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Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results. (English) Zbl 0889.34018
Zanolin, F. (ed.), Non linear analysis and boundary value problems for ordinary differential equations. Wien: Springer. CISM Courses Lect. 371, 1-78 (1996).

This article surveys results on the existence of solutions to boundary value problems for scalar ordinary differential equations of the type

$\stackrel{¨}{u}+f\left(t,u\right)=0$

by means of the method of upper and lower solutions. The notion of ${W}^{2,1}$-upper and lower solutions is introduced which allows angles in the graph of these functions. The authors study periodic boundary value problems $\left(u\left(0\right)=u\left(2\pi \right),\stackrel{˙}{u}\left(0\right)=\stackrel{˙}{u}\left(2\pi \right)\right)$ and Dirichlet boundary value problems $\left(u\left(0\right)=u\left(2\pi \right)=0\right)$. The Dirichlet problem is studied also for equations with singularities. Boundary value problems depending on parameters (Ambrosetti-Prodi problem) are included in this survey. Applications to mechanical problems with singular forces and to Landesman-Laser conditions are given. The relation with degree theory and the variational approach is used to derive multiplicity results. Together with ordered upper and lower solutions the authors consider also upper and lower solutions in the reversed order and without ordering. In this connection, monotone methods play a crucial role. Historical and bibliographical notes are given in the last section. The references contain 121 items.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34-02 Research monographs (ordinary differential equations)