Existence of regularly varying solutions with nonzero indices of half-linear differential equations with retarded arguments

Sharp conditions are established for the existence of a pair of regularly varying solutions with nonzero indices of half-linear functional differential equations of the type (|x′(t)|αsgnx′(t))′=q(t)|x(g(t))|αsgnx(g(t)),α>0,q(t)>0,g(t)