Hyers-Ulam Stability Research Articles

Hyers–Ulam stability of Nipah virus model using Atangana–Baleanu–Caputo fractional derivative with fixed point method

In this study, we present a novel investigation into the dynamics of the Nipah virus through the lens of fractional differential equations (FDEs), employing the Atangana–Baleanu–Caputo fractional derivative (ABCFD) and the fixed-point approach (FPA). The core contribution of this work lies in establishing the existence and uniqueness of solutions to the proposed FDEs, a critical step for validating the model. Furthermore, we explore the Hyers–Ulam (HU) stability of these generalized FDEs, providing a rigorous mathematical foundation for the stability analysis within the context of viral dynamics. By leveraging the ABCFD, our work extends the classical stability criteria, offering new insights into the role of memory effects in disease modeling. Additionally, we present approximate solutions across various compartments and fractional orders, highlighting the sensitivity of the system to key parameters. Numerical simulations, conducted using the Cullis method, illustrate the impact of fractional orders and validate the theoretical findings, positioning this work as a significant advancement in the application of fractional calculus to epidemiological models.

Analysis of Caputo-Type Non-Linear Fractional Differential Equations and Their Ulam–Hyers Stability

This study presents two novel frameworks, termed a quasi-modular b-metric space and a non-Archimedean quasi-modular b-metric space, and various topological properties are provided. Using comparison and simulation functions, this paper rigorously proves several fixed point theorems in the non-Archimedean quasi-modular b-metric space. As a useful application, it also establishes Ulam–Hyers stability for the fixed point problem. Finally, this study concludes with a unique solution to a non-linear fractional differential equation, making a substantial contribution to the discipline.

Hyers-Ulam stability of Davison functional equation on restricted domains

Abstract In this article, we study the Hyers-Ulam stability of Davison functional equation f ( x y ) + f ( x + y ) = f ( x y + x ) + f ( y ) f\left(xy)+f\left(x+y)=f\left(xy+x)+f(y) on some unbounded restricted domains. Using the obtained results, we study an interesting asymptotic behavior of Davison functions. We also investigate the Hyers-Ulam stability of Davison functional equation and its generalized form given by f ( x y ) + g ( x + y ) = h ( x y + x ) + k ( y ) , f\left(xy)+g\left(x+y)=h\left(xy+x)+k(y), for x , y ∈ R ⩾ 0 = { t ∈ R : t ⩾ 0 } x,y\in {{\mathbb{R}}}^{\geqslant 0}=\left\{t\in {\mathbb{R}}:t\geqslant 0\right\} .

Existence and Ulam stability of initial value problem for fractional perturbed functional q-difference equations

Abstract. In this work, we discuss the existence and uniqueness of solutions to the initial value problem for perturbed functional fractional q-difference equations involving q-derivative of the Caputo sense. By applying Banach contraction principle and Burton and Kirk’s fixed point theorems. Further, we present the Ulam-Hyers and Ulam-Hyers-Rassias stabilities results by using direct analysis methods. Finally, we give two examples illustrating of the results. Mathematics Subject Classification (2010): 26A33, 34A12, 39A13, 47H10. Keywords: Initial value problem, Caputo fractional q-derivative, Burton and Kirk’s fixed point theorem, Ulam-Hyers stability, Ulam-Hyers-Rassias stability.

Corrections to “Complex order fractional differential equation in complex domain with mixed boundary condition”

The purpose of this note is to address and correct some errors identified in the recent paper by Yadav et al. (2024). In general, the set of sufficient conditions provided for ensuring the existence and uniqueness of solutions may not be adequate. Additionally, an incorrect inequality was used in the proof to demonstrate that T is a contraction mapping on B. This mistake is then carried forward into the analysis of (generalized) Ulam–Hyers stability. We therefore revisit the problem, establish the existence and uniqueness of solutions using the Banach fixed point theorem, and analyze the Ulam–Hyers stability.

Hyers–Ulam stability of the Drygas type functional equation

Hyers–Ulam stability of the Drygas type functional equation

Existence, uniqueness and Ulam–Hyers stability result for variable order fractional predator-prey system and it's numerical solution

This study presents an approximate numerical technique for solving time fractional advection-diffusion-reaction predator-prey equations with variable order (VO), where the analyzed fractional derivatives of VO are in the Caputo sense. Results for Ulam–Hyers stability are shown, as well as the existence and uniqueness of solutions. It is suggested to use a numerical approximation based on the shifted second kind of airfoil polynomials to solve the equations under consideration. A fractional derivative operational matrix with VO is derived for shifted airfoil polynomials, which will be used to compute the unknown function. The main equations are transformed into a set of algebraic equations by substituting the aforementioned operational matrix into the equations under consideration and utilizing the properties of the shifted airfoil polynomial along with the collocation points. A numerical solution is obtained by solving the acquired set of algebraic equations. To verify the accuracy and efficiency of the discussed scheme, several illustrative examples have been considered. The results obtained by the proposed method demonstrate the efficiency and superiority of the method compared to other existing methods.

A nonlinear fractional fishery resource system model with Crowley–Martin functional response under Mittag-Leffler kernel

This article studies the action of predators and the predator-dependent functional response in the fishery resource model. We employ the Atangana–Baleanu–Caputo fractional derivative to study the proposed fractional fishery resource model in the presence of predators with Crowley–Martin functional response. We present a theoretical and numerical analysis of the governing nonlinear differential equations of the model consisting of the biomass density of the fish population inside the unrestricted fishing zone, the biomass density of the fish population inside the reserve or restricted fishing zone, and the predator population. Using the fixed point theory and nonlinear analysis, we establish the existence and uniqueness results of the proposed ABC fractional fishery resource model. We establish stability analysis of the fishery model using the Ulam–Hyers stability approach. The numerical scheme of the fractional Adams–Bashforth method is provided and the approximate solutions for the model under consideration are given and discussed. We observe that an increase in the fish capturing rates increases the size of the predator population and reduces the fish subpopulations. To maintain a high number of fish species, we recommend a control measure to reduce the fish capturing rate by the predators.

