A class of monotonicity-preserving variable-step discretizations for Volterra integral equations and time fractional ordinary differential equations

The time continuous Volterra equations valued in $\mathbb{R}$ with nonnegative resolvent kernels have two basic monotone properties. The first is that any two solution curves do not intersect with suitable given signals. 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 proposed to preserve the complete monotonicity property and thus these monotonicity properties but they are restricted to uniform meshes. In this work, through an analogue of the convolution on nonuniform meshes, we introduce the concept of ``right complementary monotone'' (R-CMM) kernels in the discrete level for nonuniform meshes, which is an analogue of the CM-preserving property but much more flexible. We prove that the discrete solutions preserve these two monotone properties if the discretized kernel satisfies R-CMM property. Technically, we highly rely on the resolvent kernels to achieve this.