Rassias Stability Research Articles

Intuitionistic Fuzzy Stability of an Euler–Lagrange Symmetry Additive Functional Equation via Direct and Fixed Point Technique (FPT)

In this article, a new class of real-valued Euler–Lagrange symmetry additive functional equations is introduced. The solution of the equation is provided, assuming the unknown function to be continuous and without any regularity conditions. The objective of this research is to derive the Hyers–Ulam–Rassias stability (HURS) in intuitionistic fuzzy normed spaces (IFNS) by applying the classical direct method and fixed point techniques (FPT). Furthermore, it is proven that the Euler–Lagrange symmetry additive functional equation and the control function, which is the IFNS of the sums and products of powers of norms, is stable. In addition, a few examples where the solution of this equation can be applied in Fourier series and Fourier transforms are demonstrated.

Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory

In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain particular theorems associated with Hyers–Ulam and Hyers–Ulam–Rassias stability to the solution, as well as the uniqueness and existence of the solution of the BVPs are studied. The results are illustrated with some particular examples, and the numerical data are analyzed for confirmation of the results. The results obtained in this work are simple and can easily be applicable to physical systems. Furthermore, symmetry analysis of fractional differential equations and HFSIDs are also presented. This is due to the fact that the aforementioned analysis plays a significant role in both the optimization and qualitative theory of fractional differential equations.

On the Analysis of a Neutral Fractional Differential System with Impulses and Delays

In this paper, we investigate the exact and approximate controllability, finite time stability, and β–Hyers–Ulam–Rassias stability of a fractional order neutral impulsive differential system. The controllability criteria is incorporated with the help of a fixed point approach. The famous generalized Grönwall inequality is used to study the finite time stability and β–Hyers–Ulam–Rassias stability. Finally, the main results are verified with the help of an example.

Existence, Uniqueness and Stability Analysis with the Multiple Exp Function Method for NPDEs

In this study, firstly, through an alternative theorem, we study the existence and uniqueness of solution of some nonlinear PDEs and then investigate the Ulam–Hyers–Rassias stability of solution. Secondly, we apply a relatively novel analytical technique, the multiple exp function method, to obtain the multiple wave solutions of presented nonlinear equations. Finally, we propose the numerical results on tables and discuss the advantages and disadvantages of the method.

Generalized Ulam–Hyers–Rassias Stability of Solution for the Caputo Fractional Non-instantaneous Impulsive Integro-differential Equation and Its Application to Fractional RLC Circuit

Generalized Ulam–Hyers–Rassias Stability of Solution for the Caputo Fractional Non-instantaneous Impulsive Integro-differential Equation and Its Application to Fractional RLC Circuit

Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

This research work is dedicated to an investigation for a new kind of boundary value problem of nonlinear fractional differential equation supplemented with general boundary condition. A full analysis of existence and uniqueness of positive solutions is respectively proved by Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. Furthermore, we prove the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of solutions. An example illustrating the validity of the existence result is also discussed.

Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρk,ϕk)-Hilfer Fractional Integro-Differential Equations

In this paper, we establish the existence and stability results for the (ρk,ϕk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρk,ϕk)-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.

Generalized β − Hyers–Ulam–Rassias Stability of Impulsive Difference Equations

This paper describes the existence and uniqueness of the solution, β-Hyers–Ulam–Rassias stability and generalized β-Hyers–Ulam–Rassias stability of an impulsive difference system on bounded and unbounded discrete intervals. At the end, an example is given to illustrate the theoretical result.

Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces

Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability theorem can be applied to numerous functional equations in investigating the stability when approximated in Banach spaces, Banach algebra, and so on. The main focus of this study is to analyse the relationship between functional equations, Hyers–Ulam–Rassias stability, Banach space, quasi-beta normed spaces, and fixed-point theory in depth. The significance of this work is the incorporation of the stability of the generalised additive functional equation in Banach space and quasi-beta normed spaces by employing concrete techniques like direct and fixed-point theory methods. They are powerful tools for narrowing down the mathematical models that describe a wide range of events. Some classes of functional equations, in particular, have lately emerged from a variety of applications, such as Fourier transforms and the Laplace transforms. This study uses linear transformation to explain our functional equations while providing suitable examples.

