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.

翻译：取值于$\mathbb{R}$且具有非负预解核的时间连续Volterra方程具有两个基本单调性质。其一为任意两条解曲线在给定适当信号条件下不相交；其二为自治方程的解具有单调性。目前已提出CM保持格式(Comm. Math. Sci., 2021,19(5), 1301-1336)用于保持完全单调性从而保留这些单调性质，但该类格式局限于均匀网格。本文通过非均匀网格上的卷积类比，在非均匀网格的离散层面引入"右互补单调"(R-CMM)核概念，该概念虽为CM保持性质的类比但更为灵活。我们证明若离散化核满足R-CMM性质，则离散解可保持这两个单调性质。在技术层面，我们高度依赖预解核来实现这一目标。