It is known that, if an equilibrium of a nonlinear system has a stability property when an external input is frozen, then the property is maintained under the input being slowly varying. In this paper, we show that the same stability property is preserved not only under slowly varying input but also under slowly-varying-average input (which is not actually slowly varying but its ‘average’ is slowly varying) The input is assumed to be periodic and to vary sufficiently fast. We prove the claim by the average theory and some previous results on the slowly varying inputs.