Old and New Results for First Order Periodic ODEs without Uniqueness: a Comprehensive Study by Lower and Upper Solutions.

An elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations under Carath\'eodory conditions. This study is carried out avoiding any uniqueness assumption, in the future, or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized