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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 20, Issue 4, 2001, pp. 1055–1063**

**DOI: 10.4171/ZAA/1059**

Published online: 2001-12-31

Differential-Functional Inequalities for Bounded Vector-Valued Functions

Gerd Herzog^{[1]}(1) Karlsruher Institut für Technologie (KIT), Germany

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 $$v'' + cv' + f v,v(_1), ..., v(h_m) ≤ u'' + cu' + f u, u(h_1), ..., u(h_m)$$ and conclude $u ≤ v$ 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 $$u'' + cu' + f u, u(h_1), ..., u(h_m) = q$$ with $u \in C^2 (\mathbb R, \mathbb R^n$ bounded.

*Keywords: *Ordered vector spaces, differential-functional inequalities, quasimonotonicity

Herzog Gerd: Differential-Functional Inequalities for Bounded Vector-Valued Functions. *Z. Anal. Anwend.* 20 (2001), 1055-1063. doi: 10.4171/ZAA/1059