On profinite groups with Engel-like conditions

Let G   be a profinite group in which for every element x∈Gx∈G there exists a natural number q=q(x)q=q(x) such that xqxq is Engel. We show that G is locally virtually nilpotent. Further, let p be a prime and G   a finitely generated profinite group in which for every γkγk-value x∈Gx∈G there exists a natural p  -power q=q(x)q=q(x) such that xqxq is Engel. We show that γk(G)γk(G) is locally virtually nilpotent.