Nonlinear Eigenvalues

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 $\mathfrak{L}$ ∈ 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

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

(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

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

.