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New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical...
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New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems

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

Deadline for manuscript submissions: 30 June 2025 | Viewed by 10456

Special Issue Editor

Prof. Dr. Iñigo Sarría

E-Mail Website
Guest Editor
Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Avenida de la Paz 123, 26006 Logroño, La Rioja, Spain
Interests: applied mathematics; dynamics; iterative methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The resolution of nonlinear equations poses a recurring challenge across various scientific domains. It is widely acknowledged that closed-form solutions for such equations are seldom attainable, necessitating the application of iterative techniques. Among these methods, Newton's method stands out as the most well-known, extensively studied, and frequently employed, owing to its favorable convergence properties and straightforward computational approach. However, there are instances where this method is impractical due to the high computational cost or even impossibility of computing the derivative's inverse. In such cases, alternative approaches like the secant method or derivative-free methods become viable options. Additionally, researchers may require higher-speed and higher-order iterative methods developed for specific applications.

The exploration of these iterative procedures includes several fields of interest:

  1. Semi-local convergence, involving conditions on the function and starting point or guess;
  2. Local convergence, requiring conditions on the solution and the function;
  3. Dynamical behavior;
  4. Optimal iterative schemes.

This Special Issue aims to showcase research developments in this discipline, particularly delving into the analysis of the dynamic behavior of nonlinear equations. Such investigations offer researchers new avenues for finding solutions to these equations or systems of equations. Manuscripts are encouraged in areas such as the following:

  • Explorations of complex dynamics through parameter and dynamical planes;
  • Multi-point iterative methods (with or without memory);
  • Study of the dynamics of higher-order methods;
  • Development of tools aiding researchers in studying dynamical behavior;
  • Optimal-order derivative-free iterative methods.

Prof. Dr. Iñigo Sarría
Guest Editor

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

  • iterative methods
  • numerical analysis
  • semi-local convergence
  • dynamical behavior
  • optimal-order iterative methods
  • higher-order methods

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Published Papers (11 papers)

