# 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

## References

- Weinberger, H.F. Variational Methods for Eigenvalue Approximation; CBMS-NSF Regional Conference Series in Applied Mathematics 15; SIAM: Philadelphia, PA, USA, 1974. [Google Scholar]
- Lindqvist, P. A Nonlinear Eigenvalue Problem. In Topics in Mathematical Analysis; World Scientific Publishing: Hackensack, NJ, USA, 2008; pp. 175–203. [Google Scholar]
- Lindqvist, P. On nonlinear Rayleigh quotients. Potential Anal.
**1993**, 2, 199–218. [Google Scholar] [CrossRef] - Appell, J.; De Pascale, E.; Vignoli, A. Nonlinear Spectral Theory; Walter de Gruyter: Berlin, Germany, 2004. [Google Scholar]
- Feng, W. A new spectral theory for nonlinear operators and its applications. Abstr. Appl. Anal.
**1997**, 2, 163–183. [Google Scholar] [CrossRef][Green Version] - Furi, M.; Martelli, M.; Vignoli, A. Contributions to the spectral theory for nonlinear operators in Banach spaces. Ann. Mater. Pura Appl.
**1978**, 118, 229–294. [Google Scholar] [CrossRef] - Edmunds, D.E.; Webb, J.R.L. Remarks on nonlinear spectral theory. Boll. UN Mater. Ital. B
**1983**, 2, 377–390. [Google Scholar] - Chiappinelli, R. An application of Ekeland’s variational principle to the spectrum of nonlinear homogeneous gradient operators. J. Math. Anal. Appl.
**2008**, 340, 511–520. [Google Scholar] [CrossRef] - Chiappinelli, R. Surjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory. Ann. Funct. Anal.
**2019**, 10, 170–179. [Google Scholar] [CrossRef] - Kato, T. Perturbation Theory for Linear Operators, 2nd ed.; Springer: Berlin, Germany, 1976. [Google Scholar]
- Taylor, A.; Lay, D. Introduction to Functional Analysis; Wiley: Hoboken, NJ, USA, 1980. [Google Scholar]
- Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations; Springer: Berlin, Germany, 2011. [Google Scholar]
- Ekeland, I. On the variational principle. J. Math. Anal. Appl.
**1974**, 47, 324–353. [Google Scholar] [CrossRef][Green Version] - De Figueiredo, D.G. Lectures on the Ekeland Variational Principle with Applications and Detours; Tata Institute of Fundamental Research: Bombay, India, 1989. [Google Scholar]
- Chiappinelli, R. What do you mean by “nonlinear eigenvalue problems”? Axioms
**2018**, 7, 39. [Google Scholar] [CrossRef] - Banaś, J.; Goebel, K. Measures of Noncompactness in Banach Spaces; Lecture Notes in Pure and Applied Mathematics 60; Marcel Dekker, Inc.: New York, NY, USA, 1980. [Google Scholar]
- Benevieri, P.; Calamai, A.; Furi, M.; Pera, M.P. On the persistence of the eigenvalues of a perturbed Fredholm operator of index zero under nonsmooth perturbations. Z. Anal. Anwend.
**2017**, 36, 99–128. [Google Scholar] [CrossRef] - Banaś, J. Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Cent. Eur. J. Math.
**2012**, 10, 2003–2011. [Google Scholar] [CrossRef] - Berger, M.S. Nonlinearity and Functional Analysis; Academic Press: Cambridge, MA, USA, 1977. [Google Scholar]
- Chiappinelli, R.; Edmunds, D.E. Measure of noncompactness, surjectivity of gradient operators and an application to the p-Laplacian. J. Math. Anal. Appl.
**2019**, 471, 712–727. [Google Scholar] [CrossRef] - Stuart, C.A. Spectrum of a self-adjoint operator and Palais-Smale conditions. J. Lond. Math. Soc.
**2000**, 61, 581–592. [Google Scholar] [CrossRef]

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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