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.     

Stability analysis of a coupled system of nonlinear implicit fractional anti‐periodic boundary value problem

In this article, we study the existence and uniqueness of solution for a coupled system of nonlinear implicit fractional anti‐periodic boundary value problem. Further, we investigate different kinds of stability such as Ulam‐Hyers stability, generalized Ulam‐Hyers stability, Ulam‐Hyers‐Rassias stability, and generalized Ulam‐Hyers‐Rassias stability. We develop conditions for existence and uniqueness by using the classical fixed point theorem. Also, two examples are provided to illustrate the obtained 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.     

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.     

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.     

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.     

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.     

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 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].     

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.     

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 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.     

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...     

On the Hyers–Ulam stabilities of functional equations on ‐Banach spaces

The purpose of this article is to generalize the theory of stability of functional equations to the case of n‐Banach spaces. In this article, we prove the generalized Hyers–Ulam stabilities of the Cauchy functional equations, Jensen functional equations and quadratic functional equations on n‐Banach spaces.     

On Hyers–Ulam stability of two functional equations in non-Archimedean spaces

In this paper we prove, using the fixed point method, the generalized Hyers–Ulam stability of two functional equations in complete non-Archimedean normed spaces. One of these equations characterizes multi-Cauchy–Jensen mappings, and the other gives a characterization of multi-additive-quadratic mappings.     

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.     

On Hyers–Ulam stability of generalized linear functional equation and its induced Hyers–Ulam programming problem

We propose a new approach called Hyers–Ulam programming to discriminate whether a generalized linear functional equation, with the form \({\sum_{i=1}^m L_if(\sum_{j=1}^n a_{ij}x_j) = 0}\) for functions from a normed space into a Banach space, has the Hyers–Ulam stability or not. Our main result is that if the induced Hyers–Ulam programming has a solution, then the corresponding functional equation possesses the Hyers–Ulam stability.     

Generalized stability of multi-additive mappings

In this paper we unify the system of Cauchy functional equations defining multi-additive mapping to obtain a single equation and prove the generalized Hyers–Ulam stability both of this system and this equation using the so-called direct method.     

On the Ulam stability of Jensen and Jensen type mappings on restricted domains

In 1941 Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 Bourgin was the second author to treat this problem for additive mappings. In 1982-1998 Rassias established the Hyers-Ulam stability of linear and nonlinear mappings. In 1983 Skof was the first author to solve the same problem on a restricted domain. In 1998 Jung investigated the Hyers-Ulam stability of more general mappings on restricted domains. In this paper we introduce additive mappings of two forms: of “Jensen” and “Jensen type,” and achieve the Ulam stability of these mappings on restricted domains. Finally, we apply our results to the asymptotic behavior of the functional equations of these types.     

On an equation characterizing multi-Jensen-quadratic mappings and its Hyers–Ulam stability via a fixed point method

In this paper, we unify the system of functional equations defining a multi-Jensen-quadratic mapping to obtain a single equation. We also prove, using the fixed point method, the generalized Hyers–Ulam stability of this equation both in Banach spaces and in complete non-Archimedean normed spaces.