and Applied Analysis 3 Generalized derivations first appeared in the context of operator algebras 7 . Later, these were introduced in the framework of pure algebra 8, 9 . Definition 1.1. LetA be an algebra and let X be anA-bimodule. A linear mapping d : A → X is called i derivation if d ab d a b ad b , for all a, b ∈ A; ii generalized derivation if there exists a derivation in the usual sense δ...

this paper is an investigation of $l$-dual frames with respect to a function-valued inner product, the so called $l$-bracket product on $l^{2}(g)$, where g is a locally compact abelian group with a uniform lattice $l$. we show that several well known theorems for dual frames and dual riesz bases in a hilbert space remain valid for $l$-dual frames and $l$-dual riesz bases in $l^{2}(g)$.

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...

The Hyers-Ulam stability, the Hyers-Ulam-Rassias stability, and also the stability in the spirit of Gǎvru̧ta for each of the following quadratic functional equations f(x+y)+ f(x−y) = 2f(x)+ 2f(y), f(x+y + z)+ f(x−y)+ f(y − z)+ f(z−x) = 3f(x)+3f(y)+3f(z), f (x+y+z)+f(x)+f(y)+f(z)= f(x+y)+f(y+z)+f(z+x) are investigated. 2000 Mathematics Subject Classification. Primary 39B52, 39B72, 39B82.

we investigate the stability of generalizedderivations on banach algebras with a bounded central approximateidentity. we show that every approximate generalized derivation inthe sense of rassias, is an exact generalized derivation. also thestability problem of generalized derivations on the faithful banachalgebras is investigated.

in this paper, we investigate the generalizedhyers--ulam stability of the functional equation