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Research

15 pages, 959 KiB  
Article
An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics
by Moin-ud-Din Junjua, Ibraheem M. Alsulami, Amer Alsulami and Sangeeta Kumari
Axioms 2025, 14(5), 360; https://doi.org/10.3390/axioms14050360 - 10 May 2025
Viewed by 270
Abstract
In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from [...] Read more.
In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from 2.414 to 3 and the efficiency index from 1.554 to 1.732, which is higher than the efficiency of optimal fourth- and eighth-order iterative schemes. The proposed method is validated through numerical and dynamic experiments concerning the absolute error, approximated computational order, regions of convergence, and CPU time (sec) on the real-world problems, including Kepler’s equation, isentropic supersonic flow, and law of population growth, demonstrating superior performance compared to some existing well-known methods. Commonly, regions of convergence of iterative methods are investigated and compared by plotting attractor basins of iteration schemes in the complex plane on polynomial functions of the type zn−1. However, in this paper, the attractor basins of the proposed method are investigated on diverse nonlinear functions. The proposed scheme creates portraits of basins of attraction faster with wider convergence areas outperforming existing well-known iteration schemes. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
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21 pages, 1074 KiB  
Article
Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics
by Mustafa R. S. Kulenović, Mehmed Nurkanović, Zehra Nurkanović and Susan Trolle
Axioms 2024, 13(11), 730; https://doi.org/10.3390/axioms13110730 - 22 Oct 2024
Viewed by 782
Abstract
This paper investigates the dynamics of non-autonomous cooperative systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to evolutionary population cooperation models. We use two methods to [...] Read more.
This paper investigates the dynamics of non-autonomous cooperative systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to evolutionary population cooperation models. We use two methods to extend the global attractivity results for autonomous cooperative systems to related non-autonomous cooperative systems which appear in recent problems in evolutionary dynamics. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
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14 pages, 275 KiB  
Article
Strong Stability for a Viscoelastic Transmission Problem Under a Nonlocal Boundary Control
by Noureddine Touati Brahim, Abderrahmane Beniani, Abderrazak Chaoui, Zayd Hajjej, Perikles Papadopoulos and Khaled Zennir
Axioms 2024, 13(10), 714; https://doi.org/10.3390/axioms13100714 - 16 Oct 2024
Viewed by 895
Abstract
The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui [...] Read more.
The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui and A. Benaissa. Namely, they focused on a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with an emphasis on rough interfaces and nonlinear dynamic (possibly, nonlocal) boundary conditions along the interface, where a transmission problem in the presence of a boundary control condition of a nonlocal type was investigated in these papers. Owing to the semigroup theory, we prove the question of well-posedness. For the very rare cases, we combined between the frequency domain approach and the Borichev–Tomilov theorem to establish strong stability results. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
17 pages, 311 KiB  
Article
Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space
by Dimple Singh, Priya Goel, Ramandeep Behl and Iñigo Sarría
Axioms 2024, 13(9), 638; https://doi.org/10.3390/axioms13090638 - 18 Sep 2024
Cited by 1 | Viewed by 641
Abstract
In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type [...] Read more.
In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type (ψ−α)-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
13 pages, 276 KiB  
Article
Connections between Some Extremization Problems and Associated Inequalities
by Wei Xu and Savin Treanţă
Axioms 2024, 13(9), 614; https://doi.org/10.3390/axioms13090614 - 11 Sep 2024
Viewed by 699
Abstract
In this paper, we investigate and characterize a family of optimization problems introduced by interval-valued functionals that are determined by curvilinear integrals. To this end, we first state the path independence and (strictly) LU convexity properties of the considered functionals. Thereafter, we [...] Read more.
In this paper, we investigate and characterize a family of optimization problems introduced by interval-valued functionals that are determined by curvilinear integrals. To this end, we first state the path independence and (strictly) LU convexity properties of the considered functionals. Thereafter, we formulate the corresponding controlled variational inequalities. The main results of this paper provide some connections for the above-mentioned variational models. Since the objective functionals have a physical importance, an illustrative application is considered and studied by using the theoretical elements obtained in this study. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
14 pages, 291 KiB  
Article
Cohen–Grossberg Neural Network Delay Models with Fractional Derivatives with Respect to Another Function—Theoretical Bounds of the Solutions
by Ravi Agarwal, Snezhana Hristova and Donal O’Regan
Axioms 2024, 13(9), 605; https://doi.org/10.3390/axioms13090605 - 5 Sep 2024
Viewed by 1143
Abstract
The Cohen–Grossberg neural network is studied in the case when the dynamics of the neurons is modeled by a Riemann–Liouville fractional derivative with respect to another function and an appropriate initial condition is set up. Some inequalities about both the quadratic function and [...] Read more.
The Cohen–Grossberg neural network is studied in the case when the dynamics of the neurons is modeled by a Riemann–Liouville fractional derivative with respect to another function and an appropriate initial condition is set up. Some inequalities about both the quadratic function and the absolute values functions and their fractional derivatives with respect to another function are proved and they are based on an appropriate modification of the Razumikhin method. These inequalities are applied to obtain the bounds of the norms of any solution of the model. In particular, we apply the squared norm and the absolute values norms. These bounds depend significantly on the function applied in the fractional derivative. We study the asymptotic behavior of the solutions of the model. In the case when the function applied in the fractional derivative is increasing without any bound, the norms of the solution of the model approach zero. In the case when the applied function in the fractional derivative is equal to the current time, the studied problem reduces to the model with the classical Riemann–Liouville fractional derivative and the obtained results gives us sufficient conditions for asymptotic behavior of the solutions for the corresponding model. In the case when the function applied in the fractional derivative is bounded, we obtain a finite bound for the solutions of the model. This bound depends on the initial function and the solution does not approach zero. An example is given illustrating the theoretical results. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
