The time continuous Volterra equations valued in R with completely
monotone kernels have two basic monotone properties. The first is that any two
solution curves do not intersect if the given signal has a monotone property.
The second is that the solutions to the autonomous equations are monotone. The
so-called CM-preserving schemes (Comm. Math. Sci., 2021,19(5), 1301-1336) have
been shown to preserve these properties but they are restricted to uniform
meshes. In this work, through the an analogue of the convolution on nonuniform
meshes, we introduce the concept of ``right quasi-completely monotone'' (R-QCM)
kernels for nonuniform meshes, which is a generalization of the CM-preserving
schemes. We prove that the discrete solutions preserve these two monotone
properties if the discretized kernel satisfies R-QCM property. Technically, we
highly rely on the so-called resolvent kernels to achieve this