The time continuous Volterra equations valued in $\mathbb{R}$ with completely positive kernels have two basic monotonicity 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. Due to the fading memory principle, we also desire the kernels to be nonincreasing. In this work, through an generalization of the convolution to nonuniform meshes, we introduce the concept of ``right complementary monotone'' (R-CMM) kernels in the discrete level for nonuniform meshes, which inherits both the nonincreasing property and complete positivity in the continuous level. We prove that the discrete solutions preserve these two monotonicity 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$\mathbb{R}$且具有完全正核的时间连续Volterra方程具有两个基本单调性性质。其一，任意两条解曲线在给定适当信号条件下不会相交。其二，自治方程的解具有单调性。基于记忆衰减原理，我们还要求核函数保持非增特性。本文通过将卷积推广至非均匀网格，在离散层面针对非均匀网格提出了"右互补单调"（R-CMM）核的概念，该核同时继承了连续层面的非增性与完全正性。我们证明，若离散核满足R-CMM性质，则离散解能保持这两个单调性性质。在技术实现上，我们高度依赖解核方法来达成这一目标。