{r}, x_{r})

such that

∥ x_{r} ∥ = r

Proof.

The proof makes use of theLyapunov–Schmidt method (see, for instance, ([3], Chapter 2) or ([29]), Chapter 11) which allows to reduce the infinite-dimensional problem in Equation (38) to a problem in the finite-dimensional space

Ker (T - λ_{0} I)

. Indeed, consider the equivalent form Equation (39) of Equation (38), and rewrite it as

L x + B (x) = δ x

(48)

where

L = T - λ_{0} I

and

δ = λ - λ_{0}

. Now recalling that

T = A^{'} (0)

, the assumptionH3 implies thatH is the orthogonal sum

H = Ker L \oplus Im L .

(49)

Then, letting

P, Q

denote the orthogonal projections ofH onto

Ker L

and

Im L

, respectively, we have

x = P x + Q x \equiv v + w

(50)

and using this in Equation (48), we obtain the equivalent system

P B (v + w) = δ v

(51)

L w + Q B (v + w) = δ w .

(52)

The restriction

{L |}_{Im L}

ofL to

Im L

is a homeomorphism of

Im L

onto itself. A standard application of the Implicit Function Theorem, together with the condition

B (x) = o (∥ x ∥)

∥ x ∥ \to 0

, then allows to solve thecomplementary equation, Equation (52) in the form

w = w (δ, v)

(53)

with

w (0, 0) = 0

, where

δ

andv belong to suitably small neighborhoodsJ andV of

δ = 0

and

v = 0

, respectively in

R

and in

Ker L

. Replacing this in Equation (51) first yields

〈 P B (v + w (δ, v)), v 〉 = {δ ∥ v ∥}^{2}

(54)

whence, applying once more the Implicit Function Theorem, one can recover

δ

as a function ofv,

δ = δ (v), δ (0) = 0,

(55)

forv in a neighborhood

V_{0} \subset V

of 0 in

Ker L

. Finally, putting

ϕ (v) = w (δ (v), v), v \in V_{0}

(56)

and replacing this in Equation (51), one is left with the finite-dimensional equation (thebifurcation equation)

F_{0} (v) \equiv P B (v + ϕ (v)) = δ (v) v .

(57)

Any solution

v \in V_{0}

v \neq 0

, of this equation will give rise to a solution

(δ, x)

x \neq 0

(δ, x) = (δ (v), v + ϕ (v))

of the original Equation (48), and the continuity (in fact,

C^{1}

regularity) of the maps

δ = δ (v), w = w (δ, v)

will ensure that this solution

(δ, x)

stays into a given small neighborhood of

(0, 0)

R \times H

provided thatv is small enough. Thus, proving bifurcation from

λ_{0}

for Equation (38)—or equivalently, bifurcation from

δ = 0

for Equation (48)—reduces to prove that Equation (57) has solutions

v \neq 0

of arbitrarily small norm.

Remark 2.

The Lyapunov–Schmidt reduction can be applied more generally, and with minor modifications, in a Banach space E whenever the basic assumptionH0 (i.e., that

L = T - λ_{0} I

is Freholm of index zero) holds and is supplemented by the transversality condition

Ker L \cap Im L = {0}

(58)

which is plainly satisfied when T is self-adjoint, as the two subspaces in Equation (58)are then orthogonal. Note that Equation (58)is in general equivalent to

Ker L = Ker L^{2}

, and thus to the fact that the algebraic and geometric multiplicities of

λ_{0}

coincide.H0 and Equation (58)imply the direct decomposition of E into (closed) subspaces as in Equation (49),and therefore allow for the same reduction on taking for

P, Q

the (continuous) projections associated with Equation (49).

Returning to the proof of Theorem 3, we let now come in the assumption that the wholeA, and therefore also its “nonlinear part”B, is a gradient. Here, we bound ourselves to give the main idea of the particularly clear demonstration provided by C. Stuart [32]. Thus, letf be such that

\nabla f = L + B

, and consider the reduced functional

f_{0} : V_{0} \subset Ker L \to R

defined putting

f_{0} (v) \equiv f (v + ϕ (v)) .

(59)

Moreover, for small

r > 0

put

M_{r} = {v \in V_{0} : g (v) \equiv {∥ v ∥}^{2} + {∥ ϕ (v) ∥}^{2} = r^{2}} .

(60)

Then,

M_{r}

is the level set of the

C^{1}

functionalg, and is compact because it is a closed and bounded subset of the finite dimensional space

Ker L

. Thus,

f_{0}

attains its minimum and its maximum on

M_{r}

, and if

v_{0} \in M_{r}

is such an extremal point we have, by the Lagrange multiplier’s rule,

\nabla f_{0} (v_{0}) = λ \nabla g (v_{0}) .

(61)

Performing the computations of

\nabla f_{0} (v_{0})

and

\nabla g (v_{0})

by the definitions in Equations (59) and (60), and using the fact that

w (δ, v)

satisfies the complementary equation, Equation (52), one checks that

λ = δ (v_{0})

and that Equation (57) is satisfied. ☐

We finally come toH1. UnlikeH2 andH3, in generalH1 is independent fromH0, and must be supplemented with it to guarantee bifurcation. Of course, whenE is finite dimensional,H0 does not play any role, and indeedH1 can in this case be viewed as a special case ofH2, because any continuous map is then compact.

Theorem 4.

IfH0 andH1 are satisfied, then

λ_{0}

is a bifurcation point of Equation (38).Moreover, if A is of class

C^{2}

in a neighborhood of

x = 0

, then near

(λ_{0}, 0)

the solution set of Equation (38)consists of the trivial solutions

{(λ, 0)}

and of a

C^{1}

curve

γ (t) = (λ (t), x (t)), t \in] - δ, δ [

with

γ (0) = (λ_{0}, 0)

and

x (t) \neq 0

for

t \neq 0

. Finally, if

Ker L = [ϕ]

, then as

t \to 0

\{\begin{matrix} x (t) & = t ϕ + o (| t |) \\ λ (t) & = λ_{0} + o (1) . \end{matrix}

(62)

The statement of Theorem 4 means that near

(λ_{0}, 0)

, the solution set of Equation (38) is topologically equivalent to the “cross”

(] - 1, 1 [\times {0}) \cup ({0} \times] - 1, 1 [) .

As to the proof, this goes for a first part along the same lines used to prove the previous Theorem 3, that is, using the Lyapounov–Schmidt decomposition in the sense indicated in Remark 2. What is specific here is that, since

\dim Ker L = 1

, one ends with an equation in

R

; a further nontrivial application of the Implicit Function Theorem then leads to the result: see, for instance, ([3], Chapter 2).

2.3. A Very Special Nonlinear Problem: Thep-Laplace Equation

Let

Ω

be a bounded open set in

R^{n}

, let

p > 1

, and letE be the Sobolev space

W_{0}^{1, p} (Ω)

, equipped with the norm

{∥ v ∥}_{W_{0}^{1, p}} = (\int_{Ω} {| \nabla v |}^{p} {d x)}^{\frac{1}{p}} .

