On some functional equation arising from $(m,n)$-Jordan derivations of prime rings

In this paper, we prove the following result. Let $m\geq1$, $n\geq 1$ be some fixed integers with $m\neq n$, and let $R$ be a prime ring with $\operatorname{char}(R)>(m+n)^{2}$. Suppose that $D:R\rightarrow R$ is an additive mapping satisfying the relation $(m+n)^{2}D(x^{4})=4m^{2}D(x)x^{3}+4mnxD(x)x^{2}+4mnx^{2}D(x)x+4n^{2}x^{3}D(x)$ for all $x\in R$. In this case, $D$ is a derivation and $R$ is commutative.