APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS

Let A1,A2,...,An be normed ternary algebras over the complex field C and let B be a Banach ternary algebra over C. A mapping δk from A1×...×An into B is called a k-th partial ternary quadratic derivation if there exists a mapping gk:Ak→B such that δ k(x1,...,[akbkck],...,xn)=[ gk(ak)gk(bk)δk(x1,...,ck,...,xn)]+[gk(ak)δk(x1,...,bk,...,xn )gk(ck)]+[δk(x1,...,ak,...,xn)gk(bk)gk(ck)]and δk(x1,...,ak +bk,...,xn)+δk(x1,...ak−bk,...,xn)=2δk(x1,...,ak,...,xn)+2δ k(x1,...,bk,...,xn)for all ak,bk,ck∈Ak and all xi∈Ai(i≠k). We prove the Hyers-Ulam- Rassias stability of the partial ternary quadratic derivations in Banach ternary algebras.