(63)

That this is actually a norm in

W_{0}^{1, p} (Ω)

, equivalent to the standard one of

W^{1, p} (Ω)

, is a consequence ofPoincaré’s inequality(see e.g., [14]), stating that

\int_{Ω} {| v |}^{p} d x \leq C \int_{Ω} {| \nabla v |}^{p} d x

(64)

for some

C > 0

and for all

v \in W_{0}^{1, p} (Ω)

. Let

E^{'} = W^{- 1, p^{'}} (Ω)

be the dual space ofE. A (weak) solution of thep-Laplace Equation (5) is a function

u \in E

such that

A_{p} (u) = λ B_{p} (u)

(65)

where

λ = μ^{- 1}

(it will soon be clear that

μ = 0

is not an eigenvalue of Equation (5)) and

A_{p}, B_{p} : E \to E^{'}

are defined by duality via the equations

〈 A_{p} (u), v 〉 = \int_{Ω} {| u |}^{p - 2} u v d x, 〈 B_{p} (u), v 〉 = \int_{Ω} {| \nabla u |}^{p - 2} \nabla u \nabla v d x

(66)

where

u, v \in E

and

〈, 〉

denotes the duality pairing betweenE and

E^{'}

The proof of the existence of countably many eigenvalues and eigenfunctions of Equation (65) relies on the Lusternik–Schnirelmann (LS) theory of critical points for an even functional on a symmetric manifold. Complete presentations of this theory, in both finite and infinite dimensional spaces, can be found, among others, in [3,4,29,33,34]. Theorem 5 below is essentially a simplified version of Theorem A in [35], save that with respect to [35] we have for expository convenience interchanged the roles of the operatorsA andB. Thus, letE be a real, infinite dimensional, uniformly convex Banach space with dual

E^{'}

, and consider the problem

A (u) = λ B (u)

(67)

where

A, B : E \to E^{'}

are continuous gradient operators with potentials

a, b

, respectively:

A = \nabla a, B = \nabla b

. Definition 1 of gradient operator extends of course to mappings ofE into

E^{'}

replacing the scalar product with the duality pairing.

Suppose that

b (u) > 0

for

u \neq 0

; then, the eigenvectors of Equation (67) satisfying a normalization condition

b (u) = r

(

r > 0

), are precisely the constrained critical points ofa on the level set

M_{r} = {u \in E : b (u) = r} .

The additional key assumptions that we make onA andB are as follows:

$A, B$ areodd(that is, $A (- u) = - A (u)$ for $u \in E$ , and similarly forB).
A isnon-negative (that is, $〈 A (u), u 〉 \geq 0$ for $u \in E$ ) andstrongly sequentially continuous (that is, if $(u_{n}) \subset E$ converges weakly to $u_{0} \in E$ , then $A (u_{n})$ converges strongly to $A (u_{0})$ in $E^{'}$ ).
B isstrongly monotone in the following sense: there exist constants $k > 0$ and $p > 1$ such that, for all $u, v \in E$ ,
$〈 B (u) - B (v), u - v 〉 \geq {k ∥ u - v ∥}^{p} .$
(68)

By the above assumptions onB,

M_{r}

issymmetric (that is,

u \in M_{r} \Rightarrow - u \in M_{r}

) andsphere-like, in the sense that each ray through the origin hits

M_{r}

in exactly one point. If

K \subset M_{r}

is compact and symmetric, then thegenus ofK, denoted

γ (K)

, is defined as

γ (K) = \inf {n \in N : there exists a continuous odd map of K into R^{n} \ {0}} .

IfV is a subspace ofE with

\dim V = n

, then

γ (M_{r} \cap V) = n

. For

n \in N

put

K_{n} (r) = {K \subset M_{r} : K compact and symmetric, γ (K) \geq n} .

(69)

Theorem 5.

Let

A, B : E \to E^{'}

be as above. Suppose moreover that

a (u) \neq 0

implies

A (u) \neq 0

. For

n \in N

and

r > 0

, put

C_{n} (r) \equiv \sup_{K_{n} (r)} \inf_{K} a (u)

(70)

where

K_{n} (r)

is as in Equation (69). Then

\sup_{M_{r}} a (u) = C_{1} (r) \geq \dots \geq C_{n} (r) \geq C_{n + 1} (r) \geq \dots \geq 0 .

(71)

Moreover,

C_{n} (r) \to 0

n \to \infty

, and if

C_{n} (r) > 0

then

C_{n} (r)

is attained and is critical value of a on

M_{r}

: thus, there exist

u_{n} (r) \in M_{r}

and

λ_{n} (r) \in R

such that

C_{n} (r) = a (u_{n} (r))

(72)

and

A (u_{n} (r)) = λ_{n} (r) B (u_{n} (r)) .

(73)

Here are a few indications for the Proof of Theorem 5:

(i) The sequence

(C_{n} (r))

is non-decreasing because, for any

n \in N

, we have

K_{n} (r) \supset K_{n + 1} (r)

as shown by Equation (69). (ii) In addition,

C_{1} (r) = \sup_{M_{r}} a (u)

because

K_{1} (r)

contains all sets of the form

{x} \cup {- x}

x \in M_{r}

. (iii) The proof that

C_{n} (r) \to 0

n \to \infty

, together with a lot of related information, can be found for instance in [34]. (iv) Finally, the assumption that

A (u) \neq 0

whenever

a (u) \neq 0

, together with the stated continuity properties ofA andB, ensures thata satisfies the crucialPalais–Smale(PS)condition on

M_{r}

at any level

C > 0

, needed to prove the final (and most important) assertion of the Theorem via the standard deformation methods of Critical Point Theory; see for this any of the above cited references.

Of special importance—with reference to the thep-Laplace equation—is the case in whichA andB have the additional property of beingpositively homogeneous of the same degree

p - 1 > 0

, meaning that

A (t u) = t^{p - 1} A (u)

for

u \in E

and

t > 0

, and similarly forB. In this case, we have from Equation (47)

a (u) = \frac{〈 A (u), u 〉}{p}, b (u) = \frac{〈 B (u), u 〉}{p}

(74)

so that

a (u) \neq 0

implies

A (u) \neq 0

. Moreover, the use of Equation (74) in Equations (72) and (73) yields at once the relation

C_{n} (r) = λ_{n} (r) r

. In fact, here,

λ_{n} (r)

isindependent of

r > 0

: to see this, it is convenient to re-parameterize the level sets on putting for

R > 0

M_{R} = {u \in E : b (u) = \frac{R^{p}}{p}} = {u \in E : 〈 B (u), u 〉 = R^{p}} .

Asa andb arep-homogeneous, it follows that

M_{R} = R M_{1}

, that each

K \in K_{n} (R)

is the image of the corresponding set in

K_{n} (1)

under the map

u \to R u

, and that

C_{n} (R) = R^{p} C_{n} (1)

. By these remarks, we thus have the equalities

λ_{n} (R) \frac{R^{p}}{p} = C_{n} (R) = R^{p} C_{n} (1)

showing as expected that

λ_{n} (R)

is independent ofR, and precisely that

λ_{n} (R) = p C_{n} (1) = \sup_{K_{n}} \inf_{K} 〈 A (u), u 〉 \equiv λ_{n}

(75)

where

K_{n} \equiv K_{n} (1)

. From Theorem 5, we then get immediately the following statements about

λ_{n}

$\sup_{M_{1}} 〈 A (u), u 〉 = λ_{1} \geq λ_{2} \geq \dots \geq λ_{n} \geq \dots$ ;
$λ_{n} \to 0$ as $n \to \infty$ ; and
if $λ_{n} > 0$ , then there exists $u_{n} \in M_{1}$ (that is, $〈 B (u_{n}), u_{n} 〉 = 1$ ) such that $A u_{n} = λ_{n} B (u_{n})$ ; in particular, $λ_{n} = 〈 A (u_{n}), u_{n} 〉$ .

Remark 3.

