Journal of Applied Mathematics
Volume 2011 (2011), Article ID 726020, 11 pages
http://dx.doi.org/10.1155/2011/726020
Research Article

Stability and Superstability of Generalized ( 𝜃 , 𝜙 )-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

M. Eshaghi Gordji,1 M. B. Ghaemi,2 G. H. Kim,3 and Badrkhan Alizadeh4

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
3Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea
4Technical and Vocational Faculty of Tabriz, Technical and Vocational University of Iran, P.O. Box 51745-135, Tabriz, Iran

Received 23 September 2011; Revised 6 October 2011; Accepted 16 October 2011

Academic Editor: Ferenc Hartung

Copyright © 2011 M. Eshaghi Gordji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐴 be an algebra, and let 𝜃 , 𝜙 be ring automorphisms of 𝐴 . An additive mapping 𝐻 ∶ 𝐴 → 𝐴 is called a ( 𝜃 , 𝜙 ) -derivation if 𝐻 ( 𝑥 𝑦 ) = 𝐻 ( 𝑥 ) 𝜃 ( 𝑦 ) + 𝜙 ( 𝑥 ) 𝐻 ( 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝐴 . Moreover, an additive mapping 𝐹 ∶ 𝐴 → 𝐴 is said to be a generalized ( 𝜃 , 𝜙 ) -derivation if there exists a ( 𝜃 , 𝜙 ) -derivation 𝐻 ∶ 𝐴 → 𝐴 such that 𝐹 ( 𝑥 𝑦 ) = 𝐹 ( 𝑥 ) 𝜃 ( 𝑦 ) + 𝜙 ( 𝑥 ) 𝐻 ( 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝐴 . In this paper, we investigate the superstability of generalized ( 𝜃 , 𝜙 ) -derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.