Journal of Inequalities and Applications
Volume 6 (2000), Issue 2, Pages 191-198
Comparison of two definitions of lower and upper functions associated to nonlinear second order differential equations
Mathematical Institute, Academy of Sciences of the Czech Republic, itná 25, 115 67 PRAHA 1, Czech Republic
Received 25 July 1999; Revised 10 October 1999
Copyright © 2000 Ivo Vrkoč. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The notions of lower and upper functions of the second order differential equations take their beginning from the classical work by C. Scorza-Dragoni and have been investigated till now because they play an important role in the theory of nonlinear boundary value problems. Most of them define lower and upper functions as solutions of the corresponding second order differential inequalities. The aim of this paper is to compare two more general approaches. One is due to Rachůnková and Tvrdý (Nonlinear systems of differential inequalities and solvability of certain boundary value problems (J. of Inequal. & Appl. (to appear))) who defined the lower and upper functions of the given equation as solutions of associated systems of two differential inequalities with solutions possibly not absolutely continuous. The second belongs to Fabry and Habets (Nonlinear Analysis, TMA 10 (1986), 985–1007) and requires the monotonicity of certain integro-differential expressions.