The situation just described contains as a more special case that of two linear operators A and B, in which the above formulae hold with

p = 2

. Suppose in particular that A acts in a real Hilbert space H and

B = I

; then

M_{1} = {u \in H : ∥ u ∥ = 1}

is the unit sphere in H, while A is a compact, self-adjoint, non-negative linear operator (strong sequential continuity and compactness are equivalent properties for a linear operator acting in a reflexive Banach space, see e.g., [15]). Then, Equation (75)and the statements following this formula yield a good part of the familiar spectral properties of such operators: indeed it is not hard to see that the LS variational characterization in Equation (75)of

λ_{n}

reduces in this case to the classical Courant’s minimax principle expressed by Equation (15),so that the sequence in Equation (75)of the LS eigenvalues of A coincides with the decreasing sequence of all the eigenvalues of A, each counted with its multiplicity.

Returning finally to thep-Laplacian, it is now a matter of applying the above information to the operators

A_{p}, B_{p}

defined in Equation (66). One can check (see [36,37], for instance) that they satisfy all the requirements for the application of Theorem 5. Moreover, they are evidently positively homogeneous of degree

p - 1

, and finally

A_{p}

is (strictly)positive, for

〈 A_{p} (v), v 〉 = \int_{Ω} {| v |}^{p} d x > 0 for v \in E, v \neq 0 .

This implies that each of the numbers

λ_{n}

defined in Equation (75) for the pair

A_{p}, B_{p}

is strictly positive, whence it follows—using the last statement of Theorem 5—that the eigenvalue problem in Equation (65) for thep-Laplacian possesses an infinite sequence of eigenvalues

λ_{n} > 0

, each given by

λ_{n} = \sup_{K_{n}} \inf_{K} \int_{Ω} {| v |}^{p} (n = 1, 2, \dots)

where

K_{n} = {K \subset {v \in W_{0}^{1, p} : \int_{Ω} {| \nabla v |}^{p} = 1}, K compact and symmetric, γ (K) \geq n} .

Setting

μ_{n} = {λ_{n}}^{- 1}

, this finally proves the properties of Equation (5) stated in the Introduction, and in particular Equation (6).

Remark 4.

For the very special properties owned by the first eigenvalue

μ_{1}

in the sequence in Equation (6)and by the associated eigenfunctions, see for instance [37]. Anyway, it follows by our discussion that

λ_{1} = μ_{1}^{- 1}

is the best constant in Poincaré’s inequality, Equation (64):

λ_{1} = \sup_{v \in W_{0}^{1, p}, v \neq 0} \frac{\int_{Ω} {| v |}^{p}}{\int_{Ω} {| \nabla v |}^{p}} .

To conclude this section, let us remark that the study and research in problems related to thep-Laplacian has grown enormously in the last decades, and even remaining in the strict context of a “spectral theory” for Equation (5), one should at least mention the following relevant points: (i) the problem of theasymptotic distribution of the LS eigenvalues (along the classical Weyl’s law for the Laplacian); (ii) the question of the existence ofother eigenvalues outside the LS sequence; and (iii) theFredholm alternative for perturbed non-homogeneous versions of Equation (5). For information on these issues, we refer the reader to [37,38,39] and to the recent and very clear review paper [36]. Related material can be found in [40].

3. Nonlinear Perturbation of an Isolated Eigenvalue

As a way to introduce and motivate the more specific content of this section, let me start recalling a famous and beautiful result of F. Rellich in perturbation theory of linear eigenvalue problems:

Theorem 6.

([41], Theorem 1). Let

A (ϵ)

be a family of Hermitian

n \times n

matrices depending analytically on the real parameter ϵ for ϵ near 0. Let

λ_{0}

be an eigenvalue of multiplicity

m > 1

A = A (0)

. Then, for ϵ near 0,

A (ϵ)

possesses m eigenvalues

λ_{1} (ϵ), \dots, λ_{m} (ϵ)

and corresponding orthonormal eigenvectors

u_{1} (ϵ), \dots, u_{m} (ϵ)

; that is, for all sufficiently small ϵ, we have

A (ϵ) u_{i} (ϵ) = λ_{i} (ϵ) u_{i} (ϵ) (i = 1, \dots, m) .

(76)

Moreover,

λ_{i} (0) = λ_{0}

for all

i = 1, \dots, m

and the functions

λ_{i}

and

u_{i}

depend analytically on ϵ near

ϵ = 0

As is well known, the “ideal" situation described by Theorem 6 for thesplitting of the multiple eigenvalue does not hold in general. In Rellich’s words, “...our question about the eigenvalues reduces to asking whether or not the zeroes of a polynomial [in the case, the characteristic polynomial of a matrix whose elements depend analytically on a parameter

ϵ

] are themselves regular analytic functions of

ϵ

for small

ϵ

. In general the answer is no; a counterexample is

λ^{2} + ϵ

. What is true is that if

λ = λ (0)

is a zero for

ϵ = 0

, then the zero

λ (ϵ)

can be written as a convergent (for small

ϵ

) power series in

ϵ^{\frac{1}{h}}

(Puiseux series) whereh is the multiplicity of

λ = λ (0)

.” The example indicated by Rellich can be displayed as

A (ϵ) \equiv \begin{matrix} (\begin{matrix} 0 & 1 \\ - ϵ & 0 \end{matrix}) \end{matrix} = \begin{matrix} (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) \end{matrix} + ϵ \begin{matrix} (\begin{matrix} 0 & 0 \\ - 1 & 0 \end{matrix}) \end{matrix} \equiv A + ϵ B

(77)

and shows the unperturbed eigenvalue

λ_{0} = 0

ofA, of multiplicity

h = 2

, splitting into the two simple eigenvalues

λ (ϵ) = \pm {(- ϵ)}^{\frac{1}{2}}

A (ϵ)

. In general, if

λ_{0}

has multiplicityh, the perturbed eigenvalue(s)

λ (ϵ)

will admit an expansion such as

λ (ϵ) = λ_{0} + ϵ^{\frac{1}{h}} λ_{1} + ϵ^{\frac{2}{h}} λ_{2} + \dots + ϵ λ_{h} + \dots = λ_{0} + \sum_{i = 1}^{\infty} ϵ^{\frac{i}{m}} λ_{i}

(78)

For the special case that

A (ϵ)

is Hermitian, using the reality of

λ (ϵ)

Rellich showed in [41] that only integral powers of

ϵ

can have non-zero coefficients in the expansion of Equation (78), thus proving the analytic dependence on

ϵ

of the perturbed eigenvalues as stated in Theorem 6. Rellich’s work was a main starting point for the very vast literature concerning the systematic analysis of the perturbation of eigenvalues of linear operators, both in finite and infinite dimensional spaces; see Kato’s book [7] and the references therein. Our aim in this section is to indicate some partial results about similar questions fornonlinear eigenvalue problems, both of typeG and of typeK, recently appearing in [8,9], respectively.

3.1. A Perturbation Problem of Type G

In the paper [8], the authors study the splitting of a multiple eigenvalue of the nonlinear eigenvalue problem, depending on the real parameter

ϵ

M (λ; ϵ) v = 0, λ \in C, v \in C^{n}, v \neq 0 .

