On the method of upper and lower solutions under Carathéodory assumptions (Q2785334)

Consider the Cauchy problem NEWLINE\[NEWLINEy'(t)=f(t,y(t),y)\tag{\(*\)}NEWLINE\]NEWLINE a.e. in \(I\backslash I_{t_0}\), \(y(t)=\eta(t)\) in \(I_{t_0}:=(-\infty,t)\), where \(t\in(-\infty,a)\). By means of the technique of lower and upper solutions, the author proves the existence of at least one solution of \((*)\) if \(f\) is a Carathéodory function satisfying some monotonicity conditions. His proof is based on a monotone fixed point technique which allows to achieve Müller's theorem in AC-setting.