Differential-Functional Inequalities for Bounded Vector-Valued Functions

Abstract

For the space ${\mathbb{R}}^n$ ordered by a cone and some functions $f:{\mathbb{R}}^{n+mn}\to {\mathbb{R}}^n$ and $h_1,...,h_m:\mathbb{R}\to \mathbb{R}$ we consider differential-functional inequalities of the type

and conclude $u\le v$ under suitable conditions on $u,v,h_k$ and $f$. The result can be applied to obtain existence and uniqueness results for differential-functional boundary value problems of the form

with $u\in C^2(\mathbb{R},{\mathbb{R}}^n)$ bounded.