The existence of solutions to nonlinear second-order periodic boundary value problems (Q715685)

The author proves the existence and location of solutions of the second-order periodic boundary value problem \[ \begin{aligned} - & x''(t)=f(t,x(t),x'(t)), \quad t \in [0,1],\\ & x(0)=x(1), \quad x'(0)=x'(1),\end{aligned} \] with \(f:[0,1] \times\mathbb R^2 \to\mathbb R\) a continuous function that satisfies some type of a one-sided Lipschitz condition. The result follows under the assumption of the existence of a pair of lower and upper solutions. In this case, in the definition of this pair of functions it is not imposed any condition on the sign of \(\alpha'(0) -\alpha'(1)\) and \( \beta'(0)-\beta'(1)\). Moreover, it is not assumed any order between both functions and the Nagumo's condition is not imposed. In the proof, for any \(a>0\), it is used a maximum principle for the linear operator \(-h''-b\, h'+ a\, h\) in the space of periodic functions. The results are applied to the forced pendulum equation with the \(\phi\)-Laplacian curvature operator and to second-order singular problems with attractive type.