Hyers-Ulam Stability Research Articles

Hyers-Ulam Stability to Linear Nonhomogeneous Quaternion-Valued Matrix Difference Equations via Complex Representation

Hyers-Ulam Stability to Linear Nonhomogeneous Quaternion-Valued Matrix Difference Equations via Complex Representation

Dynamical Transmission and Mathematical Analysis of Ebola Virus Using a Constant Proportional Operator with a Power Law Kernel

The Ebola virus continues to be the world’s biggest cause of mortality, especially in developing countries, despite the availability of safe and effective immunization. In this paper, we construct a fractional-order Ebola virus model to check the dynamical transmission of the disease as it is impacted by immunization, learning, prompt identification, sanitation regulations, isolation, and mobility limitations with a constant proportional Caputo (CPC) operator. The existence and uniqueness of the proposed model’s solutions are discussed using the results of fixed-point theory. The stability results for the fractional model are presented using Ulam–Hyers stability principles. This paper assesses the hybrid fractional operator by applying methods to invert proportional Caputo operators, calculate CPC eigenfunctions, and simulate fractional differential equations computationally. The Laplace–Adomian decomposition method is used to simulate a set of fractional differential equations. A sustainable and unique approach is applied to build numerical and analytic solutions to the model that closely satisfy the theoretical approach to the problem. The tools in this model appear to be fairly powerful, capable of generating the theoretical conditions predicted by the Ebola virus model. The analysis-based research given here will aid future analysis and the development of a control strategy to counteract the impact of the Ebola virus in a community.

A numerical study of a new non-linear fractal fractional mathematical model of malicious codes propagation in wireless sensor networks

This paper presents a novel non-linear fractal fractional mathematical model for analyzing the spread of malicious codes in wireless sensor networks (WSNs). The model incorporates fractional calculus and non-linear interactions among nodes in a WSN. It is based on the Fractal-Fractional (FF) operator, a powerful mathematical tool that combines fractional and fractal calculus. This study explores the potential of fractional derivatives in addressing computer virus-related issues and proposes a model that elucidates the spread of viruses in a vulnerable system and potential countermeasures. The Fractal-Fractional operator is utilized to fractionalize the model, and the fixed point theory of Schauder and Banach is applied to establish the existence and uniqueness of a solution. Numerical simulations, employing MATLAB and the Adams-Bashforth method, validate the effectiveness of the proposed model in capturing the propagation dynamics of malicious codes. Ulam-Hyers stability techniques are also employed for stability analysis. The model's insights contribute to enhancing the security and robustness of WSNs against malicious code propagation, paving the way for more secure network designs. This research provides a unique perspective on analyzing the dynamics of WSNs and underscores the significance of fractional derivatives in addressing security threats in network systems.

Characterization of Ulam-Hyers stability of linear differential equations with periodic coefficients

Let X be a Banach space over R or C. We study Ulam–Hyers stability on the time interval R of linear differential equations with continuous coefficients, considering solutions with values in X for higher order equations, and solutions with values in Xn for first-order systems. Roughly speaking, Ulam–Hyers stability of an equation is its property of having an exact solution close to an approximate solution. For a first-order system x′=Ax, we obtain that its Ulam–Hyers stability is equivalent to another important property, namely for each continuous and bounded function g:R→Xn there exists a bounded solution of x′=Ax+g. Using this result, theory of Jordan forms of matrices and Floquet theory, we characterize the Ulam–Hyers stability of a system x′=Ax with periodic coefficients in terms of its characteristic multipliers. Our main result states that the Ulam–Hyers stability of a linear equation with periodic coefficients of order n with unknown x is equivalent to the Ulam–Hyers stability of the corresponding system satisfied by (x,x′,…,x(n−1)).

Analysis of the dynamics of a vector-borne infection with the effect of imperfect vaccination from a fractional perspective

The burden of vector-borne infections is significant, particularly in low- and middle-income countries where vector populations are high and healthcare infrastructure may be inadequate. Further, studies are required to investigate the key factors of vector-borne infections to provide effective control measure. This study focuses on formulating a mathematical framework to characterize the spread of chikungunya infection in the presence of vaccines and treatments. The research is primarily dedicated to descriptive study and comprehension of dynamic behaviour of chikungunya dynamics. We use Banach’s and Schaefer’s fixed point theorems to investigate the existence and uniqueness of the suggested chikungunya framework resolution. Additionally, we confirm the Ulam–Hyers stability of the chikungunya system. To assess the impact of various parameters on the dynamics of chikungunya, we examine solution pathways using the Laplace-Adomian method of disintegration. Specifically, to visualise the impacts of fractional order, vaccination, bite rate and treatment computer algorithms are employed on the infection level of chikungunya. Our research identified the framework’s essential input settings for managing chikungunya infection. Notably, the intensity of chikungunya infection can be reduced by lowering mosquito bite rates in the affected area. On the other hand, vaccination, memory index or fractional order, and treatment could be used as efficient controlling variables.

