Recently the generalizedHyers-Ulam orHyers-Ulam-Rassias stability of the following functional equation ∑m j 1 f −rjxj ∑ 1≤i≤m,i / j rixi 2 ∑m i 1 rif xi mf ∑m i 1 rixi where r1, . . . , rm ∈ R, proved in Banach modules over a unital C∗-algebra. It was shown that if ∑m i 1 ri / 0, ri, rj / 0 for some 1 ≤ i < j ≤ m and a mapping f : X → Y satisfies the above mentioned functional equation then the...

‎In this paper‎, ‎using fixed point method‎, ‎we prove the generalized Hyers-Ulam stability of‎ ‎random homomorphisms in random $C^*$-algebras and random Lie $C^*$-algebras‎ ‎and of derivations on Non-Archimedean random C$^*$-algebras and Non-Archimedean random Lie C$^*$-algebras for‎ ‎the following $m$-variable additive functional equation:‎ ‎$$sum_{i=1}^m f(x_i)=frac{1}{2m}left[sum_{i=1}^mfle...

in this paper, we consider orthogonal stability of mixed type additive and cubic functional equation of the form $$f(2x+y)+f(2x-y)-f(4x)=2f (x+y)+2f(x-y)-8f(2x) +10f(x)-2f(-x),$$ with $xbot y$, where $bot$  is orthogonality in the sense of ratz.

In this paper, we define a generalized additive set-valued functional equation, which is related to the following generalized additive functional equation: f (x 1 + · · · + x l) = (l – 1)f x 1 + · · · + x l–1 l – 1 + f (x l) for a fixed integer l with l > 1, and prove the Hyers-Ulam stability of the generalized additive set-valued functional equation.

A familiar functional equation f(ax+b) = cf(x) will be solved in the class of functions f : R → R. Applying this result we will investigate the Hyers-Ulam-Rassias stability problem of the generalized additive Cauchy equation f ( a1x1+···+amxm+x0 )= m ∑ i=1 bif ( ai1x1+···+aimxm ) in connection with the question of Rassias and Tabor.

A generalized trigonometric-quadratic functional equation of the form F(x+ y) + G(x− y) = 2H(x)K(y) + L(x) +M(y) over the domain of an abelian group and the range of the complex field is considered. Its stability is established based on the assumption that the function K is unbounded. Subject to certain natural conditions, explicit shapes of the functions H and K are determined. Several existin...