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Advances in Abstract Inequalities, Partial Functional Differential Equations and Their...
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Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 29201

Special Issue Editors

Dr. Calogero Vetro

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Guest Editor
Associate Professor, Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
Interests: difference equations; flow invariance; nonlinear regularity theory; ordinary differential equations; partial differential equations; reduction methods; symmetry operators; weak symmetries
Special Issues, Collections and Topics in MDPI journals
Dr. Omar Bazighifan

E-Mail Website
Guest Editor
1. Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
2. Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen
Interests: qualitative theory; ordinary differential equations; functional differential equations; dynamical systems; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential equations, both partial (PDE) and ordinary (ODE), give key tools in understanding the mechanisms of physical systems and solving various problems of nonlinear phenomena. In particular, we mention diffusive processes, such as in problems in elasticity theory and in the study of porous media.

Abstract inequalities and functional equations are recognized as basic ingredients to carry out pure mathematics research (for example, in mathematical analysis and differential geometry) and applied mathematics research (for example, in mathematical physics and numerical analysis).

The above main fields are strongly related to each other. Indeed, the qualitative study of differential equations provides the appropriate framework setting to develop new inequalities and to consider different types of equations. On the other hand, these inequalities and equations are used to obtain useful estimates and bounds of terms in specific differential equations but also in characterizing the solutions' set.

There is a large and very active community of scientists working on these topics and focusing on their applications in dynamical programming, biology, information theory, statistics, physics, and engineering processes.

This Special Issue will collect ideas and significant contributions to the theories and applications of analytic inequalities, functional equations and differential equations. We welcome both original research articles and articles discussing the current state of the art.

Dr. Calogero Vetro
Dr. Omar Bazighifan
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • Functional, differential and difference inequalities
  • Functional and difference equations
  • ODE
  • PDE
  • Calculus of variations
  • Dynamical systems
  • Asymptotic analysis
  • Potential theory
  • Reduction and linearization techniques
  • Comparison methods
  • Differential models in engineering and physical sciences

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Published Papers (18 papers)

