Ordinary funcitonal differential equations and inequalities in the sense of carathéodory

Abstract

We consider the linear IVP (i)

and the corresponding nonlinear IVP (ii)

where A solution u belongs to the Operator P: L is linear, positive and of Volterra type. Here

The essential results are

(a) A Positivity Theorem for the linear equation (i)

(b) A Comparison Theorem for the nonlinear equation (ii)

(c) A precise characterization of inequalities like u(t)≥0 or v(t) ≤w(t) with respect to strict inequalities of components

(d) A far—reaching generalization of M.Hirsch's theorem on strongly monotone flows

e) Existence and Uniqueness Theorems under arathéodory hypotheses.

Keywords:

• Functional differential equations
• solutions in the sense of Carathéodory
• comparison theorem
• existence and uniqueness