(79)

Here,

M (λ; ϵ)

is an

n \times n

complex matrix having an eigenvalue

λ_{0}

for

ϵ = 0

(i.e., det

M (λ_{0}; 0) = 0

). As in the linear case, a perturbation theory for the eigenvalue

λ_{0}

consists in the study of the eigenvalues of Equation (79)—and of the corresponding eigenvectors—in the vicinity of

λ_{0}

, and will focus precisely on the behaviour of such eigenvalues/eigenvectors as functions

λ (ϵ), v (ϵ)

of the parameter

ϵ

for

ϵ

near 0; one assumes to know the solutions of Equation (79) for

ϵ = 0

, i.e., to know the nullspace of

M (λ_{0}; 0)

. In the linear case, we have

M (λ; ϵ) = A (ϵ) - λ I

for some assigned functionA of

ϵ

into

C^{n \times n}

, and Rellich’s theorem can be rephrased on saying that if this function is analytic and with Hermitian values, and if

\dim Ker (A (0) - λ_{0} I) = m

then there existm pairs of analytic functions

λ_{i} (ϵ), u_{i} (ϵ)

such that

λ_{i} (0) = λ_{0}, u_{i} (0) \in Ker (A (0) - λ_{0} I)

, each pair satisfying identically Equation (76) for

ϵ

near 0.

For the study of Equation (79), it is assumed that

M (λ; ϵ)

depends regularly on

λ

and

ϵ

in the following sense: there exists an open set

Ω \subset C

containing

λ_{0}

, and an open interval

I \subset R

containing zero, such that for all

ϵ \in I

the entries ofM are analytic functions of

λ

Ω

, and for all

λ \in Ω

the entries ofM are smooth functions of

ϵ

inI. In the first part of [8], the authors develop previous work on the subject and consider the case in which the geometric multiplicity of

λ_{0}

(that is, the dimension of the nullspace of

M (λ_{0}, 0)

) is one, while its algebraic multiplicity (that is, the multiplicity of

λ

as a root of the characteristic equation det

M (λ; 0) = 0)

m > 1

. Thus,

λ_{0}

in a multiple, nonsemisimple eigenvalue ofM for

ϵ = 0

. The following notations are used in the sequel:

M_{ϵ} \equiv \frac{\partial M}{\partial ϵ} (λ_{0}, 0); M_{λ} \equiv \frac{\partial M}{\partial λ} (λ_{0}, 0); M_{λ λ} \equiv \frac{\partial^{2} M}{\partial λ^{2}} (λ_{0}, 0); \dots M_{λ^{m}} \equiv \frac{\partial^{m} M}{\partial λ^{m}} (λ_{0}, 0)

Theorem 7.

[8]. Let

λ_{0}

be an eigenvalue of Equation (79)for

ϵ = 0

, with algebraic multiplicity equal to m and geometric multiplicity one, with Jordan chain

(H_{0}, \dots, H_{m - 1})

. Let

U_{0}

be the corresponding left eigenvector. Assume that the condition

U_{0}^{*} M_{ϵ} H_{0} \neq 0

holds. Then, around

ϵ = 0

, the eigenvalues in the vicinity of

λ_{0}

can be expanded as the branches of the Puiseux series in Equation (78),where

λ_{1}^{m} = - \frac{U_{0}^{*} M_{ϵ} H_{0}}{U_{0}^{*} (\frac{1}{1!} M_{λ} H_{m - 1} + \frac{1}{2!} M_{λ λ} H_{m - 2} + \dots + \frac{1}{m!} M_{λ^{m}} H_{0})} .

(80)

Remark 5.

In the classical terminology of Numerical Analysis (see e.g., ([42], p. 137), a (column) vector

U \in C^{n} = C^{n \times 1}

is a left eigenvector of a matrix M if

U^{*} M = 0

, where

U^{*}

denotes the transpose of the conjugate. “Starring both sides”, this is equivalent to

M^{*} U = 0

, that is U is a (“right”) eigenvector of the adjoint matrix

M^{*}

. With the same notations, for the scalar product in

C^{n}

we have

〈 x, y 〉 = \sum_{i = 1}^{n} x_{i} \bar{y_{i}} = y^{*} x

(the last product being the matrix product between

y^{*} \in C^{1 \times n}

and

x \in C^{n \times 1}

), and therefore Equation (80)reads

λ_{1}^{m} = - \frac{〈 M_{ϵ} H_{0}, U_{0} 〉}{〈 Z, U_{0} 〉}

with

Z = (\frac{1}{1!} M_{λ} H_{m - 1} + \frac{1}{2!} M_{λ λ} H_{m - 2} + \dots + \frac{1}{m!} M_{λ^{m}} H_{0}) .

To some extent, the proof of Theorem 7 relies on previous work by Lancaster et al. [25,43] on the perturbation of analytic matrix functions. To indicate the main idea followed to obtain Equation (80), consider that by definition the perturbed

λ (ϵ), v (ϵ)

have to satisfy for all

ϵ

the condition

M (λ (ϵ), ϵ) v (ϵ) = 0 .

(81)

Now use the Taylor expansion of

M (λ, ϵ)

around

(λ_{0}, 0)

M (λ, ϵ) = M (λ_{0}, 0) + M_{ϵ} ϵ + \frac{1}{1!} M_{λ} (λ - λ_{0}) + \frac{1}{2!} M_{λ λ} {(λ - λ_{0})}^{2} + \dots

and replace

λ

with the expansion of Equation (78) for

λ (ϵ)

. Using a similar expansion for

v (ϵ)

v (ϵ) = V_{0} + \sum_{i = 1}^{\infty} ϵ^{\frac{i}{m}} V_{i}

starting with an eigenvector

V_{0}

associated with

λ_{0}

, and putting all this in Equation (81) yields (equalling to zero the coefficients of the increasing powers of

ϵ^{\frac{1}{m}}

)

m + 1

recursive equations that contain the elements of a Jordan chain built upon the unknown vectors

V_{1}, \dots, V_{m}

. Solving these equations with the help of a technical lemma (Lemma 2.1 in [8]) that relates all possible Jordan chains corresponding to the same eigenvalue, one returns to the original chain

(H_{0}, \dots, H_{m - 1})

and finally obtains Equation (80).

Example 1.

[8]. Consider the perturbed matrix

M_{2} (λ, ϵ) = \begin{matrix} (\begin{matrix} λ - 1 + e^{- λ} + ϵ & 0 \\ 0 & λ + 1 + ϵ \end{matrix}) \end{matrix}

(82)

that for

ϵ = 0

reduces to Equation (30).For the unperturbed eigenvalue

λ_{0} = 0

, in addition to the Jordan chain in Equation (31),one has

U_{0} = \begin{matrix} (\begin{matrix} 1 \\ 0 \end{matrix}), \end{matrix} M_{ϵ} = I, M_{λ} = \begin{matrix} (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}), \end{matrix} M_{λ λ} = \begin{matrix} (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) . \end{matrix}

(83)

Substituting these values in Equation (80),one obtains

λ_{1} = \sqrt{- 2} .

In the second part of [8], the authors consider general linear functional differential equations of the form

x^{'} (t) = \int_{- τ_{m a x}}^{0} d μ (θ) x (t + θ), x (t) \in C^{n}

(84)

where

μ : [- τ_{m a x}, 0] \to C^{n \times n}

is a function of bounded variation such that

μ (0) = 0

. We refer to Chapter 7 of [26] for a thorough discussion of this kind of problems. Note that Equation (84) contains as special cases both equations with discrete delay and equations with continuous delay: indeed, Equation (84) takes the form of Equation (35) if one lets

0 = τ_{0} < τ_{1} < \dots < τ_{k} \leq τ_{m a x}

and defines

μ : [- τ_{m a x}, 0] \to C^{n \times n}

as follows:

\{\begin{matrix} μ (0) & = 0 \\ μ (θ) & = - \sum_{i = 0, - τ_{i} > θ}^{k} A_{i}, θ \in (- τ_{m a x}, 0) \\ μ (- τ_{m a x}) & = - \sum_{i = 0}^{k} A_{i} . \end{matrix}

(85)

On the other hand, taking in Equation (84)

μ (θ) = - \int_{θ}^{0} A (s) d s

for

θ \in [- τ_{m a x}, 0]

, withA a continuous function from

[- τ_{m a x}, 0]

C^{n \times n}

, yields the system with distributed delay

x^{'} (t) = \int_{- τ_{m a x}}^{0} A (θ) x (t + θ) d θ .

