lie $^*$-double derivations on lie $c^*$-algebras

a unital $c^*$ -- algebra $mathcal a,$ endowed withthe lie product $[x,y]=xy- yx$ on $mathcal a,$ is called a lie$c^*$ -- algebra. let $mathcal a$ be a lie $c^*$ -- algebra and$g,h:mathcal a to mathcal a$ be $bbb c$ -- linear mappings. a$bbb c$ -- linear mapping $f:mathcal a to mathcal a$ is calleda lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all $a,bin mathcal a.$ in this paper, our main purpose is to prove thegeneralized hyers –- ulam –- rassias stability  of lie $*$ -double derivations on lie $c^*$ - algebras associated with thefollowing additive mapping:begin{align*}sum^{n}_{k=2}(sum^{k}_{i_{1}=2} sum^{k+1}_{i_{2}=i_{1}+1}...sum^{n}_{i_{n-k+1}=i_{n-k}+1}) f( sum^{n}_{i=1, ineqi_{1},..,i_{n-k+1} } x_{i}&-sum^{n-k+1}_{ r=1}x_{i_{r}})+f(sum^{n}_{ i=1} x_{i})&=2^{n-1} f(x_{1}) end{align*} for a fixed positive integer $n$ with $n geq 2.$