Ideally, a numerical method for ordinary differential equations is stable if successive approximations decrease for each problem which has a decaying solution. Often this is interpreted in practice as requiring that a norm (usually an inner-product norm) of the difference between two approximations is non-increasing for a class of problems identified by a monotonicity condition. The authors investigate several moderated forms of such monotonicity conditions some of which admit bounded growth of the difference of two solutions. One of these identifies a system to be monotonically stable. For such a system, the conventional backward Euler method bounds the growth of a difference of two approximations by a factor which is consistent with growth which may occur in the true solution. For partitioned systems, this stability concept is restricted appropriately. Under this restriction, it is shown that a multirate backward Euler method implemented with waveform relaxation yields approximations whose growth is similarly bounded. Conditions are also given under which a waveform relaxation converges. The results are illustrated on a simple electronic circuit. This is a well-organized, clear presentation of technically challenging aspects of numerical stability for non-linear problems. The paper is an excellent reference for monotonic systems alone as well as for its application to the multirate backward Euler method.