Special Issue "Mathematical Analysis and Boundary Value Problems"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 20 December 2020.

Special Issue Editors

Prof. Dr. Alberto Cabada
Website SciProfiles
Guest Editor
Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Galicia, Spain
Interests: ordinary differential equations; boundary value problems; Green's functions; comparison results; nonlinear analysis
Prof. Dr. Rodrigo López Pouso
Website
Guest Editor
Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Galicia, Spain
Prof. Dr. José Ángel Cid
Website
Guest Editor
Departamento de Matemáticas, Universidade de Vigo, Galicia, Spain
Dr. Lucía López-Somoza

Guest Editor
Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Galicia, Spain

Special Issue Information

Dear Colleagues,

The study of the existence, nonexistence, and the uniqueness of solutions of boundary value problems, coupled to its stability, plays a fundamental role in the research of different kinds of differential equations (ordinary, fractional, and partial). One of the main tools developed in this area consists of fixed point theory and critical point theory.

The aim of this Special Issue is to study this type of problem in a broad sense. The development of theories that ensure the existence of solutions via topological or variational methods will contribute to the enrichment of this topic and will broaden the knowledge of this area.

Prof. Dr. Alberto Cabada
Prof. Dr. Rodrigo López Pouso
Prof. Dr. José Ángel Cid
Dr. Lucía López-Somoza
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 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
  • comparison principles
  • ordinary differential equations
  • stability theory
  • fractional differential equations
  • topological methods in differential equations
  • variational methods
  • fixed point theory
  • critical point theory

Published Papers (11 papers)

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Research

Open AccessArticle
Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN
Mathematics 2020, 8(10), 1792; https://doi.org/10.3390/math8101792 - 15 Oct 2020
Abstract
We are concerned with the following elliptic equations: (Δ)psv+V(x)|v|p2v=λa(x)|v|r2v+g( [...] Read more.
We are concerned with the following elliptic equations: (Δ)psv+V(x)|v|p2v=λa(x)|v|r2v+g(x,v)inRN, where (Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+, sp<N, the potential function V:RN(0,) is a continuous potential function, and g:RN×RR satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessArticle
A Closed-Form Solution for the Boundary Value Problem of Gas Pressurized Circular Membranes in Contact with Frictionless Rigid Plates
Mathematics 2020, 8(6), 1017; https://doi.org/10.3390/math8061017 - 22 Jun 2020
Abstract
In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such [...] Read more.
In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
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Open AccessArticle
Nontrivial Solutions for a System of Fractional q-Difference Equations Involving q-Integral Boundary Conditions
Mathematics 2020, 8(5), 828; https://doi.org/10.3390/math8050828 - 20 May 2020
Cited by 1
Abstract
In this paper, we study the existence of nontrivial solutions for a system of fractional q-difference equations involving q-integral boundary conditions, and we use the topological degree to establish our main results by considering the first eigenvalue of some associated linear [...] Read more.
In this paper, we study the existence of nontrivial solutions for a system of fractional q-difference equations involving q-integral boundary conditions, and we use the topological degree to establish our main results by considering the first eigenvalue of some associated linear integral operators. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessArticle
Existence of Positive Solutions to Singular φ-Laplacian Nonlocal Boundary Value Problems when φ is a Sup-multiplicative-like Function
Mathematics 2020, 8(3), 420; https://doi.org/10.3390/math8030420 - 14 Mar 2020
Cited by 2
Abstract
In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to φ-Laplacian nonlocal boundary value problems when φ is a sup-multiplicative-like function and the nonlinearity may not satisfy the [...] Read more.
In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to φ -Laplacian nonlocal boundary value problems when φ is a sup-multiplicative-like function and the nonlinearity may not satisfy the L 1 -Carath e ´ odory condition. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessArticle
Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions
Mathematics 2020, 8(2), 255; https://doi.org/10.3390/math8020255 - 14 Feb 2020
Cited by 3
Abstract
This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the α-Riemann-Liouville fractional derivative, with α(1,2]. [...] Read more.
This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the α -Riemann-Liouville fractional derivative, with α ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
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Open AccessArticle
Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian
Mathematics 2020, 8(1), 106; https://doi.org/10.3390/math8010106 - 08 Jan 2020
Abstract
In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter λ. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution [...] Read more.
In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter λ . By mountain pass theorem, we obtain that system has at least one non-trivial weak solution u λ under some local conditions for each given large parameter λ . We get a concrete lower bound of the parameter λ , and then obtain two estimates of weak solutions u λ . We also obtain that u λ 0 if λ tends to . Finally, we present an example as an application of our results. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessArticle
Nonlinear Impulsive Multi-Order Caputo-Type Generalized Fractional Differential Equations with Infinite Delay
Mathematics 2019, 7(11), 1108; https://doi.org/10.3390/math7111108 - 15 Nov 2019
Cited by 1
Abstract
We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction [...] Read more.
We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction mapping principle is employed to obtain the uniqueness of solutions for the problem at hand. The paper concludes with illustrative examples. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessFeature PaperArticle
Variational Methods for an Impulsive Fractional Differential Equations with Derivative Term
Mathematics 2019, 7(10), 880; https://doi.org/10.3390/math7100880 - 21 Sep 2019
Cited by 1
Abstract
This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main [...] Read more.
This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main results. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessArticle
A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions
Mathematics 2019, 7(9), 850; https://doi.org/10.3390/math7090850 - 14 Sep 2019
Cited by 1
Abstract
We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions. [...] Read more.
We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
Open AccessArticle
Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells
Mathematics 2019, 7(6), 508; https://doi.org/10.3390/math7060508 - 03 Jun 2019
Cited by 7
Abstract
This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied [...] Read more.
This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
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Open AccessArticle
Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System
Mathematics 2019, 7(6), 500; https://doi.org/10.3390/math7060500 - 02 Jun 2019
Cited by 2
Abstract
This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer [...] Read more.
This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
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