  In this paper a reduction and equivalence theorems for the boundedness of the
composition of a quasilinear operator $T$ with the Hardy and Copson operators
in weighted Lebesgue spaces are proved. New equivalence theorems are obtained
for the operator $T$ to be bounded in weighted Lebesgue spaces restricted to
the cones of monotone functions, which allow to change the cone of
non-decreasing functions to the cone of non-increasing functions and vice versa
not changing the operator $T$. New characterizations of the weighted Hardy-type
inequalities on the cones of monotone functions are given. The validity of
so-called weighted iterated Hardy-type inequalities are characterized.
