In this paper, we introduce and discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.

the stability problem of the functional equation was conjectured by ulam and was solved by hyers in the case of additive mapping. baker et al. investigated the superstability of the functional equation from a vector space to real numbers.in this paper, we exhibit the superstability of $m$-additive maps on complete non--archimedean spaces via a fixed point method raised by diaz and margolis.

The stability problem of the functional equation was conjectured by Ulam 1 during the conference in the University of Wisconsin in 1940. In the next year, it was solved by Hyers 2 in the case of additive mapping, which is called the Hyers-Ulam stability. Thereafter, this problem was improved by Bourgin 3 , Aoki 4 , Rassias 5 , Ger 6 , and Gǎvruţa et al. 7, 8 in which Rassias’ result is called t...

Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for nonsplitting, a particular notion of independence. We generalize their result as follows: given any abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisf...