Nonlinear Eigenvalues

Throughout this chapter, the field

$$ \mathbb{K} $$

will always be the real field ℝ; we consider a real Banach space

U

, an open interval ℭ

⊂

∝, a neighborhood

$$ \mathcal{U} $$

of 0

∈ U

, an integer number

r

≥ 0, a family

$$ \mathfrak{L} $$

∈

C

r

(Ω,

$$ \mathcal{L} $$

(

U

)), and a nonlinear map

$$ \mathfrak{N} $$

∈ C

(Ω ×

$$ \mathcal{U} $$

,

U

) satisfying the following conditions:

(AL)

$$ \mathfrak{L} $$

(

λ

) ™

I

U

∈ K

(

U

) for every

λ ∈

Ω, i.e.,

$$ \mathfrak{L} $$

(

λ

) is a compact perturbation of the identity map.

(AN)

$$ \mathfrak{N} $$

is compact, i.e., the image by

$$ \mathfrak{N} $$

of any bounded set of Ω ×

$$ \mathcal{U} $$

is relatively compact in

U

. Also, for every compact

K

⊂ Ω,

$$ \mathop {\lim }\limits_{u \to 0} \mathop {\sup }\limits_{\lambda \in K} \frac{{\left\| {\mathfrak{N}\left( {\lambda ,u} \right)} \right\|}} {{\left\| u \right\|}} = 0. $$

. From now on, we consider the operator

$$ \mathfrak{F} \in \mathcal{C}\left( {\Omega \times \mathcal{U},U} \right) $$

defined as

12.1

$$ \mathfrak{F}\left( {\lambda ,u} \right): = \mathfrak{L}\left( \lambda \right)u + \mathfrak{N}\left( {\lambda ,u} \right), $$

and the associated equation

12.2

$$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {\left( {\lambda ,u} \right) \in \Omega } \\ \end{array} \times \mathcal{U}. $$

By Assumptions (AL) and (AN), it is apparent that

$$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {D_u \mathfrak{F}\left( {\lambda ,u} \right) = \mathfrak{L}\left( \lambda \right),} & \lambda \\ \end{array} \in \Omega , $$

and, hence, (12.2) can be thought of as a

nonlinear perturbation

around (λ, 0) of the linear equation

12.3

$$ \begin{array}{*{20}c} {\mathfrak{L}\left( \lambda \right)u = 0,} & {\lambda \in \Omega ,} & u \\ \end{array} \in U. $$

Equation (12.2) can be expressed as a fixed-point equation for a compact operator. Indeed,

$$ \mathfrak{F}\left( {\lambda ,u} \right) $$

= 0 if and only if

$$ u = \left[ {I_U - \mathfrak{L}\left( \lambda \right)} \right]u - \mathfrak{N}\left( {\lambda ,u} \right). $$

.