  The main goal of this paper is to show how some monotonicity methods related
with the subdifferential of suitable convex functions and its extensions as
m-accretive operators in Banach spaces lead to new and unexpected results
showing, for instance, the continuous and monotone dependence of solutions with
the respect to the data (and coefficients) of the problem. In this way, this
paper offers `a common roof' to several methods and results concerning monotone
and non-monotone frameworks. Besides to present here some new results, this
paper offers also a peculiar survey to some topics which attracted the
attention of many specialists in elliptic and parabolic nonlinear partial
differential equations in the last years under the important influence of Haim
Brezis. To be more precise, the model problem under consideration concerns to
positive solutions of a general class of doubly nonlinear diffusion parabolic
equations with some sub-homogeneous forcing terms.
