Hyers-Ulam Stability Research Articles

Stability analysis of the corruption dynamics under fractional-order interventions

Abstract This article proposes a nonlinear deterministic mathematical model that encapsulates the dynamics of the prevailing degree of corruption in a population. The objectives are attained by exploring the dynamics of the corruption model under fractional-order derivative in the Caputo sense. The outcomes of the research are facilitated by stratifying the population into five compartments: susceptible class, exposed class, corrupted class, recovered class, and honest class. The developed model is validated by proving pivotal delicacies such as positivity, invariant region, basic reproduction number, and stability analysis. The Ulam–Hyers stability technique is used to prove the stable solution. The Adam–Bashforth numerical scheme is employed to estimate the numerical solution. Moreover, the research environment is further enriched by studying each compartment with respect to a wide range of relevant parametric settings. The realizations of this study indicate that susceptible individuals remain subject to being influenced by corrupt individuals. In addition, it is observed that the population of exposed individuals, recovered individuals, and honest individuals asymptotically approach toward the corruption equilibrium point, whereas the magnitudes of susceptible individuals and corrupted individuals decrease asymptotically to the corruption equilibrium state. The compartment dynamics are witnessed to be sensitive for various fractional-orders indicating the utility of the fractional approach. The findings of this study support the fundamental understanding of conceptualizing corruption in accordance with the viral transmission of infectious disease.

Approximating fixed points of weak enriched contractions using Kirk’s iteration scheme of higher order

In this paper, we introduce two types of weak enriched contractions, namely weak enriched F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F}$\\end{document}-contraction, weak enriched F′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F^{\\prime}}$\\end{document}-contraction, and k-fold averaged mapping based on Kirk’s iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk’s iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature.

Stability and Numerical Analysis of a Coupled System of Piecewise Atangana–Baleanu Fractional Differential Equations with Delays

This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application.

Tripled Fixed Point Approaches and Hyers-Ulam Stability With Applications

In this paper, we present tripling fixed point results for extended contractive mappings in the context of a generalized metric space. Many publications in the literature are improved, unified, and generalized by our theoretical results. Furthermore, the Ulam-Hyers stability problem for the tripled fixed point problem in vector-valued metric spaces has been examined as a stability analysis for fixed point approaches. Finally, as a type of application to support our research, the theoretical conclusions are used to explore the existence and uniqueness of solutions to a periodic boundary value problem.

Investigating a Fractal–Fractional Mathematical Model of the Third Wave of COVID-19 with Vaccination in Saudi Arabia

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders (γ) and fractal dimensions (θ), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models.

Statistical and computational analysis for corruption and poverty model using Caputo-type fractional differential equations

Since there is a clear correlation between poverty and corruption, mathematicians have been actively researching the concept of poverty and corruption in order to develop the optimal strategy of corruption control. This work aims to develop a mathematical model for the dynamics of poverty and corruption. First, we study and analyze the indicators of corruption and poverty rates by applying the linear model along with the Eviews program during the study period. Then, we present a prediction of poverty rates for 2023 and 2024 using the results of the standard problem-free model. Next, we formulate the model in the frame of Caputo fractional derivatives. Fundamental properties, including equilibrium points, basic reproduction number, and positive solutions of the considered model are obtained using nonlinear analysis. Sufficient conditions for the existence and uniqueness of solutions are studied via using fixed point theory. Numerical analysis is performed by using modified Euler method. Moreover, results about Ulam-Hyers stability are also presented. The aforementioned results are presented graphically. In addition, a comparison with real data and simulated results is also given. Finally, we conclude the work by providing a brief conclusion.

Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform.

Hyers–Ulam Stability for a Type of Discrete Hill Equation

Hyers–Ulam Stability for a Type of Discrete Hill Equation

Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems

We examine a nonlinear dynamical model that depicts the interaction between cancerous cells and an oncolytic virus. For best modelling the disease, we use the Caputo fractional derivative in piecewise approaches. By employing piecemeal techniques, we treat a compartment in the body that contains infectious and non-infectious cells. More precisely, the solvability and Ulam-Hyers (U-H) stability results are considered using standard concepts. Further, to support our investigation with numerical results, we apply the Euler method to develop an approximation solution. It connected with numerous graphical representations of the system using various arbitrary ordering and varying values of the isolation parameters. Here we remark that the multi-step behavior that certain problems exhibit, is one of important issues naturally. This paper introduces the idea of piecewise derivative with the goal of modeling real-world issues that follow multiples processes. With the help of the used approach, we investigate the cancer disease model and its transmission dynamical behavior with crossover effect.

Hyers–Ulam Stability of a Second-Order Finite Difference Scheme Using a Diamond-$$\alpha $$ Difference Operator

Hyers–Ulam Stability of a Second-Order Finite Difference Scheme Using a Diamond-$$\alpha $$ Difference Operator

Hyers–Ulam stabilities for [formula omitted]th differential operators on [formula omitted

In this paper, we introduce certain operators Tλm and Tφm on weighted Hardy spaces Hβ2, and in the following, we investigate their boundedness on Hβ2. After that, we prove the Hyers–Ulam stability for certain operators on weighted Hardy spaces Hβ2. Moreover, we show under what conditions these concepts are stable and unstable by using some examples. Finally, we investigate how differential operators are stable by using different sequences in a corollary and an example.