The relation of Equation (84) with Equation (79) is as follows: looking for solutions

x (t) = e^{λ t} v (v \in C^{n}

) of Equation (84) yields the equation

λ v = \int_{- τ_{m a x}}^{0} d μ (θ) e^{λ θ} v \equiv N (λ) v

(86)

that is a non-parametric form of Equation (79) with

M (λ) = λ I - N (λ)

. The authors then consider the infinite-dimensional vector space

X = C^{n} \times L^{2} ([- τ_{m a x}, 0], C^{n})

and a suitably defined linear operatorA acting inX and having the property that Equation (84) can be rewritten as the abstract ordinary differential equation

z^{'} (t) = A z (t)

inX. Moreover,

λ_{0}

is an eigenvalue of the nonlinear eigenvalue problem in Equation (86) if and only if it is an eigenvalue of the linear operatorA, and in Theorem 3.1 of [8] it is shown how to build an (ordinary) Jordan chain forA corresponding to

λ_{0}

starting from a Jordan chain for

λ_{0}

as an eigenvalue of the NLEVP in Equation (86), and vice versa; for this matter, see also ([26], Chapter 7, Theorem 4.2). Further exploiting this functional-analytic point of view, the authors are then able to deal with parameter-dependent forms of Equation (84)—that is, with functions

μ = μ (θ, ϵ)

—and to reformulate the sensitivity formula in Equation (80) for the eigenvalues

λ_{ϵ}

of the perturbed matrix

N (λ; ϵ)

, corresponding to

μ = μ (θ, ϵ)

as in Equation (86), in terms of eigenvectors and generalized eigenvectors of the linear operator

A (ϵ)

acting inX. This produces a more readable formula, given in Theorem 3.2 of [8], for the coefficient

λ_{1}

of the leading term in the expansion in Equation (78).

The concluding section of [8] shows applications of the theory to some numerical examples, that deal in particular with a planar time-delay system containing an uncertain delay

τ + ϵ

and with a model problem for spectral abscissa optimization.

3.2. A Perturbation Problem of Type K

In [9] we have considered the following parameter-dependent version of Equation (39),

T x + ϵ B (x) = λ x, x \in S

(87)

where—as inSection 2.2—T is a self-adjoint bounded linear operator acting in a real Hilbert spaceH and having

λ_{0} \in R

as an isolated eigenvalue of finite multiplicity. In Equation (87),S stands for the unit sphere inH, so that

S \cap Ker (T - λ_{0} I)

is the unit sphere in some

R^{n}

. As to the nonlinear termB, we shall soon give precise assumptions, but roughly speaking can say that the

ϵ

term appearing before it in Equation (87) replaces the condition

B (x) = o (∥ x ∥)

∥ x ∥ \to 0

previously considered for bifurcation in Equation (39). Indeed, rather than looking for solutions of small norm as in Equation (39), we now look fornormalized eigenvectors of the perturbed eigenvalue problem

T x + ϵ B (x) = λ x

. Here, is our result for Equation (87):

Theorem 8.

Let T be a self-adjoint bounded linear operator acting in a real Hilbert space H, and having

λ_{0}

as an isolated eigenvalue of finite multiplicity. Suppose that B is a

C^{1}

map of H into itself, and suppose moreover that at least one of the following conditions is satisfied: either

(a): the dimension of the nullspace $N \equiv Ker (T - λ_{0} I)$ is odd; or
(b): B is a gradient operator.

Then, there exist

ϵ_{0} > 0, δ_{0} > 0

such that for any

ϵ \in [- ϵ_{0}, ϵ_{0}]

, there exist

λ_{ϵ} \in [λ_{0} - δ_{0}, λ_{0} + δ_{0}]

and

x_{ϵ} \in S

such that

T x_{ϵ} + ϵ B (x_{ϵ}) = λ_{ϵ} x_{ϵ} .

(88)

If moreover B is bounded on S, then

λ_{ϵ} \to λ_{0}

ϵ \to 0

. Finally, if we suppose in addition that

B (0) = 0

and that B is Lipschitz continuous in the unit ball

U = {x \in H : ∥ x ∥ \leq 1}

of H, i.e., that there exist

k > 0

such that

∥ B (x) - B (y) ∥ \leq k ∥ x - y ∥

(89)

for

x, y \in U

, then putting

C = \inf_{0 < ∥ v ∥ \leq 1, v \in N} \frac{〈 B (v), v 〉}{{∥ v ∥}^{2}}, D = \sup_{0 < ∥ v ∥ \leq 1, v \in N} \frac{〈 B (v), v 〉}{{∥ v ∥}^{2}}

(90)

the following asymptotic estimate for

λ_{ϵ}

hold as

ϵ \to 0^{+} :

λ_{0} + ϵ C + O (ϵ^{2}) \leq λ_{ϵ} \leq λ_{0} + ϵ D + O (ϵ^{2}) .

(91)

The same estimate, with reversed inequalities, holds for

ϵ \to 0^{-}

Remark 6.

The bounds in Equation (91) are sharp in the sense that there exist perturbing operators B satisfying all the assumptions of the Theorem, and perturbed eigenvalues

λ^{+} (ϵ), λ^{-} (ϵ)

T + ϵ B

that satisfy at least one of the inequalities in Equation (91) with the equality sign. To see this, just consider a linear operator

B_{0}

acting in the finite-dimensional subspace N, and then extend it to all of H on putting

B (x) = B_{0} (v)

for all

x \in H

, with v the orthogonal projection of x onto N. If

B_{0} : N \to N

is taken to be self-adjoint, then it has n eigenvalues (counting multiplicities)

μ_{0}^{1} \leq .... \leq μ_{0}^{n}

with normalized eigenvectors

v_{1}, \dots, v_{n}

, say; that is,

B_{0} v_{i} = μ_{0}^{i} v_{i}

and

∥ v_{i} ∥ = 1

. Then, putting for each

i = 1, \dots, n

λ_{ϵ} = λ_{0} + ϵ μ_{0}^{i}, x_{ϵ} = v_{i}

we have n families of eigenvalues/eigenvectors satisfying Equation (87) for all

ϵ \in R

. We have

μ_{0}^{i} = 〈 B_{0} v_{i}, v_{i} 〉 = 〈 B v_{i}, v_{i} 〉

for each i, and the variational characterization of the eigenvalues of

B_{0}

gives in particular

μ_{0}^{1} = \inf_{0 < ∥ v ∥ \leq 1, v \in N} \frac{〈 B (v), v 〉}{{∥ v ∥}^{2}} = C

and similarly

μ_{0}^{n} = D

. Therefore taking

λ^{-} (ϵ) = λ_{0} + ϵ μ_{0}^{1}

(respectively,

λ^{+} (ϵ) = λ_{0} + ϵ μ_{0}^{n}

), the left-hand side (respectively, the right-hand side) of Equation (91) is satisfied with equality sign and

O (ϵ^{2}) = 0

The first part of Theorem 8 is proved following the track indicated inSection 2.2, that is performing the Lyapounov–Schmidt reduction of Equation (87). One non-trivial difference is that here aglobal version of the Implicit Function Theorem is employed in order to obtain a mapping