Stability of a Nonlinear ML-Nonsingular Kernel Fractional Langevin System with Distributed Lags and Integral Control

The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense.

Uniqueness and Ulam–Hyers–Rassias stability results for sequential fractional pantograph q-differential equations

We study sequential fractional pantograph q-differential equations. We establish the uniqueness of solutions via Banach’s contraction mapping principle. Further, we define and study the Ulam–Hyers stability and Ulam–Hyers–Rassias stability of solutions. We also discuss an illustrative example.

Hyers–Ulam stability for boundary value problem of fractional differential equations with κ‐Caputo fractional derivative

The purpose of this paper is to discuss basic results of boundary value problems of fractional differential equations (BVP‐FDEs) via the concept of Caputo fractional derivative with respect to another function with the order . The existence and uniqueness results of a solution for BVP‐FDEs are discussed by utilizing Banach fixed point theorem and Schaefer's fixed point theorem. We also provide new sufficient conditions to guarantee the Hyers‐Ulam stability and the Hyers–Ulam–Rassias stability of BVP‐FDEs. Furthermore, some concrete examples to consolidate the obtained results are also considered.

Hyers–Ulam Stability of a System of Hyperbolic Partial Differential Equations

In this paper, we study Hyers–Ulam and generalized Hyers–Ulam–Rassias stability of a system of hyperbolic partial differential equations using Gronwall’s lemma and Perov’s theorem.

Using Krasnoselskii's theorem to investigate the Cauchy and neutral fractionalq-integro-differential equationvianumerical technique

AbstractThis article discusses the stability results for solution of a fractionalq-integro-differential problemviaintegral conditions. Utilizing the Krasnoselskii’s, Banach fixed point theorems, we demonstrate existence and uniqueness results. Based on the results obtained, conditions are provided to ensure the generalized Ulam and Ulam–Hyers–Rassias stabilities of the original system. The results are illustrated by two examples.

Semi-Hyers–Ulam–Rassias Stability via Laplace Transform, for an Integro-Differential Equation of the Second Order

The Laplace transform method is applied to study the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of the second order. A general equation is formulated first; then, some particular cases for the function from the kernel are considered.

Ulam’s type stability of impulsive delay integrodifferential equations in Banach spaces

Abstract In this paper, by applying Pachpatte’s type integral inequality with integral impulses, we investigate Ulam–Hyers and Ulam–Hyers–Rassias stabilities for the impulsive Volterra delay integrodifferential equation Banach spaces. Suitable examples are given to support the obtained results.

Hyers Ulam Rassias Stability Analysis of First Order Partial Differential Equations

Hyers Ulam Rassias Stability Analysis of First Order Partial Differential Equations

Hyers–Ulam–Rassias Stability of Hermite’s Differential Equation

In this paper, we studied the Hyers–Ulam–Rassias stability of Hermite’s differential equation, using Pachpatte’s inequality. We compared our results with those obtained by Blaga et al. Our estimation for zx−yx, where z is an approximate solution and y is an exact solution of Hermite’s equation, was better than that obtained by the authors previously mentioned, in some parts of the domain, especially in a neighborhood of the origin.

On Ulam–Hyers–Rassias stability of the boundary value problem of Hadamard fractional differential equations of variable order

On Ulam–Hyers–Rassias stability of the boundary value problem of Hadamard fractional differential equations of variable order

Study of Nonlinear Fractional Order Delay Problem Under Mittag-Leffler Power Law

In the concerned article, the author's developed indispensable conditions for theresults of existence theory and Ulam type stability of the intended pantograph fractional differential equation. The desired results for existence of the solutions and stability analysis are investigated with a new class of fractional derivative introduced by Atangana-Baleanu-Caputo and so abbreviated as (ABC). The mentioned derivative contains Mittag-Leffler function as a nonsingular kernel instead of singular kernel. We use the tools of fixed point theory, i.e Banach's theorem and Krasnoselskii's type to develop the desired results. Furthermore, we investigate different types of stabilities results including generalized Ullam-Hyers and Rassias stabilities for the concerned problem. We also provide appropriate examples for illustrative purposes.