2007 | OriginalPaper | Buchkapitel
Nonlinear Eigenvalues
Erschienen in: Algebraic Multiplicity of Eigenvalues of Linear Operators
Verlag: Birkhäuser Basel
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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
$$ u = \left[ {I_U - \mathfrak{L}\left( \lambda \right)} \right]u - \mathfrak{N}\left( {\lambda ,u} \right). $$
.