(δ, ϵ, v) \to w (δ, ϵ, v)

defined in an open neighborhood

Y_{1} = I_{1} \times J_{1} \times V_{1} \subset R \times R \times N

{0} \times {0} \times S

by the rule that the

ϵ

-dependent complementary equation (see Equation (52)) can be solved uniquely with respect tow for each given

(δ, ϵ, v) \in Y_{1}

. Moreover,

w (0, 0, v) = 0

for any

v \in S

, and the mapping

(δ, ϵ, v) \to w (δ, ϵ, v)

Y_{1}

intoW is of class

C^{1}

. Next, expressing

δ

as a

C^{1}

function

δ (ϵ, v)

(ϵ, v)

in a possibly smaller neighborhood

J \times V \subset J_{1} \times V_{1}

, and putting for convenience

ϕ (ϵ, v) \equiv w (δ (ϵ, v), ϵ, v), (ϵ, v) \in J \times V

we arrive at the

ϵ

-dependent form of the bifurcation Equation (57), namely

ϵ P B (v + ϕ (ϵ, v)) = δ (ϵ, v) v

(92)

that is here accompanied by the norm constraint

{∥ v + ϕ (ϵ, v) ∥}^{2} = {∥ v ∥}^{2} + {∥ ϕ (ϵ, v) ∥}^{2} = 1 .

(93)

A solution

(λ, x)

of the original problem in Equation (87) will then be given by the formulae

λ = λ_{0} + δ (ϵ, v), x = v + ϕ (ϵ, v) .

(94)

In any of the two Cases (a) and (b) listed in Theorem 8, using as needed either of the methods (topological or variational) recalled in general inSection 2.2, we find for each

ϵ

small a solution

v_{ϵ}

of Equations (92) and (93). Therefore, making an appropriate choice of

δ_{0}, ϵ_{0}

for the intervals

I_{0} \equiv [- δ_{0}, δ_{0}]

J_{0} \equiv [- ϵ_{0}, ϵ_{0}]

and putting

δ_{ϵ} = δ (ϵ, v_{ϵ}) and w_{ϵ} = ϕ_{ϵ} (v_{ϵ}) = w (δ_{ϵ}, ϵ, v_{ϵ})

(95)

the first part of Theorem 8, asserting the existence of at least one solutions

(λ_{ϵ}, x_{ϵ}) \in I_{0} \times S

of Equation (87) for each

ϵ \in J_{0}

, is proved with

λ_{ϵ} = λ_{0} + δ_{ϵ}

and

x_{ϵ} = v_{ϵ} + w_{ϵ}

Some words are now in order to explain the estimates in Equation (91). One first shows that the component

w_{ϵ}

x_{ϵ}

(as defined in Equation (95)) satisfies

w_{ϵ} \to 0

ϵ \to 0

, uniformly with respect to

v_{ϵ}

, and consequently with respect to

x_{ϵ}

. This in turn implies that

λ_{ϵ} \to λ_{0}

ϵ \to 0

, uniformly with respect to

x_{ϵ}

. Indeed, using Equation (92), we have

ϵ P B (v_{ϵ} + w_{ϵ}) = δ_{ϵ} v_{ϵ}

for all

ϵ

, whence taking the scalar product with

v_{ϵ}

of both members, we obtain

ϵ 〈 B (v_{ϵ} + w_{ϵ}), v_{ϵ} 〉 = δ_{ϵ} {∥ v_{ϵ} ∥}^{2} .

(96)

Therefore,

δ_{ϵ} = ϵ \frac{〈 B (x_{ϵ}), v_{ϵ} 〉}{∥ v_{ϵ} ∥^{2}} .

(97)

Moreover, as

x_{ϵ} = v_{ϵ} + w_{ϵ} \in S

and

w_{ϵ} \to 0

as indicated above, then necessarily

∥ v_{ϵ} ∥ \to 1

ϵ \to 0

. Therefore, since

\frac{| 〈 B (x_{ϵ}), v_{ϵ} 〉 |}{∥ v_{ϵ} ∥^{2}} \leq \frac{∥ B (x_{ϵ}) ∥}{∥ v_{ϵ} ∥}

it follows, by the boundedness assumption onB, that the term multiplying

ϵ

in Equation (97) remains bounded as

ϵ \to 0

, implying that

δ_{ϵ} = O (ϵ)

ϵ \to 0

, uniformly with respect to

x_{ϵ}

We can now prove the asymptotic formula in Equation (91) on

λ_{ϵ}

ifB satisfies Equation (89). In this respect, the utility of Equation (89) is twofold. First, it permits to improve significantly the information on

w_{ϵ}

as it yields by means of straightforward computations the estimate

∥ w (δ, ϵ, v) ∥ \leq C_{1} | ϵ | ∥ v ∥

(98)

holding for some constant

C_{1} > 0

and all

(δ, ϵ, v) \in [- δ_{0}, δ_{0}] \times [- ϵ_{0}, ϵ_{0}] \times (U \cap N)

. Moreover, Equation (89) implies via the Schwarz’ inequality that, for anyv andw such that

v, v + w \in U

, one has

| 〈 B (v + w), v 〉 - 〈 B (v), v 〉 | \leq k ∥ v ∥ ∥ w ∥ .

Writing this for

w (δ, ϵ, v)

and using Equation (98), we then get the inequality

| 〈 B (v + w (δ, ϵ, v)), v 〉 - 〈 B (v), v 〉 | \leq C_{2} {| ϵ | ∥ v ∥}^{2}

(99)

with

C_{2} = k C_{1}

, valid for all the (possible) solutions

(δ, x = v + w (δ, ϵ, v))

of Equation (87) having sufficiently small

ϵ

. Using in turn this estimate in Equation (96) for the actual solutions

(δ_{ϵ}, x_{ϵ} = v_{ϵ} + w_{ϵ})

we see that as

ϵ \to 0

δ_{ϵ} ∥ v_{ϵ} ∥^{2} = ϵ 〈 B (v_{ϵ} + w_{ϵ}), v_{ϵ} 〉 = ϵ 〈 B (v_{ϵ}), v_{ϵ} 〉 + {∥ v_{ϵ} ∥}^{2} O (ϵ^{2}) .

(100)

This implies that

δ_{ϵ} = ϵ \frac{〈 B (v_{ϵ}), v_{ϵ} 〉}{∥ v_{ϵ} ∥^{2}} + O (ϵ^{2})

(101)

ϵ \to 0

, and thus for

ϵ > 0

yields immediately the estimate of Equation (91) in view of the definition in Equation (90) ofC andD.

Example 2.

Theorem 8 can be used to evaluate the convergence rate as

ϵ \to 0

of the eigenvalues

μ_{ϵ}

of the nonlinear elliptic problem

\{\begin{matrix} - Δ u & = μ (u + ϵ f (x, u)) & i n & Ω \\ u & = 0 & o n & \partial Ω \end{matrix}

(102)

near an eigenvalue

μ_{0}

of the unperturbed linear problem

- Δ u = μ u

in Ω,

u = 0

\partial Ω

. Here, Ω is a bounded domain in

R^{N} (N \geq 1)

with boundary

\partial Ω

, and

Δ = \sum_{i = 1}^{N} \frac{\partial^{2}}{\partial {x_{i}}^{2}}

is the familiar Laplace operator acting on sufficiently smooth real functions u defined in Ω. Under appropriate hypotheses on f, and assuming in particular that

m t^{2} \leq f (x, t) t \leq M t^{2} (x \in Ω, t \in R)

(103)

for some real constants

0 \leq m \leq M

, one proves that as

ϵ \to 0^{+}

μ_{0} - ϵ μ_{0} M + O (ϵ^{2}) \leq μ_{ϵ} \leq μ_{0} - ϵ μ_{0} m + O (ϵ^{2}) .

