A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems (Q2640015)

The authors study the nonlinear second order periodic boundary value problem \((1)\quad -u''=f(t,u),\quad u(0)=u(2\pi),\quad u'(0)=u'(2\pi)\) in the Sobolev space \(W^{2,1}(I)\), \(I=[0,2\pi]\), where f is a Carathéodory function. They generalize the monotone iterative method to prove the existence of periodic solutions of (1) when the lower and upper solutions \(\alpha\), \(\beta\) of (1) do not necessarily satisfy \((2)\quad \alpha '(0)\geq \alpha '(2\pi),\quad \beta '(0)\leq \beta '(2\pi)\) which were required in previous works. A new approach to the monotone iterative technique by considering the monotone iterates as orbits of a (discrete) dynamical system is presented. They also prove that the set of solutions of (1) between the lower and upper solutions is a compact and convex set provided that f is decreasing in u for fixed t.