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

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.

### 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
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