Solving an abstract nonlinear eigenvalue problem by the inverse iteration method

TL;DR

This paper applies the inverse iteration method to solve an abstract nonlinear eigenvalue problem in Banach spaces, focusing on the properties and convergence of the method in a general functional analysis setting.

Contribution

The paper introduces a novel application of the inverse iteration method to a broad class of nonlinear eigenvalue problems in Banach spaces, extending existing techniques.

Findings

Convergence of the inverse iteration method is established under certain conditions.

The method effectively approximates eigenvalues and eigenfunctions in the abstract setting.

Theoretical analysis provides insights into the spectral properties of the nonlinear problem.

Abstract

Let $\left( X,\left\Vert \cdot\right\Vert_{X}\right) $ and $\left( Y,\left\Vert \cdot\right\Vert_{Y}\right) $ be Banach spaces over $\mathbb{R},$ with $X$ uniformly convex and compactly embedded into $Y.$ The inverse iteration method is applied to solve the abstract eigenvalue problem $A(w)=\lambda\left\Vert w\right\Vert_{Y}^{p-q}B(w),$ where the maps $A:X\rightarrow X^{\star}$ and $B:Y\rightarrow Y^{\star}$ are homogeneous of degrees $p-1$ and $q-1,$ respectively.