The authors demonstrate some comparison principles for first-order ordinary differential equations with impulses at fixed moments. The results are applied to prove the existence of solutions using the method of upper and lower solutions to periodic boundary value problems for nonlinear differential equations with impulses: \[ u'(t)=f(t,u(t)),\quad t\neq t_{k};\quad u(t_{k}^{+})=I_{k}(u(t_{k})), \quad k=1,\dots , p;\quad u(0)=u(T), \] where \(f:I\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function, \(I=[0,T]\), \(0<t_{1}<\cdots <t_{p}<T\), and \(I_{k}:\mathbb{R}\to \mathbb{R}\) are continuous functions, \(k=1,\dots, p\).
To establish the existence result, new definitions of upper and lower solutions, involving the usual concepts, are considered and the function \(f\) is assumed to satisfy a one-sided Lipschitz condition.
Finally, the validity of the monotone iterative technique to approximate extremal solutions to the considered problems in a sector is proved.