Different strategies for diabetes by mathematical modeling: Applications of fractal–fractional derivatives in the sense of Atangana–Baleanu

For diabetes mellitus therapies, Bergman’s minimal model is proposed. The present study is devoted to mathematical modeling and analysis of the diabetes mellitus model without genetic factors in the fractal–fractional derivatives in the sense of Atangana–Baleanu. Diabetes mellitus disease dynamics are modeled using this modified derivative. The operator can be applied to formulate memory effects models. We proved the existence and uniqueness of the model solution using fractal–fractional derivatives in the sense of Atangana–Baleanu. The model’s solution and uniqueness, as well as the Hyers-Ulam stability analysis, are established. Moreover, we compare ordinary, Atangana–Baleanu, fractal–fractional derivatives in the sense of Atangana–Baleanu, and the real measured data of 10 subjects. Diabetes mellitus rates increase when both fractal dimensions θ1 and θ2 decrease in numerical experiments. Results from fractal–fractional derivatives in the sense of Atangana–Baleanu are closer to reality than integer-order models.

A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam–Hyers stability

A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam–Hyers stability

Different strategies for diabetes by mathematical modeling: Modified Minimal Model

A fractal-fractional-order of a modified minimal model of glucose-insulin interaction model is proposed in sense of Atangana-Baleanu operator based on the intravenous glucose tolerance test (IVGT). Based on the fractal-fractional sense, we examine the existence of the solution, Hyers-Ulam stability, uniqueness, non-negativity, positivity, and boundedness. The model is analyzed qualitatively using its equivalent integral using an iterative convergence sequence and fixed point approach. A numerical scheme for the fractal-fractional model is then applied to a case study. Modified minimal models are reasonable and acceptable. We are interested to see that both analytical and simulation results agree.

Iterative sequential approximate solutions method to Hyers–Ulam stability of time-varying delayed fractional-order neural networks

There is a key issue in the field of stability analysis for fractional-order neural networks (FONNs) concerning whether the exact solution of the FONNs exists under the condition that a series of successive approximate solutions of the models can be obtained. If so, can the state differences between the approximate solutions and the exact solution be estimated? Hyers–Ulam stability, as one of the powerful tools to solve this problem, has attracted the attention of scholars, and various related research achievements have been reported. In view of this, this paper aims to investigate the Hyers–Ulam stability of FONNs with time-varying delays. Using the successive approximation method and the modified retarded Henry–Gronwall integral inequality, several conditions guaranteeing the existence and uniqueness of the exact solution of FONNs with time-varying delays are derived. Moreover, the corresponding results on the Hyers–Ulam–Rassias stability of the addressed models are given. Finally, a numerical example is designed to illustrate the efficiency and rationality of the theoretical results.

Ulam–Hyers Stability of Pantograph Hadamard Fractional Stochastic Differential Equations

In this article, we investigate the existence and uniqueness Theorem of Pantograph Hadamard fractional stochastic differential equations (PHFSDE) using the fixed-point Theorem of Banach (BFPT). According to the generalized Gronwall inequalities, we prove the stability in the sense of Ulam–Hyers (UHS) of PHFSDE. We give some examples to show the effectiveness of our results.

Stability analysis of neutral delay fractional differential equations with Erdelyi–Kober fractional integral boundary conditions

The primary focus of this article is to provide sufficient conditions for the Ulam–Hyers stability of neutral delay fractional differential equations involving Hilfer fractional derivatives and Erdelyi–Kober fractional integral boundary conditions. The fixed point approach is utilized to prove the existence and uniqueness of mild solutions for the proposed problem. In the end, the derived results are validated through an illustrative example.

On stability of nonlocal neutral stochastic integro differential equations with random impulses and Poisson jumps

This article aims to examine the existence and Hyers-Ulam stability of non-local random impulsive neutral stochastic integrodifferential delayed equations with Poisson jumps. Initially, we prove the existence of mild solutions to the equations by using the Banach fixed point theorem. Then, we investigate stability via the continuous dependence of solutions on the initial value. Next, we study the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, we give an illustrative example of our main result.

On the Existence of the Exact Solution of Quaternion-Valued Neural Networks Based on a Sequence of Approximate Solutions.

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian

The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space.

