Foliations, associated reductions and lower and upper solutions
In this interesting paper, the authors answer \textit{J. Mawhin}'s question [Boll. Unione Mat. Ital., VI. Ser., A 3, 229-238 (1984; Zbl 0547.34032)] in establishing the existence of a solution to the periodic boundary value problem \[ u''(t) + g(t,u(t)) = 0, \quad u(0)=u(T), \quad u'(0)=u'(T), \] for \(g\) asymptotically linear and satisfying a nonresonance condition to the left of the eigenvalue \((2\pi/T)^2\) as well as an Ahmad-Lazer-Paul condition to the right of the eigenvalue \(0\); that is \[ \eta \leq \frac{g(t,s)}{s} \leq \gamma < (\frac{2\pi}{T})^2 \quad \text{for \(t \in [0,T]\) and \(|s|\geq R\)}, \] and \[ \int_0^T\int_0^s g(t,r) dr ds \to \infty \text{ as \(|s|\to \infty\)}. \] In fact, a more general result is presented taking into account the Fučik spectrum. The approach is mixed. It uses variational and topological methods with upper and lower solutions. Variational methods permit to deduce the existence of upper and lower solutions which are used to obtain a priori bounds of a suitable problem. It is worth to mention that the lower and upper solutions are not well ordered. Therefore, the solution does not lie between them.
scientific article; zbMATH DE number 1837033