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Nonlinear Eigenvalues

Part of the Operator Theory: Advances and Applications book series (OT, volume 177)

Abstract

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 \( \mathfrak{L} \) Cr(Ω,\( \mathcal{L} \)(U)), and a nonlinear map \( \mathfrak{N} \) ∈ C(Ω × \( \mathcal{U} \), U) satisfying the following conditions: (AL) \( \mathfrak{L} \)(λ) ™ IU ∈ 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
$$ \mathfrak{F}\left( {\lambda ,u} \right): = \mathfrak{L}\left( \lambda \right)u + \mathfrak{N}\left( {\lambda ,u} \right), $$
(12.1)
and the associated equation
$$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {\left( {\lambda ,u} \right) \in \Omega } \\ \end{array} \times \mathcal{U}. $$
(12.2)
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
$$ \begin{array}{*{20}c} {\mathfrak{L}\left( \lambda \right)u = 0,} & {\lambda \in \Omega ,} & u \\ \end{array} \in U. $$
(12.3)
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
$$ u = \left[ {I_U - \mathfrak{L}\left( \lambda \right)} \right]u - \mathfrak{N}\left( {\lambda ,u} \right). $$
.

Keywords

Bifurcation Point Real Banach Space Algebraic Topology Topological Degree Linear Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2007

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