Abstract
The method of lower and upper solutions deals mainly with existence results for boundary value problems. In this presentation, we will restrict attention to second order ODE problems with separated boundary conditions. Although some of the ideas can be traced back to E. Picard [16], the method of lower and upper solutions was firmly established by G. Scorza Dragoni [20]. This 1931 paper considered upper and lower solutions which are C2; in 1938, the same author extended his method to the L1;-Carathéodory case [21]. Upper and lower solutions with corners were considered by M. Nagumo in 1954 [13]. Since then a multitude of variants have been introduced. The Definitions 2.1 and 3.1 we present here tend to be general enough for applications and simple enough to model the geometric intuition built into the concept.
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De Coster, C., Habets, P. (2001). An Overview of the Method of Lower and Upper Solutions for ODEs. In: Grossinho, M.R., Ramos, M., Rebelo, C., Sanchez, L. (eds) Nonlinear Analysis and its Applications to Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 43. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0191-5_1
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