The author considers the periodic problem for a first order scalar differential equation
where are continuous, -periodic in , satisfy some assumptions of Lipschitz type, and is Carathéodory. Several results of existence are given on the basis of abstract fixed point theorems for equations of type
where are certain operators in a Banach algebra. Equation (1) is reduced to an equation of this type via Green’s function. In the presence of lower and upper solutions it is shown that there are minimal and maximal solutions. In some cases may have discontinuities provided that and a certain function formed on the basis of is increasing in the variable .