We will consider the Banach space valued differential equation y′(t) = Ay(t) , where A is an n× n complex matrix. We give a necessary and sufficient condition in order that the equation have the Hyers-Ulam stability. As a Corollary, we prove that the Banach space valued linear differential equation with constant coefficients y(n)(t) + an−1y(n−1)(t) + · · ·+ a1y′(t) + a0y(t) = 0 has the Hyers-Ul...

The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’s theorem was generalized by Aoki 3 for additive mappings. In 1978, Rassias 4 generalized Hyers theorem by obtaining a unique linear mapping near an approximate additiv...

We obtain the general solution of the generalized quartic functional equation f(x + my) + f(x - my) = 2(7m - 9)(m - 1)f(x) + 2m²(m² - 1)f(y)-(m - 1)² f(2x) + m²{f(x + y) + f(x - y)} for a fixed positive integer m. We prove the Hyers-Ulam stability for this quartic functional equation by the directed method and the fixed point method on real Banach spaces. We also investigate the Hyers-Ulam stab...

In this paper, we investigate the Hyers-Ulam stability of additive functional equations of two forms: of “Jensen” and “Jensen type” in the framework of multi-normed spaces. We therefore provide a link between multi-normed spaces and functional equations. More precisely, we establish the Hyers-Ulam stability of functional equations of these types for mappings from Abelian groups into multi-norme...

In this paper, we establish the Hyers–Ulam–Rassias stability of ring homomorphisms and ring derivations on fuzzy Banach algebras.

The Hyers-Ulam stability of the generalized trigonometric-quadratic functional equation ( ) ( ) ( ) ( ) ( ) ( ) 2 F x y G x y H x K y L x M y + − − = + + over the domain of an abelian group and the range of the complex field is established based on the assumption of the unboundedness of the function K. Subject to certain natural conditions, explicit shapes of the functions H and K are determine...

In this article, we prove the generalized Hyers–Ulam stability of the following Cauchy additive functional equation

* Correspondence: baak@hanyang. ac.kr Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea Full list of author information is available at the end of the article Abstract In this article, we prove the Hyers-Ulam stability of exact second-order linear differential equations. As a consequence, we show the Hyers-Ulam stability of the fo...

Using the Hyers-Ulam-Rassias stability method, weinvestigate isomorphisms in Banach algebras and derivations onBanach algebras associated with the following generalized additivefunctional inequalitybegin{eqnarray}|af(x)+bf(y)+cf(z)|  le  |f(alpha x+ beta y+gamma z)| .end{eqnarray}Moreover, we prove the Hyers-Ulam-Rassias stability of homomorphismsin Banach algebras and of derivations on Banach ...

We construct  a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has Hyers-Ulam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of Bèdos-Murphy-Tuset. Our main result genera...