Remarks on the Periodic Boundary Value Problems for Nonlinear Second Order Ordinary Differential Equations

TL;DR

This paper investigates the existence of solutions for nonlinear second-order periodic boundary value problems, especially when lower and upper solutions do not satisfy boundary conditions, using comparison results and monotone iterative methods.

Contribution

It extends the monotone method for second-order periodic boundary value problems to cases where boundary conditions are violated by lower and upper solutions.

Findings

Comparison results for boundary value problems.

Validation of monotone iterative method under boundary violations.

Constructive proof of solution existence.

Abstract

This paper is devoted to study the existence of solutions and the monotone method of second-order periodic boundary value problems when the lower and upper solutions $\alpha$ and $\beta$ violate the boundary conditions $ \alpha(0)=\alpha(2\pi)$ and $\beta(0)=\beta(2\pi)$. We present several comparison results. We show that the method of lower and upper solutions coupled with the monotone iterative technique is valid to obtain constuctive proof of existence of solutions.