Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 5 1 (Special Issue) 2014 03 01 Ternary $(\sigma,\tau,\xi)$-derivations on Banach ternary algebras 23 35 112 10.22075/ijnaa.2014.112 EN M. Eshaghi Gordji Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran. F. Farrokhzad Department of Mathematics, Shahid Beheshti University, Tehran, Iran. S.A.R. Hosseinioun Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, USA Journal Article 2013 08 19 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<br />$$D([xyz]_A) = [D(x)\tau(y)\xi(z)]_X + [\sigma(x)D(y)\xi(z)]_X + [\sigma(x)\tau(y)D(z)]_X$$<br />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<br />$$f(\frac{x + y + z}{4}) + f(\frac{3x - y - 4z}{4}) + f(\frac{4x + 3z}{4}) = 2f(x).$$<br />Moreover, we prove the generalized Ulam-Hyers stability of ternary $(\sigma,\tau,\xi)$-derivations on Banach ternary algebras. https://ijnaa.semnan.ac.ir/article_112_ecfffaca50a5c1a9f09e21fc58595127.pdf