Instabilities and Stabilities of Additive Functional Equation in Paranormed Spaces

In this paper, we solve the general solution in vector space and prove the Hyers-Ulam stability of the following additive functional equationin paranormed spaces by using the direct and fixed point methods. Also we present its pertinent counter examples for instabilities.

The coefficient multipliers on $ H^2 $ and $ \mathcal{D}^2 $ with Hyers–Ulam stability

<abstract><p>In this paper, we investigated the Hyers–Ulam stability of the coefficient multipliers on the Hardy space $ H^2 $ and the Dirichlet space $ \mathcal{D}^2 $. We also investigated the Hyers–Ulam stability of the coefficient multipliers between Dirichlet and Hardy spaces. We provided the necessary and sufficient conditions for the coefficient multipliers to have Hyers–Ulam stability on Hardy space $ H^2 $, on Dirichlet space $ \mathcal{D}^2 $, and between Dirichlet and Hardy spaces. We also showed that the best constant of Hyers–Ulam stability exists under different circumstances. Moreover, some illustrative examples were discussed.</p></abstract>

Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system

<abstract><p>This study aimed to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions in a nonlinear coupled system of Hilfer-Hadamard sequential fractional integrodifferential equations, which were further enhanced by nonlocal coupled Hadamard fractional integrodifferential multipoint boundary conditions. The desired conclusions were obtained by using well-known fixed-point theorems. It was emphasized that the fixed-point technique was useful in determining the existence and uniqueness of solutions to boundary value problems. In addition, we examined the solution's Ulam-Hyers stability for the suggested system. The resulting results were further demonstrated and validated using demonstration instances.</p></abstract>

Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives

<abstract><p>One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results.</p></abstract>

A mathematical model based on ABC fractional order for TB transmission with treatment interruptions in case of Bule Hora town, Ethiopia

In this study, we introduce a novel fractional-order model enriched with the Atangana–Baleanu–Caputo derivative to explore the dynamics of TB transmission in developing nations, where tuberculosis (TB) remains a pressing public health concern requiring urgent attention. We outline the methodology employed, emphasizing key aspects such as solution existence, uniqueness, Hyers-Ulam stability, and numerical simulations. Equilibrium points of the system, including the Disease-Free Equilibrium (DFE) and the Endemic Equilibrium (EE), are revealed through adopted mathematical manipulations. Following rigorous numerical simulations employing the Euler approach and a meticulously crafted scheme, the study illustrates compartmental solutions across a spectrum of fractional orders. This approach provides a comprehensive understanding of the model’s behavior and its responsiveness to different parameter configurations. Remarkably, as the fractional order tends towards one, our findings align closely with those of traditional integer-order models, offering insightful parallels into the model’s dynamics. Additionally, our work emphasizes the pivotal role of vaccination as a preventive measure, elucidating its profound impact on reducing tuberculosis incidence within communities. The promotion of widespread access to TB vaccines emerges as a potent strategy in curtailing the disease’s prevalence, underscoring the significance of proactive vaccination initiatives. As conclusion, this study presents a comprehensive examination of TB dynamics through a fractional-order modeling approach. Our results pave the way for further exploration, including the investigation of alternative fractional-order derivatives and exploration of theoretical and numerical stability. These endeavors will enhance our understanding of TB dynamics and inform more effective strategies for disease control and prevention.

Efficient results on unbounded solutions of fractional Bagley-Torvik system on the half-line

<abstract><p>The fractional Bagley-Torvik system (FBTS) is initially created by utilizing fractional calculus to study the demeanor of real materials. It can be described as the dynamics of an inflexible plate dipped in a Newtonian fluid. In the present article, we aim for the first time to discuss the existence and uniqueness (E&U) theories of an unbounded solution for the proposed generalized FBTS involving Riemann-Liouville fractional derivatives in the half-line $ (0, \infty) $, by using fixed point theorems (FPTs). Moreover, the Hyers-Ulam stability (HUS), Hyers-Ulam-Rassias stability (HURS), and semi-Hyers-Ulam-Rassias stability (sHURS) are proved. Finally, two numerical examples are given for checking the validity of major findings. By investigating unbounded solutions for the FBTS, engineers gain a deeper understanding of the underlying physics, optimize performance, improve system design, and ensure the stability of the motion of real materials in a Newtonian fluid.</p></abstract>

Differential equations with variable order generalized proportional Caputo fractional with respect to another function: existence and stability

This study is concerned with the existence and stability of solutions of a class of initial value problem for fractional differential equations with a variable order generalized proportional Caputo fractional derivative with respect to another function. The statement of the problem is discussed and compared with the existing in the literature. Existence results are studied and the Hyers-Ulam stability is defined and established. The approach is more broad-based and the same methodology can be used for a number of additional issues.

Strict fixed point and Ulam-Hyers stability of multivalued asymptotically regular mappings

In this paper, we establish the existence and uniqueness of strict fixed point for an asymptotically regular multivalued mapping in a metric space. We also study the Ulam-Hyers stability, well-posedness and data dependence of the associated strict fixed point problem. We give an example to illustrate our results. Our work extends and complements important results existing in the literature.