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Research

12 pages, 275 KiB  
Article
Instability of Standing Waves for INLS with Inverse Square Potential
by Saleh Almuthaybiri and Tarek Saanouni
Mathematics 2024, 12(19), 2999; https://doi.org/10.3390/math12192999 - 26 Sep 2024
Viewed by 423
Abstract
This work studies an inhomogeneous generalized Hartree equation with inverse square potential. The purpose is to prove the existence and strong instability of inter-critical standing waves. This means that there are infinitely many data near to the ground state, such that the associated [...] Read more.
This work studies an inhomogeneous generalized Hartree equation with inverse square potential. The purpose is to prove the existence and strong instability of inter-critical standing waves. This means that there are infinitely many data near to the ground state, such that the associated solution blows-up in finite time. The proof combines a variational analysis with the standard variance identity. The challenge is to deal with three difficulties: the singular potential |x|−2, an inhomogeneous term |x|−λ, and a non-local source term. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
15 pages, 291 KiB  
Article
Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators
by Abdulaziz Khalid Alsharidi and Ali Muhib
Mathematics 2024, 12(16), 2488; https://doi.org/10.3390/math12162488 - 12 Aug 2024
Viewed by 556
Abstract
In this study, we aim to set new criteria regarding the asymptotic behavior of the neutral differential equation of the third order. These criteria are designed to ensure that this equation is oscillatory using comparisons with first-order differential equations and Riccati substitution. The [...] Read more.
In this study, we aim to set new criteria regarding the asymptotic behavior of the neutral differential equation of the third order. These criteria are designed to ensure that this equation is oscillatory using comparisons with first-order differential equations and Riccati substitution. The results we obtained improve some of the results found in the literature. Some examples are provided to illustrate the applicability of our results and compare them with results found in some previous studies. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
12 pages, 274 KiB  
Article
Weighted Hardy–Rellich Inequality for Dunkl Operators
by Jielin Lyu, Yongyang Jin, Shoufeng Shen and Li Tang
Mathematics 2023, 11(6), 1487; https://doi.org/10.3390/math11061487 - 18 Mar 2023
Viewed by 1158
Abstract
In this paper, we proved a weighted Hardy–Rellich inequality for Dunkl operators based on the spherical h-harmonic decomposition theory of Dunkl operators. Moreover, we obtained the explicit constant of the inequalities, which is optimal in some cases. Our results extend some known inequalities. [...] Read more.
In this paper, we proved a weighted Hardy–Rellich inequality for Dunkl operators based on the spherical h-harmonic decomposition theory of Dunkl operators. Moreover, we obtained the explicit constant of the inequalities, which is optimal in some cases. Our results extend some known inequalities. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
13 pages, 296 KiB  
Article
Some Remarks on Local Fractional Integral Inequalities Involving Mittag–Leffler Kernel Using Generalized (E, h)-Convexity
by Wedad Saleh, Abdelghani Lakhdari, Ohud Almutairi and Adem Kiliçman
Mathematics 2023, 11(6), 1373; https://doi.org/10.3390/math11061373 - 12 Mar 2023
Cited by 11 | Viewed by 1103
Abstract
In the present work, we introduce two new local fractional integral operators involving Mittag–Leffler kernel on Yang’s fractal sets. Then, we study the related generalized Hermite–Hadamard-type inequality using generalized (E,h)-convexity and obtain two identities pertaining to these operators, [...] Read more.
In the present work, we introduce two new local fractional integral operators involving Mittag–Leffler kernel on Yang’s fractal sets. Then, we study the related generalized Hermite–Hadamard-type inequality using generalized (E,h)-convexity and obtain two identities pertaining to these operators, and the respective first- and second-order derivatives are given. In terms of applications, we provide some new generalized trapezoid-type inequalities for generalized (E,h)-convex functions. Finally, some special cases are deduced for different values of δ, E, and h. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
16 pages, 338 KiB  
Article
New Results Concerning Approximate Controllability of Conformable Fractional Noninstantaneous Impulsive Stochastic Evolution Equations via Poisson Jumps
by Yazid Alhojilan and Hamdy M. Ahmed
Mathematics 2023, 11(5), 1093; https://doi.org/10.3390/math11051093 - 22 Feb 2023
Cited by 1 | Viewed by 1252
Abstract
We introduce the conformable fractional (CF) noninstantaneous impulsive stochastic evolution equations with fractional Brownian motion (fBm) and Poisson jumps. The approximate controllability for the considered problem was investigated. Principles and concepts from fractional calculus, stochastic analysis, and the fixed-point theorem were used to [...] Read more.
We introduce the conformable fractional (CF) noninstantaneous impulsive stochastic evolution equations with fractional Brownian motion (fBm) and Poisson jumps. The approximate controllability for the considered problem was investigated. Principles and concepts from fractional calculus, stochastic analysis, and the fixed-point theorem were used to support the main results. An example is applied to show the established results. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
16 pages, 348 KiB  
Article
Fixed Points on Covariant and Contravariant Maps with an Application
by Rajagopalan Ramaswamy, Gunaseelan Mani, Arul Joseph Gnanaprakasam, Ola A. Ashour Abdelnaby, Vuk Stojiljković, Slobodan Radojevic and Stojan Radenović
Mathematics 2022, 10(22), 4385; https://doi.org/10.3390/math10224385 - 21 Nov 2022
Cited by 10 | Viewed by 1614
Abstract
Fixed-point results on covariant maps and contravariant maps in a C★-algebra-valued bipolar metric space are proved. Our results generalize and extend some recently obtained results in the existing literature. Our theoretical results in this paper are supported with suitable examples. We [...] Read more.
