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.

翻译：取值于$\mathbb{R}$且具有完全单调核的时间连续Volterra方程具有两个基本单调性质。其一为若给定信号具有单调性，则任意两条解曲线不相交；其二为自治方程的解具有单调性。所谓的CM保持格式（Comm. Math. Sci., 2021,19(5), 1301-1336）已被证明能保持这些性质，但其局限于均匀网格。本文通过非均匀网格上卷积的类似方法，引入了非均匀网格的“右拟完全单调”（R-QCM）核概念，这是CM保持格式的推广。我们证明，若离散核满足R-QCM性质，则离散解保持这两个单调性质。在技术上，我们高度依赖于所谓的预解核来实现这一目标。