On a Functional Equation Related to Jordan Triple Derivations in Prime Rings

A classical result of Herstein asserts that any Jordan derivation on a prime ring with $\operatorname{char}(R)\neq 2$ is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein's theorem. Let R be a prime ring with $\operatorname{char}(R) = 0$ or $\operatorname{char}(R) > 4$, and let $D:R\rightarrow R$ be an additive mapping satisfying the relation $D(x^{4})=D(x)x^{3}+xD(x^{2})x+x^{3}D(x)$ for all $x\in R$. In this case, $D$ is a derivation.