(104)

These inequalities can be used for actual computation, once an efficient approximation of the linear eigenvalue

μ_{0}

is available and the bounds in Equation (103)for f are known with accuracy. For instance, if

f (x, t) = f (t) = \frac{t}{1 + t^{2}}

, just put in Equation (104)

m = \inf_{t \neq 0} \frac{f (t) t}{t^{2}} = 0, M = \sup_{t \neq 0} \frac{f (t) t}{t^{2}} = 1 .

(105)

4. Concluding Remarks, Open Problems and Applicability

To summarize and motivate again the content of this paper, let me define it as an attempt to identify and logically re-connect (or at least give a common frame to) two important and presently distinct research areas in Mathematical Analysis and its applications that appear in the current literature under the same name ofNonlinear Eigenvalue Problems. As better explained in the Introduction, problems in these two areas are described (in abstract operator form) by the two equations

G (λ) x = 0

(106)

(“problems of type G”) and

A (x) - λ C (x) = 0

(107)

(“problems of type K”). Some basic facts and solution methods about each of the two equations are reported inSection 2.Section 3 is devoted to discuss two specific problems, one for each type, with the scope of giving samples of very recent research in either field.

While for the problem discussed inSection 3.1 there are already concrete numerical examples [8], these are still missing for the problem presented inSection 3.2 [9]. The main aim of this final section is to partially fill this gap by further commenting (inSection 4.1) on the formula

λ_{0} + ϵ C + O (ϵ^{2}) \leq λ_{ϵ} \leq λ_{0} + ϵ D + O (ϵ^{2})

(108)

proved in Theorem 8 and by finally providing, at least in a special case, a recipe ready for use in numerical simulation (Section 4.2).

4.1. Open Problems

The basic idea standing behind the formula in Equation (108) is that if the (algebraic and geometric) multiplicity

m (λ_{0})

of the unperturbed eigenvalue

λ_{0}

of the linear operatorT in Equation (87) is equal tom, then upon perturbation by a small term

ϵ B

there are potentiallym eigenvalue functions

λ_{1} (ϵ), \dots, λ_{m} (ϵ)

T + ϵ B

satisfying Equation (108). This is what actually happens for alinear operatorB (essentially under the further assumption thatT andB are self-adjoint) as described by Rellich’s Theorem 6. Our idea is that something of this persists also fornonlinear operators that have good similarity with the linear ones: the class of Lipschitz continuous maps considered in Theorem 8 is apparently quite close to that of bounded linear maps, and in fact contains properly the latter. Indeed, the conclusions of Theorem 8 point in in this direction. However, many problems remain open and we describe here three of them (in increasing order of interest and difficulty):

Verify on specific examples of nonlinear ODE/PDE/Equations in $R^{n}$ —by means of explicit computation or by means of a numerical analysis—the existence of at least one “eigenvalue branch” $λ (ϵ)$ satisfying the bounds in Equation (108) as predicted by the theory.
Verify by the same means that the bounds in Equation (108) are optimal by producing examples of nonlinear problems (in the same fields as above) where at least one eigenvalue function exists that satisfies the RHS (LHS) bound in Equation (108) with equality sign.
Exhibit examples of “nonlinear splitting of the multiple eigenvalue”, that is, of nonlinear problems in which starting from an unperturbed eigenvalue $λ_{0}$ of multiplicity $\geq 2$ there existtwo different families $λ_{+} (ϵ)$ , $λ_{-} (ϵ)$ respecting Equation (108), and possibly each satisfying the RHS (LHS) bound of it with equality sign.

Here, is a very simple example that highlights the above issues, and the last in particular. For the linear case, these questions were answered by Remark 6.

Example 3.

Consider the system

\{\begin{matrix} x + ϵ x^{3} = λ x \\ y + ϵ y^{3} = λ y . \end{matrix}

(109)

In the notations of Equation (87)and of Theorem 8, we have here

H = R^{2}

T = I

λ_{0} = 1

and

B (x, y) = (x^{3}, y^{3})

. Solving Equation (109)with the constraint

x^{2} + y^{2} = 1

gives the solutions

\{\begin{matrix} (x, y) = (0, \pm 1), (x, y) = (\pm 1, 0), λ = 1 + ϵ \\ (x, y) = (\pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}}), λ = 1 + ϵ / 2 . \end{matrix}

(110)

Therefore,

λ_{0} = 1

splits into the two eigenvalue functions (each carrying four distinct eigenvectors)

λ_{+} (ϵ) = 1 + ϵ, λ_{-} (ϵ) = 1 + ϵ / 2 .

This is in full agreement with Equation (108),for

D = \sup_{0 < ∥ v ∥ \leq 1, v \in N} \frac{〈 B (v), v 〉}{{∥ v ∥}^{2}} = \sup_{0 < x^{2} + y^{2} \leq 1} \frac{x^{4} + y^{4}}{x^{2} + y^{2}} = 1

(111)

and similarly, replacing “sup" with “inf", we find that

C = 1 / 2

4.2. Applicability

As indicated in Example 2, the bounds in Equation (108) proved in Theorem 8 for the operator Equation (87) can be used for concrete nonlinear elliptic problems in a bounded domain

Ω \subset R^{N} (N \geq 1)

such as Equation (102), or more generalized forms of it in which

- Δ

is replaced by a uniformly elliptic second order operator in divergence form. In particular for

N = 1

this applies to the Sturm–Liouville problem

\{\begin{matrix} - {(p (x) u^{'})}^{'} + q (x) u & = μ (u + ϵ f (x, u)) & in & ] a, b [ \\ u (a) = u (b) & = 0 \end{matrix}

(112)

where

p \in C^{1} ([a, b]), p > 0

and

q \in C ([a, b])

. The first remark on the applicability of Theorem 8 to such kind of problems, in one or more space variables, is that one must not worry of the multiplicity of the unperturbed eigenvalue because—as recalled for instance in [9]—the operatorB corresponding to the nonlinear termf is a gradient operator, so that the assumption

b)

of Theorem 8 is satisfied. In [9] it is previously recalled that these problems can be cast in the operator form of Equation (87) on taking as Hilbert space the Sobolev space

H_{0}^{1} (Ω) \equiv W_{0}^{1, 2} (Ω)

, equipped with the scalar product

〈 u, v 〉 = \int_{Ω} \nabla u (x) \nabla v (x) d x

(113)

and that the operatorB mentioned above is defined via the duality relation

〈 B (u), v 〉 = \int_{Ω} f (x, u (x)) v (x) d x .

