Search Results (33)

Hyers and Ulam considered the problem of whether there is a true isometry that approximates the $\epsilon$-isometry defined on a Hilbert space with a stability constant $10\epsilon$. Subsequently, Fickett considered the same question on a bounded subset of the n-dimensional Euclidean space ${\mathbb{R}}^n$ with a stability constant of $27{\epsilon }^{1/2^n}$. And Vestfrid gave a stability constant of $27n\epsilon$ as the answer for bounded subsets. In this paper, by applying singular value decomposition, we improve the previous stability constants by $C\sqrt{n}\epsilon$ for bounded subsets, where the constant C depends on the approximate linearity parameter K, which is defined later. Full article

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. We also show, using examples, how some additional assumptions imposed on the function when examining the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation. The functional equation studied in this paper has symmetric coefficients (with precision up to the sign), and it is a generalization of an equation characterizing n-quadratic functions, as well as many other functional equations with symmetric coefficients: for example, the multi-Cauchy equation and the multi-Jensen equation. Our results generalize many known outcomes. Full article

This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the linear case, we provide an explicit solution formula involving discrete Mittag-Leffler functions and establish its stability properties. In the nonlinear case, we concentrate on delayed neutral Hilfer fractional difference equations, a class of systems that appears to be unexplored in the existing literature with respect to Ulam–Hyers stability. In particular, for the linear case, the absolute difference between the solution of the linear Hilfer fractional difference equation and the solution of the corresponding perturbed equation is bounded by the function of $\epsilon$ when the perturbed term is bounded by $\epsilon$. In the case of the neutral fractional delayed Hilfer difference equation, the absolute difference is bounded by a constant multiple of $\epsilon$. Our results fill this gap by offering novel stability criteria. We support our theoretical findings with illustrative numerical examples and simulations, which visually confirm the predicted stability behavior and demonstrate the applicability of the results in discrete fractional dynamic systems. Full article

This paper introduces a novel version of the Gronwall inequality specifically related to the Hilfer–Hadamard proportional fractional derivative. By utilizing Picard’s method of successive approximations along with the definition of Mittag–Leffler functions, we derive a representation formula for the solution of the Hilfer–Hadamard proportional fractional differential equation featuring constant coefficients, expressed in the form of the Mittag–Leffler kernel. We establish the uniqueness of the solution through the application of Banach’s fixed-point theorem, leveraging several properties of the Mittag–Leffler kernel. The current study outlines optimal stability, a new Ulam-type concept based on classical special functions. It aims to improve approximation accuracy by optimizing perturbation stability, offering flexible solutions to various fractional systems. While existing Ulam stability concepts have gained interest, extending and optimizing them for control and stability analysis in science and engineering remains a new challenge. The proposed approach not only encompasses previous ideas but also emphasizes the enhancement and optimization of model stability. The numerical results, presented in tables and charts, are provided in the application section to facilitate a better understanding. Full article

This research is based on the artificial intelligence approach for the error and regression analysis of contaminants, nutrients, and oxygen level in water bodies using a Caputo’s difference model. The model is composed of four subgroups including contaminant concentration (which is denoted by C), the temperature of the fluid T, oxygen concentration O, and nutrients N. ${\xi }_C,{\xi }_T,{\xi }_O,{\xi }_N$ are assumed as diffusion constants for the respective classes. The fractional-order difference model is investigated for the existence and uniqueness of solutions as well as Hyers–Ulam stability, subjected to certain assumptions. The computational results demonstrate that the maximum contaminant concentrations reach $0.01046$ mg/L for ${\xi }_C=0.1$ and ${\mathcal{W}}_R$ = 0.1, resulting in nutrient levels as low as $4.9969$ mg/L. The model predicts that increased pollutant loads increase local temperatures to ${20.009}^{ \circ }\mathrm{C}$. Furthermore, an inverse correlation between reaction rates and contaminant concentrations is also observed, whereby an increase in ${\mathcal{W}}_R$ from $0.1$ to $0.2$ reduces concentrations to $0.0038327$ mg/L. Full article

