A class of monotonicity-preserving variable-step discretizations for Volterra integral equations with completely monotone kernels

The time continuous Volterra equations valued in $\mathbb{R}$ 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.