Some Remarks on the Spectrum of Nonlinear Continuous Operators

TL;DR

This paper investigates the spectrum of nonlinear continuous operators in Banach spaces, proposing a new approach to find eigenvalues relative to other operators and exploring nonlinear problems with parameters.

Contribution

It introduces a novel method for studying the spectrum of nonlinear operators relative to other operators, expanding the understanding of eigenvalues in nonlinear analysis.

Findings

Eigenvalues can be found relative to other nonlinear operators under certain conditions.

Examples demonstrate the applicability of the method to specific nonlinear operators.

Nonlinear problems with parameters are effectively analyzed using the proposed approach.

Abstract

In this article, the existence of the spectrum (the eigenvalues) for the nonlinear continuous operators acting in the Banach spaces is investigated. For the study, this question is used a different approach that allows the studying of all eigenvalues of the nonlinear operator relative to another nonlinear operator. Here shows that in nonlinear operators, case is necessary to seek the spectrum of the given nonlinear operator relative to another nonlinear operator satisfying certain conditions. The different examples, for which eigenvalues can be found are provided. Moreover, the nonlinear problems including parameters are studied.