Generalized Hyers-Ulam Stability Research Articles

Stability of Quadratic Mappings in 2-Banach Spaces and Related Topics

Functional analysis is an important branch of mathematics widely used to study the stability of different functional equations. This includes the stability of various quadratic functional equations, which typically involve a specific number of variables to reach results more easily in this field. One of the methods used to study the stability of functional equations is the direct method, which is known for its simplicity in proving the stability of this type of functional equation. In this research paper, we have successfully proven the Hyers-Ulam-Rassias stability of the quadratic functional equation in 2-Banach spaces. Specifically, we have shown that the equation: f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(x+z) holds true within this context. Our approach involved using either the usual or the direct method to establish this stability. Furthermore, we have also demonstrated the generalized Hyers-Ulam stability of the quadratic functional equation in 2-Banach spaces using the usual process by considering various conditions. This has led to many exciting results and revealed many related applications.

Ulam–Hyers and Generalized Ulam–Hyers Stability of Fractional Differential Equations with Deviating Arguments

In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii’s fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the fractional functional differential problem. Finally, two examples are presented to illustrate our results, one is with a pantograph-type equation and the other is numerical.

Stability Result in Lattice Random Normed Space

We use the direct method to study the generalized Hyers-Ulam stability of a mixed type(cubic-quartic) functional equation for all , in LRN-space.

Stability Of 4D alternate additive functional equation

In this paper, we confer the generalized Ulam-Hyers stability of a 4D alternate additive functional equation in Banach and Random Banach space via direct and fixed point methods.

Some new results on generalized Hyers-Ulam stability in modular function spaces

Some new results on generalized Hyers-Ulam stability in modular function spaces

Approximation and analysis regarding the structure of a multiple variable mapping

The article introduces a several variables mapping as the multimixed quadratic-cubic mapping in order to characterize such mappings. It reduces a system of equations defining the multimixed quadratic-cubic mappings to obtain a single functional equation. It is shown that under some mild conditions, every multimixed quadratic-cubic mapping can be multi-quadratic, multi-cubic and multiquadratic-cubic. Further, the generalized Hyers-Ulam stability and hyperstabilty for multimixed quadratic-cubic functional equations in quasi-$\beta$ normed spaces have been investigated.

Mathematical analysis of a coronavirus model with Caputo, Caputo–Fabrizio–Caputo fractional and Atangana–Baleanu–Caputo differential operators

This research aims to use fractional operators to analyze a fractional-order model of a coronavirus disease of 2019 (COVID-19). We use some basic results and definitions from fractional calculus and then, by using them, investigate the effects of these operators in a better elucidation of the epidemic COVID-19. We showed the existence and uniqueness of the solution of the proposed model by applying the Picard–Lindelöf theorem and Banach contraction principle. We established the generalized Hyers–Ulam stability of the fractional model using Gronwall’s inequality. We developed effective numerical scheme to solve these fractional systems, which gives a perfect approximate solution to the fractional system. Finally, the numerical simulations were performed in each case to verify all theoretical result.

A novel study for solving systems of nonlinear fractional integral equations

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of equations, we have simplified the problem to finding the solution for a set of nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam–Hyers stability and generalized Ulam–Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method.

Generalized Hyers–Ulam stability of ρ-functional inequalities

In our research work generalized Hyers-Ulam stability of the following functional inequalities is analyzed by using fixed point approach: 0.1∥f(2x+y)+f(2x−y)−2f(x+y)−2f(x−y)−12f(x)−ρ(4f(x+y2)+4(f(x−y2)−f(x+y)−f(x−y))−6f(x),r)∥≥rr+φ(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned}& \\biggl\\Vert f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x) \\\\& \\quad {}-\\rho \\biggl(4f\\biggl(x+\\frac{y}{2}\\biggr)+4\\biggl(f\\biggl(x- \\frac{y}{2}\\biggr)-f(x+y)-f(x-y)\\biggr)-6f(x),r\\biggr) \\biggr\\Vert \\geq \\frac{r}{r+\\varphi (x, y)} \\end{aligned}$$ \\end{document} and 0.2∥f(2x+y)+f(2x−y)−4f(x+y)−4f(x−y)−24f(x)+6f(y)−ρ(8f(x+y2)+8(f(x−y2)−2f(x+y)−2f(x−y))−12f(x)+3f(y),r)∥≥rr+φ(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned}& \\biggl\\Vert f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y) \\\\& \\qquad {}-\\rho \\biggl(8f\\biggl(x+\\frac{y}{2}\\biggr)+8\\biggl(f\\biggl(x- \\frac{y}{2}\\biggr)-2f(x+y)-2f(x-y)\\biggr)-12f(x)+3f(y),r\\biggr) \\biggr\\Vert \\\\& \\quad \\geq \\frac{r}{r+\\varphi (x, y)} \\end{aligned}$$ \\end{document} in the setting of fuzzy matrix, where rho neq 2 is a real number.We also discussed Hyers-Ulam stability from the application point of view.

Generalized Hyers-Ulam Stability of Additive Functional Inequality in Modular Spaces and β-Homogeneous Banach Spaces

In this work, we investigate the generalised Hyers-Ulam stability of additive functional inequality in modular spaces with ∆2-conditions and in β-homogeneous Banach spaces.

