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Axioms
Special Issues
Advances in Nonlinear Analysis and Boundary Value Problems
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Advances in Nonlinear Analysis and Boundary Value Problems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 27 December 2024 | Viewed by 7026

Special Issue Editors

Dr. Ricardo Roque Enguiça

E-Mail Website
Guest Editor
Instituto Superior de Engenharia de Lisboa, Departamento de Matemática, CEMAT, Lisbon, Portugal
Interests: ordinary differential equations; differential equations; functional analysis; mathematical analysis; real analysis; fixed point theory; numerical analysis
Dr. Clemente Cesarano

E-Mail Website
Guest Editor
Section of Mathematics, International Telematic University, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy
Interests: special functions; orthogonal polynomials; differential equations; operator theory; multivariate approximation theory; Lie algebra
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Boundary value problems are usually associated with very-well-known real-life applications on the field of differential equations, and they are also providers of breakthrough developments on the theory of nonlinear analysis.

In this Special Issue, we will cover existence and localization results, based on the properties of Green's Function, of approximation theory, maximum principles, spectral analysis, and other techniques, providing special attention to original applied problems. Numerical analysis substantiating results re also valued.

Industrial problems, not directly relatable to differential equations but solved with nonlinear arguments, also fall within the scope of this Special Issue, so works focused on these matters are welcome to be submitted.

Dr. Ricardo Roque Enguiça
Dr. Clemente Cesarano
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • boundary value problems
  • nonlinear analysis
  • numerical analysis
  • real-life problems
  • Green's function
  • industrial mathematics

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (7 papers)

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Research

16 pages, 275 KiB  
Article
Existence of Nontrivial Solutions for Boundary Value Problems of Fourth-Order Differential Equations
by Hongyu Li
Axioms 2024, 13(11), 766; https://doi.org/10.3390/axioms13110766 - 4 Nov 2024
Viewed by 498
Abstract
This article investigates the solvability problem of fourth-order differential equations with two-point boundary conditions; specifically, conclusions regarding sign-changing solutions are obtained. The methods used in this article are fixed-point theorems on lattices. Firstly, under some sublinear conditions, the existence of three nontrivial solutions [...] Read more.
This article investigates the solvability problem of fourth-order differential equations with two-point boundary conditions; specifically, conclusions regarding sign-changing solutions are obtained. The methods used in this article are fixed-point theorems on lattices. Firstly, under some sublinear conditions, the existence of three nontrivial solutions is demonstrated, including a sign-changing solution, a negative solution and a positive solution. Secondly, under some unilaterally asymptotically linear and superlinear conditions, the existence of at least one sign-changing solution is proved. Finally, this article provides several specific examples to illustrate the obtained conclusions. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
13 pages, 315 KiB  
Article
Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities
by Milena Dimova, Natalia Kolkovska and Nikolai Kutev
Axioms 2024, 13(10), 709; https://doi.org/10.3390/axioms13100709 - 14 Oct 2024
Viewed by 415
Abstract
In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we [...] Read more.
In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
11 pages, 251 KiB  
Article
The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus
by Yongxiang Li and Yanyan Wang
Axioms 2024, 13(6), 383; https://doi.org/10.3390/axioms13060383 - 4 Jun 2024
Cited by 1 | Viewed by 626
Abstract
This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain [...] Read more.
This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1<|x|<r2}(N≥2) with the boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:[r1,r2]×R×R+×R→R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ1 of the Laplace operator −▵ with boundary condition u|∂Ω=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,ξ,ζ,η) to be a superlinear growth on ξ,ζ,η as |(ξ,ζ,η)|→∞. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
19 pages, 308 KiB  
Article
Iteration with Bisection to Approximate the Solution of a Boundary Value Problem
by Richard Avery, Douglas R. Anderson and Jeffrey Lyons
Axioms 2024, 13(4), 222; https://doi.org/10.3390/axioms13040222 - 27 Mar 2024
Viewed by 1322
Abstract
Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green’s function kernel whose fixed points are solutions of a boundary value problem. In this paper, [...] Read more.
Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green’s function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
20 pages, 325 KiB  
Article
On Behavior of Solutions for Nonlinear Klein–Gordon Wave Type Models with a Logarithmic Nonlinearity and Multiple Time-Varying Delays
by Aziz Belmiloudi
Axioms 2024, 13(1), 29; https://doi.org/10.3390/axioms13010029 - 30 Dec 2023
Cited by 1 | Viewed by 1413
Abstract
In this paper, we study the existence and exponential stability of solutions to a class of nonlinear delay Klein–Gordon wave type models on a bounded domain. Such models include multiple time-varying delays, frictional damping, and nonlinear logarithmic source terms. After showing the local [...] Read more.
In this paper, we study the existence and exponential stability of solutions to a class of nonlinear delay Klein–Gordon wave type models on a bounded domain. Such models include multiple time-varying delays, frictional damping, and nonlinear logarithmic source terms. After showing the local existence result of the solutions using Faedo–Galerkin’s method and logarithmic Sobolev inequality, the global existence is analyzed. Then, under some appropriate conditions, energy decay estimates and exponential stability results of the global solutions are investigated. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
15 pages, 328 KiB  
Article
On Boundary Controllability for the Higher-Order Nonlinear Schrödinger Equation
by Andrei V. Faminskii
Axioms 2023, 12(12), 1127; https://doi.org/10.3390/axioms12121127 - 15 Dec 2023
Viewed by 1111
Abstract
A control problem with final overdetermination is considered for the higher-order nonlinear Schrödinger equation on a bounded interval. The boundary condition on the space derivative is chosen as the control. Results on the global existence of solutions under small input data are established. [...] Read more.
A control problem with final overdetermination is considered for the higher-order nonlinear Schrödinger equation on a bounded interval. The boundary condition on the space derivative is chosen as the control. Results on the global existence of solutions under small input data are established. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
14 pages, 341 KiB  
Article
Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions
by Constantin Fetecau , Costică Moroşanu and Dorin-Cătălin Stoicescu
Axioms 2023, 12(12), 1098; https://doi.org/10.3390/axioms12121098 - 30 Nov 2023
Viewed by 980
Abstract
Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems [...] Read more.
Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class Wp1,2(Q)×Wν1,2(Q). One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, alg-frac_sec-ord-varphi_PHT, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)
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