# Quasimonotonicity as a Tool for Differential and Functional Inequalities

• Peter Volkmann
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 161)

## Introduction

In the context of differential inequalities, the name “quasimonotonicity” had been introduced by Wolfgang Walter [8]. In this monograph also the basic comparison theorems involving ordinary and parabolic differential inequalities, respectively, are treated, the latter being a generalization by Mlak [3] of a theorem of Nagumo [4] to functions having values in ℝn.

For both comparison theorems versions are known, where the functions have values in ordered topological vector spaces; cf. [6] for ordinary differential inequalities and the joint paper with Simon [5] for parabolic inequalities. When restricting [5] to the semilinear case, then functions f(x,t,ξ) are involved, whereas in [6] functions f(t,ξ) occur. Here x is a variable in ℝN, t is a real variable, and ξ is a variable in an ordered topological vector space E; the values of f are in E.

Now it turns out that the comparison theorem from [6] can be considered as a special case from [5], when allowing N=0. This will be...

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