Fixed-point results on covariant maps and contravariant maps in a C★-algebra-valued bipolar metric space are proved. Our results generalize and extend some recently obtained results in the existing literature. Our theoretical results in this paper are supported with suitable examples. We have also provided an application to find an analytical solution to the integral equation and the electrical circuit differential equation. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
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22 pages, 379 KiB  
Article
Regularity for Quasi-Linear p-Laplacian Type Non-Homogeneous Equations in the Heisenberg Group
by Chengwei Yu
Mathematics 2022, 10(21), 4129; https://doi.org/10.3390/math10214129 - 5 Nov 2022
Cited by 2 | Viewed by 1328
Abstract
When 2−1/Q<p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group Hn, [...] Read more.
When 2−1/Q<p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group Hn, where Q=2n+2 is the homogeneous dimension of Hn. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
15 pages, 4088 KiB  
Article
Stochastic Analysis of a Hantavirus Infection Model
by Yousef Alnafisah and Moustafa El-Shahed
Mathematics 2022, 10(20), 3756; https://doi.org/10.3390/math10203756 - 12 Oct 2022
Viewed by 1492
Abstract
In this paper, a stochastic Hantavirus infection model is constructed. The existence, uniqueness, and boundedness of the positive solution of the stochastic Hantavirus infection model are derived. The conditions for the extinction of the Hantavirus infection from the stochastic system are obtained. Furthermore, [...] Read more.
In this paper, a stochastic Hantavirus infection model is constructed. The existence, uniqueness, and boundedness of the positive solution of the stochastic Hantavirus infection model are derived. The conditions for the extinction of the Hantavirus infection from the stochastic system are obtained. Furthermore, the criteria for the presence of a unique ergodic stationary distribution for the Hantavirus infection model are established using a suitable Lyapunov function. Finally, the importance of environmental noise in the Hantavirus infection model is illustrated using the Milstein method. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
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49 pages, 540 KiB  
Article
On the Fuzzy Solution of Linear-Nonlinear Partial Differential Equations
by Mawia Osman, Yonghui Xia, Omer Abdalrhman Omer and Ahmed Hamoud
Mathematics 2022, 10(13), 2295; https://doi.org/10.3390/math10132295 - 30 Jun 2022
Cited by 12 | Viewed by 2306
Abstract
In this article, we present the fuzzy Adomian decomposition method (ADM) and fuzzy modified Laplace decomposition method (MLDM) to obtain the solutions of fuzzy fractional Navier–Stokes equations in a tube under fuzzy fractional derivatives. We have looked at the turbulent flow of a [...] Read more.
In this article, we present the fuzzy Adomian decomposition method (ADM) and fuzzy modified Laplace decomposition method (MLDM) to obtain the solutions of fuzzy fractional Navier–Stokes equations in a tube under fuzzy fractional derivatives. We have looked at the turbulent flow of a viscous fluid in a tube, where the velocity field is a function of only one spatial coordinate, in addition to time being one of the dependent variables. Furthermore, we investigate the fuzzy Elzaki transform, and the fuzzy Elzaki decomposition method (EDM) applied to solving fuzzy linear-nonlinear Schrodinger differential equations. The proposed method worked perfectly without any need for linearization or discretization. Finally, we compared the fuzzy reduced differential transform method (RDTM) and fuzzy homotopy perturbation method (HPM) to solving fuzzy heat-like and wave-like equations with variable coefficients. The RDTM and HPM solutions are simpler than other already existing methods. Several examples are provided to illustrate the methods that have been offered. The results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. These studies are important in the context of the development of the theory of fuzzy partial differential equations. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
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8 pages, 270 KiB  
Article
Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences
by Renato Fiorenza
Mathematics 2022, 10(12), 2065; https://doi.org/10.3390/math10122065 - 15 Jun 2022
Viewed by 2125
Abstract
Let (Fn)n=1∞ be the classical Fibonacci sequence. It is well known that the limFn+1/Fn exists and equals the Golden Mean. If, more generally, [...] Read more.
Let (Fn)n=1∞ be the classical Fibonacci sequence. It is well known that the limFn+1/Fn exists and equals the Golden Mean. If, more generally, (Fn)n=1∞ is an order-k linear recurrence with real constant coefficients, i.e., Fn=∑j=1kλk+1−jFn−j with n>k, λj∈R, j=1,…,k, then the existence of the limit of ratios of consecutive terms may fail. In this paper, we show that the limit exists if the first k elements F1,F2,…,Fk of (Fn)n=1∞ are positive, λ1,…,λk−1 are all nonnegative, at least one being positive, and max(λ1,…,λk)=λk≥k. The limit is characterized as fixed point, bounded below by λk and bounded above by λ1+λ2+⋯+λk. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
12 pages, 302 KiB  
Article
Existence and Multiplicity of Solutions for a Bi-Non-Local Problem
by Jiabin Zuo, Tianqing An, Alessio Fiscella and Chungen Liu
Mathematics 2022, 10(12), 1973; https://doi.org/10.3390/math10121973 - 8 Jun 2022
Cited by 2 | Viewed by 1984
Abstract
The aim of this paper is to investigate the existence and multiplicity of solutions for a bi-non-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help [...] Read more.
The aim of this paper is to investigate the existence and multiplicity of solutions for a bi-non-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help of the fountain theorem, we obtain the existence of infinitely many positive energy solutions, assuming a symmetric condition for g. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
