a unital $c^*$ -- algebra $mathcal a,$ endowed withthe lie product $[x,y]=xy- yx$ on $mathcal a,$ is called a lie$c^*$ -- algebra. let $mathcal a$ be a lie $c^*$ -- algebra and$g,h:mathcal a to mathcal a$ be $bbb c$ -- linear mappings. a$bbb c$ -- linear mapping $f:mathcal a to mathcal a$ is calleda lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

In this paper, we investigate the existence and Ulam-Hyers-Rassias stability of solutions for stochastic differential equations with random impulses. Based on Krasnoselskii?s fixed point theorem, perform investigations to system We apply integral inequality Gronwall type study their stability.

In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some functional equation approximatelymust be close to one satisfying the equation exactly?” Next year, Hyers [2] gave an answer to this problem for additive mappings between Banach spaces. Furthermore, Aoki [3] and Rassias [4] obtained independently generalized results of Hyers’ theorem...

and Applied Analysis 3 Moreover, they also investigated the Hyers-Ulam-Rassias stability of 1.3 by using the direct method see 18 . Indeed, they tried to approximate the even and odd parts of each solution of a perturbed inequality by the even and odd parts of an “exact” solution of 1.3 , respectively. In Theorems 3.1 and 3.3 of this paper, we will apply the fixed point method and prove the Hye...

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 exact was better than that authors previously mentioned, in some parts domain, especially a neighborhood origin.

Let A be an algebra and let X be an A-bimodule. A C−linear mapping d : A → X is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ : A → X such that d(a) = ad(a) + δ(a)a for all a ∈ A. The main purpose of this paper to prove the Hyers-Ulam-Rassias stability and superstability of the generalized Jordan derivations.

The aim of the present paper is to study asymptotic properties solutions linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar those used in case when are first (integer) order) existence uniqueness initial problem for these systems discontinuous functions. As a consequence, we also unique fundamental matrix homogene...