Mathematical modeling of allelopathic stimulatory phytoplankton species using fractal–fractional derivatives

In the current study, we employ the novel fractal–fractional operator in the Atangana–Baleanu sense to investigate the dynamics of an interacting phytoplankton species model. Initially, we utilize the Picard-Lindelöf theorem to validate the uniqueness and existence of solutions for the model. We then explore equilibrium points within the phytoplankton model and conduct Hyers–Ulam stability analysis. Additionally, we present a numerical scheme utilizing the Newton polynomial to validate our analytical findings. Numerical simulations illustrate the dynamical behavior of the model across various fractal and fractional parameter values, visualized through graphical representations. Our simulations reveal that the stability of equilibrium points is not significantly impacted with the long-term memory effect, which is characterized by fractal–fractional order values. However, an increase in fractal–fractional parameters accelerates the convergence of solutions to their intended equilibrium states.

Global Analysis of a Fractional‐Order Hepatitis B Virus Model Under Immune Response in the Presence of Cytokines

AbstractThis research proposes and investigates an epidemiological model to study the dynamic behaviors of the Hepatitis B virus (HBV) under immune response and cytokine influence. The model's stability, positivity, boundedness, and equilibria are analyzed using Lyapunov functional methods and the Routh–Hurwitz criterion under Caputo fractional derivative. The study evaluates nucleoside analogues and interferon treatments, determining critical drug efficiencies. Equilibria, including infection‐free and endemic states, are analyzed using the fundamental reproduction number, , to predict disease elimination. Numerical simulations utilize the fractional Adams method and the L1 scheme, capturing memory traces as the fractional order changes. Results show the L1 scheme effectively captures memory traces, providing empirical support for the theoretical findings. Furthermore, Ulam–Hyers stability is treated according to the equilibrium point, which describes relationships between functions. Notably, the findings of the study yielded profound insights. They revealed that the HBV system remains locally asymptotic stable at disease‐free and the endemic point when . At the same time, the simulations illustrated a correlation between the rate of infection and the rise in infected individuals, indicating the feasibility of eradicating and effectively managing HBV infections through a multifaceted approach and various measures such as vaccination and effective drug administration protocols. The proposed framework can guide medical professionals and decision‐makers in developing effective strategies to limit and eliminate the spread of HBV in the population.

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

Boundary problems of sequential fractional differential equations having a monomial coefficient

The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

On the Cauchy Problem for Nonlinear Fractional Systems with Lipschitzian Matrices Under the Generalized Metric Spaces

This research paper study the existence, uniqueness and Ulam–Hyers stability of the solutions of a certain system of thegeneralized Caputo fractional differential equations in the context of the generalized metric spaces. The existence and uniqueness theorems are proved by using the Krasnoselskii’s and Perov’s fixed point theorems under the Bielecki norm with a Lipschitzian matrix in the generalized metric spaces. Moreover, the Ulam–Hyers stability analysis is conducted based on the Urs’s criterion. An example, lastly, is proposed to check the efficiency of the above-mentioned theorems. The results are novel and provide extensions to some of the findings known in the literature.

Controllability of switched integro‐neutral fractional delay systems over arbitrary time domain

In this paper, we examine various qualitative properties for some class of fractional impulsive switched integro‐neutral dynamic systems over arbitrary time domain. We establish the existence, stability, and controllability of the scrutinized models. This article is divided into three parts: The first part is concerned with the existence and uniqueness, the second part is dedicated to the Hyers–Ulam stability, and in the third part, we examine the controllability analysis. We establish our results via the Banach and Leray–Schauder fixed‐point approaches along with the Arzela–Ascoli theorem. To show the implications and effectiveness of established results, we present a simulated numerical example at the end.

β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of β-Banach spaces. Moreover, we examine the β–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained.

Hyers–Ulam stability of unbounded closable operators in Hilbert spaces

AbstractIn this paper, we discuss the Hyers–Ulam stability of closable (unbounded) operators with some examples. We also present results pertaining to the Hyers–Ulam stability of the sum and product of closable operators to have the Hyers–Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of block matrix in order to have the Hyers–Ulam stability.

Global Mittag-Leffler Attractive Sets, Boundedness, and Finite-Time Stabilization in Novel Chaotic 4D Supply Chain Models with Fractional Order Form

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM).

On the qualitative behaviors of Volterra-Fredholm integro differential equations with multiple time-varying delays

This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays.

On analysis of a system of non-homogenous boundary value problems using hausdorff derivative with exponential kernel

AbstractIn recent years, the fractals (Hausdorff) derivatives with fractional order under various types kernel have gained attention from researchers. The aforesaid area has many applications in the description of intricate and irregular geometry of various processes. Numerous studies utilizing the fractional derivatives (HFDs) for initial value problems have been carried out. But the boundary value problems using the said concepts have been very rarely studied. Thus, a coupled system with non-homogenous boundary conditions (BCs) is examined in this study by using fractals fractional derivative in Caputo Fabrizio sense. To establish the required conditions for the existence and uniqueness of solution to the considered problem, we apply the Banach and Krasnoselskii’s fixed point theorems. Furthermore, some results related to Hyers-Ulam (H-U) stability have also deduced. We have included two pertinent examples to verify our results.

The system of mixed type additive-quadratic equations and approximations

In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable.