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Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities
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Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities

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

Deadline for manuscript submissions: 30 June 2024 | Viewed by 9765

Special Issue Editors

Prof. Dr. Wei-Shih Du

E-Mail Website
Guest Editor
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
Interests: nonlinear analysis and its applications; fixed point theory; variational principles and inequalities; optimization theory; equilibrium problems; fractional calculus theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ravi P. Agarwal

grade E-Mail Website1 Website2
Guest Editor
Department of Mathematics, Texas A and M University-Kingsville, Kingsville, TX 78363, USA
Interests: boundary value problems; nonlinear analysis; differential and difference equations; fixed point theory; general inequalities
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Feng Qi

E-Mail Website
Guest Editor
1. Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China
2. School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
3. Independent Researcher, Dallas, TX 75252-8024, USA
Interests: special functions; mathematical ınequalities; mathematical means; analytic combinatorics; analytic number theory

Special Issue Information

Dear Colleagues,

Over the past century, nonlinear analysis has been widely and significantly applied in many areas of mathematics, including nonlinear ordinary and partial differential equations, functional analysis, fixed point theory, nonlinear optimization, variational analysis, convex analysis, dynamical system theory, mathematical economics, signal processing, control theory, data mining, and so forth. Typical problems in analytic number theory are enumeration problems involving prime numbers, Diophantine equations, and similar number-theoretic objects. These questions are of long-standing intrinsic interest, and the answers provided by analytic number theory are often used in applied mathematics, including applications in cryptography, asymptotic and error analysis, Fourier series and transforms, contour integrals and residues, Laplacian spectral theory, and so on. Abstract mathematical inequalities have been considered important tools for mathematical and scientific research. Classical inequalities (such as Jensen's inequality, Hermite–Hadamard's inequality, Hölder's inequality, Minkowski's inequality, Grüss' inequality, Chebyshev's inequality, etc.) have found various applications in many branches of mathematics, such as functional analysis, optimization theory, numerical analysis, probability and statistics, information theory, and so on.

This Special Issue will pay more attention to new original and real-world applications of nonlinear analysis, analytic number theory and mathematical inequalities. We cordially and earnestly invite researchers to contribute their original and high-quality research papers which will inspire advances in nonlinear analysis, analytic number theory, mathematical inequalities, and their applications. Potential topics include but are not limited to:

  • Nonlinear functional analysis;
  • Set-valued analysis;
  • Fixed point, coincidence point, and best proximity point theory;
  • Variational and topological methods for ODEs and PDEs;
  • Critical point theory;
  • Optimization;
  • Matrix theory;
  • Convex analysis;
  • Analytic number theory;
  • Diophantine approximations;
  • Analytic combinatorics;
  • Mathematical inequalities and applications;
  • Mathematical means and applications;
  • Theory and applications for special functions.

Prof. Dr. Wei-Shih Du
Prof. Dr. Ravi P. Agarwal
Prof. Dr. Feng Qi
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 analysis
  • set-valued analysis
  • fixed point theory
  • differential equations
  • optimization
  • critical point theory
  • analytic number theory
  • Diophantine approximations
  • analytic combinatorics
  • mathematical inequalities
  • mathematical means

Published Papers (10 papers)

