Some comparison and oscillation results for first-order differential equations and inequalities with a deviating argument

Abstract

In this paper we are dealing with first-order differential equations and inequalities with a deviating argument and we give some new necessary, sufficient, and necessary and sufficient conditions concerning: (a) the comparison of oscillatory and asymptotic behavior of solutions of: (1) first-order vector differential equations and, corresponding to them, operator and scalar differential equations with a deviating argument and (2) first-order differential equations and, corresponding to them, inequalities with a deviating argument; and (b) the existence of a certain type of nonoscillatory solutions of differential equations and inequalities with a deviating argument. What is important, in our opinion, is the fact that some of the obtained results concern not only the cases of constant sign coefficients, but also the cases where the coefficients of equations and inequalities under consideration are of non-constant sign.