On weak uniform normal structure in weighted Orlicz sequence spaces

It is proved that a weighted Orlicz sequence space ℓM(w), equipped with Luxemburg or Amemiya norm has weak uniform normal structure iff ℓM(w)≅hM(w) for wide class of weight sequences . An example is constructed, where M has not Δ2-condition but by choosing a suitable weight sequence limn→∞wn=∞ we get that ℓM(w) has weak uniform normal structure.