On the Hyers-Ulam Stability of Linear Mappings

Stability of generalized additive Cauchy equations

“…The case of approximately additive functions was solved by Hyers [6] under the assumption that G 1 and G 2 are Banach spaces. Later, the result of Hyers was generalized by Rassias [14] and other mathematicians (see [3,4,5,7,8,9,10,11,12,15,16,17,19,20]). The work of Rassias stimulated a number of mathematicians to investigate the stability problem of various functional equations.…”

A fixed point approach to the stability of an AQCQ-functional equation in RN-spaces

“…Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [3,5,12,13,20,26,29,45,46,47,48,49,50,51]). In [28], Jun and Kim considered the following cubic functional equation…”

Solutions and Stability of Generalized Mixed Type QCA-Functional Equations in Random Normed Spaces

“…Since then, the result of Hyers was generalized by Aoki [3] for approximate additive function in 1950 and by Rassias [44] for approximate linear functions by allowing the difference Cauchy equation ∥f (x+y)−f (x)− f (y)∥ to be controlled by ε(∥x∥ p +∥y∥ p ) in 1978. Because of a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon proved by Rassias is called the Hyers-Ulam-Rassias stability (see also [5,22,35,37,41,42,43,45,46]). In 1994, a generalization of Rassias theorem was obtained by Gǎvruta [21], who replaced ε(∥x∥ p + ∥y∥ p ) by the general control function φ(x, y).…”