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Research

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15 pages, 281 KiB  
Article
Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight
by Tao Zhang and Jie Liu
Mathematics 2024, 12(5), 785; https://0-doi-org.brum.beds.ac.uk/10.3390/math12050785 - 06 Mar 2024
Viewed by 539
Abstract
Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ωβ(x)=[−lnFo(x)|(n−1)β. This can be seen as [...] Read more.
Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ωβ(x)=[−lnFo(x)|(n−1)β. This can be seen as a generation result of the isotropic Moser–Trudinger inequality with logarithmic weight. Furthermore, we obtain the existence of extremal function when β is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
20 pages, 3375 KiB  
Article
Description and Analysis of Data Security Based on Differential Privacy in Enterprise Power Systems
by Zhaofeng Zhong, Ge Zhang, Li Yin and Yufeng Chen
Mathematics 2023, 11(23), 4829; https://0-doi-org.brum.beds.ac.uk/10.3390/math11234829 - 30 Nov 2023
Cited by 1 | Viewed by 790
Abstract
The pursuit of environmental sustainability, energy conservation, and emissions reduction has become a global focal point. Electricity is the primary source of energy in our daily lives. Through the analysis of smart power systems, we can efficiently and sustainably harness electrical energy. However, [...] Read more.
The pursuit of environmental sustainability, energy conservation, and emissions reduction has become a global focal point. Electricity is the primary source of energy in our daily lives. Through the analysis of smart power systems, we can efficiently and sustainably harness electrical energy. However, electric power system data inherently contain a wealth of sensitive user information. Therefore, our primary concern is protecting these sensitive user data while performing precise and effective analysis. To address this issue, we have innovatively proposed three granular information models based on differential privacy. In consideration of the characteristics of enterprise electricity consumption data and the imperative need for privacy protection, we implement a reasonable modeling process through data processing, information granulation expression, and the optimization analysis of information granularity. Our datasets encompass enterprise electricity consumption data and related attributes from Hitachi Building Technology (Guangzhou) Co., Ltd’s cloud computing center. Simultaneously, we have conducted experiments using publicly available datasets to underscore the model’s versatility. Our experimental results affirm that granular computation can improve the utility of differential privacy in safeguarding data privacy. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
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13 pages, 300 KiB  
Article
Characterizing q-Bessel Functions of the First Kind with Their New Summation and Integral Representations
by Mohammed Fadel, Nusrat Raza and Wei-Shih Du
Mathematics 2023, 11(18), 3831; https://0-doi-org.brum.beds.ac.uk/10.3390/math11183831 - 07 Sep 2023
Cited by 3 | Viewed by 596
Abstract
As a powerful tool for models of quantum computing, q-calculus has drawn the attention of many researchers in the discipline of special functions. In this paper, we present new properties and characterize q-Bessel functions of the first kind using some identities [...] Read more.
As a powerful tool for models of quantum computing, q-calculus has drawn the attention of many researchers in the discipline of special functions. In this paper, we present new properties and characterize q-Bessel functions of the first kind using some identities of q-calculus. The results presented in this article help us to obtain new expression results related to q-special functions. New summation and integral representations for q-Bessel functions of the first kind are also established. A few examples are also provided to demonstrate the effectiveness of the proposed strategy. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
12 pages, 285 KiB  
Article
A Series Expansion of a Logarithmic Expression and a Decreasing Property of the Ratio of Two Logarithmic Expressions Containing Sine
by Xin-Le Liu, Hai-Xia Long and Feng Qi
Mathematics 2023, 11(14), 3107; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143107 - 14 Jul 2023
Cited by 3 | Viewed by 817
Abstract
In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms of specific [...] Read more.
In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the sine function. These results are interesting purely in pure mathematics. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
12 pages, 296 KiB  
Article
Non-Emptiness, Relative Coincidences and Axiomatic Results for the Precore
by Yan-An Hwang and Yu-Hsien Liao
Mathematics 2023, 11(13), 2812; https://0-doi-org.brum.beds.ac.uk/10.3390/math11132812 - 22 Jun 2023
Viewed by 530
Abstract
Taking into account the definition of the precore, we initially analyze its non-emptiness using duality results from linear programming theory. We then introduce the dominance core to investigate the coincident relations between the precore and dominance core. Finally, we propose specific reduction approaches [...] Read more.
Taking into account the definition of the precore, we initially analyze its non-emptiness using duality results from linear programming theory. We then introduce the dominance core to investigate the coincident relations between the precore and dominance core. Finally, we propose specific reduction approaches to axiomatize the precore. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
