Differential inequalities via maximal element techniques

Abstract

A VERY useful and effective tool of solving a wide class of problems concerning differential equations is that offered by the so-called differential inequalities (see, e.g., as classical references, Bellman [ 11, Coddington and Levinson [2], Corduneanu [3], Lakshmikantham and Leela [4], Walter [5], among many others). Taking into account the great importance of this topic, a considerable number of extensions of these classical results were obtained in the last years. In this direction, an important way ofextension seems to be that employed by Bebernes and Meisters [6], in which, an enumerable exceptional set is allowed in the formulation of the problem. However, the “supremum” technique used by the above quoted authors depends essentially on the continuity properties of the considered functions and therefore, it is a natural question to ask whether this continuity assumption cannot be removed. It is the main aim of this note to give a positive answer to this question, extending the above Bebernes-Meisters’ results to a class of order-semi-continuous functions and, in this context, it is not without importance to emphasize that the basic instrument in proving our main result is represented by a maximal element theorem on ordered metric spaces that may be considered as a metric variant of a general ordering principle due to Brbis and Browder [7]. Finally, it should be noted that our main results have a number of important applications to mean value theorems; this question will be treated in a forthcoming paper, under preparation.