New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization

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

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 9092

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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
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Special Issue Information

Dear Colleagues,

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more besides. The rapid development of fractional calculus and its applications during the past thirty years or more has led to a number of scholarly essays that on the importance of its promotion and application in physical chemistry, probability and statistics, electromagnetic theory, financial economics, biological engineering, electronic networks, and so forth. Almost all areas of modern science and engineering have been influenced by the theory of fractional calculus. Due to the complexity of the various problems that arise in nonlinear analysis, fractional calculus and optimization, it is not always easy to find exact solutions, which often leads researchers to resort to approximate solutions. Over the past eighty years, optimization problems have been intensively studied, and many scholars have developed various feasible methods with which to analyze the convergence of algorithms and find approximate solutions.

This Special Issue will pay greater attention to the new originality and real-world applications of nonlinear analysis, fractional calculus, optimization and their applications. We are pleased to invite researchers to contribute their original and high-quality research papers in order to inspire advances in nonlinear analysis, fractional calculus, optimization and their applications. Potential topics include, but are not limited to:

  • Nonlinear functional analysis;
  • Fixed point, coincidence point, and best proximity point theory;
  • Set-valued analysis;
  • Critical point theory;
  • Matrix theory;
  • Convex analysis;
  • Boundary value problems;
  • Singular and impulsive fractional differential and integral equations;
  • Well-posedness of fractional system;
  • Fractional epidemic model;
  • Modeling biological phenomena;
  • Nonsmooth analysis and optimization;
  • Stability analysis;
  • Dynamics and chaos.

Prof. Dr. Wei-Shih Du
Guest Editor

Manuscript Submission Information

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Keywords

  • functional analysis
  • fixed point theory and its applications
  • set-valued analysis
  • critical point theory
  • matrix theory
  • convex analysis
  • fractional differential equation
  • well-posedness of fractional system
  • fractional epidemic model
  • nonsmooth analysis and optimization
  • graph theory and optimization
  • stability analysis
  • dynamics and chaos

Related Special Issue

Published Papers (11 papers)

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Editorial

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4 pages, 170 KiB  
Editorial
Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”
by Wei-Shih Du
Axioms 2024, 13(6), 350; https://doi.org/10.3390/axioms13060350 - 24 May 2024
Viewed by 139
Abstract
Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, [...] Read more.
Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...] Full article