14 pages, 302 KiB  
Article
Remarks on Nonlocal Dirichlet Problems
by Kholoud Saad Albalawi, Mona Bin-Asfour and Francesca Vetro
Mathematics 2022, 10(9), 1546; https://doi.org/10.3390/math10091546 - 4 May 2022
Viewed by 1505
Abstract
We study a nonlocal Dirichlet problem with the (p(b(u)),q(b(u)))-Laplacian operator and integrable data on a bounded domain with smooth boundary. We establish the existence of at [...] Read more.
We study a nonlocal Dirichlet problem with the (p(b(u)),q(b(u)))-Laplacian operator and integrable data on a bounded domain with smooth boundary. We establish the existence of at least one weak solution in the case the variable exponents of the leading operator depend on the solution u, without assuming any growth conditions on g. The proof is based on the characterization of the energy functional associated to the problem, using the methods of the calculus of variations. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
16 pages, 323 KiB  
Article
The Existence and Multiplicity of Homoclinic Solutions for a Fractional Discrete p−Laplacian Equation
by Yong Wu, Bouali Tahar, Guefaifia Rafik, Abita Rahmoune and Libo Yang
Mathematics 2022, 10(9), 1400; https://doi.org/10.3390/math10091400 - 22 Apr 2022
Cited by 3 | Viewed by 1451
Abstract
In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on Z. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain [...] Read more.
In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on Z. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain pass lemma, we obtain that this equation exists at least two nonnegative and nontrivial homoclinic solutions when the real parameter λ>0 is large enough. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
15 pages, 326 KiB  
Article
Gradient and Parameter Dependent Dirichlet (p(x),q(x))-Laplace Type Problem
by Kholoud Saad Albalawi, Nadiyah Hussain Alharthi and Francesca Vetro
Mathematics 2022, 10(8), 1336; https://doi.org/10.3390/math10081336 - 18 Apr 2022
Cited by 12 | Viewed by 1636
Abstract
We analyze a Dirichlet (p(x),μq(x))-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter μ varies on [...] Read more.
We analyze a Dirichlet (p(x),μq(x))-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter μ varies on the non-negative real axis. The results are obtained by applying the properties of pseudomonotone operators, jointly with certain a priori estimates. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
19 pages, 310 KiB  
Article
A Krasnoselskii–Ishikawa Iterative Algorithm for Monotone Reich Contractions in Partially Ordered Banach Spaces with an Application
by Nawab Hussain, Saud M. Alsulami and Hind Alamri
Mathematics 2022, 10(1), 76; https://doi.org/10.3390/math10010076 - 27 Dec 2021
Cited by 2 | Viewed by 2212
Abstract
Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe [...] Read more.
Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe sufficient conditions for the existence of an approximate fixed-point sequence (AFPS) and prove certain fixed-point results using the Krasnoselskii–Ishikawa iterative algorithm. Moreover, we present some interesting examples to highlight the superiority of our results. Lastly, we provide both weak and strong convergence results for such mappings and consider an application of our results to prove the existence of a solution to an initial value problem. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
10 pages, 267 KiB  
Article
On the Oscillation of Solutions of Differential Equations with Neutral Term
by Fatemah Mofarreh, Alanoud Almutairi, Omar Bazighifan, Mohammed A. Aiyashi and Alina-Daniela Vîlcu
Mathematics 2021, 9(21), 2709; https://doi.org/10.3390/math9212709 - 25 Oct 2021
Cited by 4 | Viewed by 1590
Abstract
In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the [...] Read more.
In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
17 pages, 310 KiB  
Article
On Some New Weighted Steffensen-Type Inequalities on Time Scales
by Ahmed A. El-Deeb, Omar Bazighifan and Jan Awrejcewicz
Mathematics 2021, 9(21), 2670; https://doi.org/10.3390/math9212670 - 21 Oct 2021
Cited by 8 | Viewed by 1234
Abstract
The motivation of this paper is to explore some new inequalities of Steffensen-type which were demonstrated by Pečarić and Kalamir in 2014. The main idea is to investigate a class of certain inequalities by employing diamond-α dynamic integral on time scales. In [...] Read more.
The motivation of this paper is to explore some new inequalities of Steffensen-type which were demonstrated by Pečarić and Kalamir in 2014. The main idea is to investigate a class of certain inequalities by employing diamond-α dynamic integral on time scales. In addition, to obtain some new inequalities as special cases, we also extend our results to continuous and discrete calculations. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
14 pages, 1061 KiB  
Article
Convergence and Error Estimation of a New Formulation of Homotopy Perturbation Method for Classes of Nonlinear Integral/Integro-Differential Equations
by Mohamed M. Mousa and Fahad Alsharari
Mathematics 2021, 9(18), 2244; https://doi.org/10.3390/math9182244 - 12 Sep 2021
Cited by 3 | Viewed by 1929
Abstract
In this work, the main concept of the homotopy perturbation method (HPM) was outlined and convergence theorems of the HPM for solving some classes of nonlinear integral, integro-differential and differential equations were proved. A theorem for estimating the error in the approximate solution [...] Read more.
In this work, the main concept of the homotopy perturbation method (HPM) was outlined and convergence theorems of the HPM for solving some classes of nonlinear integral, integro-differential and differential equations were proved. A theorem for estimating the error in the approximate solution was proved as well. The proposed HPM convergence theorems were confirmed and the efficiency of the technique was explored by applying the HPM for solving several classes of nonlinear integral/integro-differential equations. Full article
(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)
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