(114)

The representation in Equation (114) ofB is the key formula to be used to gain information on the constants

C, D

appearing in Equation (108), for these are defined by the formulae in Equation (90) that involve precisely the nonlinear Rayleigh quotient ofB. Indeed, using Equation (114) and the fact (see e.g., [9]) that for

v \in N

we have

{∥ v ∥}^{2} = \int_{Ω} \nabla v^{2} (x) d x = μ_{0} \int_{Ω} v^{2} (x) d x

(115)

where

μ_{0}

is the unperturbed eigenvalue andN the corresponding eigenspace, yields the following quite readable expression forD:

D = \sup_{0 < ∥ v ∥ \leq 1, v \in N} \frac{〈 B (v), v 〉}{{∥ v ∥}^{2}} = \sup_{0 < ∥ v ∥ \leq 1, v \in N} \frac{\int_{Ω} f (x, v (x)) v (x) d x}{μ_{0} \int_{Ω} v^{2} (x) d x}

(116)

Thus, essentially, in the applications of the theory to elliptic PDE or ODE, estimating the Rayleigh quotient ofB reduces to estimating the ratio appearing in the RHS of Equation (116). In turn, this can be easily obtained bypointwise bounds onf: for clearly iff satisfies Equation (103), then it follows that for every

v \in H

(and in fact for every

v \in L^{2} (Ω)

)

m \leq \frac{\int_{Ω} f (x, v (x)) v (x) d x}{\int_{Ω} v^{2} (x) d x} \leq M .

We conclude by Equation (116) and the dual formula forC that

\frac{m}{μ_{0}} \leq C, D \leq \frac{M}{μ_{0}} .

(117)

Using the inequalities in Equations (117) and (108) and putting

λ_{ϵ} = 1 / μ_{ϵ}

yield bounds on the perturbed eigenvalues

μ_{ϵ}

of Equation (102). Considering for instance the right-hand side of in Equation (108), we obtain

\frac{1}{μ_{ϵ}} \leq \frac{1}{μ_{0}} + \frac{ϵ M}{μ_{0}} + O (ϵ^{2}) = \frac{1}{μ_{0}} (1 + ϵ M + O (ϵ^{2}))

(118)

and doing the same with the lower bound thus yields

\frac{μ_{0}}{1 + ϵ m + O (ϵ^{2})} \geq μ_{ϵ} \geq \frac{μ_{0}}{1 + ϵ M + O (ϵ^{2})} .

(119)

Remark 7.

Note that Equation (119)contains—as due—the equality

μ_{ϵ} = \frac{μ_{0}}{1 + ϵ a}

which plainly holds for the eigenvalues of Equation (102)in the linear case

f (x, s) = a s

a =

const.

Finally, using in Equation (119) the asymptotic relation

{(1 + x)}^{- 1} = 1 - x + O (x^{2})

for

x \to 0

, we obtain as

ϵ \to 0^{+}

the formula in Equation (104), which as remarked is ready for use in numerical experiments once

μ_{0}, m

andM are known. For instance, taking

f (x, s) = f (s) = \frac{s}{1 + s^{2}}

and using the bounds

m = 0, M = 1

(see Equation (105)) yields, for

ϵ \to 0^{+}

μ_{0} - ϵ μ_{0} + O (ϵ^{2}) \leq μ_{ϵ} \leq μ_{0} + O (ϵ^{2}) .

(120)

The case of the simple eigenvalue. More information can be gained in the case that

\dim N = 1

, so that

N = {t ϕ, t \in R}

for some

ϕ

that we normalize taking

∥ ϕ ∥ = 1

. Then, the Rayleigh quotient ofB simplifies as

\frac{〈 B (v), v 〉}{{∥ v ∥}^{2}} = \frac{〈 B (t ϕ), t ϕ 〉}{t^{2} {∥ ϕ ∥}^{2}} = \frac{1}{t} 〈 B (t ϕ), ϕ 〉 \equiv h (t), 0 < | t | \leq 1 .

(121)

Note thath is bounded sinceB is sublinear (that is,

∥ B (u) ∥ \leq k ∥ u ∥

for all

u \in H

) as follows from Equation (89) and the condition

B (0) = 0

. It follows by Equations (114) and (121) that

h (t) = \frac{1}{t} \int_{Ω} f (x, t ϕ (x)) ϕ (x) d x .

(122)

Considering as above the example

f (x, s) = f (s) = \frac{s}{1 + s^{2}}

, we obtain

h (t) = \int_{Ω} \frac{ϕ^{2} (x)}{1 + t^{2} ϕ^{2} (x)} d x

showing thath can be extended continuously to

t = 0

and that it is an even function oft in

[- 1, 1]

. Therefore, using also Equation (115) and the condition

∥ ϕ ∥ = 1

, we get

D = \sup_{- 1 \leq t \leq 1} h (t) = \sup_{0 \leq t \leq 1} h (t) = \int_{Ω} ϕ^{2} (x) d x = \frac{1}{μ_{0}}

(123)

while

C = \inf_{- 1 \leq t \leq 1} h (t) = \inf_{0 \leq t \leq 1} h (t) = \int_{Ω} \frac{ϕ^{2} (x)}{1 + ϕ^{2} (x)} d x \equiv K .

(124)

These computations show that in the present case:

The upper bound $\frac{M}{μ_{0}} = \frac{1}{μ_{0}}$ given by Equation (117) forD is optimal.
The lower bound 0 given by Equation (117) forC can be improved to $C = K > 0$ .

Proceeding in the same way as before (see Equations (118) and (119)) and using as before the asymptotic expansion for

\frac{1}{1 + x}

x \to 0

, we see that, as

ϵ \to 0^{+}

, Equation (120) can be replaced by the more precise formula

μ_{0} - ϵ μ_{0} + O (ϵ^{2}) \leq μ_{ϵ} \leq μ_{0} - ϵ μ_{0}^{2} K + O (ϵ^{2}) .

(125)

The practical use of Equation (125) for numerical purposes requires explicit knowledge of the eigenvalue

μ_{0}

and of the corresponding eigenfunction

ϕ

. Typical cases in which these data are available are:

$N = 2$ , $Ω$ a rectangle or a circle, and $μ_{0}$ the first eigenvalue of the Dirichlet Laplacian in $Ω$ (see e.g., [44].
$N = 1$ , $Ω =] a, b [$ and $μ_{0} = μ_{n}$ any eigenvalue of the Sturm–Liouville problem in Equation (112) with simple forms of the coefficientsp andq. For instance, if $] a, b [=] 0, π [$ , $p \equiv 1$ and $q \equiv 0$ we have
$μ_{n} = n^{2}, ϕ_{n} (x) = \frac{1}{n} \sqrt{\frac{2}{π}} sin n x (0 \leq x \leq π) .$
As to the expression of $ϕ_{n}$ , recall that we have normed $H_{0}^{1} (a, b)$ via the formula in Equation (113), which in this case reduces to $(u, v) = \int_{0}^{π} u^{'} (x) v^{'} (x) d x$ .

Acknowledgments

The author wishes to thank the Referees for their valuable suggestions that have helped to improve the quality of the present paper.

Conflicts of Interest

The authors declare no conflict of interest.

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Chiappinelli, R. What Do You Mean by “Nonlinear Eigenvalue Problems”?Axioms2018,7, 39. https://doi.org/10.3390/axioms7020039

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Chiappinelli, R. (2018). What Do You Mean by “Nonlinear Eigenvalue Problems”?Axioms,7(2), 39. https://doi.org/10.3390/axioms7020039

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What Do You Mean by “Nonlinear Eigenvalue Problems”?

Abstract

1. Introduction

2. The Two Types of NLEVP

2.1. Problems of Type G: (Linear) Operator- and Matrix-Valued Functions

2.2. Problems of Type K: Nonlinear Operators and Bifurcation

2.3. A Very Special Nonlinear Problem: Thep-Laplace Equation

3. Nonlinear Perturbation of an Isolated Eigenvalue

3.1. A Perturbation Problem of Type G

3.2. A Perturbation Problem of Type K

4. Concluding Remarks, Open Problems and Applicability

4.1. Open Problems

4.2. Applicability

Acknowledgments

Conflicts of Interest

References

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