Completeness Problem via Fixed Point Theory

The purpose of this paper is to present the notion of MR-Kannan type contractions using the generalized averaged operator. Several examples are provided to illustrate the concept presented herein. We provide a characterisation of normed spaces using MR-Kannan type contractions with a fixed point. We investigate the Ulam–Hyers stability and well-posedness result for the mappings presented here.

Mathematical analysis of fractional order alcoholism model

In this manuscript, we are going to study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under the concept of conformable fractional order derivative (CFOD). Currently, most of real-world problems are considered under fractional order derivatives because of their stable and global behavior. First, we will investigate the model for qualitative theory including existence and uniqueness of solution and Ulam-Hyers stability. For qualitative theory, we will use fixed point theory. In addition, we use a numerical method to find the approximate solution of the proposed model. In the final part of the paper, we give a detailed discussion of its numerical results and its graphical presentation.

Analysis of age wise fractional order problems for the Covid-19 under non-singular kernel of Mittag-Leffler law

The developed article considers SIR problems for the recent COVID-19 pandemic, in which each component is divided into two subgroups: young and adults. These subgroups are distributed among two classes in each compartment, and the effect of COVID-19 is observed in each class. The fractional problem is investigated using the non-singular operator of Atangana Baleanu in the Caputo sense (ABC). The existence and uniqueness of the solution are calculated using the fundamental theorems of fixed point theory. The stability development is also determined using the Ulam-Hyers stability technique. The approximate solution is evaluated using the fractional Adams-Bashforth technique, providing a wide range of choices for selecting fractional order parameters. The simulation is plotted against available data to verify the obtained scheme. Different fractional-order approximations are compared to integer-order curves of various orders. Therefore, this analysis represents the recent COVID-19 pandemic, differentiated by age at different fractional orders. The analysis reveals the impact of COVID-19 on young and adult populations. Adults, who typically have weaker immune systems, are more susceptible to infection compared to young people. Similarly, recovery from infection is higher among young infected individuals compared to infected cases in adults.

Numerical solutions of fractional order rabies mathematical model via Newton polynomial

The recent outbreak of rabies virus has affected numerous individuals in the community, highlighting the importance of studying the disease mathematically in epidemiology. In this paper, we develop a mathematical model for the spread of rabies disease using the harmonic mean incidence rate and determine the reproduction number R0 using the next generation matrix approach. The fractional dynamics of the model incorporate the interaction between infected individuals and environmental factors. We analyze the model using both Atangana-Baleanu-Caputo (ABC) and Caputo-Fabrizio (CF) techniques, drawing on both modern and classical approaches. For qualitative investigation using fractional operators, we utilize the Banach fixed-point theory and derive the Hyers-Ulam stability concept. We assign values to the model parameters and utilize the Newton interpolation technique to obtain a numerical scheme. Additionally, we perform sensitivity analysis on the model. In conclusion, our findings provide important insights for the study of epidemics.

Parameterized shadowing for nonautonomous dynamics

For nonautonomous and nonlinear differential and difference equations depending on a parameter, we formulate sufficient conditions under which they exhibit Ck, k∈N shadowing with respect to a parameter. Our results are applicable to situations when the linear part is not hyperbolic. In the case when the linear part is hyperbolic, we obtain results dealing with parameterized Hyers-Ulam stability.

Population Dynamics on Fractional Tumor System Using Laplace Transform and Stability Analysis

Modeling is an effective way of using mathematical concepts and tools to represent natural systems and phenomena. Fractional calculus is an essential part of modeling a biological system. Recently, many researchers have been interested in modeling real-time problems mathematically and analyzing them. In this paper, the tumor system under fractional order is considered, and it comprises normal cells, tumor cells and effector-immune cells. By taking chemotherapy drugs into account, the toxicity of the drug and concentration of the drug is also studied in the model. The main objective of this work is to establish the solution for the model using Laplace transform and analyze the stability of the model. Laplace transform, a simple and efficient method, is used in solving the system that proves the existence and uniqueness of the solution. The boundedness of the system is also verified using the Lipschitz condition. Further, the system is solved for numerical values, and the population dynamics of cells are provided for different values of $\alpha$ as a graphical representation. Also, after analyzing the effect of chemotherapy drugs on tumor cells for different $\alpha$'s, which signifies that $\alpha$ = 0.9 provides a sufficient decrease in the dynamics of tumor cells. The main and significant part of this work is presenting that the usage of chemotherapy drugs reduces the number of tumor cells. The importance of the work is that apart from the immune system, chemotherapy drugs play a significant role in destroying tumor cells. The Hyers Ulam stability has a significant application that one need not find the exact solution to when analyzing a Hyers Ulam stable system. Thus, the stability of this tumor model under Caputo fractional order is presented using Hyers-Ulam stability and Hyers-Ulam-Rassias stability.