Homomorphisms and derivations in proper JCQ∗JCQ∗-triples

In this paper, we investigate homomorphisms in proper JCQ∗JCQ∗-triples and derivations on proper JCQ∗JCQ∗-triples associated with the following functional equation1kf(kx+ky+kz)=f(x)+f(y)+f(z) for a fixed positive integer k  . We moreover prove the generalized Hyers–Ulam stability of homomorphisms in proper JCQ∗JCQ∗-triples and of derivations on proper JCQ∗JCQ∗-triples. This is applied to investigate isomorphisms between proper JCQ∗JCQ∗-triples.