Dominant and subdominant positive solutions to generalized Dickman equation

摘要

The paper considers a generalized Dickman equationtx˙(t)=−∑i=1saix(t−τi)for t → ∞ where s∈N, ai > 0, τi > 0, i=1,…,s and ∑i=1sai=1. It is proved that there are two mutually disjoint sets of positive decreasing solutions such that, for every two solutions from different sets, the limit of their ratio for t → ∞ equals 0 or ∞. The asymptotic behavior of such solutions is derived and a structure formula utilizing such solutions and describing all the solutions of a given equation is discussed. In addition, a criterion is proved giving sufficient conditions for initial functions to generate solutions falling into the first or the second set. Illustrative examples are given. Some open problems are suggested to be solved.