Differential inequalities for functional perturbations of first-order ordinary differential equations

The theory of differential inequalities plays a central role in the qualitative and quantitative study of differential equations. In this paper, we present several comparison results for a class of functional differential equations of first order with periodic boundary value conditions. The inequalities obtained are, generally speaking, of the following type: Pv 5 0 implies that v 5 0, where P is a functional differential operator subject to some boundary conditions, and v is an element of a prescribed space of functions. We first obtain several new results for the linear problem. Then, we consider a nonlinear differential equation as a functional perturbation of the original differential equation and give different comparison results. Our results improve and generalize previous estimates described in the literature. @ 2002 Elsevier Science Ltd. All rights reserved. Keywords-Differential inequalities, Comparison result, Functional differential equation, Periodic boundary value problem.