  We study in this paper the monotonicity properties of the numerical solutions
to Volterra integral equations with nonincreasing completely positive kernels
on nonuniform meshes. There is a duality between the complete positivity and
the properties of the complementary kernel being nonnegative and nonincreasing.
Based on this, we propose the ``complementary monotonicity'' to describe the
nonincreasing completely positive kernels, and the ``right complementary
monotone'' (R-CMM) kernels as the analogue for nonuniform meshes. We then
establish the monotonicity properties of the numerical solutions inherited from
the continuous equation if the discretization has the R-CMM property. Such a
property seems weaker than being log-convex and there is no resctriction on the
step size ratio of the discretization for the R-CMM property to hold.
