General problems of explicit integration of differential inequalities

Abstract

We consider the problem of the explicit search for all solutions of a first-order nonstrict differential inequality. We use the formula of the general solution of the corresponding differential equation. Using an analog of the method of arbitrary constant variation or, in other words, a straightening diffeomorphism, we reduce the original inequality to the simplest form ẋ ≤ 0 or ẋ ≥ 0. Even if the equation is considered in the existence and uniqueness region, theoretical and practical problems arise. The first problem is related to the extension of solutions (i.e., to the interval of determination). The second problem is that the general solution may consist of several functions given on different intervals of the equation domain. As a result, the resulting inequality also may have a solution that is composed of different functions. The situation becomes more complicated when the equation has points of branching. In this case, the method of comparison of theorems cannot be used. In this paper, we describe a method for solving differential inequalities and estimating their solutions for this case as well. The result obtained in this study provides a unified approach to many theorems on differential inequalities available in the literature.