This study presents a fresh perspective on the existence, uniqueness, and stability of solutions for initial value problems involving variable-order differential equations with finite delay. Departing from conventional techniques that utilize generalized intervals and piecewise constant functions, we introduce a novel fractional operator tailored for this specific problem. Our methodology integrates sophisticated mathematical analysis, including the Schauder fixed-point theorem and Banach’s contraction principle, with an examination of the Ulam–Hyers stability of the problem. The strength of our approach is in its simplicity, requiring fewer restrictive assumptions. We conclude with a practical application to illustrate our findings. These results are valuable for understanding complex dynamical systems with time delays, offering applications in diverse fields such as engineering, economics, and medicine, and enhancing numerical methods for solving delay equations. Full article

This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach’s contraction principle. Our study also addresses the Ulam–Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios. Full article

The problem of Ulam stability for equations can be stated in terms of how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. One of the best known results in this area is the following: Let g be a mapping from a normed space V into a Banach space B. Let $\xi \ge 0$ and $t\ne 1$ be fixed real numbers and $g:V\to B$ satisfy the inequality $\parallel g(u+v)-g\left(u\right)-g\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)$ for $u,v\in V\setminus \left\{0\right\}$. Then, there exists a unique additive $f:V\to B$ fulfilling the inequality $\parallel g\left(u\right)-f\left(u\right)\parallel \le \xi |1-2^{t-1}{|}^{-1}{\parallel u\parallel }^t$ for $u\in V\setminus \left\{0\right\}$. There arises a natural problem if the constant, on the right hand side of the latter inequality, is the best possible. It is known as the problem of the best Ulam constant. We discuss this problem, as well as several related issues, show possible generalizations of the existing results, and indicate open problems. To make this publication more accessible to a wider audience, we limit the related information, avoid advanced generalizations, and mainly focus only on the additive Cauchy equation $f(x+y)=f\left(x\right)+f\left(y\right)$ and on the general linear difference equation $x_{n+p}=a_1{x}_{n+p-1}+\dots +a_p{x}_n+b_n$ (considered for sequences in a Banach space). In particular, we show that there is a significant symmetry between Ulam constants of several functional equations and of their inhomogeneous or radical forms. We hope that in this way we will stimulate further research in this area. Full article

We addressed the Ulam stability of second-order periodic linear differential equations with periodic damping. The necessary and sufficient conditions for the Ulam stability of second-order periodic linear differential equations were obtained by providing an Ulam stability theorem and an Ulam instability theorem. In particular, when the elastic coefficient remained constant at 0, a very general conclusion was obtained. These results expand on the conclusions in the relevant literature. In addition, for the situation of constant coefficients, the minimum Ulam constant is provided. Full article

Deep neural network models are vulnerable to attacks from adversarial methods, such as gradient attacks. Evening small perturbations can cause significant differences in their predictions. Adversarial training (AT) aims to improve the model’s adversarial robustness against gradient attacks by generating adversarial samples and optimizing the adversarial training objective function of the model. Existing methods mainly focus on improving robust accuracy, balancing natural and robust accuracy and suppressing robust overfitting. They rarely consider the AT problem from the characteristics of deep neural networks themselves, such as the stability properties under certain conditions. From a mathematical perspective, deep neural networks with stable training processes may have a better ability to suppress overfitting, as their training process is smoother and avoids sudden drops in performance. We provide a proof of the existence of Ulam stability for deep neural networks. Ulam stability not only determines the existence of the solution for an operator inequality, but it also provides an error bound between the exact and approximate solutions. The feature subspace of a deep neural network with Ulam stability can be accurately characterized and constrained by a function with special properties and a controlled error boundary constant. This restricted feature subspace leads to a more stable training process. Based on these properties, we propose an adversarial training framework called Ulam stability adversarial training (US-AT). This framework can incorporate different Ulam stability conditions and benchmark AT models, optimize the construction of the optimal feature subspace, and consistently improve the model’s robustness and training stability. US-AT is simple and easy to use, and it can be easily integrated with existing multi-class AT models, such as GradAlign and TRADES. Experimental results show that US-AT methods can consistently improve the robust accuracy and training stability of benchmark models. Full article

This study investigates the initial value problem of high-order variable-order $\phi$-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions. Full article

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied. Full article

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity. Full article

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. Full article

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues. Full article