We study in this paper the monotonicity properties of the numerical solutions to Volterra integral equations with nonincreasing completely positive kernels on nonuniform meshes. There is a duality between the complete positivity and the properties of the complementary kernel being nonnegative and nonincreasing. Based on this, we propose the ``complementary monotonicity'' to describe the nonincreasing completely positive kernels, and the ``right complementary monotone'' (R-CMM) kernels as the analogue for nonuniform meshes. We then establish the monotonicity properties of the numerical solutions inherited from the continuous equation if the discretization has the R-CMM property. Such a property seems weaker than being log-convex and there is no resctriction on the step size ratio of the discretization for the R-CMM property to hold.

翻译：本文研究非均匀网格上具有非增完全正核的Volterra积分方程数值解的单调性性质。完全正性与互补核非负且非增的性质之间存在对偶性。基于此，我们提出"互补单调性"来描述非增完全正核，并引入"右互补单调"(R-CMM)核作为非均匀网格的对应概念。随后证明：若离散格式具有R-CMM性质，则数值解可继承连续方程的单调性。该性质弱于对数凸性，且R-CMM性质的成立对离散格式的步长比无限制。