Generalized Hyers-Ulam stability of derivations on Lie * C-algebras

In this paper, we investigate new generalized Hyers-Ulam stability results for (®; ¯; °)- derivations on Lie C¤-algebras associated with the following (m; n)-Cauchy-Jensen additive functional equation:


Introduction
The theory of finite dimensional complex Lie algebras is an important part of Lie theory. Lie algebras have many applications in physics and connections with other parts of mathematics. With an increasing amount of theory and applications concerning Lie algebras of various dimensions, it is becoming necessary to ascertain which tools are applicable for handling them. The miscellaneous characteristics of Lie algebras constitute such tools and have also found applications in Casimir operators [1], derived, lower central and upper central sequences, the Lie algebra of derivations, radicals, nilradicals, ideals, subalgebras [11], [20] and megaideals [19]. These characteristics are particularly crucial when considering possible affinities among Lie algebras. Recently, some authors have studied the stability problems of some functional equations in the setting of Lie algebras.
The stability problem concerning the stability of group homomorphisms of functional equations was originally introduced by Ulam [ : for all 1 , G y x  , then there is a homomorphism 2 1 : If the answer is affirmative, we would say that the equation of a homomorphism The famous Ulam stability problem was partially solved by Hyers [10] for linear functional equation of Banach spaces. Later, the results of Hyers were generalized by Aoki [2], G a  vruta [8] and Rassias [23]. C a  dariu and Radu [4] applied the fixed point method to investigation of the stability for a Jensen functional equation. They could present a short and a simple proof, which is different from the direct method initiated by Hyers in 1941, for the generalized Hyers-Ulam stability for a Jensen functional equation. In 2008, Novotny and Hrivnak [15] investigated generalizing the concept of Lie derivations via certain complex parameters and obtained various Lie and established the structure and properties of (see [15] and an expansively superadditive mapping if there exists a constant L with

 
and positive integer  , then we say that  is a n -contractively subhomogeneous mapping if satisfying the following functional equation: We observe that, in case In particular, we have Recently, Asgari et al. [3] established the generalized Hyers-Ulam-Rassias stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras associated to the functional equation For more details about the stability for various types of derivations, refer to [6], [12], [13], [14], [18] and [22].
In this paper, using some strategies from [9], [15] and [21], we investigate new stability of

Fixed point method
Let us recall that a mapping The following fixed point theorem proved by Diaz and Margolis [7] plays an important role in proving our theorem: Proof. Letting Let  be a set of all mappings from A into A and introduce a generalized metric on  as follows: is a generalized complete metric space ( [5]). We consider the mapping . Then T is a strictly contractive self-mapping on  with the Lipschitz constant L . It follows from (2.5) that Proof. It follows from (2.4) and an expansively superadditive mapping of  that

Direct method
In this subsection, we apply the direct method to investigate the new generalized Hyers-Ulam stability results for Proof. If we replace x by x j  and divide j  both sides of (2.5) , then we have