Research

18 pages, 345 KiB

Open AccessArticle

Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case

by Pierluigi Benevieri, Alessandro Calamai, Massimo Furi and Maria Patrizia Pera

Mathematics 2021, 9(5), 561; https://0-doi-org.brum.beds.ac.uk/10.3390/math9050561 - 06 Mar 2021

Viewed by 1210

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)

9 pages, 242 KiB

Open AccessArticle

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

by Shugui Kang, Yanlei Zhang and Wenying Feng

Mathematics 2021, 9(3), 278; https://0-doi-org.brum.beds.ac.uk/10.3390/math9030278 - 31 Jan 2021

Cited by 1 | Viewed by 1751

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)

8 pages, 740 KiB

Open AccessArticle

Eigenvalues of Elliptic Functional Differential Systems via a Birkhoff–Kellogg Type Theorem

by Gennaro Infante

Mathematics 2021, 9(1), 4; https://0-doi-org.brum.beds.ac.uk/10.3390/math9010004 - 22 Dec 2020

Cited by 19 | Viewed by 1509

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)

20 pages, 810 KiB

Open AccessEditor’s ChoiceArticle

Multiple Solutions for Partial Discrete Dirichlet Problems Involving the p-Laplacian

by Sijia Du and Zhan Zhou

Mathematics 2020, 8(11), 2030; https://0-doi-org.brum.beds.ac.uk/10.3390/math8112030 - 14 Nov 2020

Cited by 17 | Viewed by 1509

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)

10 pages, 293 KiB

Open AccessArticle

Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

by Bin-Sheng Wang, Gang-Ling Hou and Bin Ge

Mathematics 2020, 8(10), 1768; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101768 - 13 Oct 2020

Cited by 10 | Viewed by 1986

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)

13 pages, 300 KiB

Open AccessArticle

Remarks on Surjectivity of Gradient Operators

by Raffaele Chiappinelli and David E. Edmunds

Mathematics 2020, 8(9), 1538; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091538 - 08 Sep 2020

Cited by 3 | Viewed by 1751

Abstract

Let X be a real Banach space with dual

X^{*}

and suppose that

F : X \to X^{*}

. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent [...] Read more.

Let X be a real Banach space with dual

X^{*}

and suppose that

F : X \to X^{*}

. 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 < \infty)

, completing our previous result which was limited to the case

p \geq 2

. Full article

(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)

15 pages, 254 KiB

Open AccessArticle

Precise Asymptotics for Bifurcation Curve of Nonlinear Ordinary Differential Equation

by Tetsutaro Shibata

Mathematics 2020, 8(8), 1272; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081272 - 03 Aug 2020

Cited by 2 | Viewed by 1384

Abstract

We study the following nonlinear eigenvalue problem [...] Read more.

We study the following nonlinear eigenvalue problem

- u^{^{″}} (t) = λ f (u (t)), u (t) > 0, t \in I : = (- 1, 1), u (\pm 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 = ∥ u_{λ} ∥_{\infty}

of the solution

u_{λ}

associated with

λ

. We establish the precise asymptotic formula for

λ = λ (α)

as

α \to \infty

up to the third term of

λ (α)

. Full article

(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)

12 pages, 275 KiB

Open AccessArticle

Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

by Jianxia Wang and Zhan Zhou

Mathematics 2020, 8(3), 381; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030381 - 09 Mar 2020

Cited by 6 | Viewed by 1593

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 [...] Read more.

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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