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

• Published in: Applied mathematics letters Vol. 15; no. 2; pp. 173 - 179
• Main Author: Nieto, J.J.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.02.2002; Elsevier
• Subjects: Comparison result; Differential inequalities; Exact sciences and technology; Functional differential equation; Mathematical analysis; Mathematics; Ordinary differential equations; Periodic boundary value problem; Real functions; Sciences and techniques of general use; Functional differential equation; Periodic boundary value problem; Comparison result; Differential inequalities; Differential equation; Functional equation; Periodicity; Perturbation; Boundary condition; Periodic solution; Differential inequality; First order equation; Boundary value problem; Functional perturbation; Differential operator; Non linear effect; Approximate solution; Inequality
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/S0893-9659(01)00114-8

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 ≤ 0 implies that v ≤ 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.