On Spectrum of Nonlinear Continuous Operators

TL;DR

This paper introduces a new method for determining the spectrum of nonlinear continuous operators in Banach spaces, emphasizing the importance of relative spectra and operator nonlinearity order, with applications to eigenvalues and solvability.

Contribution

It presents a novel approach to spectrum analysis for nonlinear operators, highlighting the necessity of relative spectra and matching nonlinearity orders.

Findings

Established a method for spectrum determination relative to another operator

Showed the importance of matching nonlinearity orders in spectrum analysis

Provided examples for eigenvalue computation and solvability issues

Abstract

This article proposed a new approach to the determination of the spectrum for nonlinear continuous operators in the Banach spaces and using it investigated the spectrum of some classes of operators. Here shows that in nonlinear operators case is necessary to seek the spectrum of a given nonlinear operator relatively to another nonlinear operator. Moreover, the order of nonlinearity of examined operator and operator relatively to which seek the spectrum must be identical. Here provided different examples relative to how one can find the eigenvalue and also studied solvability problems.