Regularity and selecting principles for implicit ordinary differential equations
The authors study implicit scalar differential equations of the form \(f(t,u(t),u'(t))=0\) on some interval \((a,b)\). Solutions are sought in the space of absolute continuous functions with bounded derivatives and which are zero at the boundaries. As can be seen from the simple example \(|u'(t)|=1\) on \((0,1)\), there exist in general infinitely many solutions. For selecting a meaningful solution, the authors propose a selecting principle which singles out the most regular solution. The regularity of a solution is measured by the number of discontinuity points of its derivative. The authors give sufficient conditions which ensure the existence of such most regular solutions and they also formulate conditions which ensure that there exists exactly one positive and one negative most regular solution. The selecting principle is compared with two other selecting principle, the so called viscosity method and the maximal solution principle. Furthermore, several examples are given which also show the limits of the selecting principles. The paper is well structured and nicely written.
scientific article; zbMATH DE number 5509382