Optimal existence conditions for the periodic delay \(\phi\)-Laplace equation with upper and lower solutions in the reverse order (Q1430557)

This paper deals with the scalar periodic delay \(\phi\)-Laplace equation \[ -\frac{d}{dt}\phi(u'(t))=f(t,u(t),u(t-\tau)),\quad t\in\mathbb{R},\tag{1} \] where \(f: \mathbb{R}^3\to \mathbb{R}\) is a continuous function and \(f(t,u,v)=f(t+\tau,u,v)\), \(\tau>0\); \(\phi:\mathbb{R}\to \mathbb{R}\) is an increasing homeomorphism such that \(\phi(0)=0\). The authors show that the monotone iterative technique generates two monotone sequences that converge uniformly to extremal solutions of (1). Moreover, the authors obtain optimal existence conditions with upper and lower solutions in the reverse order.