Research

Jump to: Editorial

15 pages, 275 KiB  
Article
Existence of Solutions: Investigating Fredholm Integral Equations via a Fixed-Point Theorem
by Faruk Özger, Merve Temizer Ersoy and Zeynep Ödemiş Özger
Axioms 2024, 13(4), 261; https://doi.org/10.3390/axioms13040261 - 14 Apr 2024
Viewed by 627
Abstract
Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer [...] Read more.
Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by x(s)=1+λx(s)01st+sx(t)dt, can be very often encountered in many applications, where x is the function to be determined, λ is a parameter, and t,s[0,1]. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form χ(l)=ϱ(l)+χ(l)pqk(l,z)(Vχ)(z)dz are investigated in the space Cωp,q, where χ is the unknown function to be determined, V is a given operator, and ϱ,k are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided. Full article
14 pages, 297 KiB  
Article
New Summation and Integral Representations for 2-Variable (p,q)-Hermite Polynomials
by Nusrat Raza, Mohammed Fadel and Wei-Shih Du
Axioms 2024, 13(3), 196; https://doi.org/10.3390/axioms13030196 - 15 Mar 2024
Cited by 2 | Viewed by 945
Abstract
In this paper, we introduce and study new features for 2-variable (p,q)-Hermite polynomials, such as the (p,q)-diffusion equation, (p,q)-differential formula and integral representations. In addition, we establish some [...] Read more.
In this paper, we introduce and study new features for 2-variable (p,q)-Hermite polynomials, such as the (p,q)-diffusion equation, (p,q)-differential formula and integral representations. In addition, we establish some summation models and their (p,q)-derivatives. Certain parting remarks and nontrivial examples are also provided. Full article
18 pages, 1090 KiB  
Article
Finite-Time Stability of Impulsive Fractional Differential Equations with Pure Delays
by Tingting Xie and Mengmeng Li
Axioms 2023, 12(12), 1129; https://doi.org/10.3390/axioms12121129 - 15 Dec 2023
Viewed by 952
Abstract
This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the [...] Read more.
This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the solutions, the finite-time stability results of impulsive fractional differential delay equations are presented. Finally, we present four examples to illustrate the validity of our theoretical results. Full article
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15 pages, 336 KiB  
Article
On Local Unique Solvability for a Class of Nonlinear Identification Problems
by Vladimir E. Fedorov, Marina V. Plekhanova and Daria V. Melekhina
Axioms 2023, 12(11), 1013; https://doi.org/10.3390/axioms12111013 - 27 Oct 2023
Viewed by 669
Abstract
Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous [...] Read more.
Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders. Full article
15 pages, 338 KiB  
Article
Common Fixed Point of (ψ, β, L)-Generalized Contractive Mapping in Partially Ordered b-Metric Spaces
by Binghua Jiang, Huaping Huang and Stojan Radenović
Axioms 2023, 12(11), 1008; https://doi.org/10.3390/axioms12111008 - 26 Oct 2023
Viewed by 874
Abstract
The purpose of this paper is to attain the existence of coincidences and common fixed points in four mappings satisfying (ψ,β,L)-generalized contractive conditions in the framework of partially ordered b-metric spaces. The main results presented [...] Read more.
The purpose of this paper is to attain the existence of coincidences and common fixed points in four mappings satisfying (ψ,β,L)-generalized contractive conditions in the framework of partially ordered b-metric spaces. The main results presented in this paper generalize some recent results in the existing literature. Furthermore, a nontrivial example is presented to support the obtained results. Full article
25 pages, 362 KiB  
Article
On the Study of Pseudo 𝒮-Asymptotically Periodic Mild Solutions for a Class of Neutral Fractional Delayed Evolution Equations
by Naceur Chegloufa, Belkacem Chaouchi, Marko Kostić and Wei-Shih Du
Axioms 2023, 12(8), 800; https://doi.org/10.3390/axioms12080800 - 20 Aug 2023
Viewed by 602
Abstract
The goal of this paper is to investigate the existence and uniqueness of pseudo S-asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory [...] Read more.
The goal of this paper is to investigate the existence and uniqueness of pseudo S-asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory and some classical fixed point theorems. Furthermore, we provide an example to illustrate the applications of our abstract results. Full article
15 pages, 510 KiB  
Article
Pricing of Credit Risk Derivatives with Stochastic Interest Rate
by Wujun Lv and Linlin Tian
Axioms 2023, 12(8), 782; https://doi.org/10.3390/axioms12080782 - 12 Aug 2023
Cited by 1 | Viewed by 881
Abstract
This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that [...] Read more.
This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that the stochastic interest rate follows a jump-diffusion Cox–Ingersoll–Ross process. First, we derive the joint Laplace transform of the distribution of the vector process (rt,Rt) by applying piecewise deterministic Markov process theory and martingale theory. Then, using the joint Laplace transform, we obtain the explicit pricing of defaultable bonds and a credit default swap. Lastly, numerical examples are presented to illustrate the dynamic relationships between defaultable securities (defaultable bonds, credit default swap) and the maturity date. Full article
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14 pages, 291 KiB  
Article
Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent
by Zhen-Ying Sun, Bai-Ni Guo and Feng Qi
Axioms 2023, 12(7), 665; https://doi.org/10.3390/axioms12070665 - 5 Jul 2023
Cited by 3 | Viewed by 1060
Abstract
In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula [...] Read more.
In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients. Full article
16 pages, 350 KiB  
Article
Fixed Point Results in Generalized Menger Probabilistic Metric Spaces with Applications to Decomposable Measures
by Huaping Huang, Tatjana Došenović, Dušan Rakić and Stojan Radenović
Axioms 2023, 12(7), 660; https://doi.org/10.3390/axioms12070660 - 3 Jul 2023
Viewed by 775
Abstract
The aim of this paper is to give some fixed point results in generalized Menger probabilistic metric spaces. Moreover, some nontrivial examples are presented to illustrate the superiority of the obtained results. In addition, several interesting applications are given to show that our [...] Read more.
The aim of this paper is to give some fixed point results in generalized Menger probabilistic metric spaces. Moreover, some nontrivial examples are presented to illustrate the superiority of the obtained results. In addition, several interesting applications are given to show that our results are meaningful and valuable. Full article
14 pages, 356 KiB  
Article
Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems
by Liu Gao, Guolin Yu and Wenyan Han
Axioms 2023, 12(7), 635; https://doi.org/10.3390/axioms12070635 - 27 Jun 2023
Cited by 1 | Viewed by 678
Abstract
This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the [...] Read more.
This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz’s function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems. Full article
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