Generalized Hyers-Ulam stability of a 3D additive-quadratic functional equation in Banach spaces: A study with counterexamples

In this research, we focus on solving a mixed type additive-quadratic functional equation expressed as:\begin{align*}h(3s_1+2s_2+s_3) + h(3s_1+2s_2-s_3) + h(3s_1-2s_2+s_3)+&h(3s_1-2s_2-s_3)\\= &12\tilde{h}(s_1) +8\tilde{h}(s_2)+2\tilde{h}(s_3)+12h(s_1)\end{align*}where $\tilde{h}(s_1)=h(s_1)+h(-s_1)$ is derived. We proceed to investigate the generalized Hyers-Ulam stability of this equation within the framework of Banach spaces, employing the Hyers direct method. Additionally, examples of non-stable cases are also provided.

Generalized Hyers-Ulam stability of a bi-quadratic mapping in non-Archimedean spaces

Generalized Hyers-Ulam stability of a bi-quadratic mapping in non-Archimedean spaces

A coupled system of $ p $-Laplacian implicit fractional differential equations depending on boundary conditions of integral type

<abstract><p>The objective of this article is to investigate a coupled implicit Caputo fractional $ p $-Laplacian system, depending on boundary conditions of integral type, by the substitution method. The Avery-Peterson fixed point theorem is utilized for finding at least three solutions of the proposed coupled system. Furthermore, different types of Ulam stability, i.e., Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability, are achieved. Finally, an example is provided to authenticate the theoretical result.</p></abstract>

Analysis of Turberculosis‐COVID‐19 Coinfection Using Fractional Derivatives

Fractional‐order derivative modeling continues to receive great interest among researchers across the globe. In this study, Tuberculosis‐COVID‐19 coinfection is studied using Atangana–Baleanu fractional‐order derivatives defined in Caputo sense. We confirmed the existence and singularity of the solution and investigated the model’s equilibrium points. Additionally, we examined the model’s stability in terms of the Ulam–Hyers and generalized Ulam–Hyers stability criteria. The basic reproduction number R0 was calculated using the next‐generation matrix approach. We also looked into the model’s disease‐free equilibrium point’s regional stability. Numerical scheme for simulating the fractional‐order system with Mittag–Leffler Kernels are presented. Numerical simulations are given to validate the model. Results of the simulation showed a decline in the number of COVID‐19 infections within the population when the fractional operator was reduced.

ON STABILITY OF A-QUARTIC FUNCTIONAL EQUATIONS IN RANDOM NORMED SPACES

In this paper, we shall prove the generalized Hyers-Ulam stability of the additive-quartic functional equation introduced by C. Muthamilarasi et al. [11] in Random Normed spaces by using direct and fixed-point methods.

Solvability and stability analysis of a coupled system involving generalized fractional derivatives

<abstract><p>In this article, we investigate the existence of unique maximal and minimal solutions for a coupled differential system in terms of generalized fractional derivative with arbitrary order. The iterative technique of a fixed point operator together with the properties of green's function are used basically. Moreover, we investigate the generalized Ulam-Hyers stability of the solution for the given coupled system. Finally, some examples are given to illustrate the theoretic results.</p></abstract>

Further Stability Results of the Functional Equation in Non-Archimedean Space

In this paper, we investigate the generalized Hyers-Ulam stability of the following reciprocal functional equation in non-Archimedean space using a direct method.

Chaotic dynamics in tritrophic interaction with volatile compounds in plants with power law kernel

In this article, we study a fractional-order mathematical model representing tritrophic interaction amongst plants, herbivores, and carnivores with Caputo derivative. The existence and uniqueness of the system are investigated by fixed point theory, while the stability is studied by Hyers-Ulam and generalized Hyers-Ulam stability analysis. The Adams-Bashforth-Moulton scheme is used for numerical calculations. From numerical simulations, it is observed that when the fractional order decreases the system converges to a stable state. It is observed that for a small value of fractional order, the system approaches a stable state rapidly as compared to the integer order. The chaotic behaviour of the system is studied using the Lyapunov spectrum. It is noted that two positive exponents of the proposed model show that the system is hyper-chaotic. It is also observed that a small value of attraction constant disrupts the system due to volatile organic compounds.

The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient

In this paper, we prove the Hyers–Ulam stability and generalized Hyers–Ulam stability of ut(x,t)=a(t)Δu(x,t) with an initial condition u(x,0)=f(x) for x∈Rn and 0<t<T; the corresponding conclusions of the standard heat equation can be also derived as corollaries. All of the above results are proved by using the properties of the fundamental solution of the equation.

General solution and generalized Hyers-Ulam stability for additive functional equations

In this paper, we introduce new types of additive functional equations and obtain the solutions to these additive functional equations. Furthermore, we investigate the Hyers-Ulam stability for the additive functional equations in fuzzy normed spaces and random normed spaces using the direct and fixed point approaches. Also, we will present some applications of functional equations in physics. Through these examples, we explain how the functional equations appear in the physical problem, how we use them to solve it, and we talk about solutions that are not used for solving the problem, but which can be of interest. We provide an example to show how functional equations may be used to solve geometry difficulties.