General problems of integration of differential inequalities in explicit form

Abstract

This article examines the problem of finding in explicit form all solutions of the first-order nonstrict differential inequality. We use the formula of the general solution of the corresponding differential equation. Using the analogue of the method of arbitrary constant variation or, in other words, the straightening diffeomorphism, we reduce initial inequality to the simplest form x˙ 6 0 or x˙ > 0. Even in case when the equation is considered in a region of existence and uniqueness, we encounter several theoretical and practical problems. Firstly, there is the problem with the extension of solutions. Secondly, the general solution may consist of several functions that are set on different intervals of the region of definition of the equation, therefore the resulting inequality may have the solution, composed of different functions. In this case there are problems of the connection of solutions. The situation becomes more complicated when the equation has points of nonuniqueness. For such inequalities the method of comparison theorems is not applicable. We show how to solve such inequality and obtain some estimates on its solutions for this case. The result obtained in the article, provides unified approach to many theorems about differential inequalities existing in literature. Refs 10.