12 pages, 339 KiB  
Article
Stability Analysis of a Credit Risk Contagion Model with Distributed Delay
by Martin Anokye, Luca Guerrini, Albert L. Sackitey, Samuel E. Assabil and Henry Amankwah
Axioms 2024, 13(7), 483; https://doi.org/10.3390/axioms13070483 - 18 Jul 2024
Viewed by 907
Abstract
This research investigates the stability and occurrence of Hopf bifurcation in a credit risk contagion model, which includes distributed delay, using the chain trick method. The model is a generalized version of those previously examined. The model is an expanded version of those [...] Read more.
This research investigates the stability and occurrence of Hopf bifurcation in a credit risk contagion model, which includes distributed delay, using the chain trick method. The model is a generalized version of those previously examined. The model is an expanded version of those previously studied. Comparative analysis showed that unlike earlier models, which only used the nonlinear resistance coefficient to determine the rate of credit risk infection, the credit risk contagion rate is also affected by the weight given to past behaviors of credit risk participants. Therefore, it is recommended to model the transmission of credit risk contagion using dispersed delays. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
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14 pages, 280 KiB  
Article
Topological Degree via a Degree of Nondensifiability and Applications
by Noureddine Ouahab, Juan J. Nieto and Abdelghani Ouahab
Axioms 2024, 13(7), 482; https://doi.org/10.3390/axioms13070482 - 18 Jul 2024
Cited by 1 | Viewed by 845
Abstract
The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer’s fixed point theorem and the nonlinear alternative of Leray–Schauder type. [...] Read more.
The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer’s fixed point theorem and the nonlinear alternative of Leray–Schauder type. As applications, we study the existence of mild solution of functional semilinear integro-differential equations. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
12 pages, 250 KiB  
Article
Right Quantum Calculus on Finite Intervals with Respect to Another Function and Quantum Hermite–Hadamard Inequalities
by Asawathep Cuntavepanit, Sotiris K. Ntouyas and Jessada Tariboon
Axioms 2024, 13(7), 466; https://doi.org/10.3390/axioms13070466 - 10 Jul 2024
Viewed by 802
Abstract
In this paper, we study right quantum calculus on finite intervals with respect to another function. We present new definitions on the right quantum derivative and right quantum integral of a function with respect to another function and study their basic properties. The [...] Read more.
In this paper, we study right quantum calculus on finite intervals with respect to another function. We present new definitions on the right quantum derivative and right quantum integral of a function with respect to another function and study their basic properties. The new definitions generalize the previous existing results in the literature. We provide applications of the newly defined quantum calculus by obtaining new Hermite–Hadamard-type inequalities for convex, h-convex, and modified h-convex functions. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
31 pages, 1393 KiB  
Article
Achieving Optimal Order in a Novel Family of Numerical Methods: Insights from Convergence and Dynamical Analysis Results
by Marlon Moscoso-Martínez, Francisco I. Chicharro, Alicia Cordero, Juan R. Torregrosa and Gabriela Ureña-Callay
Axioms 2024, 13(7), 458; https://doi.org/10.3390/axioms13070458 - 7 Jul 2024
Cited by 2 | Viewed by 1114
Abstract
In this manuscript, we introduce a novel parametric family of multistep iterative methods designed to solve nonlinear equations. This family is derived from a damped Newton’s scheme but includes an additional Newton step with a weight function and a “frozen” derivative, that is, [...] Read more.
In this manuscript, we introduce a novel parametric family of multistep iterative methods designed to solve nonlinear equations. This family is derived from a damped Newton’s scheme but includes an additional Newton step with a weight function and a “frozen” derivative, that is, the same derivative than in the previous step. Initially, we develop a quad-parametric class with a first-order convergence rate. Subsequently, by restricting one of its parameters, we accelerate the convergence to achieve a third-order uni-parametric family. We thoroughly investigate the convergence properties of this final class of iterative methods, assess its stability through dynamical tools, and evaluate its performance on a set of test problems. We conclude that there exists one optimal fourth-order member of this class, in the sense of Kung–Traub’s conjecture. Our analysis includes stability surfaces and dynamical planes, revealing the intricate nature of this family. Notably, our exploration of stability surfaces enables the identification of specific family members suitable for scalar functions with a challenging convergence behavior, as they may exhibit periodical orbits and fixed points with attracting behavior in their corresponding dynamical planes. Furthermore, our dynamical study finds members of the family of iterative methods with exceptional stability. This property allows us to converge to the solution of practical problem-solving applications even from initial estimations very far from the solution. We confirm our findings with various numerical tests, demonstrating the efficiency and reliability of the presented family of iterative methods. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
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26 pages, 864 KiB  
Article
A New Optimal Numerical Root-Solver for Solving Systems of Nonlinear Equations Using Local, Semi-Local, and Stability Analysis
by Sania Qureshi, Francisco I. Chicharro, Ioannis K. Argyros, Amanullah Soomro, Jihan Alahmadi and Evren Hincal
Axioms 2024, 13(6), 341; https://doi.org/10.3390/axioms13060341 - 21 May 2024
Cited by 12 | Viewed by 1333
Abstract
This paper introduces an iterative method with a remarkable level of accuracy, namely fourth-order convergence. The method is specifically tailored to meet the optimality condition under the Kung–Traub conjecture by linear combination. This method, with an efficiency index of approximately 1.5874, employs [...] Read more.
This paper introduces an iterative method with a remarkable level of accuracy, namely fourth-order convergence. The method is specifically tailored to meet the optimality condition under the Kung–Traub conjecture by linear combination. This method, with an efficiency index of approximately 1.5874, employs a blend of localized and semi-localized analysis to improve both efficiency and convergence. This study aims to investigate semi-local convergence, dynamical analysis to assess stability and convergence rate, and the use of the proposed solver for systems of nonlinear equations. The results underscore the potential of the proposed method for several applications in polynomiography and other areas of mathematical research. The improved performance of the proposed optimal method is demonstrated with mathematical models taken from many domains, such as physics, mechanics, chemistry, and combustion, to name a few. Full article
(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)
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