12 pages, 277 KiB  
Article
On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings
by Huaping Huang, Subhadip Pal, Ashis Bera and Lakshmi Kanta Dey
Mathematics 2023, 11(8), 1852; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081852 - 13 Apr 2023
Viewed by 817
Abstract
The purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation T(u,C(u))=u using Kannan-type equicontractive mappings, [...] Read more.
The purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation T(u,C(u))=u using Kannan-type equicontractive mappings, T:A×C(A)¯→Y, where C is a compact mapping, A is a bounded, closed and convex subset of a Banach space Y. To achieve this objective, the authors have presented Sadovskii’s theorem, which utilizes the measure of noncompactness. The relevance of the obtained results has been illustrated through the consideration of various initial value problems. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
21 pages, 444 KiB  
Article
Two New Modified Regularized Methods for Solving the Variational Inclusion and Null Point Problems
by Yuanheng Wang, Miaoqing Li, Chengru Yao and Bingnan Jiang
Mathematics 2023, 11(6), 1469; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061469 - 17 Mar 2023
Viewed by 951
Abstract
In this article, based on the regularization techniques, we construct two new algorithms combining the forward-backward splitting algorithm and the proximal contraction algorithm, respectively. Iterative sequences of the new algorithms can converge strongly to a common solution of the variational inclusion and null [...] Read more.
In this article, based on the regularization techniques, we construct two new algorithms combining the forward-backward splitting algorithm and the proximal contraction algorithm, respectively. Iterative sequences of the new algorithms can converge strongly to a common solution of the variational inclusion and null point problems in real Hilbert spaces. Multi-inertial extrapolation steps are applied to expedite their convergence rate. We also give some numerical experiments to certify that our algorithms are viable and efficient. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
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12 pages, 314 KiB  
Article
Cash Flow Optimization on Insurance: An Application of Fixed-Point Theory
by Yangmin Zhong and Huaping Huang
Mathematics 2023, 11(4), 902; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040902 - 10 Feb 2023
Cited by 1 | Viewed by 1320
Abstract
The purpose of this paper is to explore a discrete-time cash flow optimization problem of the insurance company with time value of ruin under different interest rates. For the sake of considering the time value of ruin, we assume that the shareholders can [...] Read more.
The purpose of this paper is to explore a discrete-time cash flow optimization problem of the insurance company with time value of ruin under different interest rates. For the sake of considering the time value of ruin, we assume that the shareholders can get subsidies per unit time, as long as the insurance company is not bankrupt. The switching of different interest rates on the market is controlled by a stationary Markov chain. The dynamic programming principle is used to solve this optimization problem. By using the method of fixed-point theory, we show that the value function is the unique solution of the dynamic programming equation and a numerical algorithm is proposed to solve the value function as well as the optimal policy. Furthermore, two examples are revealed to illustrate the application of the main results obtained in the presented paper. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
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Review

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21 pages, 392 KiB  
Review
Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions
by Feng Qi and Ravi Prakash Agarwal
Mathematics 2024, 12(1), 44; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010044 - 22 Dec 2023
Cited by 1 | Viewed by 503
Abstract
In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao [...] Read more.
In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao geometry of the Dirichlet distributions in the two-dimensional case. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
16 pages, 395 KiB  
Review
A Brief Survey and an Analytic Generalization of the Catalan Numbers and Their Integral Representations
by Jian Cao, Wen-Hui Li, Da-Wei Niu, Feng Qi and Jiao-Lian Zhao
Mathematics 2023, 11(8), 1870; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081870 - 14 Apr 2023
Cited by 1 | Viewed by 1476
Abstract
In the paper, the authors briefly survey several generalizations of the Catalan numbers in combinatorial number theory, analytically generalize the Catalan numbers, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy’s integral formula in the theory [...] Read more.
In the paper, the authors briefly survey several generalizations of the Catalan numbers in combinatorial number theory, analytically generalize the Catalan numbers, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy’s integral formula in the theory of complex functions, and point out potential directions for further study. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)
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