# Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory

## Abstract

**:**

## 1. Introduction

## 2. A Simplified Spectrum

- the number $\Vert F\Vert $ given by the first equality in (6) defines a norm in the vector space just described, and $\Vert F\Vert $ coincides with the usual linear operator norm when restricted to $L\left(X\right)$;
- the definition of $b\left(F\right)$ given by the second equality in (6) implies that$$\Vert F\left(x\right)\Vert \ge b\left(F\right)\Vert x\Vert \phantom{\rule{2.em}{0ex}}(x\in X)$$

**Remark**

**1.**

**Definition**

**1.**

**Remark**

**2.**

**Definition**

**2.**

**Remark**

**3.**

**Definition**

**3.**

**Theorem**

**1.**

**Theorem**

**2.**

**(Darbo’s**

**Fixed**

**Point**

**Theorem)**Let C be a closed, bounded, convex subset of the Banach space X, and let$F:C\to C$be continuous and$\alpha \u2014$Lipschitz with$\alpha \left(F\right)<1$. Then there exists$x\in C:F\left(x\right)=x$.

**Corollary**

**1.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

## 3. Gradient Operators

**Theorem**

**3.**

**(Ekeland**

**Variational**

**principle)**Let$(X,d)$be a complete metric space. Let$f:X\to \mathbb{R}$be lower semicontinuous and bounded below. Put$c={inf}_{x\in X}f\left(x\right)$; then given any$\u03f5>0$, there exists${x}_{\u03f5}\in X$such that

**Corollary**

**2.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**4.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Theorem**

**5.**

**Proof.**

**Remark**

**10.**

**Theorem**

**6.**

- $m\left(F\right),M\left(F\right)\in {\sigma}_{S}\left(F\right)$;
- If moreover$m\left(F\right)<-\alpha \left(F\right)$, then$m\left(F\right)\in {\sigma}_{p}\left(F\right)$. Furthermore,$m\left(F\right)$is the smallest eigenvalue of F and is a compact eigenvalue. A similar conclusion holds for $M\left(F\right)$ in case $M\left(F\right)>\alpha \left(F\right)$.

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

## 4. Compact Operators

**Theorem**

**7.**

**Corollary**

**5.**

**Proof.**

## Funding

## Conflicts of Interest

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Chiappinelli, R. Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory. *Symmetry* **2019**, *11*, 928.
https://doi.org/10.3390/sym11070928

**AMA Style**

Chiappinelli R. Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory. *Symmetry*. 2019; 11(7):928.
https://doi.org/10.3390/sym11070928

**Chicago/Turabian Style**

Chiappinelli, Raffaele. 2019. "Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory" *Symmetry* 11, no. 7: 928.
https://doi.org/10.3390/sym11070928