Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators

Shugui Kang,Wenying Feng,Yanlei Zhang

doi:10.3390/math9030278

Shugui Kang, Wenying Feng + Show 1 more

Open Access

https://doi.org/10.3390/math9030278

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Abstract

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

Highlights

Nonlinear spectral theory has been shown to have applications in the study of existence of solutions for operator equations, in integral equations [1,2]
On the other hand, fixed point index is well known as a popular technique to prove existence and multiplicity of positive solutions for Boundary Value Problems (BVPs)
We obtain results on fixed point index of the nonlinear operator LF based on parameters that are related to the nonlinear spectra

Summary

Introduction

Nonlinear spectral theory has been shown to have applications in the study of existence of solutions for operator equations, in integral equations [1,2]. We prove existence of positive solutions for a second-order differential equation with separated boundary conditions [5] and obtain a spectral interval for the Hammerstein integral operator. Let E, F be Banach spaces and f : E → F be a continuous nonlinear map. The following two lemmas on fixed point index [7] have been applied to prove existence of solutions for boundary value problems [8] and many other applications [7,9].

Stably-Solvable Maps and Fixed Point Index

Positive Solutions and Spectral Interval for BVPs