Special Issue "Advances in Nonlinear Spectral Theory"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 December 2020).

Special Issue Editor

Prof. Raffaele Chiappinelli
E-Mail Website
Guest Editor
Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy
Interests: nonlinear functional analysis; nonlinear operators; nonlinear eigenvalue problems; nonlinear spectral theory

Special Issue Information

Dear Colleagues,

Since its appearance in the late Sixties of the last Century, the spectral theory of nonlinear operators acting in Banach spaces has made many advances, and in various different directions, including the applications of the theory itself to boundary value problems for ordinary and partial differential equations. Discovering the similarities and the differences existing with the realm of bounded linear operators, known to us from linear functional analysis, is an active and exciting field of research, so much if we deal with nonlinear perturbations of linear operators.

Nonlinear Spectral Theory is closely related to (and in a sense contains properly) the extremely vast and popular field of Nonlinear Eigenvalue Problems, and as such it employs and develops practically all methods of nonlinear analysis, notably Fixed point theory, Degree theory and topological methods, Bifurcation theory, Non-compact operators, Minimization methods and critical point theory for gradient operators, as well as the applications of these methods to differential equations.

This Special Issue of the Journal Mathematics aims at collecting new ideas, methods and/or specific results (but also well organized reviews of known results) from any researcher sharing the interest in the field and working in the areas cited above, or nearby.

Prof. Raffaele Chiappinelli
Guest Editor

Manuscript Submission Information

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Keywords

  • Nonlinear eigenvalue problems
  • Degree theory and topological methods
  • Abstract bifurcation theory
  • Non-compact operators
  • Variational methods for nonlinear operators
  • Critical point theory

Published Papers (8 papers)

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Research

Open AccessArticle
Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case
Mathematics 2021, 9(5), 561; https://doi.org/10.3390/math9050561 - 06 Mar 2021
Viewed by 262
Abstract
We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a [...] Read more.
We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators
Mathematics 2021, 9(3), 278; https://doi.org/10.3390/math9030278 - 31 Jan 2021
Viewed by 310
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Eigenvalues of Elliptic Functional Differential Systems via a Birkhoff–Kellogg Type Theorem
Mathematics 2021, 9(1), 4; https://doi.org/10.3390/math9010004 - 22 Dec 2020
Cited by 2 | Viewed by 340
Abstract
Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of elliptic functional differential equations subject to functional boundary [...] Read more.
Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of elliptic functional differential equations subject to functional boundary conditions. We obtain a localization of the corresponding non-negative eigenfunctions in terms of their norm. Under additional growth conditions, we also prove the existence of an unbounded set of eigenfunctions for these systems. The class of equations that we study is fairly general and our approach covers some systems of nonlocal elliptic differential equations subject to nonlocal boundary conditions. An example is presented to illustrate the theory. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Multiple Solutions for Partial Discrete Dirichlet Problems Involving the p-Laplacian
Mathematics 2020, 8(11), 2030; https://doi.org/10.3390/math8112030 - 14 Nov 2020
Cited by 1 | Viewed by 378
Abstract
Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value [...] Read more.
Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value problem with the p-Laplacian by using critical point theory. Moreover, under appropriate assumptions on the nonlinear term, we determine open intervals of the parameter such that at least two positive solutions and an unbounded sequence of positive solutions are obtained by using the maximum principle. We also show two examples to illustrate our results. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term
Mathematics 2020, 8(10), 1768; https://doi.org/10.3390/math8101768 - 13 Oct 2020
Viewed by 419
Abstract
In this paper, we consider the existence and uniqueness of solutions for a quasilinear elliptic equation with a variable exponent and a reaction term depending on the gradient. Based on the surjectivity result for pseudomonotone operators, we prove the existence of at least [...] Read more.
In this paper, we consider the existence and uniqueness of solutions for a quasilinear elliptic equation with a variable exponent and a reaction term depending on the gradient. Based on the surjectivity result for pseudomonotone operators, we prove the existence of at least one weak solution of such a problem. Furthermore, we obtain the uniqueness of the solution for the above problem under some considerations. Our results generalize and improve the existing results. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Remarks on Surjectivity of Gradient Operators
Mathematics 2020, 8(9), 1538; https://doi.org/10.3390/math8091538 - 08 Sep 2020
Viewed by 477
Abstract
Let X be a real Banach space with dual X and suppose that F:XX. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent result of ours, we prove that any gradient map that has this property and is additionally bounded, coercive and continuous is surjective. As before, the main tool for the proof is the Ekeland Variational Principle. Comparison with known surjectivity results is made; finally, as an application, we discuss a Dirichlet boundary-value problem for the p-Laplacian (1<p<), completing our previous result which was limited to the case p2. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Precise Asymptotics for Bifurcation Curve of Nonlinear Ordinary Differential Equation
Mathematics 2020, 8(8), 1272; https://doi.org/10.3390/math8081272 - 03 Aug 2020
Cited by 1 | Viewed by 494
Abstract
We study the following nonlinear eigenvalue problem u(t)=λf(u(t)),u(t)>0,tI:=(1,1),u(±1)=0, where f(u)=log(1+u) and λ>0 is a parameter. Then λ is a continuous function of α>0, where α is the maximum norm α=uλ of the solution uλ associated with λ. We establish the precise asymptotic formula for λ=λ(α) as α up to the third term of λ(α). Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
Open AccessArticle
Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator
Mathematics 2020, 8(3), 381; https://doi.org/10.3390/math8030381 - 09 Mar 2020
Cited by 2 | Viewed by 516
Abstract
In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
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