Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions (Q1586267)

Consider the periodic boundary value problem \[ du(t)/dt =g(t,u(t), u(\Theta(t))), \;u (0)=u(T), \;0 \leq t \leq T,\tag{*} \] where \(g:[0,T] \times \mathbb{R}^2 \rightarrow \mathbb{R}\) and \(\Theta: [0,T] \rightarrow \mathbb{R}\) are continuous, \(0 \leq \Theta (t) \leq t.\) First, the authors study the linear problem \(du(t)/dt+M u(t) +Nu(\Theta (t)) = \sigma (t)\) with \(\sigma \in C([0,T], \mathbb{R})\) and prove maximum principles for different boundary inequality conditions. Using these principles, the authors define upper and lower solutions to \((*)\) and derive conditions such that the existence of ordered lower and upper solutions imply the existence of a solution to \((*)\). Finally, a result for monotone iteration is proved.