Stability analysis of higher order nonlinear differential equations in β–normed spaces

In this paper, we interrogate different Ulam type stabilities, ie, β–Ulam–Hyers stability, generalized β–Ulam–Hyers stability, β–Ulam–Hyers–Rassias stability, and generalized β–Ulam–Hyers–Rassias stability, for nth order nonlinear differential equations with integrable impulses of fractional type. The existence and uniqueness of solutions are investigated by using the Banach contraction principle. In the end, we give an example to support our main result.     

Existence and uniqueness of solutions to a fractional difference equation with <Emphasis Type="Italic">p</Emphasis>-Laplacian operator

In this paper, we investigate existence and generalized Hyers–Ulam–Rassias stability of Stieltjes quadratic functional integral equations. Firstly, we show some basic properties of the composite function of bounded variation. Secondly, we derive the generalized Hyers–Ulam–Rassias stability result after examining the existence and uniqueness results via the theory of measure of noncompactness and a fixed point theorem of Darbo type. Finally, two examples of functional integral equations of fractional order are given to demonstrate the applicability of our results.     

Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,T], for a class of higher‐order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.     

A Uniform Method to Ulam–Hyers Stability for Some Linear Fractional Equations

In this paper, we first utilize fractional calculus, the properties of classical and generalized Mittag-Leffler functions to prove the Ulam–Hyers stability of linear fractional differential equations using Laplace transform method. Meanwhile, Ulam–Hyers–Rassias stability result is obtained as a direct corollary. Finally, we apply the same techniques to discuss the Ulam’s type stability of fractional evolution equations, impulsive fractional evolutions equations and Sobolev-type fractional evolution equations.     

New concepts and results in stability of fractional differential equations

In this paper, some new concepts in stability of fractional differential equations are offered from different perspectives. Hyers–Ulam–Rassias stability as well as Hyers–Ulam stability of a certain fractional differential equation are presented. The techniques rely on a fixed point theorem in a generalized complete metric space. Some applications of our results are also provided.     

On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation

In this paper, by means of Banach fixed point theorem, we investigate the existence and Ulam–Hyers–Rassias stability of the noninstantaneous impulsive integrodifferential equation by means of ψ‐Hilfer fractional derivative. In this sense, some examples are presented, in order to consolidate the results obtained.     

Study of implicit type coupled system of non‐integer order differential equations with antiperiodic boundary conditions

In this paper, the first purpose is to study existence and uniqueness of solutions to a system of implicit fractional differential equations (IFDEs) equipped with antiperiodic boundary conditions (BCs). To obtain the mentioned results, we use Schauder's and Banach fixed point theorem. The second purpose is discussing the Ulam‐Hyers (UH) and generalized Ulam‐Hyers (GUH) stabilities for the respective solutions. An example is provided to illustrate the established results.     

Stability of preserving lattice cubic functional equation in Menger probabilistic normed Riesz spaces

The aim of this paper is to investigate Hyers–Ulam–Rassias stability of preserving lattice functional equation in various spaces. First, we prove stability of generalized preserving lattice functional equation in Banach lattices. Next, we show stability of preserving lattice cubic functional equation in Menger probabilistic normed Riesz spaces.     

A fixed point method for almost partial derivations of Jensen type n-variable functional equations

In this paper, we will apply a fixed point method for proving the generalized Hyers–Ulam–Rassias stability of the partial derivations for Jensen type n-variable functional equations.     

A note on linear θ-derivations on JB1-triples

In this paper we prove a generalization of the stability of the functional equation in the spirit of Hyers, Ulam and Rassias. Also we introduce the concept of linear θ-derivations on JB¹-triple, and prove the generalization of the stability of the functional equation in the spirit of Hyers, Ulam and Rassias of linear θ-derivations on JB¹-triple. For resent results see [1], [2], [3].     

Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations

In this paper, we derive the equivalent fractional integral equation to the nonlinear implicit fractional differential equations involving Ψ-Hilfer fractional derivative subject to nonlocal fractional integral boundary conditions. The existence of a solution, Ulam–Hyers, and Ulam–Hyers–Rassias stability have been acquired by means of an equivalent fractional integral equation. Our investigations depend on the fixed-point theorem due to Krasnoselskii and the Gronwall inequality involving Ψ-Riemann–Liouville fractional integral. Finally, examples are provided to show the utilization of primary outcomes.     

Hyers–Ulam Stability of Euler’s Differential Equation

We obtain a result on generalized Hyers–Ulam stability for Euler’s differential equation in Banach spaces. Our result extends and improves some recent results of Mortici, Jung and Rassias concerning the stability of Euler’s equation on a bounded domain.     

On the stability of approximately additive mappings

In this paper we prove a generalization of the stability of approximately additive mappings in the spirit of Hyers, Ulam and Rassias.

     

A class of nonlinear differential equations with fractional integrable impulses

In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki’s normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki–Ulam’s type stability are introduced and generalized Ulam–Hyers–Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.     

Stability Property of Functional Equations in Modular Spaces: A Fixed-Point Approach

Mathematical Notes - We investigate the Hyers–Ulam–Rassias stability property of a quadratic functional equation. The analysis is done in the context of modular spaces. The type of...     

Stability of a cubic functional equation in fuzzy normed space

In this paper, the authors investigate the general solution of a new cubic functional equation \(\begin{equation*} 3f(x+3y)-f(3x+y)=12[f(x+y)+f(x-y)]+80f(y)-48f(x) \end{equation*}\) and discuss its generalized Hyers - Ulam - Rassias stability in Banach spaces and stability in fuzzy normed spaces.     

Ulam‐Hyers stability of Caputo fractional difference equations

We study the Ulam‐Hyers stability of linear and nonlinear nabla fractional Caputo difference equations on finite intervals. Our main tool used is a recently established generalized Gronwall inequality, which allows us to give some Ulam‐Hyers stability results of discrete fractional Caputo equations. We present two examples to illustrate our main results.     

A class of impulsive nonautonomous differential equations and Ulam–Hyers–Rassias stability

In this paper, we study a model described by a class of impulsive nonautonomous differential equations. This new impulsive model is more suitable to show dynamics of evolution processes in pharmacotherapy than the classical one. We apply Krasnoselskii's fixed point theorem to obtain existence of solutions. Meanwhile, we mainly present the sufficient conditions on Ulam–Hyers–Rassias stability on both compact and unbounded intervals. Many analysis techniques are used to derive our results. Copyright © 2014 John Wiley & Sons, Ltd.     

On Hyers-Ulam-Rassias stability of functional equations

In this paper, we investigate the stability of functional equation given by the pseudoadditive mappings of the mixed quadratic and Pexider type in the spirit of Hyers, Ulam, Rassias and Gavruta.     

On the Ulam‐Hyers stabilities of the solutions of Ψ‐Hilfer fractional differential equation with abstract Volterra operator

In this paper, we consider the new class of the fractional differential equation involving the abstract Volterra operator in the Banach space and investigate existence, uniqueness and stabilities of Ulam‐Hyers on the compact interval Δ = [a,b] and on the infinite interval I = [a,∞). Our analysis is based on the application of the Banach fixed‐point theorem and the Gronwall inequality involving generalized Ψ‐fractional integral. At last, we performed out an application to elucidate the outcomes got.