Regularly varying solutions of half-linear functional differential equations with retarded arguments

We consider half-linear retarded functional differential equations of the form $$ (|x'(t)|^alpha operatorname{sgn} x'(t))' = q(t)|x(g(t))|^alpha operatorname{sgn} x(g(t)), alpha > 0, g(t) < t $$ and obtain a necessary and sufficient condition to possess a slowly varying solution and a regularly varying solution of index 1 at the same time.