Document Type : Research Paper

Authors

• M. Eshaghi Gordji ¹
• F. Farrokhzad ²
• S.A.R. Hosseinioun ³

¹ Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.

² Department of Mathematics, Shahid Beheshti University, Tehran, Iran.

³ Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, USA

Abstract

Let $A$ be a Banach ternary algebra over a scalar field $\mathbb{R}$ or $\mathbb{C}$ and $X$ be a Banach ternary $A$-module. Let $\sigma ,\tau$ and $\xi$ be linear mappings on $A$, a linear mapping $D:(A,[{]}_A)\to (X,[{]}_X)$ is called a ternary $(\sigma ,\tau ,\xi )$-derivation, if
$$D([xyz{]}_A)=[D(x)\tau (y)\xi (z){]}_X+[\sigma (x)D(y)\xi (z){]}_X+[\sigma (x)\tau (y)D(z){]}_X$$
for all $x,y,z\in A$. In this paper, we investigate ternary $(\sigma ,\tau ,\xi )$-derivation on Banach ternary algebras, associated with the following functional equation
$$f(\frac{x+y+z}{4})+f(\frac{3x-y-4z}{4})+f(\frac{4x+3z}{4})=2f(x).$$
Moreover, we prove the generalized Ulam-Hyers stability of ternary $(\sigma ,\tau ,\xi )$-derivations on Banach ternary algebras.

Keywords

• Banach ternary algebra
• Banach ternary $A$-module
• Ternary $(\sigma ,\tau ,\xi )$-derivation