A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle. (English) Zbl 1024.34030

Torres, Pedro J.; Zhang, Meirong

Math. Nachr. 251, 101-107 (2003).

The existence of solutions to the following periodic boundary value problem \[ x''+ g(t,x) = 0,\quad x(0) = x(2 \pi), \quad x^\prime(0) = x^\prime (2 \pi), \] where \(g:[0,2 \pi] \times\mathbb R \rightarrow \mathbb R\) is \(L^1\)-Carathéodory function, is studied. The authors prove a generalized anti-maximum principle corresponding to the above problem and use it to develop a monotone iterative scheme for proving the existence of solutions.

Reviewer: Emil Minchev (Chiba)

MSC:

34C25	Periodic solutions to ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
47J25	Iterative procedures involving nonlinear operators
65L15	Numerical solution of eigenvalue problems involving ordinary differential equations

Keywords:

anti-maximum principle; lower and upper solutions; monotone iterative scheme